CHAPTER 16 ANALOG INTEGRATED CIRCUITS Chapter Outline 16.1 16.2 16.3 16.4 16.5 16.6 16.7 16.8 16.9 Circuit Element Matching Current Mirrors High-Output-Resistance Current Mirrors Reference Current Generation The Bandgap Reference The Current Mirror as an Active Load Active Loads in Operational Amplifiers The A741 Operational Amplifier The Gilbert Analog Multiplier Summary Key Terms References Problems Chapter Goals In Chapter 16 we concentrate on understanding integrated circuit design techniques that are based upon the characteristics of closely matched devices and look at a number of key building blocks of operational amplifiers and other ICs. Our goals are to: • Understand bipolar and MOS current mirror operation and mirror ratio errors • Explore high output resistance current sources including cascode and Wilson current source circuits • Learn to design current sources for use in both discrete and integrated circuits • Add reference current circuit techniques to our kit of circuit building blocks. These circuits produce currents that exhibit a substantial degree of independence from power supply voltage including the VBE -based reference and the Widlar current source. • Investigate the operation and design of bandgap reference circuits, one of the most important 1178 • • • • • techniques for providing an accurate reference voltage that is independent of power supply voltages and temperature Use current mirrors as active loads in differential amplifiers to increase the voltage gain of single-stage amplifiers to the amplification factor µ f Learn how to include the effects of device mismatch in the calculation of amplifier performance measures such as CMRR Analyze the design of the classic A741 operational amplifier Understand the techniques used to realize fourquadrant analog multipliers with large input signal range Continue to increase our understanding of SPICE simulation techniques In Chapter 16, we explore several extremely clever and exciting circuits designed by two of the legends of integrated circuit design, Robert Widlar and Barrie Gilbert. Widlar developed the LM101 operational amplifier and many of the circuits that led to the design of the classic A741 op amp. Widlar was also responsible for the bandgap reference. Gilbert invented a four-quadrant analog multiplier circuit referred to today as the Gilbert multiplier. The A741 circuit techniques spawned a broad range of follow-on designs that are still in use today. The bandgap reference forms the heart of most precision voltage references and voltage regulator circuits, and is also used as a temperature sensor in digital thermometry. Circuits related to the analog multiplier are used in RF mixers (the Gilbert mixer) and phase detectors in phase-locked loops. Integrated circuit (IC) technology allows the realization of large numbers of virtually identical transistors. Although the absolute parameter tolerances of these devices are relatively poor, device characteristics can be matched to Chapter 16 (a) 1179 Analog Integrated Circuits (b) Legends of Analog Design (a) Robert J. Widlar. (b) Barrie Gilbert (a) Courtesy of National Semiconductor. (b) Courtesy of Analog Devices within 1 percent or better. The ability to build devices with nearly identical characteristics has led to the development of special circuit techniques that take advantage of the tight matching of the device characteristics. Figures 16.1 and 16.2 show an example of the use of four matched transistors to improve the performance of the differential amplifier that we studied in the last chapter. The four devices are cross-connected to further improve the overall parameter matching and temperature tracking of the circuit. Chapter 16 begins by exploring the use of matched transistors in the design of current sources, called current mirrors, in both MOS and bipolar technology. The cascode and Wilson current B24 B13 Q1 Q3 Q4 Q1 E1 E2 Q2 Q4 E4 E3 Q3 Q2 C24 C13 Figure 16.1 Differential amplifier Figure 16.2 Layout of the formed with a cross-connected quad of identical transistors. cross-coupled transistor quad in Fig. 16.1. 1180 Chapter 16 Analog Integrated Circuits sources are subsequently added to our repertoire of high-output-resistance current source circuits. Circuit techniques that can be used to achieve power supply independent biasing are also introduced. We will also study the bandgap reference circuit which uses the well defined behavior of the pn junction to produce a precise output voltage that is highly independent of power supply voltage and temperature. The bandgap circuit is widely used in voltage references and voltage regulators. The current mirror is often used to bias analog circuits and to replace load resistors in differential and operational amplifiers. This active-load circuit can substantially enhance the voltage gain capability of many amplifiers, and a number of MOS and bipolar circuit examples are presented. The chapter then discusses circuit techniques used in IC operational amplifiers, including the classic 741 amplifier. This design provides a robust, high-performance, generalpurpose operational amplifier with breakdown-voltage protection of the input stage and shortcircuit protection of the output stage. The final section looks at the precision four-quadrant analog multiplier design of Gilbert. 16.1 CIRCUIT ELEMENT MATCHING Integrated circuit design is based directly on the ability to realize large numbers of transistors with nearly identical characteristics. Transistors are said to be matched when they have identical sets of device parameters: (I S , βFO , V A ) for the BJT, (VTN , K , λ) for the MOSFET, or (IDSS , V P , λ) for the JFET. The planar geometry of the devices can easily be changed in integrated designs, and so the emitter area A E of the BJT and the W/L ratio of the MOSFET become important circuit design parameters. (Remember from our study of MOS digital circuits in Part II that W/L represents a fundamental circuit design parameter.) In integrated circuits, absolute parameter values may vary widely from fabrication process run to process run, with ±25 to 30 percent tolerances not uncommon (see Table 16.1). However, the matching between nearby circuit elements on a given IC chip is typically within a fraction of a percent. Thus, IC design techniques have been invented that rely heavily on matched device characteristics and resistor ratios rather than absolute parameter values. The circuits described in this chapter depend, for proper operation, on the tight device matching that can be realized through IC fabrication processes, and many will not operate correctly if built with mismatched discrete components. However, many of these circuits can be used in discrete circuit design if integrated transistor arrays are used in the implementation. TABLE 16.1 IC Tolerances and Matching [1] Diffused resistors Ion-implanted resistors VBE IS , βF , VA VTN , VTP K , λ ABSOLUTE TOLERANCE, % MISMATCH, % 30 5 10 30 15 30 ≤2 ≤1 ≤1 ≤1 ≤1 ≤1 16.2 1181 Current Mirrors Exercise: An IC resistor has a nominal value of 10 k and a tolerance of ±30 percent. A particular process run has produced resistors with an average value 20 percent higher than the nominal value, and the resistors are found to be matched within 2 percent. What range of resistor values will occur in this process run? Answer: 11.88 k–12.12 k 16.2 CURRENT MIRRORS Current mirror biasing is an extremely important technique in integrated circuit design. Not only is it heavily used in analog applications, it also appears routinely in digital circuit design as well. Figure 16.3 shows the circuits for basic MOS and bipolar current mirrors. In Fig. 16.3(a), MOSFETs M1 and M2 are assumed to have identical characteristics (VT N , K n , λ) and W/L ratios; in Fig. 16.3(b), the characteristics of Q 1 and Q 2 are assumed to be identical (I S , β F O , V A ). In both circuits, a reference current IREF provides operating bias to the mirror, and the output current is represented by current I O . These basic circuits are designed to have I O = IREF ; that is, the output current mirrors the reference current — hence, the name “current mirror.” IREF ID1 IREF IO ID2 0 M1 M2 IO IC1 IC2 Q1 Q2 IB1 IB2 VGS VBE −VSS −VEE (a) (b) Figure 16.3 (a) MOS and (b) BJT current mirror circuits. 16.2.1 dc Analysis of the MOS Transistor Current Mirror Although the MOS current mirror was introduced in Chapter 4, a review of the basic analysis is repeated here so we can easily refer to the equations. Because the gate currents are zero for the MOSFETs, reference current IREF must flow into the drain of M1 , which is forced to operate in pinch-off by the circuit connection because VDS1 = VG S 1 = VG S . VG S must equal the value required for I D1 = IREF . Assuming matched devices:1 IREF Kn (VG S 1 − VT N )2 (1 + λVDS1 ) = 2 or VG S 1 = VT N + 2IREF K n1 (1 + λVDS1 ) (16.1) 1 Matching between elements in the current mirror is very important; this is a case in which the ( 1 + λVD S) term is included in the dc, as well as ac, calculations. 1182 Chapter 16 Analog Integrated Circuits Current I O is equal to the drain current of M2 : I O = I D2 = Kn (VG S2 − VT N )2 (1 + λVDS2 ) 2 (16.2) but the circuit connection forces VG S2 = VG S 1 and VDS1 = VG S 1 . Substituting Eq. (16.1) into Eq. (16.2) yields I O = IREF (1 + λVDS2 ) ∼ = IREF (1 + λVDS1 ) (16.3) For equal values of VDS , the output current is identical to the reference current (that is, the output VDS1 , and there mirrors the reference current). Unfortunately in most circuit applications, VDS2 = is a slight mismatch between the output current and the reference current, as demonstrated in Ex. 16.1. For convenience, we define the ratio of I O to IREF to be the mirror ratio MR given by MR = EXAMPLE 16.1 IO (1 + λVDS2 ) = IREF (1 + λVDS1 ) (16.4) OUTPUT CURRENT OF THE MOS CURRENT MIRROR In this example, we find the output current for the standard current mirror configuration. PROBLEM Calculate the output current I O for the MOS current mirror in Fig. 16.3(a) if VSS = 10 V, K n = 250 A/V2 , VT N = 1 V, λ = 0.0133 V−1 , and IREF = 150 A. SOLUTION Known Information and Given Data: Current mirror circuit in Fig. 16.3(a); VSS = 10 V; transistor parameters are given as K n = 250 A/V2 , VT N = 1 V, λ = 0.0133 V−1 , and IREF = 150 A Unknowns: Output current I O Approach: Find VG S 1 and VDS2 and then evaluate Eq. (16.3) to give the output current. Assumptions: Transistors are identical and operating in the active region of operation Analysis: We need to evaluate Eq. (16.3) and must find the value of VG S 1 using Eq. (16.1). Since VDS1 = VG S 1 , we can write 2(150 A) 2IREF VDS1 = VT N + =1+ = 2.10 V A Kn 250 2 V in which we have neglected the (1 + λVDS1 ) term to simplify the dc bias calculation. Substituting this value and VDS2 = 10 V in Eq. (16.3): I O = (150 A) [1 + 0.0133(10)] = 165 A [1 + 0.0133(2.10)] The ideal output current would be 150 A, whereas the actual currents are mismatched by approximately 10 percent. 16.2 1183 Current Mirrors Check of Results: A double check shows the calculations to be correct. M1 is pinched-off by connection, and M2 will also be active as long as its drain-source voltage exceeds (VG S 1 − VT N ), which is easily met in Fig. 16.3 since VDS2 = 10 V. Discussion: We could attempt to improve the precision of our answer slightly by including the (1 + λVDS1 ) term in the evaluation of VG S 1 . The solution then requires an iterative analysis that barely changes the value of I O . Computer-Aided Analysis: We can check our analysis directly with SPICE by setting the MOS transistor parameters to KP = 250 A/V2 , VTO = 1 V, LEVEL = 1, and LAMBDA = 0. SPICE yields an output current of 150 A with VG S = 2.095 V. With nonzero λ, LAMBDA = 0.0133 V−1 , SPICE yields I O = 165 A with VG S = 2.081 V. The values are in agreement with our hand calculations. Exercise: Suppose we include the (1 + λVDS1 ) term in the evaluation of VGS1 . Show that the equation to be solved is VDS1 = VT N + 2I REF K n (1 + λVDS1 ) Find the new value of VDS1 using the numbers in Ex. 16.1. What is the new value of I O ? Answers: 2.08 V; 165 A Exercise: Based on the numbers in Ex. 16.1, what is the minimum value of the drain voltage required to keep M2 saturated in Fig. 16.3(a)? Answer: −8.9 V 16.2.2 Changing the MOS Mirror Ratio The power of the current mirror is greatly increased if the mirror ratio can be changed from unity. For the MOS current mirror, the ratio can easily be modified by changing the W/L ratios of the two transistors forming the mirror. In Fig. 16.4, for example, remembering that K n = K n (W/L) for the MOSFET, the K n values of the two transistors are given by K n1 = K n W L and K n2 = K n 1 W L (16.5) 2 Substituting these two different values of K n in Eqs. (16.1) and (16.2) yields W K n2 (1 + λVDS2 ) (1 + λVDS2 ) L I O = IREF = IREF 2 W K n1 (1 + λVDS 1 ) (1 + λVDS 1 ) L 1 (16.6) 1184 Chapter 16 Analog Integrated Circuits IREF IO ID1 2 M 1 1 ID2 M2 10 1 VGS −VSS Figure 16.4 MOS current mirror with unequal (W/L) ratios. The mirror ratio is given by W (1 + λVDS2 ) L MR = 2 W (1 + λVDS 1 ) L 1 (16.7) In the ideal case (λ = 0) or for VDS2 = VDS 1 , the mirror ratio is set by the ratio of the W/L values of the two transistors. For the particular values in Fig. 16.4, this design value of the mirror ratio would be 5, and the output current would be I O = 5IREF . However, the differences in VDS will again create an error in the mirror ratio. Exercise: (a) Calculate the mirror ratio for the MOS current mirrors in the figure here for λ = 0. (b) For λ = 0.02 V−1 if VT N = 1 V, K n = 25 A/V2 , and I REF = 50 A. +10 V IREF IREF IO ID1 3 1 ID2 25 1 IO ID1 5 1 VGS ID2 2 1 VGS −15 V Answers: 8.33, 0.400; 10.4, 0.462 16.2.3 dc Analysis of the Bipolar Transistor Current Mirror Analysis of the BJT current mirror in Fig. 16.3(b) is similar to that of the FET. Applying KCL at the collector of “diode-connected” transistor Q 1 yields IREF = IC1 + I B1 + I B2 and I O = IC2 (16.8) 16.2 1185 Current Mirrors The currents needed to relate I O to IREF can be found using the transport model, noting that the circuit connection forces the two transistors to have the same base-emitter voltage VB E : IC1 = I S exp VB E VT 1+ VC E1 VA IC2 = I S exp β F1 = β F O I B1 = IS βF O VC E1 1+ VA VB E exp VT VB E VT 1+ β F2 = β F O I B2 = IS βF O VC E2 1+ VA VB E exp VT VC E2 VA (16.9) Substituting Eq. (16.9) into Eq. (16.7) and solving for I O = IC2 yields VC E2 VC E2 1+ 1+ VA VA = IREF I O = IREF VC E1 VB E 2 2 1+ 1+ + + VA βF O VA βF O (16.10) If the Early voltage were infinite, Eq. (16.10) would give a mirror ratio of MR = IO = IREF 1 1+ 2 (16.11) βF O and the output current would mirror the reference current, except for a small error due to the finite current gain of the BJT. For example, if β F O = 100, the currents would match within 2 percent. As for the FET case, however, the collector-emitter voltage mismatch in Eq. (16.10) is generally more significant than the current gain defect term, as indicated in Ex. 16.2. EXAMPLE 16.2 MIRROR RATIO CALCULATIONS Compare the mirror ratios for MOS and BJT current mirrors operating with similar bias conditions and output resistances (V A = 1/λ). PROBLEM Calculate the mirror ratio for the MOS and BJT current mirrors in Fig. 16.3 for VG S = 2 V, VDS2 = 10 V = VC E2 , λ = 0.02 V−1 , V A = 50 V, and β F O = 100. Assume M1 = M2 and Q1 = Q2. SOLUTION Known Information and Given Data: Current mirror circuits in Fig. 16.3 with M2 = M1 and Q 2 = Q 1 ; VSS = 10 V; operating voltages: VG S = 2 V, VDS2 = VC E2 = 10 V and VB E = 0.7 V; transistor parameters: λ = 0.02 V−1 , V A = 50 V, and β F O = 100 Unknowns: Mirror ratio MR for each current mirror Approach: Use Eqs. (16.7) and (16.10) to determine the mirror ratios. 1186 Chapter 16 Analog Integrated Circuits Assumptions: BJTs and MOSFETs are in the active region of operation respectively. Assume VB E = 0.7 V for the BJTs and the MOSFETs are enhancement-mode devices. Analysis: For the MOS current mirror, MRMOS 0.02 (10 V) 1+ (1 + λVDS2 ) V = 1.15 = = 0.02 (1 + λVDS 1 ) 1+ (2 V) V and for the BJT case 10 V VC E2 1+ VA 50 V = = 1.16 = 2 2 VC E1 0.7 V 1+ 1+ + + βF O VA 100 50 V 1+ MRBJT Check of Results: A double check shows our calculations to be correct. M1 is forced to be active by connection. M2 has VDS2 > VG S2 and will be pinched-off for VT N > 0 (enhancement-mode transistor). Q 1 has VC E = VB E , so it is forced to be in the active region. Q 2 has VC E2 > VB E2 and is also in the active region. The assumed regions of operation are valid. Discussion: The FET and BJT mismatches are very similar — 15 percent and 16 percent, respectively. The current gain error is a small contributor to the overall error in the BJT mirror ratio. Computer-Aided Analysis: We can easily perform an analysis of the current mirrors using SPICE, which will be done shortly as part of Ex. 16.3. Exercise: What is the actual value of VBE in the bipolar current mirror in Ex. 16.2 if I S = 0.1 fA and I REF = 100 A? What is the minimum value of the collector voltage required to maintain Q2 in the active region in Fig. 16.3(b)? Answers: 0.691 V; −VE E + 0.691 V 16.2.4 Altering the BJT Current Mirror Ratio In bipolar IC technology, the designer is free to modify the emitter area of the transistors, just as the W/L ratio can be chosen in MOS design. To alter the BJT mirror ratio, we use the fact that the saturation current of the bipolar transistor is proportional to its emitter area A E and can be written as IS = IS O AE A (16.12) I S O represents the saturation current of a bipolar transistor with one unit of emitter area: A E = 1 × A. The actual dimensions associated with A are technology-dependent. 16.2 IREF Current Mirrors 1187 IO IC1 IC2 Q1 AE1 Q2 IB1 + IB2 VBE – AE 2 AE 2 = nAE1 −VEE Figure 16.5 BJT current mirror with unequal emitter area. By changing the relative sizes of the emitters (emitter area scaling) of the BJTs in the current mirror, the IC designer can modify the mirror ratio. For the modified mirror in Fig. 16.5, IC1 = I S O I B1 = VC E1 VB E 1+ VT VA VB E A E1 exp A VT A E1 exp A IS O βF O IC2 = I S O I B2 = VC E2 VB E 1+ VT VA (16.13) VB E A E2 exp A VT A E2 exp A IS O βF O Substituting these equations in Eq. (16.7) and then solving for I O yields VC E2 VA I O = n IREF VB E 1+n 1+ + VA βF O 1+ where n = A E2 A E1 (16.14) In the ideal case of infinite current gain and identical collector-emitter voltages, the mirror ratio would be determined only by the ratio of the two emitter areas: MR = n (16.15) However, for finite current gain, MR = n 1+n 1+ βF O (16.16) For example, suppose A E2 /A E1 = 10 and β F O = 100; then the mirror ratio becomes MR = 10 1 1+ 11 100 = 9.01 (16.17) A relatively large error (10 percent) is occurring even though the effect of collector-emitter voltage mismatch has been ignored. For high mirror ratios, the current gain error term can become quite important because the total number of units of base current increases directly with the mirror ratio. 1188 Chapter 16 Analog Integrated Circuits Exercise: (a) Calculate the ideal mirror ratio for the BJT current mirrors in the figure below if V A = ∞ and β F O = ∞. (b) If V A = ∞ and β F O = 75. (c) If V A = 60 V, β F O = 75, and VBE = 0.7 V. +15 V IREF +15 V IREF IO A 0.5 A IO 5A 2A Answers: 0.500, 2.50; 0.490, 2.39; 0.606, 2.95 16.2.5 Multiple Current Sources Analog circuits often require a number of different current sources to bias the various stages of the design. A single reference transistor, M1 or Q 1 , can be used to generate multiple output currents using the circuits in Figs. 16.6 and 16.7. In Fig. 16.6, the unusual connection of the gate terminals through the MOSFETs is being used as a “short-hand” method to indicate that all the gates are connected together. Circuit operation is similar to that of the basic current mirror. The reference current enters the “diode-connected” transistor — here, the MOSFET M1 — establishing gatesource voltage VG S , which is then used to bias transistors M2 through M5 , each having a different W/L ratio. Because there is no current gain defect in MOS technology, a large number of output transistors can be driven from one reference transistor. Exercise: What are the four output currents in the circuit in Fig. 16.6 if I REF = 100 A and λ = 0 for all the FETs? Answers: 200 A; 400 A; 800 A; 50.0 A +10 V +5 V +12 V +10 V + +8 V VEB IREF 5 1 M1 + VGS – IO2 IO3 IO4 IO5 10 1 20 1 40 1 5 2 M2 M3 M4 M5 A − Q1 IREF IO2 A 5A 10 A Q2 Q3 Q4 IO3 IO4 Figure 16.6 Multiple MOS current sources generated from Figure 16.7 Multiple bipolar sources biased by one one reference voltage. reference device. 16.2 Current Mirrors 1189 Exercise: Recalculate the four output currents in the circuit in Fig. 16.6 if λ = 0.02 for all the FETs. Assume VGS = 2 V. Answers: 231 A; 423 A; 954 A; 55.8 A The situation is very similar in the pnp bipolar mirror in Fig. 16.7. Here again, the base terminals of the BJTs are extended through the transistors to simplify the drawing. In this circuit, reference current IREF is supplied by diode-connected BJT Q 1 to establish the emitter-base reference voltage VE B . VE B is then used to bias transistors Q 2 to Q 4 , each having a different emitter area relative to that of the reference transistor. Because the total base current increases with the addition of each output transistor, the base current error term gets worse as more transistors are added, which limits the number of outputs that can be used with the basic bipolar current mirror. The buffered current mirror in Sec. 16.2.6 was invented to solve this problem. An expression for the output current from a given collector can be derived following the steps that led to Eq. (16.14): 1+ I Oi = n i IREF VECi VA VE B 1+ + VA 1+ where n i = m ni A Ei A E1 (16.18) i=2 βF O Exercise: (a) What are the three output currents in the circuit in Fig. 16.5 if I REF = 10 A, β F O = 50, and V A = ∞ for all the BJTs? (b) Repeat for V A = 50 V and VE B = 0.7 V. Use Eq. (16.18). Answers: 7.46 A, 37.3 A, 74.6 A; 8.86 A, 44.3 A, 88.6 A 16.2.6 Buffered Current Mirror The current gain defect in the bipolar current mirror can become substantial when a large mirror ratio is used or if many source currents are generated from one reference transistor. However, this error can be reduced greatly by using the circuit in Fig. 16.8, called a buffered current mirror. The current gain of transistor Q 3 is used to reduce the base current that is subtracted from the reference current. Applying KCL at the collector of transistor Q 1 , and assuming that V A = ∞ IB3 IREF Q3 Q1 A IO Q2 IB1 IB2 VBE nA –VEE Figure 16.8 Buffered current mirror. 1190 Chapter 16 Analog Integrated Circuits for simplicity, IC1 is expressed as IC1 = IREF − I B3 = IREF − IC1 β F O1 β F O3 + 1 (1 + n) (16.19) and solving for the collector current yields I O = n IC1 = n IREF 1 (1 + n) 1+ β F O1 (β F O3 + 1) (16.20) The error term in the denominator has been reduced by a factor of (β F O3 + 1) from the error in Eq. (16.16). Exercise: What is the mirror ratio and the percent error for the buffered current mirror in Fig. 16.8 if β F O = 50, n = 10, and V A = ∞ for all the BJTs? (b) What is that value of VC E2 required to balance the mirror if β F O = ∞? Answers: 9.96, 0.430 percent; 1.4 V 16.2.7 Output Resistance of the Current Mirrors Now that we have found the dc output current of the current mirror, we will focus on the second important parameter that characterizes the electronic current source — the output resistance. The output resistance of the basic current mirror can be found by referring to the ac model of Fig. 16.9. Diode-connected bipolar transistor Q 1 represents a simple two-terminal device, and its smallsignal model is easily found using nodal analysis of Fig. 16.10: i = gπ v + gm v + go v = (gm + gπ + go )v (16.21) By factoring out gm , an approximate result for the diode conductance is i 1 1 ∼ = gm 1 + + = gm v βo µf and 1 R∼ = gm (16.22) for βo and µ f 1. The small-signal model for the diode-connected BJT is simply a resistor of value 1/gm . Note that this result is the same as the small-signal resistance rd of an actual diode that was developed in Sec. 13.4. i Q1 Q2 i Rout v rπ gmv v Figure 16.9 ac Model for the output resistance of the bipolar current mirror. Figure 16.10 Model for “diode-connected” transistor. ro 16.2 Q2 1 gm1 Rout = ro2 M2 M1 Rout = ro2 1 gm1 Rout M2 1191 Current Mirrors Figure 16.11 Simplified small-signal model for the bipolar current mirror. Figure 16.12 Output resistance of the MOS current mirror. Using this diode model simplifies the ac model for the current mirror to that shown in Fig. 16.11. This circuit should be recognized as a common-emitter transistor with a Thévenin equivalent resistance Rth = 1/gm connected to its base; the output resistance just equals the output resistance ro2 of transistor Q 2 . The equation describing the small-signal model for the two-terminal “diode-connected” MOSFET is similar to that in Eq. (16.22) except that the current gain is infinite. Therefore, the two-terminal MOSFET is also represented by a resistor of value 1/gm , as in Fig. 16.12; the output resistance of the MOS current mirror is equal to ro2 of MOSFET M2 . Thus, the output resistance and figure of merit (see Chapter 15) for the basic current mirror circuits are determined by output transistors Q 2 and M2 : Rout = ro2 and VC S ∼ = V A2 or 1 λ2 (16.23) Exercise: What are the output resistances of sources I O2 and I O3 in Fig. 16.4 for I REF = 100 A and Fig. 16.5 for I REF = 10 A if V A = 1/λ = 50 V? Answers: 260 k, 130 k; 6.77 M, 1.35 M 16.2.8 Two-Port Model for the Current Mirror We shall see shortly that the current mirror can be used not only as a dc current source but, in more complex circuits, as a current amplifier and active load. It will be useful to understand the small-signal behavior of the current mirror, redrawn as a two-port in Fig. 16.13. Because the current mirror has a current input and current output, the h-parameters represent a convenient parameter set to model the current mirror: v1 = h 11 i1 + h 12 v2 (16.24) i2 = h 21 i1 + h 22 v2 i2 i1 v1 Q1 A Q2 v2 nA Figure 16.13 Current mirror as a two-port. 1192 Chapter 16 Analog Integrated Circuits i1 v1 1 gm1 i1 i2 gm2v1 rπ 2 ro2 v2 v1 i2 gm2 i gm1 1 1 gm1 ro2 v2 Figure 16.14 Small-signal model for the Figure 16.15 Simplified small-signal current mirror. model for the current mirror. The small-signal model for the current mirror is in Fig. 16.14, in which diode-connected transistor Q 1 is represented in its simplified form by 1/gm1 . From the circuit in Fig. 16.14, v1 1 1 1 ∼ h 11 = = = = n i1 v2 =0 (gm1 + gπ 2 ) gm1 gm1 1 + βo2 v1 h 12 = =0 v2 i1 =0 (16.25) i2 gm2rπ 2 βo2 gm2 ∼ IC2 ∼ h 21 = = = =n = = gm1 i1 v2 =0 1 + gm1rπ 2 gm1 IC1 1+ βo2 gm2 i2 1 h 22 = = v2 i1 =0 ro2 Figure 16.15 shows the two-port model representation for these h-parameters. The bipolar current mirror has an input resistance of 1/gm1 determined by diode Q 1 and an output resistance equal to ro2 of Q 2 . The current gain is determined approximately by the emitter-area ratio n = A E2 /A E1 . Be sure to remember to use the correct values of IC1 and IC2 when calculating the values of the small-signal parameters. Analysis of the MOS current mirror yields similar results [or by simply setting βo2 = ∞ in Eq. (16.25)]: h 11 = h 21 1 gm1 W gm2 ∼ L 2 ∼ = = =n W gm1 L 1 h 12 = 0 h 22 = 1 ro2 (16.26) In this case, the current gain h 21 is determined by the W/L ratios of the two FETs rather than by the bipolar emitter-area ratio. Exercise: What are the values of I C1 and I C2 and the small-signal parameters for the current mirror in Fig. 16.5 if I REF = 100 A, β F O = 50, V A = 50 V, VBE = 0.7 V, VC E2 = 10 V, and n = 5? Answers: 89.4 A; 529 A; 280 ; 0; 5.92; 113 k 16.2 EXAMPLE 16.3 Current Mirrors 1193 CALCULATING THE TWO-PORT PARAMETERS OF A CURRENT MIRROR USING SPICE Transfer function analysis is used to find the two-port parameters of the BJT current mirror. PROBLEM Use the transfer function capability of SPICE to find the two-port parameters of the BJT current mirror biased by a reference current of 100 A and a power supply of +10 V. SOLUTION Known Information and Given Data: A current mirror using bipolar transistors; IREF = 100 A and VCC = 10 V Unknowns: Output current I O , VB E , h 11 , h 12 , h 21 , and h 22 for the current mirror Approach: Construct the circuit using the schematic editor in SPICE. Use the transfer function analysis to find the forward transfer function from IREF to I (VCC ), and reverse transfer function from VCC to node 1. The SPICE transfer function analysis automatically calculates three values: the requested transfer function, the resistance at the input source node, and the resistance at the output source node. However, since the output node is connected to VCC , the output resistance calculated at that node will be zero, and two analyses will be required to find all the two-port parameters. Assumptions: Use the current mirror with a single positive supply VCC biased by current source IREF as shown in the figure here. V A = 50 V, β F O = 100, and I S = 0.1 fA. I(VCC) VCC IREF 100 UA 10 V IO 1 2 Q1 Q2 Analysis: First, we must set the BJT parameters to the desired values: BF = 100, VAF = 50 V, and IS = 0.1 fA. An operating point and two transfer function analyses are used in this example. The first asks for the transfer function from input source IREF to output variable I (VCC ). The operating point analysis yields V (1) = 0.719 V and I O = 116 A. The transfer function analysis gives input resistance h 11 = (1/259 ) and current gain h 21 = +1.16. The second analysis requests the transfer function from voltage source VCC to node 1. SPICE analysis gives h 22 = 510 k, and h 12 = 2.59 × 10−12 . Check of Results: Based on equation set (16.25) and the operating point results, we expect h 11 = (1/250 ) h 12 = 0 h 21 = +1.16 h 22 = 517 k and we see that agreement with theory is very good. 1194 Chapter 16 Analog Integrated Circuits Discussion: One should always try to understand and account for the differences between our theory and SPICE. In this example, the input resistance difference can be traced to the use of VT = 25.9 mV. The nonzero value for h 12 simply resulted from numerical noise in the calculation and is as close to zero as the computer could achieve in this particular case. Be careful not to make a sign error in interpreting the data for h 21 . A negative sign appears in the SPICE output because of the assumed polarity of VCC and I (VCC ). Finally, the SPICE model uses ro = (V A + VC B )/IC = 511 k, accounting for the small difference in the values of h 22 . Exercise: Use the transfer function capability of SPICE to find the two-port parameters for a MOS current mirror biased by a reference current of 100 A and a power supply of +10 V. Assume K n = 1 mA/V2 , VT N = 0.75 V, and λ = 0.02/V. Answers: I O = 117 A, VGS = 1.19 V; 4.55 mS, 0, 1.17, 512 k Exercise: Compare the answers in the previous exercise to hand calculations. Answers: I O = 117 A with VGS = 1.20 V; 4.47 mS, 0, 1.17, 513 k 16.2.9 The Widlar Current Source Resistor R in the Widlar2 current source circuit shown in the schematic in Fig. 16.16 gives the designer an additional degree of freedom in adjusting the mirror ratio of the current mirror. In this circuit, the difference in the base-emitter voltages of transistors Q 1 and Q 2 appears across resistor R and determines the output current I O . Transistor Q 3 buffers the mirror reference transistor in Fig. 16.16(b) to minimize the effect of finite current gain. IREF Q1 + AE1 – VBE1 (a) + IO Q2 + VBE2 – AE2 + VBE1 – VBE2 R – + IREF + IO Q3 Q1 AE1 Q2 + VBE1 – VBE2 AE2 R – (b) Figure 16.16 (a) Basic Widlar current source and (b) buffered Widlar source. 2 Robert Widlar was a famous IC designer who made many lasting contributions to analog IC design. For examples, see references 2 and 3. 16.2 Current Mirrors 1195 An expression for the output current may be determined from the standard expressions for the base-emitter voltage of the two bipolar transistors. In this analysis, we must accurately calculate the individual values of VB E1 and VB E2 because the behavior of the circuit depends on small differences in the values of these two voltages. Assuming high current gain, IREF ∼ IREF VB E1 = VT ln 1 + = VT ln I S1 I S1 and (16.27) VB E2 = VT ln 1 + IO I S2 IO ∼ = VT ln I S2 The current in resistor R is equal to I E2 VB E1 − VB E2 VT = = ln R R IREF I S2 I O I S1 (16.28) If the transistors are matched, then I S1 = (A E1 /A)I S O and I S2 = (A E2 /A)I S O , and Eq. (16.28) can be rewritten as VT IREF A E2 ∼ I O = α F I E2 = (16.29) ln R I O A E1 If IREF , R, and the emitter-area ratio are all known, then Eq. (16.29) represents a transcendental equation that must be solved for I O . The solution can be obtained by iterative trial and error or by using Newton’s method. Widlar Source Output Resistance The ac model for the Widlar source in Fig. 16.16(a) represents a common-emitter transistor with resistor R in its emitter and a small value of Rth (= 1/gm1 ) from diode Q 1 in its base, as indicated in Fig. 16.17. In normal operation, the voltage developed across resistor R is usually small (≤ 10VT ). Referring to Table 14.1, or by simplifying Eq. (15.182) for this case, we can reduce the output resistance of the source to IO R Rout ∼ (16.30) = ro2 [1 + gm2 R] = ro2 1 + VT in which I O R can be found from Eq. (16.28): IREF A E2 ∼ Rout = ro2 1 + ln = K ro2 I O A E1 and VC S ∼ = K V A2 Q2 1 gm1 Rout = Kro2 R Figure 16.17 Widlar source output resistance – K = 1 + ln[(IREF /IC2 )(A E2 /A E1 )] (16.31) 1196 Chapter 16 Analog Integrated Circuits where IREF A E2 K = 1 + ln I O A E1 For typical values, 1 < K < 10. Exercise: What value of R is required to set I O = 25 A if I REF = 100 A and AE2 /AE1 = 5? What are the values of output resistance and K in Eq. (16.33) for this source if V A + VC E = 75 V? Answers: 3000 ; 12 M, 4 Exercise: Find the output current in the Widlar source if I REF = 100 A, R = 100 , and AE2 = 10AE1 . What are the values of output resistance and K in Eq. (16.31) for this source if V A + VC E = 75 V? Answers: 301 A; 551 k, 2.20 16.2.10 The PTAT Voltage The voltage developed across resistor R in Fig. 16.16 represents an extremely useful quantity because it is directly proportional to absolute temperature (referred to as PTAT). VPTAT is equal to the difference in the two base-emitter voltages described by Eq. (16.27): kT IC1 A E2 IC1 A E2 VPTAT = VB E1 − VB E2 = VT ln = (16.32) ln IC2 A E1 q IC2 A E1 and the change of VPTAT with temperature is ∂ VPTAT k = + ln ∂T q IC1 A E2 IC2 A E1 =+ VPTAT T (16.33) For example, suppose T = 300 K, IC1 = IC2 and A E2 = 10A E1 . Then VPTAT = 59.6 mV with a temperature coefficient of slightly less than +0.2 mV/K. The PTAT voltage developed in the Widlar cell, combined with an analog-to-digital converter, forms the heart of all of today’s highly accurate electronic thermometers. We will see the PTAT voltage again shortly in the form of the bandgap voltage reference, another extremely important circuit building block. ELECTRONICS IN ACTION PTAT Voltage Based Digital Thermometry The PTAT generator produces a well-defined output voltage that is used in many of today’s digital thermometers as in the block diagram on the next page. The output of the PTAT circuit 16.2 Current Mirrors 1197 + A PTAT generator A/D converter n – Display is scaled and the offset voltage is shifted to provide an output voltage that directly represents either the Fahrenheit or Celsius temperature scales. The analog voltage is converted to a digital representation by an A/D converter and the digital output is sent to an alphanumeric display. The scaling and offset shift can also be easily done in digital form after the A/D conversion operation is performed. Scaling and offset Bandgap reference Block diagram of a digital thermometer. Wireless digital thermometer. 16.2.11 The MOS Version of the Widlar Source Figure 16.18 is the MOS version of the Widlar source. In this circuit, the difference between the gate-source voltages of transistors M1 and M2 appears across resistor R, and I O can be expressed as IO = VG S 1 − VG S2 = R 2IREF − K n1 2I O K n2 R or (16.34) IO = 1 R 2IREF K n1 1− I O (W/L)1 IREF (W/L)2 1198 Chapter 16 Analog Integrated Circuits +VDD IREF IO M1 M2 M4 Rout 1 + gm1 VGS1 − VGS2 R R – (a) (b) Figure 16.18 (a) MOS Widlar source and (b) small-signal model. Dividing through by IREF , IO 1 = IREF R 2 K n1 IREF 1− I O (W/L)1 IREF (W/L)2 (16.35) If I O is known, then IREF can be calculated directly from Eq. (16.34). If IREF , R, and √ the W/L ratios are known, then Eq. (16.35) can be written as a quadratic equation in terms of I O /IREF : IO IREF 2 1 + R 2 K n1 IREF (W/L)1 (W/L)2 IO IREF 1 − R 2 =0 K n1 IREF (16.36) MOS Widlar Source Output Resistance In Fig. 16.18(b), the small-signal model for the MOS Widlar source is recognized as a commonsource stage with resistor R in its source. Therefore, from Table 14.1, Rout = ro2 (1 + gm2 R) (16.37) Exercise: (a) Find the output current in Fig. 16.18(a) if I REF = 200 A, R = 2 k, and K n2 = 10K n1 = 250 A/V2 . (b) What is Rout if λ = 0.02/V and VDS = 10 V? Answers: 764 A; 176 k 16.3 HIGH-OUTPUT-RESISTANCE CURRENT MIRRORS In the discussion of differential amplifiers in Chapter 15, we found that current sources with very high output resistances are needed to achieve good CMRR. The basic current mirrors discussed in the previous sections have a figure of merit VC S equal to V A or 1/λ; that for the Widlar source is only a few times higher. This section continues our introduction to current mirrors by discussing two additional circuits, the Wilson current source and the cascode current source, which enhance the value of VC S to the order of βo V A or µ f /λ. 16.3 1199 High-Output-Resistance Current Mirrors IO IO 0 M3 IB3 VGS + VBE – ID2 IREF M2 IC2 IREF ID1 0 + IB1 IB2 VCE2 + – A Q2 VBE Q1 – M1 VGS –VSS Q3 A + VCE1 – A −VEE Figure 16.19 MOS Wilson current Figure 16.20 Original Wilson current source. source circuit using BJTs. 16.3.1 The Wilson Current Sources The Wilson current sources [4] depicted in Figs. 16.19 and 16.20 use the same number of transistors as the buffered current mirror but achieve much higher output resistance; it is often used in applications requiring precisely matched current sources. In the MOS version, the output current is taken from the drain of M3 , and M1 and M2 form a current mirror. During circuit operation, the three transistors are all pinched-off and in the active region. Because the gate current of M3 is zero, I D2 must equal reference current IREF . If the transistors all have the same W/L ratios, then VG S3 = VG S 1 = VG S because I D3 = I D1 The current mirror requires I D2 = I D1 1 + 2λVG S 1 + λVG S and because I O = I D3 and I D3 = I D1 , the output current is given by 1 + λVG S I O = IREF 1 + 2λVG S where VG S ∼ = VT N + 2IREF Kn (16.38) For small λ, I O ∼ = IREF . For example, if λ = 0.02/V and VG S = 2 V, then I O and IREF differ by 3.7 percent. The Wilson source actually appeared first in bipolar form as drawn in Fig. 16.20. The circuit operates in a manner similar to the MOS source, except for the loss of current from IREF to the base of Q 3 and the current gain error in the mirror formed by Q 1 and Q 2 . Applying KCL at the base of Q 3 , IREF = IC2 + I B3 (16.39) in which IC2 and I B3 are related through the current mirror formed by Q 1 and Q 2 : IC2 = 1+ 1+ 2VB E VA VB E 2 + VA βF O I E3 = 1+ 1+ 2VB E VA VB E 2 + VA βF O Note in Fig. 16.20 that VC E1 = VB E and VC E2 = 2VB E . (β F O + 1)I B3 (16.40) 1200 Chapter 16 Analog Integrated Circuits Directly combining Eqs. (16.37) and (16.38) yields a messy expression that is difficult to interpret. However, if we assume the error terms are small, then we can eventually reduce (with considerable effort) the expression to the following approximate result: IO ∼ = IREF 1+ 1+ VB E VA 2 β F O (β F O + 2) + (16.41) 2VB E VA For β F O = 50, V A = 60 V, and VB E = 0.7 V, the mirror ratio is 0.988. The primary source of error results from the collector-emitter voltage mismatch between transistors Q 1 and Q 2 . The base current error has been reduced to less than 0.1 percent of IREF . The errors due to drain-source voltage mismatch in Fig. 16.19, or collector-emitter voltage mismatch in Fig. 16.20, may still be too large for use in precision circuits, but this problem can be significantly reduced by adding one more transistor to balance the circuit. In Fig. 16.21, transistor Q 4 reduces the collector-emitter voltage of Q 2 by one VB E drop and balances the collector-emitter voltages of Q 1 and Q 2 : VC E2 = VB E1 + VB E3 − VB E4 = VB E + VB E − VB E = VB E All four transistors are operating at approximately the same value of collector current, and the values of VB E are all the same if the devices are matched with equal emitter areas. IREF C4 C3 B4 B3 E4 IO IREF Q4 A Q2 A + VBE – + VBE – (a) Q3 A VEE Q1 A IO E3 C2 C1 B2 B1 E2 E1 −VEE (b) Figure 16.21 (a) Wilson source using balanced collector-emitter voltages. (b) Layout of Wilson source. Exercise: Draw a voltage-balanced version of the MOS Wilson source by adding one additional transistor to the circuit in Fig. 16.19. Answer: See Prob. 16.43. 16.3 High-Output-Resistance Current Mirrors ro3 v3 gm3vgs vgs ix ro2 v2 i 1 gm1 1201 vx v1 i = ix Figure 16.22 Small-signal model for the MOS version of the Wilson source. 16.3.2 Output Resistance of the Wilson Source The primary advantage of the Wilson source over the standard current mirror is its greatly increased output resistance. The small-signal model for the MOS version of the Wilson source is given in Fig. 16.22, in which test current i x is applied to determine the output resistance. The current mirror formed by transistors M1 and M2 is represented by its simplified two-port model assuming n = 1. Voltage vx is determined from vx = v3 + v1 = [ix − gm3 vgs ]ro3 + v1 (16.42) where vgs = v2 − v1 with v1 = ix and v2 = −µ f 2 v1 gm1 Combining these equations, and recognizing that gm1 = gm2 for n = 1 yields vx 1 ∼ Rout = = ro3 µ f 2 + 2 + = µ f 2ro3 ix µf2 (16.43) (16.44) and VC S = I D3 µ f 2 1 + λ3 VDS3 ∼ µ f 2 = λ3 I D3 λ3 (16.45) Analysis of the bipolar source is somewhat more complex because of the finite current gain of the BJT and yields the following result: βo3ro3 Rout ∼ = 2 and βo V A VC S ∼ = 2 (16.46) Derivation of this equation is left for Prob. 16.41. Exercise: Calculate Rout for the Wilson source in Fig. 16.20 if β F = 150, V A = 50 V, VE E = 15 V, and I O = I REF = 50 A. What would be the output resistance of a standard current mirror operating at the same current? Answer: 96.6 M versus 1.30 M 1202 Chapter 16 Analog Integrated Circuits Exercise: Use SPICE to find the output current and output resistance of the Wilson source in the previous exercise. Answers: I O = 49.5 A; 118 M 16.3.3 Cascode Current Sources We learned in Chapter 15 that the output resistance of the cascode connection (C-E/C-B cascade) of two transistors is very high, approaching µ f ro for the FET case and βo ro for the BJT circuit. Figure 16.23 shows the implementation of the MOS and BJT cascode current sources using current mirrors. +VDD ID3 IREF IO M3 I + D1 VDS1 M1 – (a) +VCC IC3 + VGS – IREF M4 ID2 + + VGS – Q3 + VCE1 Q1 – M2 VDS2 – IO + VBE – IC1 + VBE – Q4 IC2 + Q2 VCE2 – (b) Figure 16.23 (a) MOS and (b) BJT cascode current sources. In the MOS circuit in Fig. 16.23(a), I D1 = I D3 = IREF . The current mirror formed by M1 and M2 forces the output current to be approximately equal to the reference current because I O = I D4 = I D2 . Diode-connected transistor M3 provides a dc bias voltage to the gate of M4 and balances VDS 1 and VDS2 . If all transistors are matched with the same W/L ratios, then the values of VG S are all the same, and VDS2 equals VDS 1 : VDS2 = VG S 1 + VG S3 − VG S4 = VG S + VG S − VG S = VG S = VDS 1 Thus the M1 -M2 current mirror is precisely balanced, and I O = IREF . The BJT source in Fig. 16.23(b) operates in the same manner. For β F = ∞, IREF = IC3 = IC1 on the reference side of the source. Q 1 and Q 2 form a current mirror, which sets I O = IC4 = IC2 = IC1 = IREF . Diode Q 3 provides the bias voltage at the base of Q 4 needed to keep Q 2 in the active region and balances the collector-emitter voltages of the current mirror: VC E2 = VB E1 + VB E3 − VB E4 = 2VB E − VB E = VB E = VC E1 16.3.4 Output Resistance of the Cascode Sources Figure 16.24 shows the small-signal model for the MOS cascode source; the two-port model has been used for the current mirror formed of transistors M1 and M2 . Because current i represents the gate current of M4 , which is zero, the circuit can be reduced to that on the right in Fig. 16.24, which should be recognized as a common-source stage with resistor ro2 in its source. Thus, its output 16.3 ib M4 M4 gm3 Rth Rout i=0 1 gm2 i Rth Q4 1 gm3 Rout 1 1203 High-Output-Resistance Current Mirrors Rout i ro2 1 gm2 i ro2 ro2 Figure 16.25 Small-signal model Figure 16.24 Small-signal model for the MOS cascode source. for the BJT cascode source. resistance is Rout = ro4 (1 + gm4ro2 ) ∼ = µ f 4ro2 and µf4 ∼ µf4 VC S ∼ = = λ2 λ4 (16.47) Analysis of the output resistance of the BJT source in Fig. 16.25 is again more complex because of the finite current gain of the BJT. If the base of Q 4 were grounded, then the output resistance would be just equal to that of the cascode stage, βo ro . However, the base current i b of Q 4 enters the current mirror, doubles the output current, and causes the overall output resistance to be reduced by a factor of 2: βo4ro4 Rout ∼ = 2 and βo4 V A4 VC S ∼ = 2 (16.48) Detailed calculation of this result is left as Prob. 16.67. Exercise: Calculate the output resistance of the MOS cascode current source in Fig. 16.23(a) and compare it to that of a standard current mirror if I O = I REF = 50 A, VD D = 15 V, K n = 250 A/V2 , VT N = 0.8 V, and λ = 0.015 V−1 . Answer: 379 M versus 1.63 M including all λVDS terms Exercise: Use SPICE to find the output current and output resistance of the cascode current source in the previous exercise. Answers: I O = 50.0 A; 382 M Exercise: Calculate the output resistance of the BJT cascode current source in Fig. 16.23(b) and compare it to that of a standard current mirror if I O = I REF = 50 A, VCC = 15 V, β o = 100, and V A = 67 V. Answer: 81.3 M versus 1.63 M 1204 Chapter 16 Analog Integrated Circuits 16.3.5 Current Mirror Summary Table 16.2 is a summary of the current mirror circuits discussed in this chapter. The cascode and Wilson sources can achieve very high values of VC S and often find use in the design of differential and operational amplifiers as well as in many other analog circuits. TABLE 16.2 Comparison of the Basic Current Mirrors TYPE OF SOURCE Resistor Two-transistor mirror DESIGN EXAMPLE 16.4 Rout VCS TYPICAL VALUES OF VCS R VE E 15 V ro Cascode BJT βo ro 2 Cascode FET µ f ro BJT Wilson βo ro 2 FET Wilson µ f ro 1 V A or λ βo V A 2 µf λ βo V A 2 µf λ 75 V 5000 V >5000 V 5000 V >5000 V ELECTRONIC CURRENT SOURCE DESIGN Design an IC current source to meet a given set of specifications. PROBLEM Design a 1:1 current mirror with a reference current of 25 A and a mirror ratio error of less than 0.1 percent when the output is operating from a 20-V supply. Devices with these parameters are available: β F O = 100, V A = 75 V, I S O = 0.5 fA; K n = 50 A/V2 , VT N = 0.75 V, and λ = 0.02/V. SOLUTION Known Information and Given Data: IREF = 25 A. A mirror ratio error of less than 0.1 percent requires an output current of 25 A ± 25 nA when the output voltage is 20 V. Either a bipolar or MOS realization is acceptable. Unknowns: Current source configuration; transistor sizes Approach: The specifications define the required values of Rout and VC S . Use this information to choose a circuit topology. Complete the design by choosing device sizes based on the output resistance expressions for the selected circuit topology. Assumptions: Room temperature operation; devices are in the active region of operation. Analysis: The output resistance of the current source must be large enough that 20 V applied across the output does not change (increase) the current by more than 25 nA. Thus, the output resistance must satisfy Rout ≥ 20 V/25 nA = 800 M. Let us choose Rout = 1 G to provide some safety margin. The effective current source voltage is then VC S = 25 A (1 G) = 25,000 V! From Table 16.2 we see that either a cascode or Wilson source will be required to 16.3 High-Output-Resistance Current Mirrors 1205 meet this value of VC S . In fact, the source must be an MOS version, since our BJTs can at best reach VC S = 100(75 V)/2 = 3750 V. The choice between the Wilson and cascode sources is arbitrary at this point. Let us pick the cascode source, which does not involve an internal feedback loop. In order to achieve the small mirror error, a voltage-balanced version is required. Our final circuit choice is therefore the circuit shown in Fig. 16.23(a). Now we must choose the device sizes. In this case, the W/L ratios are all the same since we require MR = 1. Again referring to Table 16.2, the required amplification factor for the transistor is 0.02 µ f = λVC S = (25,000 V) = 500 V and the MOS transistor’s amplification factor is given approximately by µ f = gm ro ∼ = 2K n I D 1 λI D Using µ f = 500, λ = 0.02/V, and I D = 25 A gives a value of K n = 1.25 mS. Since K n = K n (W/L), we need a W/L ratio of 25/1 for the given technology. (This W/L ratio is easy to achieve in integrated circuit form.) In this circuit all the transistors are operating at the same current, so that the W/L ratios should all be the same size in order to maintain the required voltage balance. Check of Results: Let us check the calculations by directly calculating the output resistance of the source. Rout ∼ = gm4ro4ro2 gm4 = 2K n I D (1 + λVDS4 ) ro = (1/λ) + VDS ID We can either neglect the values of VDS in these expressions, or we can calculate them. In order to best compare with simulation, let us find VDS and the corresponding values of gm and ro . VDS2 = VG S2 = VT N + 2I D = 0.75 + Kn 50 A = 0.95 V 1.25 mS VDS4 = 20 − VDS2 = 19.0 V gm4 = 2K n I D (1 + λVDS4 ) = 2(1.25 mA/V2 )(25 A)[1 + .02(19)] = 0.294 mS ro2 = (1/λ) + VDS2 51.0 V = 2.04 M = ID 25 A ro4 = (1/λ) + VDS4 69.0 = 2.76 M = ID 25 A Multiplying the small-signal parameters together produces an output resistance estimate of 1.65 G, which exceeds the design requirement that we originally calculated from the design specifications. Discussion: Note that our ability to set the amplification factor of the MOS transistor was very important in achieving the design goals. In this case µ f 4 = 811. A possible layout for the cascode current source is presented in the figure. The four 25/1 NMOS transistors are stacked 1206 Chapter 16 Analog Integrated Circuits vertically. G 1 and G 2 are the gates of the current mirror transistors. Gates G 1 and G 3 are connected directly to their respective drains. The drain of M1 and the source of M3 are merged as are those of M2 and M4 . However, there are no contacts required to the connection between the drain of M2 and the source of M4 . VDD G4 G2 G1 G3 IREF Computer-Aided Analysis: SPICE represents a good way to double check the results. First, we must set the MOS device parameters: KP = 50 A/V2 , VTO = 0.75 V, LAMBDA = 0.02/V, W = 25 m, and L = 1 m. A dc simulation of the final circuit (shown next) with the given device parameters yields an output current of 25.014 A. In addition, the voltages at the drains of M1 and M2 are 0.948 V and 0.976 V, respectively, indicating that the voltage balancing is working as desired. A transfer function analysis from source VD D to the output node yields an output resistance of 1.66 G, easily meeting the specifications with a satisfactory safety margin. We also have good agreement with the value of Rout that we calculated by hand. M3 M4 M1 M2 VDD IREF 25 UA 20 V Exercise: In the SPICE results in Design Ex. 16.4, I O = 25.015 A at VD D = 20 V. If Rout = 1.66 G, what will be the output current at VD D = 10 V? Answer: 25.008 A 16.4 Reference Current Generation 1207 Exercise: What is the minimum value of VD D for which M4 remains in the active region of operation? Answer: 1.15 V Exercise: Repeat the design in Design Ex. 16.4 for a current source with a mirror ratio of 2 ± 0.1 percent. Answers: ( W/L) 3 = ( W/L) 1 = 25/1; ( W/L) 4 = ( W/L) 2 = 50/1 16.4 REFERENCE CURRENT GENERATION A reference current is required by all the current mirrors that have been discussed. The least complicated method for establishing this reference current is to use resistor R, as shown in Fig. 16.26(a). +VDD M4 R M3 IO IREF IO IREF Q1 Q2 M1 VBE M2 −VSS −VEE (a) (b) Figure 16.26 Reference current generation for current mirrors: (a) resistor reference and (b) series-connected MOSFETs. However, the source’s output current is directly proportional to the supply voltage VE E : IREF = VE E − VB E R (16.49) In MOS technology, the gate-source voltages of MOSFETs can be designed to be large, and several MOS devices can be connected in series between the power supplies to eliminate the need for large-value resistors. An example of this technique is given in Fig. 16.26(b), in which VD D + VSS = VSG4 + VG S3 + VG S 1 and the drain currents must satisfy I D1 = I D3 = I D4 . However, any change in the supply voltages directly alters the values of the gate-source voltages of the three MOS transistors and again changes the reference current. Note that the series device technique is not usable in bipolar technology because of the small fixed voltage (∼ = 0.7 V) developed across each diode as well as the exponential relationship between voltage and current in the diode. 1208 Chapter 16 Analog Integrated Circuits Exercise: What is the reference current in Fig. 16.26(a) if R = 43 k and VE E = −5 V? (b) If VE E = −7.5 V? Answers: 100 A; 158 A Exercise: What is the reference current in Fig. 16.26(b) if K n = K p = 400 A/V2 , VT N = −VT P = 1 V, and VSS = −5 V? (b) If VSS = −7.5 V? Answers: 88.9 A; 450 A. Note: the variation is worse than in the resistor bias case because of the square-law MOSFET characteristic. 16.4.1 Supply-Independent Biasing In most cases, the supply voltage dependence of IREF is undesirable. For example, we would like to fix the bias points of the devices in general-purpose op amps, even though they must operate from power supply voltages ranging from ±3 V to ±22 V. In addition, Eq. (16.47) indicates that relatively large values of resistance are required to achieve small operating currents, and these resistors use significant area in integrated circuits, as was discussed in detail in Sec. 6.6.9. Thus, a number of circuit techniques that yield currents relatively independent of the power supply voltages have been invented. Some bipolar technologies offer the capability of fabricating p-channel JFETs, which can be used to set a fixed reference current, as shown in Fig. 16.27. For this circuit, the JFET is operating with VSG = 0, and therefore I D = I DSS , assuming that VS D is large enough to pinch off the JFET. In MOS technology, depletion-mode devices can be used in a similar manner, if available. However, because both these circuit techniques require special IC processes and therefore lack generality, other methods are preferred. A VB E -Based Reference One possibility is the VB E -based reference, shown in Fig. 16.28, in which the output current is determined by the base-emitter voltage of Q 1 . For high current gain, the collector current of Q 1 is equal to the current through resistor R1 , IC1 = J3 VE E − VB E1 − VB E2 ∼ VE E − 1.4 V = R1 R1 (16.50) IO IO IDSS Q2 Q1 Q2 R1 Q1 VBE1 −VEE R2 VEE Figure 16.27 Constant reference Figure 16.28 VB E -based current from a JFET. current source. 16.4 Reference Current Generation and the output current I O is approximately equal to the current in R2 : VB E1 VB E1 ∼ 0.7 V I O = α F2 I E2 = α F2 + I B1 ∼ = = R2 R2 R2 1209 (16.51) Rewriting VB E1 in terms of VE E , VE E − 1.4 V VT IO ∼ ln = R2 I S1 R1 (16.52) A substantial degree of supply-voltage independence has been achieved because the output current is now only logarithmically dependent on changes in the supply voltage VE E . However, the output current is temperature dependent due to the temperature coefficients of both VB E and resistor R. Exercise: (a) Calculate I O in Fig. 16.28 for I S = 10−16 A, R1 = 39 k, R2 = 6.8 k, and VE E = −5 V. Assume infinite current gains. (b) Repeat for VE E = −7.5 V. Answers: 101 A; 103 A The Widlar Source Actually, we already discussed another source that achieves a similar independence from power supply voltage variations. The expression for the output current of the Widlar source given in Fig. 16.16 and Eq. (16.29) is IREF A E2 VT ∼ I O = α F I E2 = (16.53) ln R I O A E1 Here again, the output current is only logarithmically dependent on the reference current IREF (which may be proportional to VCC ). Power-Supply-Independent Bias Cell Bias circuits with an even greater degree of power supply voltage independence can be obtained by combining the Widlar source with a standard current mirror, as indicated in the circuit in Fig. 16.29. Assuming high current gain, the pnp current mirror forces the currents on the two A A +VCC Q3 Q4 IC1 IC2 Q1 Q2 A 20 A R –VEE Figure 16.29 Power-supply-independent bias circuit using the Widlar source and a current mirror. 1210 Chapter 16 Analog Integrated Circuits sides of the reference cell to be equal — that is, IC1 = IC2 . In addition, the emitter-area ratio of the Widlar source in Fig. 16.29 is equal to 20. With these constraints, Eq. (16.53) can be satisfied by an operating point of VT 0.0749 V IC2 ∼ ln(20) = = R R (16.54) In this example, a fixed voltage of approximately 75 mV is developed across resistor R, and this voltage is independent of the power supply voltages. Resistor R can then be chosen to yield the desired operating current. Obviously, a wide range of mirror ratios and emitter-area ratios can be used in the design of the circuit in Fig. 16.29. Although the current, once established, is independent of supply voltage, the actual value of IC still depends on temperature as well as the absolute value of R and varies with run-to-run process variations. Unfortunately, IC1 = IC2 = 0 is also a stable operating point for the circuit in Fig. 16.29. Start-up circuits must be included in IC realizations of this reference to ensure that the circuit reaches the desired operating point. Exercise: Find the output current in the current source in Fig. 16.29 if AE3 = 10AE4 , AE2 = 10AE1 , and R = 1 k. Answer: 115 A Exercise: What is the minimum power supply voltage for proper operation of the supplyindependent bias circuit in Fig. 16.29? Answer: 2VBE ∼ = 1.4 V Once the current has been established in the reference cell consisting of Q 1 –Q 4 in Fig. 16.29, the base-emitter voltages of Q 1 and Q 4 can be used as reference voltages for other current mirrors, as shown in Fig. 16.30. In this figure, buffered current mirrors have been used in the reference +VCC R8 AE 3 Q3 AE 4 AE 7 Q4 Q7 AE 8 Q8 Q9 Q10 Q6 Q5 AE6 R6 Q1 AE 5 Q2 AE1 AE 2 R – VEE Figure 16.30 Multiple source currents generated from the supply-independent cell. 16.4 Reference Current Generation 1211 cell to minimize errors associated with finite current gains of the npn and pnp transistors. Output currents are shown generated from basic mirror transistors Q 5 and Q 7 and from Widlar sources, Q 6 and Q 8 . 16.4.2 A Supply-Independent MOS Reference Cell The MOS analog of the circuit in Fig. 16.29 appears in Fig. 16.31. In this circuit, the PMOS current mirror forces a fixed relationship between drain currents I D3 and I D4 . For the particular case in Fig. 16.31, I D3 = I D4 , and so I D1 = I D2 . Substituting this constraint into Eq. (16.35) yields an equation for the value of R required to establish a given current I D2 : R= 2 K n1 I D2 1− (W/L)1 (W/L)2 (16.55) Based on Eq. (16.55), we see that the MOS source is independent of supply voltage but is a function of the absolute values of R and K n . +VDD 5 M 3 1 M4 5 1 ID3 ID4 ID1 ID2 5 M 1 1 M2 50 1 R –VSS Figure 16.31 Supply-independent current source using MOS transistors. Exercise: What value of R is required in the current source in Fig. 16.31 if I D2 is to be designed to be 100 A and K n = 25 A/V2 ? Answer: 8.65 k 16.4.3 Variation of Reference Cell Current with Power Supply Variations (Advanced Topic) Analysis of the bias circuits in this section has ignored the influence of the output resistance of the transistors, and the current is actually affected somewhat by changes in the power supply voltages. For small changes in supply voltages, the small-signal models of the circuit in Fig. 16.32(a) can be used to relate the changes in cell currents to the changes in power supply voltages. These two 1212 Chapter 16 Analog Integrated Circuits 1 gm4 1 gm4 M3 ro3 gm4 ni Q3 i ∆V ∆V v1 Q2 M2 1 gm1 1 gm1 R ix 1 1 gm1 v2 g'm2v1 gm2 1 + gm2R ro2 = ro2 (1 + gm2R) gm2 = vx ro2 ' R (b) (a) Figure 16.32 (a) Small-signal models for the reference cell current variations. (b) Simplified small-signal model. circuits are redrawn in simplified form in Fig. 16.32(b), in which resistor R has been absorbed into the model for transistor M2 or Q 2 (see Prob. 14.56). Source vx represents the total change in the supply voltages vx = VCC + VE E vx = VD D + VSS (16.56) ix = gm1 v1 + gm2 v1 + go2 v2 = (gm1 + gm2 )v1 + go2 v2 (16.57) or and the current i x is expressed as Node voltages v1 and v2 must both be found in order to determine ix . Writing the nodal equations for the circuit using i = gm4 (vx − v2 ) and collecting terms, (ngm4 + go3 )vx = (gm1 + go3 )v1 + ngm4 v2 gm4 vx = gm2 v1 + (gm4 + go2 )v2 (16.58) Calculating the determinant of this system of equations yields 2 3 g g = gm1 gm4 1 − n m2 + O m gm1 µF (16.59) Solving Eq. (16.58) for v1 and v2 yields: vx gm go v1 = gm4 (go3 + +O µf 2 gm2 gm vx v2 = gm1 gm4 1 − n +O gm1 µf ngo2 ) 3 O( x) = terms of the order of x. (16.60) 16.4 Reference Current Generation 1213 Substituting the results from Eqs. (16.59) and (16.60) into Eq. (16.57) produces ix = vx go3 g 1 + m2 gm1 1−n + go2 (1 + n) (16.61) gm2 gm1 Equation (16.61) represents the sum of two conductance terms, and therefore the output resistance can be represented as the parallel combination of two equivalent resistances: ro3 ro2 gm2 1 − n = g 1 + n gm1 1 + m2 gm1 Rout for n gm2 <1 gm1 (16.62) Equations (16.61) and (16.62) can be interpreted by referring to Fig. 16.32(b). In the lefthand branch of the circuit, voltage change vx appears almost entirely across ro3 . The current in ro3 is amplified by the current gain of the Widlar source (gm2 /gm1 ), yielding a total current i = go3 (1 + gm2 /gm1 )vx . In the middle branch of the circuit, the voltage change appears almost entirely across ro2 , and this current is amplified by the gain n of the upper current mirror, producing a total current of i = go2 (1 + n)vx . In addition, this circuit represents a positive feedback amplifier with a loop gain n(gm2 /gm1 ). Because of this positive feedback, the overall output resistance is reduced by the factor /gm1 )]. Note carefully that circuit stability requires n(gm2 /gm1 ) < 1. Negative resis[1 − n(gm2 tance would cause instability, and the circuit designer must be careful not to violate this condition. Further discussion of feedback circuits is postponed to Chapter 18. Exercise: What is the output resistance of the source in Fig. 16.31 if R = 8.65 k and I D2 = 100 A? Assume K n = 25 A/V2 , K p = 10 A/V2 , VT N = −VT P = 0.75 V, and 1/λ = 80 V. Answer: 148 k Exercise: Simulate the circuit in Fig. 16.31 and determine I D2 and the output resistance of the source. (Suggestion: Make use of SPICE transfer function analysis.) Answer: 97.8 A, 184 k Equation (16.62) indicates that the output resistances ro2 and ro3 of the Widlar source and current mirror determine the sensitivity to power supply variations. The supply voltage dependence can be improved by increasing these two resistance values. Figure 16.33 shows improved versions of the supply-independent reference current cells in which cascode sources have been used to improve the output resistance of the Widlar portion of the cell, and Wilson sources have been used to improve the output resistance of the current mirrors. 1214 Chapter 16 Analog Integrated Circuits VCC Q4 VDD Q3 M4 M3 Wilson source Q5 M5 Q6 Q7 M6 M7 Q1 Q2 M1 M2 R (a) Cascode Widlar source R (b) Figure 16.33 (a) Bipolar and (b) MOS reference current cells with improved power supply rejection. DESIGN EXAMPLE 16.5 REFERENCE CURRENT DESIGN Design a supply-independent current source using bipolar technology. PROBLEM Design a supply-independent current source to provide an output current of 45 A at T = 300 K using the circuit topology in Fig. 16.29 with symmetrical 5-V power supplies. The circuit should use no more than 1 k of resistance or 60 A of total current. Use SPICE to determine the sensitivity of the design current to power supply voltage variations. Assume that a unit-area BJT has the following parameters: β F O = 100, V A = 75 V, and I S O = 0.1 fA for both npn and pnp transistors. SOLUTION Known Information and Given Data: Circuit topology in Fig. 16.29; β F O = 100, V A = 75 V, I S O = 0.1 fA. Total current ≤ 60 A. Unknowns: R and the area ratio between Q 1 and Q 2 Approach: The current in the circuit is described by Eq. (16.53). Use the maximum resistance values to select the area ratio. Select a current ratio in the sides of the reference to satisfy the total supply current requirement. Assumptions: Transistors operate in the active region. IC2 = 45 A. Analysis: At T = 300 K and VT = 25.88 mV, and from Eq. (16.53), we have (45 A)(1 k) IC1 A E2 IC2 R IC1 A E2 ≤ ≤ 5.69 ln = = 1.739 or IC2 A E1 VT 25.88 mV IC2 A E1 In addition, the maximum current specification requires IC2 45 A 3 ≥ = IC1 15 A 1 16.4 Reference Current Generation 1215 Let’s choose IC2 = 5IC1 . Then A E2 /A E1 ≤ 28.45. Choosing A E2 /A E1 = 20, we obtain 25.88 mV ln(4) = 797 45 A R= The final design is R = 797 , A E1 = A, A E2 = 20 A, A E3 = A, A E4 = 5 A with 35.88 mV across resistor R. Check of Results: Since we need to use SPICE to find the power supply sensitivity, let us use it to also check our design. Computer-Aided Analysis: The circuit shown is drawn using the schematic editor. Zero-valued sources VIC2 and VIC3 function as ammeters to measure the collector currents of transistors Q 2 and Q 3 , and IVR serves as a voltmeter to measure the voltage across R. First we must remember to set the npn and pnp BJT parameters to BF = 100, VAF = 75 V, IS = 0.1 fA, and TEMP = 27 C. We must also specify AREA = 1, AREA = 20, AREA = 1 and AREA = 5 for Q 1 through Q 4 , respectively. SPICE then gives IC2 = 49.6 A and IC3 = 10.94 A with 39.89 mV across R. The currents and voltage are slightly higher than predicted, and this is primarily due to having neglected the mirror ratio error due to the different values of VEC4 and VEC3 . (Try the exercise after this example.) We can correct for this error by modifying the emitter area ratio: 9.34 − 0.65 VEC3 − VEC4 A E4 = 5 1 + =5 1+ = 5.58 VA 75 SPICE now yields IC2 = 45.9 A, IC3 = 9.08 A and VE2 = 36.9 mV. A transfer function analysis from VCC to VI C2 gives a total output resistance of 928 k for the current source, and the sensitivity of IC2 to changes in VCC is 0.808 A/V. VCC 5V Q3 Q4 VIC3 0 VIC2 0 Q1 VEE 5V Q2 R 797 IVR 0 Discussion: The current source meets the specifications. If desired, the power supply sensitivity could be reduced by using the circuit in Fig. 16.33(a). 1216 Chapter 16 Analog Integrated Circuits Exercise: Explore the errors caused by finite current gain and Early voltage by simulating the circuit with BF = 10,000 and VAF = 10,000 V. What are the new values of I C2 , I C3 , and the voltage developed across R? Answers: 45.0 A; 9.01 A; 35.88 mV Exercise: What are the new design values if we choose AE2 /AE1 = 25? Answers: R = 925 ; AE1 = A; AE2 = 25 A; AE3 = A; AE4 = 5.57 A 16.5 THE BANDGAP REFERENCE Precision voltage references need to not only be independent of power supply voltage, but also be independent of temperature. Although the circuits described in Sec. 16.4 can produce reference currents and voltages that are substantially independent of power supply voltage, they all still vary with temperature. Robert Widlar solved this problem with his invention of the elegant bandgap reference circuit, and today, the bandgap reference is the most common technique used to generate a precision voltage. It has supplanted Zener reference diodes in the majority of applications. Based on his detailed understanding of bipolar transistor characteristics, Widlar realized that the negative temperature coefficient associated with the base-emitter junction could be canceled out by the positive temperature dependence of a scaled PTAT voltage as indicated conceptually in Fig. 16.34. The output voltage of the circuit in Fig. 16.34 can be written as VBG = VB E + GVPTAT (16.63) We desire this output voltage to have a zero temperature coefficient: ∂ VBG ∂ VB E ∂ VPTAT = +G =0 (16.64) ∂T ∂T ∂T The dependence of VB E and VPTAT on temperature were developed previously in Eqs. (3.15) and (16.33), respectively. Substituting these values into Eq. (16.64) gives ∂ VBG VB E − VG O − 3VT VPTAT = +G =0 ∂T T T or (16.65) GVPTAT = VG O + 3VT − VB E where VG O is the silicon bandgap voltage at 0 K (1.12 V). Substituting this result into Eq. (16.65) reduces the output voltage to VBG = VG O + 3VT (16.66) The output voltage at which zero temperature coefficient is achieved is slightly above the bandgap voltage of silicon. Hence, this circuit is referred to as a “bandgap reference.” At room temperature, the output voltage is approximately 1.20 V. A circuit realization of the bandgap reference is shown in Fig. 16.35. This circuit is attributed to another talented designer, Paul Brokaw of Analog Devices [5], and is easier to understand 16.5 1217 The Bandgap Reference VCC R R – + Q1 Q2 IREF AE2 PTAT voltage generator Q1 VBE GVPTAT AE1 VPTAT R1 VBG VBE1 VBG R2 R 2 R2 VPTAT 1 Figure 16.34 Concept for the Figure 16.35 Brokaw version of the bandgap reference. bandgap reference. than the original circuit of Widlar. In this circuit, the output voltage is equal to the sum of the base-emitter voltage of Q 1 plus the voltage across resistor R2 , which is a scaled replica of the PTAT voltage being developed across resistor R1 . The scaling factor is controlled by the op amp and resistors R. The ideal op amp forces the voltage across the two matched collector resistors to be the same, thereby setting IC2 = IC1 and I E2 = I E1 . Thus the PTAT voltage is equal to VT ln(A E2 /A E1 ), and the emitter current of Q 2 equals VPTAT /R1 . The current in R2 is twice that in R1 , since I E2 = I E1 . Combining these results yields an expression for the output voltage VBG : VBG = VB E1 + 2 R2 A E2 VT ln R1 A E1 (16.67) For this circuit, the gain G = 2R2 /R1 , and based on Eq. (16.65), the resistor ratio is given by ∂ VB E1 R2 1 ∂T VG O + 3VT − VB E1 =− = R1 2 ∂ VPTAT 2VPTAT ∂T (16.68) Often we want an output voltage that is not equal to 1.2 V, and other voltages are easy to achieve by adding a two-resistor voltage divider to the Brokaw circuit as in Fig. 16.36. In this case the op amp output voltage becomes VO = R4 1+ R3 VBG (16.69) which can be scaled up to any desired value (e.g., 2.5 or 5 V). A word of caution is needed here. In most bandgap reference designs, zero-output voltage is a valid operating point, and some additional circuitry must be added to ensure that the circuit “starts up” and reaches the desired operating point. In many simple circuit cases, SPICE will have considerable difficulty converging to the desired operating point. 1218 Chapter 16 Analog Integrated Circuits VCC R R – VO + Q2 AE2 Q1 AE1 R4 VBG R3 R1 R2 Figure 16.36 Bandgap reference with VO > VBG . EXAMPLE 16.6 BANDGAP REFERENCE ANALYSIS In this example, we determine the operating point for a specific bandgap reference circuit. PROBLEM Find IC , VPTAT , VB E , and VBG for the circuit in Fig. 16.35 if R = 30 k, R1 = 1 k, and R2 = 4.16 k. Assume I S = 0.1 fA and A E2 = 10A E2 . SOLUTION Known Information and Given Data: The circuit is the Brokaw reference in Fig. 16.35 with R = 30 k, R1 = 1 k, and R2 = 4.16 k. Transistor parameters are specified as I S = 0.1 fA and A E2 = 10A E2 . Unknowns: IC , VPTAT , VB E , and VBG Approach: Find VT and VPTAT which determine IC . Use IC to find VB E1 . Use VB E1 and VPTAT to find the output voltage. Assumptions: T = 300 K; BJTs are in the active region of operation; the current gain is large, say β F O = 10,000 and V A = ∞. Analysis: Because of the precision involved, we should carry more digits in our calculations than normal. Following the sequence of calculations outlined earlier, VT = VPTAT = IC = VB E1 = VBG = kT 1.380 × 10−23 (300) = 25.84 mV = q 1.602 × 10−19 A E2 VT ln = VT ln(10) = 59.50 mV A E1 VPTAT IE = = 59.50 A R1 59.50 A IC1 = (25.84 mV) ln = 0.7006 V VT ln I S1 0.1 fA R2 4.16 k (59.50 mV) = 1.196 V VB E1 + 2 VPTAT = 0.7006 + 2 R1 1 k 16.5 The Bandgap Reference 1219 Check of Results: VBG is approximately 1.20 V so our calculation appears correct. Our analysis showed that the output voltage should also be VG O + 3VT = 1.198 V, which also checks. Discussion: Note that the voltage drop across the collector resistors must be enough to bring the inputs of the op amp into its common-mode operating range. In this circuit, the drop across the collector resistors is only 1.5 V. Computer-Aided Analysis: We first set the npn parameters to BF = 10,000, IS = 0.1 fA, and let VAF default to infinity. Set AREA = 1 for Q 1 and AREA = 10 for Q 2 . In the circuit shown here, the ideal op amp is modeled by EOPAMP, whose controlling voltage appears across zero-value current source IOP. The gain is set to 106 . Source VSTART may be needed in certain versions of SPICE to help the circuit start up. (Remember that VO = 0 is a valid operating point.) Sweeping VCC from 0 to 10 V can also help the startup problem. SPICE simulation produces VBG = 1.197 V and VPTAT = 59.55 mV for the default temperature of 27◦ C. VCC 10 V RC1 RC2 30 K 30 K IOP 0 Q1 Q2 R1 1K R2 4.16 K EOPAMP VSTART 0.8 V Exercise: Suppose β F O = 100 and V A = 75. Use SPICE to find the new output voltage of the bandgap reference in Ex. 16.6? Answer: 1.194 V DESIGN EXAMPLE 16.7 BANDGAP REFERENCE DESIGN Design of the bandgap reference requires a slightly different sequence of calculations than the analysis in the previous example. PROBLEM Design the bandgap reference in Fig. 16.36 to produce an output voltage of 5.000 V with zero temperature coefficient at a temperature of 47◦ C. Design for a collector current of 25 A, and assume I S = 0.5 fA. 1220 Chapter 16 Analog Integrated Circuits SOLUTION Known Information and Given Data: The circuit is the Brokaw reference with amplified output given in Fig. 16.36. VO = 5.000 V with a zero temperature coefficient (TC) at T = 320 K. Collector currents are to be 25 A, and the transistor saturation current is 0.5 fA. Unknowns: Values of resistors R, R1 , R2 , R3 , and R4 Approach: Find VT and VPTAT . Then use IC to determine R1 . Use IC to find VB E1 . Determine R2 using Eq. (16.68). Choose R4 and R3 to set VO = 5 V. Choose R to provide operating voltage to the op amp. Assumptions: BJTs are in the active region of operation. β F O = ∞ and V A = ∞. A E2 = 10A E1 represents a reasonable emitter area ratio. Drop 2 V across R. Analysis: Because of the precision involved, we will carry four digits in our calculations. kT 1.380 × 10−23 (320) = 27.57 mV = q 1.602 × 10−19 A E2 = VT ln(10) = 63.47 mV = VT ln A E1 VT = VPTAT VPTAT 63.47 mV = 2.539 k = IE 25 A 25 A IC1 = (27.57 mV) ln = VT ln = 0.6792 V I S1 0.5 fA R1 = VB E1 R2 VG O + 3VT − VB E1 1.12 + 3(0.02757) − 0.6792 = 4.124 = = R1 2VPTAT 2(0.06347) R2 = 4.124R1 = 10.47 k VBG = VB E1 + 2 R2 VPTAT = 0.6792 + 2(4.124)(63.47 mV) = 1.203 V R1 R4 VO = − 1 = 3.157 R3 VBG We should not waste an excessive amount of current in the output voltage divider, so let us choose I3 = I4 = 50 A. Also, set the voltage drop across R to 2 V. R3 = VBG 1.203 V = = 24.0 k I3 50 A and R4 = VO − VBG 3.797 V = = 75.9 k I3 50 A 2V = 80 k 25 A Check of Results: VBG is approximately 1.20 V so our calculation appears correct. Our analysis showed that the output voltage should also be VG O + 3VT = 1.203 V, which also checks. R= Discussion: Note that the voltage drop across the collector resistors must be enough to bring the inputs of the op amp into its common-mode operating range. In this circuit, the drop across the collector resistors is designed to be 2 V. Computer-Aided Analysis: We first set the npn parameters to BF = 10,000 and IS = 0.5 fA and let VAF default to infinity. Set AREA = 1 for Q 1 , AREA = 10 for Q 2 , and TEMP = 47◦ C. 16.6 The Current Mirror as an Active Load 1221 In the circuit shown here, the ideal op amp is modeled by EOPAMP whose controlling voltage appears across zero-value current source IOP. The gain is set to 106 . Source VSTART may be needed in some versions of SPICE to help the circuit start up. Another help is to sweep VCC from 0 to 10 V. (Remember that VO = 0 is a valid operating point.) SPICE simulation produces VBG = 1.204 V and VPTAT = 63.52 mV and VO = 5.01 V. With BF = 100 and VAF = 75 V, the values are VBG = 1.201 V and VPTAT = 63.52 mV and VO = 5.03 V. VCC RC2 RC1 80 K 10 V 80 K IOP R4 0 Q1 75.9 K EOPAMP Q2 R3 R1 2.539 K R2 10.47 K 24 K VSTART 6V Exercise: Redesign the reference in Ex. 16.7 using AE2 = 20AE1 . Answer: 3.17 k, 10.5 k, 24.0 k, 75.9 k, 80 k 16.6 THE CURRENT MIRROR AS AN ACTIVE LOAD One of the most important applications of the current mirror4 is as a replacement for the load resistors of differential amplifier stages in IC operational amplifiers. This elegant application of the current mirror can greatly improve amplifier voltage gain while maintaining the operatingpoint balance necessary for good common-mode rejection and low offset voltage. When used in this manner, the current mirror is referred to as an active load because the passive load resistors have been replaced with active transistor circuit elements. 16.6.1 CMOS Differential Amplifier with Active Load Figure 16.37 shows a CMOS differential amplifier with an active load; the load resistors have been replaced by a PMOS current mirror. Let us first study the quiescent operating point of this circuit and then look at its small-signal characteristics. 4 In addition to its role as a current source. 1222 Chapter 16 Analog Integrated Circuits VDD VSG3 M3 v1 M4 ID3 ID4 ID1 ID2 M1 vO v2 M2 vS ISS –VSS Figure 16.37 CMOS differential amplifier with PMOS active load. dc Analysis Assume for the moment that the amplifier is voltage balanced (in fact, it will turn out that it is balanced). Then bias current I SS divides equally between transistors M1 and M2 , and I D1 and I D2 are each equal to I SS /2. Current I D3 must equal I D1 and is mirrored as I D4 at the output of the PMOS current mirror. Thus, I D3 and I D4 are also equal to I SS /2, and the current in the drain of M4 is exactly the current required to satisfy M2 . The mirror ratio set by M3 and M4 is exactly unity when VS D4 = VS D3 and hence VDS 1 = VDS2 . Thus, the differential amplifier is completely balanced at dc when the quiescent output voltage is I SS VO = VD D − VS D4 = VD D − VSG3 = VD D − (16.70) − VT P Kp Q-Points The drain-source voltages of M1 and M2 are VDS 1 = VO − VS = VD D − I SS − VT P Kp or VDS 1 = VD D + VT N + VT P + and those of M3 and M4 are VS D3 = VSG3 = I SS − Kn VT N + + I SS ∼ = VD D Kp I SS − VT P Kp I SS Kn (16.71) (16.72) (Remember that VTP < 0 for p-channel enhancement-mode devices.) The drain currents of all the transistors are equal: I DS 1 = I DS2 = I S D3 = I S D4 = I SS 2 (16.73) 16.6 The Current Mirror as an Active Load 1223 Small-Signal Analysis Now that we have found the operating points of the transistors, we can proceed to analyze the small-signal characteristics of the amplifier including differential-mode gain, differential-mode input and output resistances, common-mode gain, CMRR, and common-mode input and output resistances. Differential-Mode Signal Analysis Analysis of the ac behavior of the differential amplifier begins with the differential-mode input applied in the ac circuit model in Fig. 16.38. Upon studying the circuit in Fig. 16.38, we realize that it is a two terminal network and can be represented by its Norton equivalent circuit consisting of the short-circuit output current and Thévenin equivalent output resistance. With the output terminals short circuited, the NMOS differential pair produces equal and opposite currents with amplitude gm2 vid /2 at the drains of M1 and M2 . Drain current i d1 is supplied by current mirror transistor M3 and is replicated at the output of M4 . Thus the total short circuit output current is io = 2 gm2 vid = gm2 vid 2 (16.74) The current mirror provides a single-ended output but with a transconductance equal to the full value of the C-S amplifier. M3 M4 id1 io gmvid 2 M1 vid 2 vo M2 vS 0 isc vid 2 RSS isc (a) Rth (b) Figure 16.38 (a) CMOS differential amplifier with differential-mode input. (b) The circuit is a one port and can be represented by its Norton equivalent circuit. The Thévenin equivalent output resistance will be found using the circuit in Fig. 16.39 in which the internal output resistances of M2 and M4 are shown next to their respective transistors. In the next section we will show that Rth is equal to the parallel combination of ro2 and ro4 : Rth = ro2 ro4 (16.75) The differential-mode voltage gain of the open-circuited differential amplifier is simply the product of i sc and Rth : µf2 ∼ µf2 Adm = gm2 (ro2 ro4 ) = (16.76) ro2 = 2 1+ ro4 1224 Chapter 16 Analog Integrated Circuits M3 M3 ro 4 M4 M4 ro 4 1 Rth M1 ro2 M1 M2 1 gm1 RSS vx 2ro 2 M2 vx ro 2 vx RSS Figure 16.39 Simple CMOS op amp with Figure 16.40 Output resistance component due active load in the first stage. to ro2 . Equation (16.76) indicates that the gain of the input stage of the amplifier approaches one-half the amplification factor of the transistors forming the differential pair. We are now within a factor of 2 of the theoretical voltage gain limit for a single-transistor amplifier! Output Resistance of the Differential Amplifier The origin of the output resistance expression in Eq. (16.75) can be thought of conceptually in the following (although technically incorrect) manner. At node 1 in Fig. 16.39, ro4 is connected directly to ac ground at the positive power supply, whereas ro2 appears connected to virtual ground at the sources of M2 and M1 . Thus ro2 and ro4 are effectively in parallel. Although this argument gives the correct answer, it is not precisely correct. Because the differential amplifier with active load no longer represents a symmetric circuit, the node at the sources of M1 and M2 is not truly a virtual ground. Exact Analysis A more precise analysis can be obtained from the circuit in Fig. 16.40. The output resistance ro4 of M4 is indeed connected directly to ac ground and represents one component of the output resistance. However, the current from vx due to ro2 is more complicated. The actual behavior can be determined from Fig. 16.40, in which R SS is assumed to be negligible with respect to 1/gm1 , R SS 1/gm1 . Transistor M2 is operating as a common-gate transistor with an effective resistance in its source of R S = 1/gm1 . Based on the results in Table 14.1, the resistance looking into the drain of M2 is 1 Ro2 = ro2 (1 + gm2 R S ) = ro2 1 + gm2 = 2ro2 (16.77) gm1 Therefore, the drain current of M2 is equal to vx /2ro2 . However, the current goes around the differential pair and into the input of the current mirror at M3 . The current is replicated by the mirror to become the drain current of M4 . The total current from source vx becomes 2(vx /2ro2 ) = vx /ro2 . Combining this current with the current through ro4 yields a total current of iTx = vx vx + ro2 ro4 and Rod = ro2 ro4 (16.78) 16.6 The Current Mirror as an Active Load 1225 The equivalent resistance at the output node is, in fact, exactly equal to the parallel combination of the output resistances of M2 and M4 . Exercise: Find the Q-points of the transistors in Fig. 16.34 if I SS = 250 A, K n = 250 A/V2 , K p = 200 A/V2 , VT N = −VT P = 0.75 V, and VD D = VSS = 5 V. What are the transconduc- tance, output resistance, and voltage gain of the amplifier if λ = 0.0133 V−1 ? Answers: (125 A, 4.88 V), (125 A, 1.87 V); 250 S, 314 k, 78.5 Common-Mode Input Signals Figure 16.41 is the CMOS differential amplifier with a common-mode input signal. The commonmode input voltage causes a common-mode current i oc in both sides of the differential pair consisting of M1 and M2 . The common-mode current (i oc ) in M1 is mirrored at the output of M4 with a small error since no current can appear in ro4 with the output shorted. In addition, the small voltage difference developed between the drains of M1 and M2 causes a current in the differential output resistance (2ro2 ) of the pair that is then doubled by the action of the current mirror. An expression for the short-circuit output current can be found using the small-signal model for the circuit in Fig. 16.41(b). The differential pair with common-mode input is represented by the two-port model from Sec. 15.3.15 with vic i oc ∼ = 2R SS Rod = 2ro2 Roc = 2µ f R SS (16.79) With the output short-circuited, we have a one-node problem. Solving for v3 , v3 = M3 −i oc go2 gm3 + go3 + + G oc 2 io vo i oc M1 i sc M2 vs ≈ vic 2i ic RSS (a) (16.80) M4 i oc vic go2 v3 i sc = − i oc + gm4 v3 − 2 and 1 gm3 v3 ro3 ro4 gm4 v3 isc v3 vic 2ro2 Roc ioc ioc (b) Figure 16.41 CMOS differential amplifier with common-mode input. Roc 1226 Chapter 16 Analog Integrated Circuits which together with Eq. (16.79) yield ro3 1+ vic go3 + go2 ro2 ∼ i sc = − i oc = − go2 µf3 2R SS gm3 + go3 + + G oc 2 (16.81) where it is assumed that gm4 = gm3 and G oc gm3 . The Thévenin equivalent output resistance is exactly the same as found in the previous section, Rth = ro2 ro4 . Thus the common-mode gain is ro3 1+ i sc Rth ro2 = =− (ro2 ro4 ) vic 2µ f 3 R SS Acm (16.82) where µ f 3 1 has been assumed. The common-mode rejection ratio is Adm 2µ f 3 gm2 R SS = ∼ CMRR = = µ f 3 gm2 R SS ro3 Acm 1+ ro2 for ro3 ∼ = ro2 (16.83) which is improved by a factor of approximately µ f 3 over that of the pair with a resistor load! Exercise: Evaluate Eq. (16.83) for K p = K n = 5 mA/V2 , λ = 0.0167 V−1 , I SS = 200 A, and RSS = 10 M. Answer: 6.00 × 106 or 136 dB In the last exercise, we find that the CMRR predicted by Eq. (16.83) is quite large, whereas typical op amp specs are 80 to 100 dB. We need to look deeper. In reality, this level will not be achieved, but will be limited by mismatches between the devices in the circuit. Mismatch Contributions to CMRR Analysis In this section we explore the techniques used to calculate the effects of device mismatches on CMRR. Figure 16.42 presents the small-signal model for the differential amplifier with mismatches in transistors M1 and M2 in which we assume gm1 = gm + gm 2 gm2 = gm − gm 2 go1 = go + go 2 go2 = go − go 2 (16.84) In this analysis, M3 and M4 are still identical. We desire to find the short circuit output current i sc = (i d1 − i d2 ) in which i d1 is replicated by the current mirror. Let us use our knowledge of the gross behavior of the circuit to simplify the analysis. We have vd2 = 0, since we are finding the short-circuit output current, and based on previous common-mode analyses, we expect the signal at vd1 to be small. So let us assume that vd1 ∼ = 0. With this assumption, and noting that the two gate-source voltages are identical, i sc = i d1 − i d2 = (gm1 − gm2 )vgs − (go1 − go2 )vs = gm vgs − go vs (16.85) 16.6 M3 1227 The Current Mirror as an Active Load M4 i sc i d1 vic vgs gm1vgs i d2 vd1 vd2 ro1 ro2 vgs vic gm2vgs vs RSS Figure 16.42 CMOS differential amplifier in which M1 and M2 are no longer matched. To evaluate this expression, we need to find source voltage vs and gate-source voltage vgs . Writing a nodal equation for vs with vgs = vic − vs , vd1 = 0 and vd2 = 0, yields gm + gm gm + gm − 2 2 (vic − vs ) = go + go go + go − + G SS vs 2 2 in which we may be surprised to see all the mismatch terms cancel out! Thus, for common-mode inputs, vs and vgs are not affected by the transistor mismatches5 : vs ∼ = 2gm R SS vic ∼ = vic 1 + 2gm R SS and 1 + 2go R SS vgs ∼ vic ∼ = = 1 + 2gm R SS 1 1 + 2gm R SS µf vic (16.86) The short-circuit output current goes through the Thévenin output resistance Rth = ro2 ro4 to produce the output voltage, and Acm = 1 i sc Rth 1 − go (ro2 ro4 ) = gm + vic 2gm R SS µf The CMRR is then Acm 1 go 1 Acm 1 −1 = gm CMRR = − = + Adm gm (ro2 ro4 ) gm 2gm R SS µf go µ f (16.87) (16.88) For very large R SS , we see that CMRR is now limited by the transistor mismatches and value of the amplification factor. For example, a 1 percent mismatch with an amplification factor of 500 limits the individual terms in Eq. (16.88) to 2 × 10−5 . Since we cannot predict the signs on the g/g terms, the expected CMRR is 2.5 × 104 or 88 dB. This is much more consistent with observed values of CMRR. 5 An exact analysis without assuming that vd1 = 0 shows that a negligibly small change actually occurs. 1228 Chapter 16 Analog Integrated Circuits 16.6.2 Bipolar Differential Amplifier with Active Load The bipolar differential amplifier with an active load formed from a pnp current mirror is depicted in Fig. 16.43 with v1 = 0 = v2 . If we assume that the circuit is balanced with β F O = ∞, then the bias current I E E divides equally between transistors Q 1 and Q 2 , and IC1 and IC2 are equal to I E E /2. Current IC1 is supplied by transistor Q 3 and is mirrored as IC4 at the output of pnp transistor Q 4 . Thus, IC3 and IC4 are both also equal to I E E /2, and the dc current in the collector of Q 4 is exactly the current required to satisfy Q 2 . Q3 Q4 IC4 IC3 v1 +VCC + VEB – IC1 IC2 Q1 Q2 v2 IEE –VEE Figure 16.43 Bipolar differential amplifier with active load. If β F O is very large, then the current mirror ratio is exactly 1 when VEC4 = VEC3 = VE B , and the differential amplifier is completely balanced when the quiescent output voltage is VO = VCC − VE B (16.89) Q-Points The collector currents of all the transistors are equal: IC1 = IC2 = IC3 = IC4 = IE E 2 (16.90) The collector-emitter voltages of Q 1 and Q 2 are VC E1 = VC E2 = VC − VE = (VCC − VE B ) − (−VB E ) ∼ = VCC (16.91) and for Q 3 and Q 4 , VEC3 = VEC4 = VE B (16.92) Finite Current Gain The current gain defect in the current mirror upsets the dc balance of the circuit. However, as long as the transistors remain in the forward-active region, the collector current of Q 4 must equal the collector current of Q 2 , and the collector-emitter voltage of Q 4 adjusts itself to make up for the current-gain defect of the current mirror. The required value of VEC4 can be found using the 16.6 1229 The Current Mirror as an Active Load current mirror expression from Eq. (16.10): IC4 VEC4 1+ VA = IC1 VE B 2 1+ + VA β F O4 (16.93) However, because IC4 = IC2 and IC2 = IC1 , the mirror ratio must be unity, which requires VEC4 = VE B + 2V A β F O4 (16.94) For β F O3 = 50, V A = 60 V, and VE B = 0.7 V, VEC4 = 3.10 V. This collector-emitter voltage difference represents a substantial offset at the amplifier output and translates to an equivalent input offset voltage of VO S = VEC4 − VEC3 VEC4 − VE B = Add Add (16.95) VO S represents the input voltage needed to force the output voltage differential to be zero. For Add = 100, VO S would be 24.0 mV. To eliminate this error, a buffered current mirror is usually used as the active load, as shown in Fig. 16.44. Q3 Q4 io Q11 iC1 v1 vid 2 Q1 Q2 gm2vid 2 REE (a) v2 RL vid 2 isc Rth RL (b) Figure 16.44 (a) BJT differential amplifier with differential-mode input. (b) Equivalent circuit. Exercise: Calculate the dc value of VEC4 if the circuit buffered current mirror replaces the active load in Fig. 16.43. What is VOS if A dd = 100? Answers: VEC4 = 1.25 V and VEC = 47 mV; VOS = 0.47 mV It should be noted that Eq. (16.94) actually overestimates the value of VEC4 because the increase in VEC4 decreases VC E2 and thereby reduces IC2 . 1230 Chapter 16 Analog Integrated Circuits Differential-Mode Signal Analysis Analysis of the ac behavior of the differential amplifier begins with the differential-mode input applied in the ac circuit model in Fig. 16.44. The differential input pair produces equal and opposite currents with amplitude gm2 vid /2 at the collectors of Q 1 and Q 2 . Collector current i c1 is supplied by Q 3 and is replicated at the output of Q 4 . Thus the total short circuit output current is equal to isc = 2 gm2 vid = gm2 vid 2 (16.96) The output resistance is identical to Eq. (16.75) Rth = ro2 ro4 (16.97) and Add = i sc (R L Rth ) = gm2 (R L ro2 ro4 ) = −gm2 R L vdm (16.98) The current mirror provides a single-ended output but with a voltage equal to the full gain of the C-E amplifier, just as for the FET case. Here we have included R L which models the loading of the next stage in a multistage amplifier. The power of the current mirror is again most apparent when additional stages are added, as in the prototype operational amplifier in Fig. 16.45. The resistance at the output of the differential input stage, node 1, is now equivalent to the parallel combination of the output resistances of transistors Q 2 and Q 4 and the input resistance of Q 5 (R L = rπ 5 ): Req = ro2 ro4 rπ 5 ∼ = rπ 5 (16.99) and the gain of the differential input stage becomes IC2 Adm = gm2 Req ∼ = gm2rπ 5 = βo5 IC5 (16.100) +VCC Q3 Q4 Q11 v1 Q5 1 v2 Q1 Q2 Q6 vO I1 I2 I3 – VEE Figure 16.45 Bipolar op amp with active load in first stage. 16.6 Q3 Q4 Q3 Q4 isc Q1 vo Q11 v1 Q2 1231 The Current Mirror as an Active Load ioc ioc Q1 Q2 io v2 isc ve ≈ vic vic vic vic 2iic REE vic REE (a) (b) Figure 16.46 Bipolar differential amplifiers with common-mode input. Exercise: What is the approximate differential-mode voltage gain of the amplifier in Fig. 16.45 if β F O = 150, V A = 75 V, and I C5 = 3 I C2 ? Answer: 50 Common-Mode Input Signals The circuits in Fig. 16.46 represent the bipolar differential amplifier with current mirror load and a buffered current mirror load. The detailed analysis is quite involved and tedious, particularly for the buffered mirror, so here we will argue the result based on earlier analyses. The common-mode current i oc in Q 1 and Q 2 is found with the help of Eq. (15.87): i oc Acc vic = = vic RC 1 1 − 2R E E βo r o (16.101) The current from Q 1 is mirrored at the output of Q 4 with a mirror error of 2/βo . Thus the shortcircuit output current is 2 i sc = vic βo 1 1 − βo r o 2R E E (16.102) In a manner similar to that of the FET pair, the voltage developed at the collector of Q 1 , i oc /gm3 , forces a current in the differential output resistance of the pair (2ro2 ), which is doubled by the action of the current mirror: vic 1 1 1 1 1 ∼ i sc = 2vic (16.103) − − = βo r o 2R E E gm3 (2ro2 ) µ f 2 βo ro 2R E E Since µ f βo for the BJT, the output will be dominated by Eq. (16.102) and the CMRR is −1 gm2 Rth 1 2 1 ∼ CMRR = − = i sc Rth /vic βo3 βo2 µ f 2 2gm2 R E E (16.104) 1232 Chapter 16 Analog Integrated Circuits Exercise: Evaluate Eq. (16.104) for β F = 100, V A = 75 V, I E E = 200 A, and RE E = 10 M. Answer: 5.45 × 106 or 135 dB The expression in Eq. (16.104) yields a very large CMRR that is almost impossible to achieve. The CMRR predicted for the buffered current mirror is even larger, since the mirror error is approximately 2/βo11 βo3 . In both these circuits, however, the CMRR will actually be limited to much smaller levels by small mismatches between the various transistors: CMRR−1 = gm gπ + gm gπ 1 1 + 2gm R SS µf − go 1 go µ f (16.105) Equation (16.105) is similar to the results for the FET from Eq. (16.88) with the addition of the gπ /gπ term. In an actual amplifier, the common-mode gain is determined by small imbalances in the bipolar transistors and overall symmetry of the amplifier. 16.7 ACTIVE LOADS IN OPERATIONAL AMPLIFIERS Let us now explore more fully the use of active loads in MOS and bipolar operational amplifiers. Figure 16.47 shows a complete three-stage MOS operational amplifier. The input stage consists of NMOS differential pair M1 and M2 with PMOS current mirror load, M3 and M4 , followed by a second common-source gain stage M5 loaded by current source M10 . The output stage is a class-AB amplifier consisting of transistors M6 and M7 . Bias currents I1 and I2 for the two gain stages are set by the current mirrors formed by transistors M8 , M9 , and M10 , and class-AB bias for the output stage is set by the voltage developed across resistor RGG . At most, only two resistors are required: RGG and one for the current mirror reference current. +VDD M3 M4 M5 va M1 v2 M2 v1 vb M6 vO RGG IREF I1 M7 I2 M8 M9 M10 −VSS Figure 16.47 Complete CMOS op amp with current mirror bias. 16.7 Active Loads in Operational Amplifiers 1233 16.7.1 CMOS Op Amp Voltage Gain Assuming that the gain of the output stage is approximately 1, then the overall differential-mode gain Adm of the three-stage operational amplifier is approximately equal to the product of the terminal gains of the first two stages: Adm = va vb vo = Avt1 Avt2 (1) ∼ = Avt1 Avt2 vid va vb (16.106) As discussed earlier, the input stage provides a gain of µf2 Avt1 = gm2 (ro2 ro4 ) ∼ = 2 (16.107) The terminal gain of the second stage is equal to Avt2 = gm5 (ro5 (RGG + ro10 )) ∼ = gm5 (ro5 ro10 ) ∼ = gm5 (ro5 ro5 ) = µf5 2 (16.108) assuming that the output resistances of M5 and M10 are similar in value and RGG ro10 . Combining the three equations above yields µ f 2µ f 5 Adm ∼ = 4 (16.109) The gain approaches one-quarter of the product of the amplification factors of the two gain stages. The factor of 4 in the denominator of Eq. (16.109) can be eliminated by improved design. If a Wilson source is used in the first-stage active load, then the output resistance of the current mirror is much greater than ro2 , and Av1 becomes equal to µ f 2 . The gain of the second stage can also be increased to the full amplification factor of M5 if the current source M10 is replaced by a Wilson or cascode source. If both these circuit changes are used (see Prob. 16.111), then the gain of the op amp can be increased to Adm ∼ = µ f 2µ f 5 (16.110) This discussion has only scratched the surface of the many techniques available for increasing the gain of the CMOS op amp. Several examples appear in the problems at the end of this chapter; further discussion can be found in the bibliography. 16.7.2 dc Design Considerations When the circuit in Fig. 16.47 is operating in a closed-loop op amp configuration, the drain current of M5 must be equal to the output current I2 of current source transistor M10 . For the amplifier to have a minimum offset voltage, the (W/L) ratio of M5 must be carefully selected so the source-gate bias of M5 , VSG5 = VS D4 = VSG3 , is precisely the proper voltage to set I D5 = I2 . The W/L ratio of M5 is also usually adjusted to account for VDS and λ differences between M5 and M10 . RGG and the (W/L) ratios of M6 and M7 determine the quiescent current in the class-AB output stage. Even resistor RGG has been eliminated from the op amp in Fig. 16.48 by using the gate-source voltage of FET M11 to bias the output stage. The current in the class-AB stage is determined by the W/L ratios of the output transistors and the matching diode-connected MOSFET M11 . 1234 Chapter 16 Analog Integrated Circuits +VDD M4 50 1 50 M3 1 M5 100 1 M1 v2 M2 20 1 20 1 IREF M8 10 1 M6 10 1 vO v1 5 M11 1 I1 M7 25 1 I2 M9 M10 20 1 20 1 −VSS Figure 16.48 Op amp with current mirror bias of the class-AB output stage. EXAMPLE 16.8 CMOS OP AMP ANALYSIS Find the small-signal characteristics of a CMOS operational amplifier. PROBLEM Find the voltage gain, input resistance, and output resistance of the amplifier in Fig. 16.48 if K n = 25 A/V2 , K p = 10 A/V2 , VT N = 0.75 V, VT P = −0.75 V, λ = 0.0125 V−1 , VD D = VSS = 5 V, and IREF = 100 A. SOLUTION Known Information and Given Data: The schematic for the operational amplifier appears in Fig. 10.48; VD D = VSS = 5 V, and IREF = 100 A; device parameters are given as K n = 25 A/V2 , K p = 10 A/V2 , VT N = 0.75 V, VT P = −0.75 V, λ = 0.0125 V−1 . Unknowns: Q-points, Adm , Rid , and Rout Approach: Find the Q-point currents and use the device parameters to evaluate Eq. (16.109) for Adm . Since we have MOSFETS at the input, Rid = Ric = ∞. Rout is set by M6 and M7 : Rout = (1/gm6 ) (1/gm7 ). Assumptions: MOSFETs operate in the active region. Analysis: The gain can be estimated using Eq. (16.109). µ 2K 1 2K p5 1 1 µ f2 f5 n2 ∼ Adm = = 4 4 λ2 I D2 λ5 I D5 For the amplifier in Fig. 16.50, 2IREF I1 = = 100 A 2 2 A = 20K n = 500 2 V I D2 = I D5 = I2 = 2IREF = 200 A K n2 K p5 = 100K p = 1000 A V2 16.7 and µ f 2µ f 5 Adm ∼ = 4 Active Loads in Operational Amplifiers 1235 A A 2 2 500 2 2 1000 2 1 1 V V V2 = = 16,000 4 0.0125 100 A 200 A The input resistance is twice the input resistance of M1 , which is infinite: Rid = ∞. The output resistance is determined by the parallel combination of the output resistances of M6 and M7 , which act as two source followers operating in parallel: 1 1 1 1 Rout = =√ gm6 gm7 2K n6 I D6 2K p7 I D7 To evaluate this expression, the current in the output stage must be found. The gate-source voltage of M11 is 2(200 A) 2I D11 = 2.54 V VG S 11 = VT N 11 + = 0.75 V + A K n11 125 V2 In this design, VT P = −VT N and the W/L ratios of M6 and M7 have been chosen so that K p7 = K n6 . Because I D6 must equal I D7 , VG S6 = VSG7 . Thus, both VG S6 and VSG7 are equal to one-half VG S 11 , and I D7 = I D6 = 250 A (1.27 V − 0.75 V)2 = 33.7 A 2 V2 The transconductances of M6 and M7 are also equal, gm7 = gm6 A −4 (33.7 × 10−6 A) = 1.30 × 10−4 S = 2 2.50 × 10 V2 and the output resistance at the Q-point is Rout = 3.85 k. Check of Results: A double check of our hand calculations indicates they are correct. Because of the complexity of the circuit, SPICE simulation represents an excellent check of hand calculations. The simulation results appear in the next exercise. Discussion: Simulation of the open-loop characteristics of high-gain amplifiers in SPICE can be difficult. The open-loop gain will amplify the offset voltage of the amplifier and may saturate the output. One approach is to first determine the offset voltage and then to apply a compensating voltage to the amplifier input to bring the output near zero. The steps are outlined next. In very high gain cases, SPICE may still be unable to converge because numerical “noise” during the simulation steps is amplified just as an input voltage. The successive voltage and current injection method discussed in Chapter 18 solves this problem. Computer-Aided Analysis: After drawing the circuit of Fig. 16.48 with the schematic editor, be sure to set the device parameters to the desired values. For the NMOS devices, KP = 25 A/V2 , VTO = 0.75 V, and LAMBDA = 0.0125 V−1 . For the PMOS devices, KP = 10 A/V2 , VTO = −0.75 V, and LAMBDA = 0.0125 V−1 . W and L must be specified for each individual transistor. For example, use W = 5 m and L = 1 m for a 5/1 device. 1236 Chapter 16 Analog Integrated Circuits –VOS VO = VOS VO VIC (a) (b) Figure 16.49 Op amp setups for SPICE simulation. (a) Offset voltage determination. (b) Circuit for open-loop analysis using SPICE transfer functions. The first step in the simulation is to find the offset voltage by operating the op amp in a voltage-follower configuration for which VO = VO S , as in Fig. 16.49(a). VO S is then applied as a differential input to the amplifier in Fig. 16.49(b) with a common-mode input VI C = 0. If the value of VO S is correct, an operating point analysis should yield a value of approximately 0 for VO . A transfer function analysis from VO S to the output will give values of Adm , Rid , and Rout . A transfer function analysis from VI C to the output will give Acm , Ric , and Rout . The SPICE results are given as the answers to the next exercise. Exercise: Simulate the amplifier in Fig. 16.48 using SPICE and compare the results to the answers in Ex. 16.6. Which terminal is the noninverting input? What are the offset voltage, common-mode and differential-mode gains, CMRR, common-mode and differential-mode input resistances, and output resistance? Answers: v1 ; 64.164 mV; 17,800; 0.052; 90.7 dB; ∞; ∞; 3.63 k 16.7.3 Bipolar Operational Amplifiers Active-load techniques can be applied equally well to bipolar op amps. In fact, most of the techniques discussed thus far were developed first for bipolar amplifiers and later applied to MOS circuits as NMOS and CMOS technologies matured. In the circuit in Fig. 16.50, a differential input stage with active load is formed by transistors Q 1 to Q 4 . The first stage is followed by a high gain C-E amplifier formed of Q 5 and its current source load Q 8 . Load resistance R L is driven by the class-AB output stage, consisting of transistors Q 6 and Q 7 biased by current I2 and diodes Q 11 and Q 12 . (The diodes will actually be implemented with BJTs, in this case with emitter areas five times those of Q 6 and Q 7 .) Based on our understanding of multistage amplifiers, the gain of this circuit is approximately Adm = Avt1 Avt2 Avt3 and IC2 ro5 µf5 gm2 Adm ∼ = gm5rπ 5 gm5 βo5 = = [gm2rπ 5 ][gm5 (ro5 ro8 (βo6 + 1)R L )][1] ∼ gm5 2 IC5 2 (16.111) in which it has been assumed that the input resistance of the class-AB output stage is much larger than the parallel combination of ro5 and ro8 . Note that the upper limit to Eq. (16.111) is set by the βo V A product of Q 5 , because IC2 is typically less than or equal to IC5 . 16.7 A A Q3 Q4 Active Loads in Operational Amplifiers 1237 +VCC AE5 Q5 v1 v2 Q1 Q2 Q6 5A Q11 A 5A Q12 A I1 IREF vO RL Q7 I2 Q9 Q10 A A Q8 5A –VEE Figure 16.50 Complete bipolar operational amplifier. Exercise: Estimate the voltage gain of the amplifier in Fig. 16.50 using Eq. (16.111) if I REF = 100 A, V A5 = 60 V, β o1 = 150, β o5 = 50, RL = 2 M, and VCC = VE E = 15 V. What is the gain of the first stage? The second stage? What should be the emitter area of Q5 ? What is RI D ? Which terminal is the inverting input? Answers: 7500; 5; 1500; 10 A; 150 k; v1 Exercise: Simulate the amplifier in the previous exercise using SPICE and determine the offset voltage, voltage gain, differential-mode input resistance, CMRR, and common-mode input resistance. Answers: 3.28 mV; 8440; 165 k; 84.7 dB; 59.1 M 16.7.4 A BJT Amplifier with Improved Voltage Gain To improve the gain of the amplifier in Fig. 16.50, Eq. (16.107) indicates that we need a transistor with an improved βo V A product. We also see from the exercise that the first-stage gain is low because rπ 5 is small (the ratio IC2 /IC5 is too low). Mentally searching through our bag of basic circuit tools, we should discover the two-transistor Darlington circuit, which has a current gain of βo1 βo2 , an amplification factor of µ f 2 /4, an output resistance of ro2 /2, and an input resistance of 2βo1rπ 2 . This configuration has been used to replace Q 5 in the circuit in Fig. 16.51. The pnp Darlington circuit requires an emitter-base bias of 2VE B , and the buffered current mirror provides proper dc balance at the collectors of Q 3 and Q 4 . Let us now determine an expression for the voltage gain of the amplifier in Fig. 16.51. Writing the voltage gain as a product of the gains of the individual stages and assuming the output stage has unity gain, Adm = va vb vo = Avt1 Avt2 (1) ∼ = Avt1 Avt2 vid va vb (16.112) 1238 Chapter 16 Analog Integrated Circuits +VCC A A 4A va Q5 v1 Q7 Q4 Q3 Q1 Q2 4A Q6 vb Q8 v2 4A Q10 4A Q11 A vo A I1 IB I2 Q13 Q12 A Q9 Q14 2A A –VEE Figure 16.51 Op amp with buffered current mirror and second-stage Darlington circuit. The input stage provides a gain of Avt1 ro2 ∼ = gm2 (ro2 ro4 2βo6rπ 7 ) = gm2 2βo6rπ 7 2 (16.113) in which the load resistance represents the parallel combination of the output resistances of transistors Q 2 and Q 4 and the input resistance of the Darlington stage. We expect ro4 ∼ = ro2 , and comparing the input resistance of the Darlington stage to ro2 yields βo7 VT 2βo6 2βo6rπ 7 IC7 ∼ I1 0.025βo6 βo7 = = V A2 + VC E2 ro2 I2 V A2 + VC E2 IC2 (16.114) Using I2 = 2I1 , βo = 50, V A = 60 V, and VC E = 15 V, we find that the value of Eq. (16.114) is approximately 0.42. Therefore, an estimate for the gain of the first stage is Avt1 ∼ = gm2 (0.5ro2 0.42ro2 ) = 0.23µ f 2 (16.115) For large values of R L , we can assume the resistance at node vb is dominated by the output resistances of the Darlington stage and current source I2 . For this case, the gain of the second stage is equal to gm7 Avt2 ∼ = 2 2ro7 µf7 gm7 2ro7 gm7 2ro7 = ro14 ∼ ro7 = = 3 2 3 2 5 5 (16.116) Combining Eqs. (16.112), (16.114), and (16.116) and assuming that the output stage provides a gain of unity yields a final estimate for the voltage gain of the amplifier in Fig. 16.51: µ f 2µ f 7 40(75) 40(75) Adm ∼ = = 4.15 × 105 = 22 22 (16.117) 16.7 Active Loads in Operational Amplifiers 1239 Exercise: Calculate the voltage gain of the circuit in Fig. 16.51 including the effect of a 2-k load resistor on the output if the input resistance of the output stage is (β o8 + 1) RL . Assume β o = 100, V A + VC E = 75 V, and I B = 100 A. Which terminal is the noninverting input? Answers: 2.14 × 105 ; v2 Exercise: Use SPICE to determine the offset voltage, voltage gain, differential-mode input resistance, output resistance, CMRR, and common-mode input resistance of the amplifier in Fig. 16.51. Assume β on = 150, β op = 50, V A = 60 V, RL = 2 k, and I B = 100 A. Answers: 11.6 V, 2.40 × 105 , 128 k, 822 , 120 dB, 57.3 M We can come up with an almost endless array of circuit permutations to modify the various characteristics of the amplifier in Figs. 16.50 and 16.51. Cascode circuits can be used in the input stage and second stage. In BIMOS technology, FETs can be used to increase the input resistance at Q 5 as well as that of the output stages. A FET input stage will offer higher input resistance but lower voltage gain. 16.7.5 Input Stage Breakdown Although the bipolar amplifier designs discussed thus far have provided excellent voltage gain, input resistance, and output resistance, the amplifiers all have a significant flaw. The input stage does not offer overvoltage protection and can easily be destroyed by the large input voltage differences that can occur, not only under fault conditions but also during unavoidable transients during normal use of the amplifier. For example, the voltage across the input of an op amp can temporarily be equal to the total supply voltage span during slew-rate limited recovery. Consider the worst-case fault condition applied to the differential pair in Fig. 16.52. Under the conditions shown, the base-emitter junction of Q 1 will be forward-biased, and that of Q 2 reverse-biased by a voltage of (VCC + VE E − VB E1 ). If VCC = VE E = 22 V, the reverse voltage exceeds 41 V. Because of heavy doping in the emitter, the typical Zener breakdown voltage of the base-emitter junction of an npn transistor is only 5 to 7 V. Thus any voltage exceeding this value by more than one diode drop may destroy at least one of the transistors in the differential input pair. +VCC VBE1 Q1 Q2 R −VEE VBE1 VBE 2 (a) VBE 2 = − (VCC + VEE − VBE1 ) Q1 Q2 R VBE 2 (b) Figure 16.52 (a) Differential input stage voltages under a fault condition. (b) Simple diode input protection circuit. 1240 Chapter 16 Analog Integrated Circuits Early IC op amps required circuit designers to add external diode protection across the input terminals, as shown Fig. 16.52(b). The diodes prevent the differential input voltage from exceeding approximately 1.4 V, but this technique adds extra components and cost to the design. The two resistors limit the current through the diodes. The A741 described in the next section was the first commercial IC op amp to solve this problem by providing a fully protected input, as well as output, stage. 16.8 THE A741 OPERATIONAL AMPLIFIER The now classic Fairchild A741 operational-amplifier design was the first to provide a highly robust amplifier from the application engineer’s point of view. The amplifier provides excellent overall characteristics (high gain, input resistance and CMRR, low output resistance, and good frequency response) while providing overvoltage protection for the input stage and short-circuit current limiting of the output stage. The 741 style of amplifier design quickly became the industry standard and spawned many related designs. By studying the 741 design, we will find a number of new amplifier circuit design and bias techniques. Figure 16.53 is a simplified schematic of the A741 operational amplifier. The three bias sources shown in symbolic form are discussed in more detail following a description of the overall circuit. The op amp has two stages of voltage gain followed by a class-AB output stage. In the first stage, transistors Q 1 to Q 4 form a differential amplifier with a buffered current mirror active load, Q 5 to Q 7 . Practical operational amplifiers offer an offset voltage adjustment port, which is provided in the 741 through the addition of 1-k resistors R1 and R2 and an external potentiometer REXT . VCC Q8 I3 I2 220 µA 670 µA Q9 Q15 Q17 Q1 Q14 I1 18 µ A VCC VCC Q7 Q10 R4 R3 R2 50 kΩ 50 kΩ 1 kΩ 1 kΩ 22 Ω 40 kΩ Q16 Q12 Q11 Q6 R1 R8 Q18 R6 Q4 Q5 27 Ω Q13 Q2 Q3 R7 R5 100 Ω – VEE = –15 V REXT Input stage Second stage Output stage Figure 16.53 Overall schematic of the classic Fairchild A741 operational amplifier (the bias network appears in Fig. 16.54). 16.8 The µA741 Operational Amplifier 1241 The second stage consists of emitter follower Q 10 driving common-emitter amplifier Q 11 with current source I2 and transistor Q 12 as load. Transistors Q 13 to Q 18 form a short-circuit protected class-AB push-pull output stage that is buffered from the second gain stage by emitter follower Q 12 . Exercise: Reread this section and be sure you understand the function of each individual transistor in Fig. 16.53. Make a table listing the function of each transistor. 16.8.1 Bias Circuitry The three current sources shown symbolically in Fig. 16.53 are generated by the bias circuitry in Fig. 16.54. The value of the current in the two diode-connected reference transistors Q 20 and Q 22 is determined by the power supply voltage and resistor R5 : IREF = VCC + VE E − 2VB E 15 + 15 − 1.4 = 0.733 mA = R5 39 k (16.118) assuming ±15-V supplies. Current I1 is derived from the Widlar source formed of Q 20 and Q 21 . The output current for this design is VT IREF I1 = ln 5000 I1 (16.119) Using the reference current calculated in Eq. (16.118) and iteratively solving for I1 in Eq. (16.119) yields I1 = 18.4 A. The currents in mirror transistors Q 23 and Q 24 are related to the reference current IREF by their emitter areas using Eq. (16.17). Assuming VO = 0 and VCC = 15 V, and neglecting the voltage drop across R7 and R8 in Fig. 16.53, VEC23 = 15 + 1.4 = 16.4 V and VEC24 = 15 − 0.7 = 14.3 V. +VCC 0.25 A A Q22 IREF 0.75 A R5 −1.4 V 39 kΩ −1.4 V Q23 I2 Q24 I3 +0.7 V I1 Q21 Q20 A A 5 kΩ −VEE Figure 16.54 741 bias circuitry with voltages corresponding to VO = 0 V. 1242 Chapter 16 Analog Integrated Circuits Using these values with β F = 50 and V A = 60 V, the two source currents are 16.4 V 60 V I2 = 0.75(733 A) = 666 A 0.7 V 2 1+ + 60 V 50 14.4 V 1+ 60 V I3 = 0.25(733 A) = 216 A 0.7 V 2 1+ + 60 V 50 1+ (16.120) and the two output resistances are R2 = V A23 + VEC23 60 V + 16.4 V = 115 k = I2 0.666 mA V A24 + VEC24 60 V + 14.3 V R3 = = = 344 k I3 0.216 mA (16.121) Exercise: What are the values of I REF , I 1 , I 2 , and I 3 in the circuit in Fig. 16.54 for VCC = VE E = 22 V? Answers: 1.09 mA, 20.0 A, 1.08 mA, 351 A Exercise: What is the output resistance of the Widlar source in Fig. 16.54 operating at 18.4 A for V A = 60 V and VE E = 15 V? Answer: 18.8 M 16.8.2 dc Analysis of the 741 Input Stage The input stage of the A741 amplifier is redrawn in the schematic in Fig. 16.55. As noted earlier, Q 1 , Q 2 , Q 3 , and Q 4 form a differential input stage with an active load consisting of the buffered current mirror formed by Q 5 , Q 6 , and Q 7 . In this input stage there are four base-emitter junctions between inputs v1 and v2 , two from the npn transistors and, more importantly, two from the pnp transistors, and (v1 − v2 ) = (VB E1 + VE B3 − VE B4 − VB E2 ). In standard bipolar IC processes, pnp transistors are formed from lateral structures in which both junctions exhibit breakdown voltages equal to that of the collector-base junction of the npn transistor. This breakdown voltage typically exceeds 50 V. Because most general-purpose op amp specifications limit the power supply voltages to less than ±22 V, the emitter-base junctions of Q 3 and Q 4 provide sufficient breakdown voltage to fully protect the input stage of the amplifier, even under a worst-case fault condition, such as that depicted in Fig. 16.52(a). Q-Point Analysis In the 741 input stage in Fig. 16.55, the current mirror formed by transistors Q 8 and Q 9 operates with transistors Q 1 to Q 4 to establish the bias currents for the input stage. Bias current I1 represents the output of the Widlar source discussed previously (18 A) and must be equal to the collector The µA741 Operational Amplifier 16.8 1243 VCC Q8 IC8 Q9 v1 IC1 IC2 Q1 Q2 Q3 v2 Q4 2IB4 I1 IC3 IC4 Q7 IC6 IC5 Q5 vO Q6 R1 R3 R2 50 kΩ 1 kΩ 1 kΩ –VEE Figure 16.55 A741 input stage. current of Q 8 plus the base currents of matched transistors Q 3 and Q 4 : I1 = IC8 + I B3 + I B4 = IC8 + 2I B4 (16.122) For high current gain, the base currents are small and IC8 ∼ = I1 . The collector current of Q 8 mirrors the collector currents of Q 1 and Q 2 , which are summed together in mirror reference transistor Q 9 . Assuming high current gain and ignoring the collectorvoltage mismatch between Q 7 and Q 8 , IC8 = IC1 + IC2 = 2IC2 (16.123) Combining Eqs. (16.122) and (16.123) yields the ideal bias relationships for the input stage I1 IC1 = IC2 ∼ = 2 and I1 IC3 = IC4 ∼ = 2 (16.124) because the emitter currents of Q 1 and Q 3 and Q 2 and Q 4 must be equal. The collector current of Q 3 establishes a current equal to I1 /2 in current mirror transistors Q 5 and Q 6 as well. Thus, transistors Q 1 to Q 6 all operate at a nominal collector current equal to one-half the value of source I1 . Now that we understand the basic ideas behind the input stage bias circuit, let us perform a more exact analysis. Expanding Eq. (16.122) using the current mirror expression from Eq. (16.9), VEC8 V A8 I1 = 2IC2 + 2I B4 2 VE B8 1+ + β F O8 V A8 1+ (16.125) 1244 Chapter 16 Analog Integrated Circuits IC2 is related to I B4 through the current gains of Q 2 and Q 4 : IC2 = α F2 I E2 = α F2 (β F O4 + 1)I B4 = β F O2 (β F O4 + 1)I B4 β F O2 + 1 (16.126) Combining Eqs. (16.125) and (16.126) and solving for IC2 assuming small errors yields I1 IC2 ∼ × = 2 1 1+ VEC8 − VE B8 2 1 − + V A8 β F O8 β F O4 (16.127) which is equal to the ideal value of I1 /2 but reduced by the nonideal current mirror effects because of finite current gain and Early voltage. The emitter current of Q 4 must equal the emitter current of Q 2 , and so the collector current of Q 4 is IC4 = α F4 I E4 = α F4 IC2 β F O4 β F O2 + 1 = IC2 α F2 β F O4 + 1 β F O2 (16.128) The use of buffer transistor Q 7 essentially eliminates the current gain defect in the current mirror. Note from the full amplifier circuit in Fig. 16.53 that the base current of transistor Q 10 , with its 50-k emitter resistor R4 , is designed to be approximately equal to the base current of Q 7 , and VC E6 ∼ = VC E5 as well. Thus, the current mirror ratio is quite accurate and I1 IC5 = IC6 = IC3 ∼ = 2 (16.129) If 50-k resistor R3 were omitted, then the emitter current of Q 7 would be equal only to the sum of the base currents of transistors Q 5 and Q 6 and would be quite small. Because of the Q-point dependence of β F , the current gain of Q 7 would be poor. R3 increases the operating current of Q 7 to improve its current gain as well as to improve the dc balance and transient response of the amplifier. The value of R3 is chosen to approximately match I B7 to I B10 . To complete the Q-point analysis, the various collector-emitter voltages must be determined. The collectors of Q 1 and Q 2 are 1VE B below the positive power supply, whereas the emitters are 1VB E below ground potential. Hence, VC E1 = VC E2 = VCC − VE B9 + VB E2 ∼ = VCC (16.130) The collector and emitter of Q 3 are approximately 2VB E above the negative power supply voltage and 1VB E below ground, respectively: VEC3 = VE3 − VC3 = −0.7 V − (−VE E + 1.4 V) = VE E − 2.1 V (16.131) The buffered current mirror effectively minimizes the error due to the finite current gain of the transistors, and VC E6 = VC E5 ∼ = 2VB E = 1.4 V, neglecting the small voltage drop (<10 mV) across R1 and R2 . Finally, the collector of Q 8 is 2VB E below zero so that VEC8 = VCC + 1.4 V (16.132) and the emitter of Q 7 is 1VB E above −VE E : VC E7 = VE E − 0.7 V (16.133) 16.8 EXAMPLE 16.9 The µA741 Operational Amplifier 1245 A741 INPUT STAGE BIAS CURRENTS Find the currents in the 741 input stage. PROBLEM Calculate the bias currents in the 741 input stage if I1 = 18 A, β F Onpn = 150, V Anpn = 75 V, β F O pnp = 60, V Apnp = 60 V, and VCC = VE E = 15 V. SOLUTION Known Information and Given Data: A741 input stage depicted in Fig. 16.55. I1 = 18A, β F Onpn = 150, V Anpn = 75 V, β F O pnp = 60, V Apnp = 60 V, and VCC = VEE = 15 V. Unknowns: IC1 , IC2 , IC3 , IC4 , IC5 , and IC6 Approach: Use given data to evaluate Eqs. (16.127) through (16.129). Assumptions: Transistors are in the active region; use default values of I S . Analysis: From Fig. 16.55, we find that the collector-emitter voltage of Q 8 is equal to VCC + VB E1 + VE B3 ∼ = 16.4 V. Substituting the known values into Eq. (16.127) gives IC2 = 18 A 2 1 = 7.32 A 16.4 V 1+ 1 60 V + 2 150 0.7 V 1+ + (60 + 1) 50 60 V 150 + 1 IC1 = IC2 = 7.32 A Equation (16.128) yields IC3 = IC4 IC2 β F O4 = α F4 = α F2 β F O4 + 1 IC5 ∼ = IC3 = 7.25 A β F O2 + 1 β F O2 and IC2 60 = 61 151 150 IC2 = 7.25 A IC6 = IC4 = 7.25 A Check of Results: The basic objective of the bias circuit would be to set all currents to 18 A/2 or 9 A. Our calculations are close to this value and appear correct. Discussion: The actual bias currents are slightly greater than 7 A, whereas the ideal value would be 9 A. The dominant source of error arises from the collector-emitter voltage mismatch of the pnp current mirror. Computer-Aided Analysis: We draw the circuit using the schematic editor and set the BJT parameters. For the npn devices, BF = 150 and VAF = 75 V. For the pnp transistors, BF = 60 and VAF = 60 V. Source VO is added to balance the circuit by forcing the output voltage to the same voltage as that which will appear at the collector of Q 5 . Otherwise, the voltage at the collectors of Q 4 and Q 6 will float to a value determined by the difference in overall output resistances of transistors Q 4 and Q 6 . When balance is achieved, the current in source VO will be nearly zero. Table 16.3 summarizes the Q-points based on these calculations and Eqs. (16.124) to (16.129) and compares them with the SPICE operating point simulation results. 1246 Chapter 16 Analog Integrated Circuits Q8 VCC Q9 15 V Q1 Q2 Q3 Q4 I1 18 UA Q7 VEE VO 15 V Q5 Q6 R1 R3 R2 1K 50 K 1K 13.69 V TABLE 16.3 Q-points of 741 Input Stage Transistors for I 1 = 18 A and VCC = VEE = 15 V TRANSISTORS Q-POINT SPICE RESULTS Q 1 and Q 2 Q 3 and Q 4 Q 5 and Q 6 Q7 Q8 Q9 7.32 A, 15 V 7.25 A, 12.9 V 7.25 A, 1.4 V 12.2 A, 14.3 V 17.7 A, 16.4 V 14.0 A, 0.7 V 7.30 A, 15.0 V 7.24 A, 13.0 V 7.16 A, 1.30 V 13.1 A, 14.3 V 17.8 A, 16.3 V 14.1 A, 0.66 V Exercise: Remove VO and simulate the 741 input stage amplifier. What are the new collector currents? What are the voltages at the collectors or Q5 and Q6 ? Answers: 7.31 A, 7.28 A, 7.25 A, 7.22 A, 7.18 A, 7.22 A, 13.1 A, 17.8 A, 14.1 A; −13.7 V, −13.1 V Exercise: Suppose buffer transistor Q7 and resistor R3 are eliminated from the amplifier in Fig. 16.55 and Q5 and Q6 were connected as a standard current mirror. What would be the collector-emitter voltage of Q6 if VBE6 = 0.7 V, β F O6 = 100, and V A6 = 60 V? Use Eq. (16.89). Answer: 1.90 V 16.8 +VCC A 2 A 2 Q7 Q8 v1 Q1 The µA741 Operational Amplifier +VCC Q7 A 2 A 2 Q8 v2 Q2 ib Q4 Q3 I1 2 1247 I1 2 R Q2 vid 2 ie Q4 io R Rin4 –VEE –VEE Figure 16.57 Differential-mode half-circuit Figure 16.56 Symmetry in the 741 input stage. for the 741 input stage. 16.8.3 ac Analysis of the 741 Input Stage The 741 input stage is redrawn in symmetric form in Fig. 16.56, with its active load temporarily replaced by two resistors. From Fig. 16.56, we see that the collectors of Q 1 and Q 2 as well as the bases of Q 3 and Q 4 lie on the line of symmetry of the amplifier and represent virtual grounds for differential-mode input signals. The corresponding differential-mode half-circuit shown in Fig. 16.57 is a common-collector stage followed by a common-base stage, a C-C/C-B cascade. The characteristics of the C-C/C-B cascade can be determined from Fig. 16.57 and our knowledge of single-stage amplifiers. The emitter current of Q 2 is equal to its base current i b multiplied by (βo2 + 1), and the collector current of Q 4 is αo4 times the emitter current. Thus, the output current can be written as io = αo4 ie = αo4 (βo2 + 1)ib ∼ = βo2 ib (16.134) The base current is determined by the input resistance to Q 2 : vid 2 ib = = rπ 2 + (βo2 + 1)Rin4 vid vid 2 2 ∼ vid = = rπ 4 rπ 2 + rπ 4 4rπ 2 rπ 2 + (βo2 + 1) βo4 + 1 (16.135) in which Rin4 = rπ 4 /(βo4 + 1) represents the input resistance of the common-base stage. Combining Eqs. (16.134) and (16.135) yields gm2 vid io ∼ = vid = βo2 4rπ 2 4 (16.136) Each side of the C-C/C-B input stage has a transconductance equal to one-half of the transconductance of the standard differential pair. From Eq. (16.135) we can also see that the differential-mode 1248 Chapter 16 Analog Integrated Circuits io Q4 Q2 Rout4 1 gm2 io Q4 Rout4 1 R= g m2 Figure 16.58 Output resistance of C-C/C-B cascade. input resistance is twice the value of the corresponding C-E stage: Rid = vid = 4rπ 2 ib (16.137) From Fig. 16.58, we can see that the output resistance is equivalent to that of a common-base stage with a resistor of value 1/gm2 in its emitter: 1 ∼ Rout = ro4 (1 + gm4 R) = ro4 1 + gm4 = 2ro4 (16.138) gm2 16.8.4 Voltage Gain of the Complete Amplifier We now use the results from the previous section to analyze the overall ac performance of the op amp. We find a Norton equivalent circuit for the input stage and then couple it with a two-port model for the second stage. Norton Equivalent of the Input Stage Figure 16.59 is the simplified differential-mode ac equivalent circuit for the input stage. We use Figure 16.59(a) to find the short-circuit output current of the first stage. Based on our analysis of Fig. 16.57, the differential-mode input signal establishes equal and opposite currents in the two v1 vid 2 Q3 i Q7 Q5 (a) R3 Q3 Q4 Rout4 io Q7 i Q6 50 kΩ R1 Q2 Q1 vid 2 Q4 i 1 kΩ v2 Q2 Q1 R2 Q5 1 kΩ 1 kΩ Rout6 Q6 50 kΩ R1 R3 R2 1 kΩ (b) Figure 16.59 Circuits for finding the Norton equivalent of the input stage. vx The µA741 Operational Amplifier 16.8 1249 sides of the differential amplifier where i = (gm2 /4)vid . Current i, exiting the collector of Q 3 , is mirrored by the buffered current mirror so that a total signal current equal to 2i flows in the output terminal: gm2 vid io = −2i = − = (−20IC2 )vid 2 (16.139) 20 =− 7.32 × 10−6 A vid = (−1.46 × 10−4 S)vid V The Thévenin equivalent resistance at the output is found using the circuit in Fig. 16.59(b) and is equal to Rth = Rout6 Rout4 (16.140) Because only a small dc voltage is developed across R2 , the output resistance of Q 6 can be calculated from IC6 R2 0.0073 V ∼ ∼ Rout6 = ro6 [1 + gm6 R2 ] = ro6 1 + = ro6 1 + (16.141) = 1.3ro6 VT 0.025 V The output resistance of Q 4 was already found in Eq. (16.138) to be 2ro4 . Substituting the results from Eqs. (16.138) and (16.141) into Eq. (16.140), Rth = 2ro4 1.3ro6 = 0.79ro4 ∼ = 0.79 60 V = 6.54 M 7.25 × 10−6 A (16.142) in which ro4 = ro2 has been assumed for simplicity with V A + VC E = 60 V. The resulting Norton equivalent circuit for the input stage appears in Fig. 16.60. Based on the values in this figure, the open-circuit voltage gain of the first stage is −955. SPICE simulations yield values very similar to those in Fig. 16.60: (1.40 × 10−4 S)vid , 6.95 M, and Adm = −973. Exercise: Improve the estimate of Rth using the actual values of VC E6 and VC E4 if VCC = VE E = 15 V and V A = 60 V. What are the values of Rout4 and Rout6 ? Answers: 7.12 M; 20.2 M, 11.0 M Model for the Second Stage Figure 16.61 is a two-port representation for the second stage of the amplifier. Q 10 is an emitter follower that provides high input resistance and drives a common-emitter amplifier consisting i1 + v1 1.46 × 10 –4vid 6.54 MΩ i2 Q10 Q11 50 kΩ 100 Ω R2 + v2 115 kΩ – Figure 16.60 Norton equivalent of the Figure 16.61 Two-port representation for the 741 input stage. second stage. – 1250 Chapter 16 Analog Integrated Circuits i2 i1 i1 Q10 ve v1 Q10 Q11 50 k Ω ve v1 100 Ω Rin11 (a) 50 k Ω 20.7 k Ω (b) Figure 16.62 Network for finding y11 and y21 . of Q 11 and its current source load represented by output resistance R2 . A y-parameter model is constructed for this network. From Fig. 16.53 and the bias current analysis, we can see that the collector current of Q 11 is approximately equal to I2 or 666 A. Calculating the collector current of Q 10 yields IC11 VB11 666 A 0.7 + (0.67 mA)(0.1 k) IC10 ∼ + = + = 19.8 A (16.143) = I E10 = β F11 50 k 150 50 k Using these values to find the small-signal parameters with (βon = 150) gives rπ 10 = βo10 VT 3.75 V = 189 k = IC10 19.8 A rπ 11 = and 3.75 V = 5.63 k 0.666 mA (16.144) Parameters y11 and y21 are calculated by applying a voltage v1 to the input port and setting v2 = 0, as in Fig. 16.62. The input resistance to Q 11 is that of a common-emitter stage with a 100- emitter resistor: Rin11 = rπ 11 + (βo11 + 1)100 ∼ = 5630 + (151)100 = 20.7 k (16.145) This value is used to simplify the circuit, as in Fig. 16.62(b), and the input resistance to Q 10 is [y11 ]−1 = rπ 10 + (βo10 + 1)(50 k Rin11 ) = 189 k + (151)(50 k 20.7 k) = 2.40 M (16.146) The gain of emitter follower Q 10 is: ve = v1 (βo10 + 1)(50 k Rin11 ) rπ 10 + (βo10 + 1)(50 k Rin11 ) (151)(50 k 20.7 k) = = 0.921v1 189 k + (151)(50 k 20.7 k) (16.147) The output current i2 in Fig. 16.60(a) is given by i2 = ve 1 gm11 + 100 = 0.921v1 1 40 (0.666 mA) V + 100 = 0.00670v1 (16.148) 16.8 The µA741 Operational Amplifier 1251 Rout11 i1 i2 i2 Q10 Q11 Q11 v2 R2 50 kΩ 1 100 Ω 50 kΩ gm10 v2 R2 100 Ω Figure 16.63 Network for finding y12 and y22 . yielding a forward transconductance of y21 = 6.70 mS (16.149) Parameters y12 and y22 can be found from the network in Fig. 16.63. We assume that the reverse transconductance y12 is negligible and reserve its calculation for Prob. 16.133. The output conductance y22 can be determined from Fig. 16.63(b). [y22 ]−1 = R2 Rout11 (16.150) where R2 = 115 k was calculated during the analysis of the bias circuit. Because the voltage drop across the 100- resistor is small, the output resistance of Q11 is approximately IC11 R E V A11 + VC E11 Rout11 = ro11 [1 + gm11 R E ] = 1+ IC11 VT (16.151) 60 V + 13.6 V 0.067 V = 1+ = 407 k 0.666 mA 0.025 V and [y22 ]−1 = 115 k 407 k = 89.1 k (16.152) Figure 16.64 depicts the completed two-port model for the second stage, driven by the Norton equivalent of the input stage. Using this model, the open-circuit voltage gain for the first two stages of the amplifier is v2 = −0.00670(89.1 k)v1 = −597v1 v1 = −1.46 × 10−4 (6.54 M 2.40 M)vid = −256vid (16.153) v2 = −597(−256vid ) = 153,000vid Note from Eq. (16.152) that the 2.42-M input resistance of Q 10 reduces the voltage gain of the first stage by a factor of almost 4. 1.46 × 10–4vid 6.54 MΩ v1 89.1 kΩ 2.40 MΩ 0.00670 v1 First stage Second stage Figure 16.64 Combined model for first and second stages. v2 1252 Chapter 16 Analog Integrated Circuits Exercise: What would be the voltage gain of the input stage if transistor Q10 and its 50-k emitter resistor were omitted so that the output of the first stage would be connected directly to the base of Q11 ? Use the small-signal element values already calculated. Answer: −3.00 16.8.5 The 741 Output Stage Figure 16.65 shows simplified models for the 741 output stage. Transistor Q 12 is the emitter follower that buffers the high impedance node at the output of the second stage and drives the push-pull output stage composed of transistors Q 15 and Q 16 . Class-AB bias is provided by the sum of the base-emitter voltages of Q 13 and Q 14 , represented as diodes in Fig. 16.65(b). The 40-k resistor is used to increase the value of IC13 . Without this resistor, IC13 would only be equal to the base current of Q 14 . The short-circuit protection circuitry in Fig. 16.53 is not shown in Fig. 16.65 in order to simplify the diagram. +V + CC = +15 + V 219 µA I3 VCC 219 µA Q15 I3 Q15 Q13 iout Q14 Q13 RL RL Q14 40 kΩ Q16 Q16 Q12 Q12 vS (a) + vO – –VEE –VEE = –15 V (b) Figure 16.65 (a) 741 output stage without short-circuit protection. (b) Simplified output stage. The input and output resistances of the class-AB output stage are actually complicated functions of the signal voltage because the operating current in Q 15 and Q 16 changes greatly as the output voltage changes. However, because only one transistor conducts strongly at any given time in the class-AB stage, separate circuit models can be used for positive and negative output signals. The model for positive signal voltages is shown in Fig. 16.66. (The model for negative signal swings is similar except npn transistor Q 15 is replaced by pnp transistor Q 16 connected to the emitter of Q 12 .) Let us first determine the input resistance of transistor Q 12 . If Rin12 is much larger than the 89-k output resistance of the two-port in Fig. 16.66, then it does not significantly affect the 16.8 The µA741 Operational Amplifier 1253 overall voltage gain of the amplifier. Using single-stage amplifier theory, Rin12 = rπ 12 + (βo12 + 1)Req1 (16.154) Req1 = rd14 + rd13 + R3 Req2 (16.155) Req2 = rπ 15 + (βo15 + 1)R L ∼ = (βo15 + 1)R L (16.156) where and The value of R3 (344 k) was calculated in the bias circuit section. For IC12 = 216 A, and assuming a representative collector current in Q 15 of 2 mA, Req2 = rπ 15 + (βo15 + 1)R L = 3.75 V + (151)2 k = 304 k 2 mA (16.157) Note that the value of Req2 is dominated by the reflected load resistance βo15 R L . Resistor rπ 15 represents a small part of Req2 , and knowing the exact value of IC15 is not critical. Req1 = rd14 + rd13 + R3 Req2 = 2 0.025 V + 344 k 304 k = 162 k 0.216 mA (16.158) 1.25 V + (51)162 k = 8.27 M 0.216 mA (16.159) and Rin12 = rπ12 + (βo12 + 1)Req1 = −1 Because Rin12 is approximately 100 times the output resistance (y22 ) of the second stage, Rin12 has little effect on the gain of the second stage. Although the value of Rin12 changes for different values of load resistance, the overall op amp gain is not affected because the value of Rin12 is so −1 much larger than the value of y22 in Fig. 16.66. Similar results are obtained for negative signal voltages. The values are slightly different because the current gain of the pnp transistor Q 16 differs from that of the npn transistor Q 15 . Req2 Req3 R3 R3 Q15 Q15 rd13 rd13 RL = 2 kΩ Req1 rd14 rd14 Q12 Rin12 (a) Rout Q12 = 89.1 kΩ y–1 22 (b) Figure 16.66 Circuits for determining input and output resistance of the output stage. 1254 Chapter 16 Analog Integrated Circuits 16.8.6 Output Resistance The output resistance of the amplifier for positive output voltages is determined by transistor Q 15 rπ 15 + Req3 βo15 + 1 (16.160) −1 rπ 12 + y22 = R3 rd13 + rd14 + βo12 + 1 0.025 V 5.71 k + 89.1 k = 304 k 2 + = 2.08 k 0.219 mA 51 (16.161) Ro = in which Req3 Substituting the values from Eq. (16.161) into Eq. (16.160) yields Ro = 1.88 k + 2.08 k = 26.2 151 (16.162) From Fig. 16.56, we can see that the 27- resistor R7 , which determines the short-circuit current limit, adds directly to the overall output resistance of the amplifier so that actual op-amp output resistance is Rout = Ro + R7 = 53 (16.163) Exercise: Repeat the calculation of Rin12 and Rout if pnp transistor Q16 has a current gain of 50, I C16 = 2 mA, and I C15 = 0. Be sure to draw the new equivalent circuit of the output stage for negative output voltages. Answers: 3.94 M ( 89.1 k), 53 + 22 = 75 16.8.7 Short Circuit Protection For simplicity, the output short-circuit protection circuitry was not shown in Fig. 16.65. Referring back to the complete op amp schematic in Fig. 16.53, we see that short-circuit protection is provided by resistors R7 and R8 and transistors Q 17 and Q 18 . The circuit is identical to the one presented in the previous chapter in Fig. 15.60(a). Transistors Q 17 and Q 18 are normally off, but if the current in resistor R7 becomes too high, then transistor Q 17 turns on and steals the base current from Q 15 . Likewise, if the current in resistor R8 becomes too large, then transistor Q 18 turns on and removes the base current from Q 16 . The positive and negative short-circuit current levels will be limited to approximately VB E17 /R7 and −VE B18 /R8 , respectively. As already mentioned, resistors R7 and R8 increase the output resistance of the amplifier since they appear directly in series with the output terminal. Exercise: Estimate the positive and negative short-circuit output current in the 741 op amp in Fig. 16.53. Answers: 26 mA; −32 mA 16.9 The Gilbert Analog Multiplier 1255 16.8.8 Summary of the A741 Operational Amplifier Characteristics Table 16.4 is a summary of the characteristics of the A741 operational amplifier. Column 2 gives our calculated values; column 3 presents values typically found in the actual commercial product. The observed values depend on the exact values of current gain and Early voltage of the npn and pnp transistors and vary from process run to process run. TABLE 16.4 A741 Characteristics CALCULATION TYPICAL VALUES 153,000 2.05 M 53 49 nA — 200,000 2 M 75 80 nA 2 mV Voltage gain Input resistance Output resistance Input bias current Input offset voltage 16.9 THE GILBERT ANALOG MULTIPLIER In Chapter 11 we saw how operational amplifiers could be used to perform scaling, addition, subtraction, integration, and differentiation of electronic signals. However, one of the more difficult operations to realize is accurate multiplication of two analog signals. Barrie Gilbert, another of the “legends” of integrated circuit design, discovered a solution to this problem using the characteristics of the bipolar transistor. The basic multiplier “core” in Fig. 16.67 consists of three differential pairs. The Q 1 –Q 2 pair has significant emitter degeneration so that the transconductance of the pair is approximately6 1/R1 . Under this assumption, the collector currents of the lower pair can VCC R ic4 ic3 + Q3 R + vo – ic5 Q4 ic6 Q5 Q6 v2 – + v1 – ic2 ic1 Q1 R1 R1 Q2 IBB –VEE Figure 16.67 Gilbert multiplier core. 6 More sophisticated voltage-to-current converters (transconductance amplifiers) can also be used. 1256 Chapter 16 Analog Integrated Circuits be written as IB B v1 i c1 ∼ + = 2 2R1 IB B v1 i c2 ∼ − = 2 2R1 for |v1 | ≤ I B B R1 (16.164) The bound on v1 is determined by the requirement that neither collector current can be negative. Multiplier output current vo is taken from the upper two differential pairs and can be written as vo = [(i c3 + i c5 ) − (i c4 + i c6 )]R = [(i c3 − i c4 ) + (i c5 − i c6 )]R (16.165) Using Eq. (15.63), we can write expressions for the collector current differences in this equation: v2 v2 i c3 − i c4 = i c1 tanh and i c5 − i c6 = −i c2 tanh (16.166) 2VT 2VT Using these equations, the output voltage can be reduced to R v2 v2 vo = (i c1 − i c2 )R tanh = v1 tanh 2VT R1 2VT (16.167) At this point, one approach to multiplication is to expand the hyperbolic tangent as a series, and then keep only the first term: x3 R v2 x3 tanh(x) = x − for v + ··· and vo ∼ x (16.168) = 1 3 R1 2VT 3 where x = v2 /2VT . However, this approach greatly restricts the input signal range of v2 to only a few tens of mV [see discussion following Eq. (15.64)]. The key to the full range Gilbert multiplier is to use another pair of pn junctions to “predistort” the input signal as in Fig. 16.68. Diode connected transistors Q 9 and Q 10 are driven by a second transconductance stage formed by Q 7 and Q 8 for which IE E v3 i c7 ∼ + = 2 2R3 IE E v3 i c8 ∼ − = 2 2R3 |v3 | ≤ I E E R3 (16.169) v2 = (VB B − v B E10 ) − (VB B − v B E9 ) = v B E9 − v B E10 (16.170) for to develop voltage v2 : VBB Q9 Q10 – v2 + ic7 + v3 – Q7 ic8 R3 R3 Q8 IEE –VEE Figure 16.68 Inverse hyperbolic tangent “predistortion” circuit. 16.9 The Gilbert Analog Multiplier 1257 Using the standard expressions for the base-emitter voltages and assuming the two transistors are matched gives IE E IE E v3 v3 v3 1 + + − 2 2R3 2R3 I E E R3 − VT ln 2 = VT ln v2 = VT ln (16.171) v3 IS IS 1− I E E R3 Searching our math tables, we might stumble on this identity: 1+x ln = 2 tanh−1 (x) 1−x which can be used to rewrite the expression for v2 as v3 −1 v2 = 2VT tanh I E E R3 (16.172) Combining Eq. (16.172) with (16.167) gives the final result for the analog multiplier R vo = v1 v3 I E E R1 R3 (16.173) The circuit described by Eq. (16.173) is known as a four-quadrant multiplier since both input voltages are permitted to take on both positive and negative values. One common design sets the scaling constant to be 0.1 so that the input and output signals can all have a 10-V range. An example of operation of an analog multiplier appears in Fig. 16.69. Input v3 = 5 sin 20000πt V. Signal v1 is a ramp starting at −5 V at t = 0 and reaching +5 V at t = 2 ms. The product of the two waveforms appears in the figure. The product is zero and changes sign as v1 crosses through 0 V at t = 1 ms. (V) +6.000 v3 v1 +4.000 v0 +2.000 +0.000 – 2.000 – 4.000 – 6.000 +0.000 +500u +1.000m Time (s) +1.500m +2.000m Figure 16.69 Gilbert multiplier simulation results for v3 = 5 sin 20000πt and v1 ramping between −5 and +5 V with a scale factor of 0.1. Exercise: What should the scale factor be in Eq. (16.173) if all voltages are to have a 5-V range? A 1-V range? Answers: 0.2; 1.0 1258 Chapter 16 Analog Integrated Circuits Exercise: Simulate the full Gilbert multiplier with a 5-V, 1-kHz sine wave for v1 with v3 = 5 sin 20000π t V. ELECTRONICS IN ACTION G m -C Integrated Filters The design of integrated circuit filters is complicated by the lack of well-controlled resistive components in most mainstream CMOS processes. One approach to overcome this is the use of G m -C filter topologies based on the operational transconductance amplifier (OTA). The OTA is characterized by both a high input and high output impedance. A simple form of an OTA is shown below. The high impedance output is a small-signal current given by the product of the differential pair gm and the differential input voltage vid . Typically, commercial OTA designs include additional devices to improve output resistance and voltage swing. +VDD M3 M4 iout = gmvid + vid – M1 M2 + vid + – – iout = gm vid ibias –VSS Equivalent schematic symbol for operational transconductance amplifier (OTA). vin + – vout Vout = C g Iout = (Vout – Vin) m sC sC Av(s) = Vout (s) 1 = Vin (s) 1 + s C g m Single pole gm -C low-pass filter. Summary 1259 A simple low-pass filter formed with an OTA and a capacitor is shown above. The transfer characteristic is also included and indicates that the upper cutoff frequency occurs at f H = gm /2πC. One of the more useful characteristics of the gm -C filter approach is the ease with which the characteristics can be tuned. Recalling that gm is a function of the differential pair current, we see that the cutoff frequency of the filter is easily modified by adjusting the bias current. A second-order version (a biquad topology) is shown below. This version allows for the adjustment of cutoff frequency with constant Q, and still requires no resistors. High-pass, band-pass, and band-reject are also readily derived from this basic form. Because of their compatibility with standard CMOS processes and excellent power efficiency, gm -C filters have become prevalent in communication circuits, A/D converter anti-alias filters, noise shaping, and many other applications. vin + + – C1 – vout Vout(s) gm2 = 2 Vin(s) s C1C2 + sC1gm + gm2 o = C2 gm √C1C2 Q= C2 √ C1 Two pole biquadratic gm -C low-pass filter. SUMMARY Integrated circuit (IC) technology permits the realization of large numbers of virtually identical transistors. Although the absolute parameter tolerances of these devices are relatively poor, device characteristics can actually be matched to within less than 1 percent. The availability of large numbers of such closely matched devices has led to the development of special circuit techniques that depend on the similarity of device characteristics for proper operation. These matched circuit design techniques are used throughout analog circuit design and produce high-performance circuits that require very few resistors. • One of the most important of the IC techniques is the current mirror circuit, in which the output current replicates, or mirrors, the input current. Multiple copies of the replicated current can be generated, and the gain of the current mirror can be controlled by scaling the emitter areas of bipolar transistors or the W/L ratios of FETs. Errors in the mirror ratio of current mirrors are related directly to the finite output resistance and/or current gain of the transistors through the parameters λ, V A , and β F . • In bipolar current mirrors, the finite current gain of the BJT causes an error in the mirror ratio, which the buffered current mirror circuit is designed to minimize. In both FET and BJT circuits, the ideal balance of the current mirror is disturbed by the mismatch in dc voltages between the input and output sections of the mirror. The degree of mismatch is determined by the output resistance of the current sources. 1260 Chapter 16 Analog Integrated Circuits • The figure of merit VC S for the basic current mirror is approximately equal to V A for the BJT or 1/λ for the MOS version. However, the value of VC S can be improved by up to two orders of magnitude through the use of either the cascode or Wilson current sources. • Current mirrors can also be used to generate currents that are independent of the power supply voltages. The VB E -based reference and the Widlar reference produce currents that depend only on the logarithm of the supply voltage. By combining a Widlar source with a current mirror, a reference is realized that exhibits first-order independence of the power supply voltages. The only variation is due to the finite output resistance of the current mirror and Widlar source used in the supply-independent cell. Even this variation can be significantly reduced through the use of cascode and Wilson current mirror circuits within the reference cell. Once generated, the stabilized currents of the reference cell can be replicated using standard current mirror techniques. • The Widlar cell produces a PTAT voltage (proportional to absolute temperature) which is used as the basic sensing element in most electronic thermometers. • The bandgap reference combines a PTAT cell with a base-emitter voltage to produce a highly precise output voltage that is independent of temperature and supply voltage. The typical output voltage of the basic bandgap cell is 1.20 V at room temperature and is approximately equal to the silicon bandgap voltage. The 1.20-V output is easily scaled up to any desired reference voltage. • An extremely important application of the current mirror is as a replacement for the load resistors in differential and operational amplifiers. This active-load circuit can substantially enhance the voltage gain capability of most amplifiers while maintaining the operatingpoint balance necessary for low offset voltage and good common-mode rejection. Amplifiers with active loads can achieve single-stage voltage gains that approach the amplification factor of the transistor. Analysis of the ac behavior of circuits employing current mirrors can often be simplified using a two-port model for the mirror. • Active current mirror loads are used to enhance the performance of both bipolar and MOS operational amplifiers. The classic A741 operational amplifier, introduced in the late 1960s, was the first highly robust design combining excellent overall amplifier performance with input-stage breakdown-voltage protection and short-circuit protection of the output stage. Active loads are used to achieve a voltage gain in excess of 100 dB in an amplifier with two stages of gain. This operational amplifier design immediately became the industry standard op amp and spawned many similar designs. • Four-quadrant multiplication of analog signals can be accurately obtained using the Gilbert multiplier circuit. KEY TERMS Active load Buffered current mirror Cascode current source Current gain defect Current mirror “Diode-connected” transistor Emitter area scaling Four-quadrant multiplier Gilbert multiplier Matched devices Matched transistors A741 Mirror ratio Overvoltage protection Power-supply independent biasing Reference current Short-circuit protection Start-up circuit VB E -based reference Voltage reference Widlar current source Wilson current source Problems 1261 REFERENCES 1. P. R. Gray, P. J. Hurst, S. H. Lewis, and R. G. Meyer, Analysis and Design of Analog Integrated Circuits, 4th ed., John Wiley and Sons, New York: 2001. 2. R. J. Widlar, “Some circuit design techniques for linear integrated circuits,” IEEE Transactions on Circuit Theory, vol. CT-12, no. 12, pp. 586–590, December 1965. 3. R. J. Widlar, “Design techniques for monolithic operational amplifiers,” IEEE Journal of Solid-State Circuits, vol. SC-4, no. 4, pp. 184–191, August 1969. 4. G. R. Wilson, “A monolithic junction FET-NPN operational amplifier,” IEEE Journal of Solid-State Circuits, vol. SC-3, no. 6, pp. 341–348, December 1968. 5. A. Paul Brokaw, “A simple three-terminal IC bandgap reference,” IEEE Journal of Solid-State Circuits, vol. SC-9, no. 6, pp. 388–393, December 1994. 6. Robert J. Widlar, “New developments in IC voltage regulators,” IEEE Journal of Solid-State Circuits, vol. SC-6, no. 1, pp. 2-7, January 1991. PROBLEMS 16.1 Circuit Element Matching 16.1. An integrated circuit resistor has a nominal value of 4.02 k. A given process run has produced resistors with a mean value 15 percent higher than the nominal value, and the resistors are found to be matched within 3 percent. What are the maximum and minimum resistor values that will occur? 16.2. (a) The emitter areas of two bipolar transistors are mismatched by 10 percent. What will be the baseemitter voltage difference between these two transistors when their collector currents are identical? (Assume V A = ∞.) (b) Repeat for a 20 percent area mismatch. (c) What degree of matching is required for a base-emitter voltage difference of less than 1 mV? 16.3. The bipolar transistors in the differential pair in Fig. 15.19(a) are mismatched. (a) What will be the offset voltage if the current gains are mismatched by 5 percent? (b) If the saturation currents are mismatched by 5 percent? (c) If the Early voltages are mismatched by 5 percent? (d) If the collector resistors are mismatched by 5 percent? (Remember, the offset voltage is the input voltage required to force the differential output voltage to be zero.) 16.4. What is the worst-case fractional mismatch I D /I D in drain currents in two MOSFETs if K n = 250 A/V2 ± 5 percent and VT N = 1 V± 25 mV for (a) VG S = 2 V? (b) VG S = 4 V? Assume I D1 = I D + I D /2 and I D2 = I D − I D /2. 16.5. (a) A layout design error causes the W/L ratios of the two NMOSFETs in a differential amplifier to differ by 10 percent. What will be the gate-source voltage difference between these two transistors when their drain currents are identical if the nominal value of (VG S − VT N ) = 0.5 V? (Assume VT N = 1 V, λ = 0 and identical values of K n ). (b) What degree of matching is required for a gatesource voltage difference of less than 3 mV? (c) For 1 mV? ∗ 16.6. The collector currents of two BJTs are equal when the base-emitter voltages differ by 2 mV. What is the fractional mismatch I S /I S in the saturation current of the two transistors if I S1 = I S + I S /2 and I S2 = I S − I S /2? Assume that the collectoremitter voltages and Early voltages are matched. If β F O /β F O = 5 percent, what are the values of I B1 and I B2 for the transistors at a Q-point of (100 A, 10 V)? Assume β F O = 100 and V A = 50 V. 16.7. The MOS transistors in the differential pair in Fig. 15.20(b) are mismatched. The nominal value of (VG S − VT N ) = 0.75 V. (a) What will be the offset voltage if the (W/L) ratios are mismatched by 5 percent? (b) If the threshold voltages are mismatched by 5 percent? (c) If the values of λ are mismatched by 5 percent? (d) If the drain resistors are mismatched by 5 percent? (Remember, the offset voltage is the input voltage required to force the differential output voltage to be zero.) 16.2 Current Mirrors 16.8. (a) What are the output currents and output resistances for the current sources in Fig. 16.70 if IREF = 30 A, K n = 25 A/V2 , VT N = 0.75 V and λ = 0.015 V−1 ? (b) What are the currents if 1262 Chapter 16 Analog Integrated Circuits +10 V IREF +8 V +12 V IO2 IO3 IO4 10 1 20 1 40 1 M2 M3 M4 4 1 M1 16.15. What value of R is required in Fig. 16.71 to have I O2 = 35 A? Use device data from Prob. 16.11. 16.16. (a) What are the output currents and output resistances for the current sources in Fig. 16.72(a) if R = 50 k, β F O = 50, and V A = 60 V? (b) Repeat part (a) if the emitter areas of all the transistors are doubled. (c) Repeat for Fig. 16.72(b). 12 V 5V R IO2 Figure 16.70 ∗ 16.11. What are the output currents and output resistances for the current sources in Fig. 16.71 if R = 30 k, K p = 15 A/V2 , VT P = −0.90 V, and λ = 0.01 V−1 ? +5 V M1 M2 M3 8 1 16 1 IO2 IO3 Q3 8.3 A 5A A (a) 12 V 5V 3V IO2 R IO3 Q5 Q1 A A Q2 5A Q3 8.3 A (b) Figure 16.72 16.17. Simulate the current source array in Fig. 16.72(a) and compare the results to the hand calculations in Prob. 16.16. (b) Repeat for Fig. 16.72(b). 2 1 R IO3 Q2 Q1 IREF is changed to 50 A? (c) What would be the values if λ = 0? 16.9. (a) What are the output currents for the circuit in Prob. 16.8 if the W/L ratio of M1 is changed to 2.5/1? (b) If IREF = 20 A and (W/L)1 = 6/1? 16.10. The current sources in Prob. 16.8 could represent the binary weighted currents needed for a 3-bit D/A converter. (a) What are the ideal values of the three output currents (i.e., λ = 0)? (b) Express the current errors from Prob. 16.8 in terms of LSBs. 3V –5 V Figure 16.71 16.12. (a) What are the output currents for the circuit in Prob. 16.11 if the W/L ratio of M1 is changed to 3.3/1? (b) R = 50 k and (W/L)1 = 4/1? 16.13. Simulate the current source array in Fig. 16.70 and compare the results to the hand calculations in Prob. 16.8. 16.14. Simulate the current source array in Fig. 16.71 and compare the results to the hand calculations in Prob. 16.11. 16.18. What value of R is required in Fig. 16.72(a) to have I O3 = 166 A? What is the value of I O2 ? Assume β F O = 50 and V A = 60 V. (b) Repeat for the circuit in Fig. 16.72(b). 16.19. (a) What are the output currents in the circuit in Fig. 16.72(a) if the area of transistor Q 1 is changed to 2A, and R = 75 k? Use β F O = 125 and V A = 75 V. (b) Repeat for Fig. 16.72(b). 16.20. What are the output currents in the circuit in Fig. 16.72(b) if the area of transistor Q 1 is changed to 3A, and R = 100 k? Use β F O = 100 and V A = 75 V. 16.21. (a) What are the output currents in the circuit in Fig. 16.72(a) if R = 140 k? Use β F O = 125 and V A = 75 V. (b) What value of R is required to produce the same output currents in Fig. 16.72(b). 1263 Problems 16.22. (a) What are the output currents in Fig. 16.72(a) if R = 100 k? (b) What are the output currents if the 5-V supply increases to 6 V? (c) What are the output currents if the 12-V supply decreases to 11 V? (d) Show that the change in I O2 in part (b) is equal to go2 V . 16.23. (a) What are the output currents in Fig. 16.72(b) if R = 43 k? (b) What are the output currents if the 5-V supply increases to 6 V? (c) What are the output currents if the 12-V supply decreases to 11 V? (d) Show that the change in I O2 in part (b) is equal to go2 V . 16.24. What are the output currents and output resistances for the current sources in Fig. 16.73 if R = 60 k, β F O = 50, and V A = 60 V? ∗ 16.28. What are the output currents and output resistances for the current sources in Fig. 16.74 if R = 10 k, R1 = 10 k, R2 = 5 k, R3 = 2.5 k, n = 4, β F O = 75, and V A = 60 V? ∗ 16.29. What values of n and R3 would be required in Prob. 16.28 so that I O2 = 3I O3 ? 16.30. Repeat Prob. 16.28 if the area of transistor Q 1 is changed to 0.5 A and R1 is changed to 20 k? 16.31. What are the output current I O and output resistance in the circuit in Fig. 16.75 if −VE E = −5 V, n = 7.2, K n = 50 A/V2 , VT N = 0.75 V, IREF = 15 A, β F O = 100, and V A = 75 V? 2A Q1 A 6A 9A Q2 Q3 Q4 IO IREF 15 V M3 Q2 nA Q1 A R IO2 VEE IO4 IO3 +6 V −12 V Figure 16.75 Figure 16.73 16.25. What value of R is required in the circuit in Fig. 16.73 to set I O3 = 50 A? What are the values of I O2 and I O4 ? ∗ 16.26. Draw a buffered current mirror version of the source in Fig. 16.73 and find the value of R required to set IREF = 25 A if β F O = 50 and V A = 60 V. What are the values of the three output currents? What is the collector current of the additional transistor? 16.27. In Fig. 16.74, R2 = 5R3 . What value of n is required to set I E3 to be equal to exactly 5I E2 ? 12 V 10 V 10 V IO2 IO3 16.32. Use SPICE to simulate the circuit in Prob. 16.31 and compare the results to hand calculations. 16.33. (a) What is the input resistance presented to source IREF at the gate of transistor M3 in Fig. 16.75 if n = 1? Use the other parameters from Prob. 16.31? (b) Use transfer function analysis in SPICE to verify your result. Widlar Sources 16.34. (a) What are the output current and output resistance for the Widlar current source I O2 in Fig. 16.76 if R = R2 = 10 k and V A = 60 V? (b) For I O3 if R3 = 5 k and n = 12? 10 V R 5V IO2 R Q2 Q1 2A A A Q1 Q2 A R1 R2 Figure 16.74 R3 IO3 Q4 Q3 nA 5V 2A R2 Figure 16.76 Q3 nA R3 1264 Chapter 16 Analog Integrated Circuits 16.35. What value of R is required to set IREF = 75 A in Fig. 16.76? If IREF = 75 A, what value of R2 is needed to set I O2 = 5 A? If R3 = 2 k, what value of n is required to set I O3 = 10 A? 16.36. Simulate the source of Prob. 16.35 and compare the results to hand calculations. 16.37. (a) What are the output current and output resistance for the Widlar current source I O2 in Fig. 16.77 if R = 40 k and R2 = 5 k? Use V A = 70 V and β F = 100. (b) For I O3 if R3 = 2.5 k and n = 20? value of VC S for the current source in (b)? (d) What is the minimum value of VE E ? ∗ 16.40. Derive an expression for the output resistance of the Wilson source in Fig. 16.78 as a function of the area ratio n. ∗∗ 16.41. Derive an expression for the output resistance of the BJT Wilson source in Fig. 16.21 and show that it can be reduced to Eq. (16.46). What assumptions were used in this simplification? 16.42. What is the minimum voltage that can be applied to the collector of Q 3 in Fig. 16.78 and have the transistor remain in the active region if IREF = 15 A, n = 5, β F O = 125, and I S O = 3 fA? Calculate an exact value based on the value of I S O . 16.43. R = 30 k in the Wilson source in Fig. 16.79. (a) What is the output current if (W/L)1 = 5/1, (W/L)2 = 20/1, (W/L)3 = 20/1, K n = 25 A/V2 , VT N = 0.75 V, λ = 0 V−1 , and VSS = −5 V. What value of (W/L)4 is required to balance the drain voltages of M1 and M2 ? (*b) Repeat if λ = 0.015 V−1 . (c) Check your results in (b) with SPICE simulation. +5 V R3 R2 nA 10 A A Q1 A Q2 IO2 R Q3 Q2 IO3 IO −5 V Figure 16.77 R 16.38. What value of R is required to set IREF = 50 A in Fig. 16.77. If IREF = 50 A, what value of R2 is needed to set I O2 = 10 A? If R3 = 2 k, what value of n is required to set I O3 = 10 A? M4 M3 M2 M1 16.3 High-Output-Resistance Current Mirrors –VSS Wilson Sources 16.39. IREF = 50 A, −VE E = −5 V, β F O = 125, and V A = 40 V in the Wilson source in Fig. 16.78. (a) What are the output current and output resistance for n = 1? (b) For n = 3? (c) What is the IO IREF Q4 Q3 A A Q2 Q1 A nA –VEE Figure 16.78 Figure 16.79 ∗ 16.44. Derive an expression for the output resistance of the Wilson source in Fig. 16.79 as a function of (W/L)1 , (W/L)2 , (W/L)3 , (W/L)4 , and the reference current IREF . Assume R = ∞. 16.45. Derive an expression for the equivalent resistance presented to IREF in the Wilson source in Fig. 16.19. 16.46. Derive an expression for the equivalent resistance presented to IREF in the Wilson source in Fig. 16.20. 16.47. What is the minimum voltage required on the drain of M3 to maintain it in pinch-off in the circuit in Fig. 16.79 if IREF = 150 A, (W/L)1 = 5/1, (W/L)2 = 20/1, (W/L)3 = 20/1, K n = 1265 Problems 25 A/V2 , VT N = 0.75 V, λ = 0 V−1 , and −VSS = −10 V? 16.48. In Fig. 16.79, (W/L)3 = 5/1, (W/L)4 = 5/1, and IREF = 50 A. What value of (W/L)2 is required for Rout = 250 M if K n = 25 A/V2 , VT N = 0.75 V, λ = 0.0125 V−1 . Assume (W/L)2 = (W/L)1 , R = ∞, and VSS = 5 V. Neglect VDS . ∗∗ 16.55. (a) What are the output current and output resistance for the cascode current source in Fig. 16.81 if IREF = 17.5 A, β F O = 110, and V A = 50 V? (b) What is the value of VC S for this current source? (c) What is the minimum value of VCC ? +VCC IREF IO 16.49. Redraw the equivalent circuit used to calculate the output resistance of the MOS Wilson source in Figs. 16.21 and 16.22 including a finite output resistance RREF for the reference source. Based on this circuit, how large must RREF be to keep from degrading the output resistance of the Wilson source? What type of current source could be used to implement IREF to meet this requirement? 16.50. (a) What are the output current and output resistance for the cascode current source in Fig. 16.80 if IREF = 17.5 A, VD D = 5 V, K n = 75 A/V2 , VT N = 0.75 V, and λ = 0.0125 V−1 . (b) What is the value of VC S for this current source? (c) What is the minimum value of VD D ? +VDD IO M3 M1 Q4 Q1 Q2 Figure 16.81 Cascode Current Sources IREF Q3 M4 M2 Figure 16.80 16.51. Use SPICE to simulate the current source in Prob. 16.50 and compare the results to your calculations. 16.52. (a) A layout error causes the W/L ratio of M2 to be 5 percent larger than that of M1 in Prob. 16.50. What is the error in the output current I O ? (a) Repeat if M1 = M2 , but M4 is 5 percent larger than that of M3 . 16.53. (a) Repeat Prob. 16.50 for IREF = 25 A. (b) Repeat Prob. 16.50 for IREF = 50 A. 16.54. What is the equivalent resistance presented to IREF in the cascode current source in Prob. 16.50? 16.56. Simulate the current source in Prob. 16.81 and compare the results to hand calculations. 16.57. In Fig. 16.80, (W/L)1 = 5/1, (W/L)2 = 5/1, (W/L)3 = 5/1, and IREF = 50 A. What value of (W/L)4 is required for Rout = 250 M if K n = 25 A/V2 , VT N = 0.75 V, and λ = 0.0125 V−1 ? 16.58. What is the equivalent resistance presented to IREF in the cascode current source in Prob. 16.55? 16.4 Reference Current Generation 16.59. What are the output current and output resistance for the Widlar source in Fig. 16.82 if IREF = 80 A and R2 = 500 ? (b) What is the new value of the output current if a layout error causes the area of Q 2 to be 5 percent larger than desired? (c) What are the new values or output current and resistance if the emitter area of Q 2 is reduced to 14A? +10 V IREF Q1 A IO2 20 A Q2 R2 Figure 16.82 1266 Chapter 16 Analog Integrated Circuits 16.60. What are the output current and output resistance for the Widlar source in Fig. 16.82 if IREF = 35 A and R2 = 935 ? (b) What are the new values if the emitter area of Q 1 is increased to 2A? 16.61. IREF = 73 A in Fig. 16.82. (a) What value of R2 is required to set I O2 = 22 A? (b) To set I O2 = 5.7 A? (c) To set I O2 = 5.7 A if the area of Q 2 is changed to 10A? 16.62. IREF = 62 A in Fig. 16.82. (a) What value of R2 is required to set I O2 = 12 A if the area Q 1 is changed to 2A? (b) If the area of Q 2 is changed to 10A? 16.63. Plot the variation of the output current vs. IREF for the Widlar source in Fig. 16.82 for 50 A ≤ IREF ≤ 5 mA if R2 = 4 k and β F O = 100. 16.64. (a) Find the PTAT voltage across R2 in Fig. 16.84 if T = 300 K, IREF = 75 A, and R2 = 2 k? (b) What is the temperature coefficient of the PTAT voltage? (c) If R2 has a temperature coefficient of 1500 ppm/◦ C, what is the temperature coefficient of the collector current of Q 2 at 300 K? 16.65. (a) What is the output current of the VB E -based reference in Fig. 16.83(a) if I S = 10−15 A, β F = ∞, R1 = 10 k, R2 = 2.2 k, and VE E = 15 V? (b) For VE E = 3.3 V? (c) What is the output current of the VB E -based reference in Fig. 16.83(b) if R1 = 10 k, R2 = 10 k, and VCC = 5 V? +VCC ∗ 16.68. What is the output current of the NMOS reference in Fig. 16.84 if R1 = 10 k, R2 = 15 k, K n = 250 A/V2 , VT N = 0.75 V, λ = 0.017 V−1 , and VD D = 10 V? +VDD IO R1 M1 M2 R2 Figure 16.84 16.69. Design the reference in Fig. 16.84 to produce an output current I O = 75 A. Assume VD D = 6 V and use the transistor parameters from Prob. 16.68. ∗ 16.70. What is the output current of the PMOS reference in Fig. 16.85 if R1 = 10 k, R2 = 18 k, K p = 100 A/V2 , VT P = −0.75 V, λ = 0.02 V−1 , and VD D = 5 V? +VDD R2 M1 IO M2 R2 R1 Q1 Q2 R1 Q1 Q2 R1 R2 IO −VEE (a) (b) Figure 16.83 16.66. (a) Design the reference in Fig. 16.83(a) to produce an output current I O = 30 A. Assume −VE E = −3.3 V and I S = 0.1 fA and β F O = 130 for both transistors. (b) Repeat for the circuit in Fig. 16.83(b) if VCC = 3.3 V. ∗ 16.67. Derive an expression for the output resistance of the cascode current source in Fig. 16.25. IO Figure 16.85 16.71. Design the reference in Fig. 16.85 to produce an output current I O = 125 A. Assume VD D = 9 V and use the transistor parameters from Prob. 16.70. 16.72. What are the collector currents in Q 1 and Q 2 in the reference in Fig. 16.86 if VCC = VE E = 1.5 V, n = 20, and R = 2.2 k? Assume β F O = ∞ and V A = ∞. 16.73. Simulate the reference in Fig. 16.86 using SPICE, assuming β F O = 100 and V A = 50 V. Compare the currents to hand calculations and discuss the source of any discrepancies. Use SPICE to determine the sensitivity of the reference currents to power supply voltage changes. 1267 Problems transistor types. (b) Repeat for γn = 0.5 V0.5 and γ p = 0.75 V0.5 and compare the results. +VCC 3A A Q3 Q4 Q1 Q2 A 16.79. Simulate the references in Prob. 16.78(a) and (b) using SPICE with λ = 0.017 V−1 . Compare the currents to hand calculations (with γ = 0 and λ = 0) and discuss the source of any discrepancies. Use SPICE to determine the sensitivity of the reference currents to power supply voltage changes. 16.80. What are the collector currents in Q 1 to Q 8 in the reference in Fig. 16.88 if VCC = 0 V, VE E = 3.3 V, R = 11 k, R6 = 3 k, R8 = 4 k, and A E2 = 5 A, A E3 = 2 A, A E4 = A, A E5 = 2.5 A, A E6 = A, A E7 = 5 A, and A E8 = 3 A? nA R –VEE Figure 16.86 16.74. What are the collector currents in the four transistors in Fig. 16.86 if VCC = VE E = 3.3 V, n = 8, and R = 4 k? 16.75. What is the smallest value of n required for the circuit in Fig. 16.86 to operate properly based on Eq. 16.32 (i.e., VPTAT > 0)? +VCC R8 AE3 Q3 16.76. (a) What value of R is required to set IC2 = 35 A in Fig. 16.86 if n = 5 and T = 50◦ C? (b) For n = 10 and T = 0◦ C? 16.77. What are the drain currents in M1 and M2 in the reference in Fig. 16.87 if R = 5.1 k and VD D = VSS = 5 V? Use K n = 25 A/V2 , VT N = 0.75 V, K p = 10 A/V2 , and VT P = −0.75 V. Assume γ = 0 and λ = 0 for both transistor types. M3 M4 Q5 Q6 AE6 R6 Q4 Q1 AE5 AE7 Q7 AE8 Q8 Q2 A AE2 R –VEE +VDD 10 1 AE4 Figure 16.88 10 1 16.81. Repeat Prob. 16.80 if A E2 = 10A and A E3 = A. ∗ M2 M1 10 1 20 1 ∗ R –VSS Figure 16.87 16.78. (a) Find the currents in both sides of the reference cell in Fig. 16.87 if R = 10 k and VD D = VSS = 5 V, using K n = 25 A/V2 , VT on = 0.75 V, K p = 10 A/V2 , VT op = −0.75 V, γn = 0 and γ p = 0. Use 2φ F = 0.6 V and λ = 0 for both 16.82. (a) What are the collector currents in Q 1 to Q 7 in the reference in Fig. 16.89 if VCC = 5 V and R = 4300 ? Assume β F = ∞ = V A . (b) Repeat part (a) if the emitter areas of transistors Q 5 , Q 6 , and Q 7 are all changed to 2A. 16.83. (a) Simulate the reference in Prob. 16.82 using SPICE. Assume β F On = 100, β F O p = 50, and both Early voltages = 50 V. Compare the currents to hand calculations and discuss the source of any discrepancies. Use SPICE to determine the sensitivity of the reference currents to power supply voltage changes. 16.84. Repeat Prob. 16.82 assuming the emitter area of transistor Q 3 is changed to 2A. ∗ 16.85. (a) What are the drain currents in M1 and M2 in the reference in Fig. 16.90 if R = 3300 , 1268 Chapter 16 Analog Integrated Circuits Q4 Q5 VCC 2A A Compare the currents to those in Prob. 16.85 and discuss the source of any discrepancies. Use SPICE to determine the sensitivity of the reference currents to power supply voltage changes. 16.87. Repeat Prob. 16.85 assuming the W/L ratio of transistor M3 is changed to 15/1. Q3 A 16.5 The Bandgap Reference Q6 Q1 A A A 7A 16.88. A layout error caused A E2 = 9A E1 in the bandgap reference in Ex. 16.6. What is the new output voltage? What temperature corresponds to zero TC? 16.89. (a) Process variations cause the value of the two collector resistors in the circuit in Ex. 16.6 to increase to 35 k. What is the new value of VBG ? What temperature corresponds to zero TC? (b) Repeat for R = 25 k. 16.90. What are the bandgap reference output voltage and temperature coefficient of the reference in Ex. 16.6 if I S changes to 0.5 fA? 16.91. What are the bandgap reference output voltage and the temperature coefficient of the reference in Design Ex. 16.7 at 320 K if I S changes to 0.3 fA? 16.92. Process variations cause the values of the two collector resistors in the circuit in Design Ex. 16.7 to be mismatched. If R1 = 82 k and R2 = 78 k, what is the new value of VBG ? What temperature corresponds to zero TC? 16.93. The bandgap reference in Design Ex. 16.7 was designed to have zero temperature coefficient at 320 K. What will be the temperature coefficient at 280 K? At 320 K? 16.94. Redesign the bandgap reference in Design Ex. 16.7 to use A E2 = 8A E1 . 16.95. Redesign the bandgap reference in Design Ex. 16.7 to produce an output voltage of 7.500 V with zero TC at 10◦ C. Assume VCC = 10 V. Q7 Q2 R Figure 16.89 15 M4 1 VDD M3 10 1 10 1 M5 10 M6 1 M7 10 1 10 M1 1 M2 30 1 R 16.6 The Current Mirror as an Active Load ∗ Figure 16.90 VD D = 15 V, K n = 25 A/V2 , VT N = 0.75 V, K p = 10 A/V2 , VT P = −0.75 V, and λ = 0 for both transistor types? (b) Repeat part (a) if the W/L ratios of transistors M5 , M6 , and M7 are all increased to 15/1. 16.86. Simulate the reference in Prob. 16.85 with SPICE using λ = 0.017 V−1 for both transistor types. 16.96. What are the values of Add , Acd , and CMRR for the amplifier in Fig. 16.37 if I SS = 200 A, R SS = 25 M, K n = K p = 500 A/V2 , VT N = 1 V, and VT P = −1 V and λ = 0.02 V−1 for both transistors? 16.97. Use SPICE to simulate the amplifier in Prob. 16.96 and compare the results to the hand calculations. Use symmetrical 12-V supplies. 16.98. What are the values of Add , Acd , and CMRR for the amplifier in Fig. 16.37 if I SS = 1 mA, R SS = 10 M, K n = K p = 500 A/V2 , 1269 Problems VT N = −VT P = 1 V, and λ = 0.015/V for both transistors? What are the minimum power supply voltages if the common-mode input range must be ±5 V? Assume symmetrical supply voltages. 16.99. Use SPICE to simulate the amplifier in Prob. 16.98 and compare the results to hand calculations. Use symmetrical 12-V power supplies. ∗∗ of the amplifier? (c) Compare this result to the gain of the amplifier in Fig. 16.37 if the Q-point and W/L ratios of M1 to M4 are the same. 16.105. Use SPICE to simulate the amplifier in Prob. 16.104(a,b) and compare the results to hand calculations. ∗ 16.100. (a) What are Add and Acd for the bipolar differential amplifier in Fig. 16.43 (R L = ∞) if βop = 70, βon = 125, I E E = 200 A, R E E = 25 M, and the Early voltages for both transistors are 60 V? What is the CMRR for vC1 = vC2 ? (b) What are the minimum power supply voltages if the commonmode input range must be ±1.5 V? Assume symmetrical supply voltages. 16.101. Use SPICE to calculate Add and Acd for the differential amplifier in Prob. 16.100. Compare the results to hand calculations. 16.106. Find the Q-points of the transistors in the foldedcascode CMOS differential amplifier in Fig. 16.92 if VD D = VSS = 5 V, I1 = 250 A, I2 = 250 A, (W/L) = 40/1 for all transistors, K n = 25 A/V2 , VT N = 0.75 V, K p = 10 A/V2 , VT P = −0.75 V, and λ = 0.017 V−1 for both transistor types. Draw the differential-mode halfcircuit for transistors M1 to M4 and show that the circuit is in fact a cascode amplifier. What is the differential-mode voltage gain of the amplifier? +VDD 16.102. (a) Repeat Prob. 16.100 if I E E is changed to 50 A, R E E = 100 M, and V A = 75 V. (b) Repeat part (a) for V A = 100 V. I2 16.103. Use SPICE to simulate the amplifier in Prob. 16.102 and compare the results to hand calculations. Use symmetrical 3-V power supplies. ∗ v1 I2 M1 16.104. (a) Find the Q-points of the transistors in the CMOS differential amplifier in Fig. 16.91 if VD D = VSS = 10 V and I SS = 200 A. Assume K n = 25 A/V2 , VT N = 0.75 V, K p = 10 A/V2 , VT P = −0.75 V, and λ = 0.017 V−1 for both transistor types. (b) What is the voltage gain Add M3 –VSS M5 M4 v2 vO I1 +VDD 80 1 M2 M4 M3 M7 M6 –VSS 80 1 Figure 16.92 M5 80 1 16.107. Use SPICE to simulate the amplifier in Prob. 16.106 and determine its voltage gain, output resistance, and CMRR. Compare to hand calculations. vO ∗ v1 M1 40 1 40 M2 1 v2 16.108. Design a current mirror bias network to supply the three currents needed by the amplifier in Prob 16.106. Output Stages ISS –VSS Figure 16.91 16.109. What are the currents in Q 3 and Q 4 in the class-AB output stage in Fig. 16.93 if R1 = 20 k, R2 = 20 k, and I S4 = I S3 = I S2 = 10−14 A. Assume β F = ∞. 1270 Chapter 16 Analog Integrated Circuits +VDD +VCC 200 µA 80 M 3 1 M4 80 1 80 M 13 1 M5 80 1 Q3 R2 Q2 + VO = 0 V dc – R1 M6 Q4 vS M1 40 1 v1 40 M 1 2 v2 vO –VEE Figure 16.93 ∗ 16.110. (a) Show that the currents in Q 3 and Q 4 in the class-AB output stage in Fig. 16.94 are equal √ to Io = I2 (A E3 A E4 )/(A E1 A E2 ). (b) What are the currents in Q 3 and Q 4 if A E1 = 3A E3 , A E2 = 3A E4 , I2 = 300 A, I S O pnp = 4 fA, and I S Onpn = 10 fA? M9 M11 5 1 IREF M7 15 1 5 1 5 1 M10 5 M12 1 M8 15 1 –VSS Figure 16.95 +VCC I2 calculations in Prob. 16.111. (b) Use SPICE to calculate the offset voltage and CMRR of the amplifier. Q1 Q3 Q2 Q4 ∗ 16.113. What is the differential-mode gain of the amplifier in Fig. 16.96 if VD D = VSS = 10 V, VO = 0 V dc +VDD 50 M 3 1 M4 50 1 vS M5 100 1 –VEE M1 –VSS M2 Figure 16.94 v2 16.7 Active Loads in Operational Amplifiers ∗ 16.111. (a) Find the Q-points of the transistors in Fig. 16.95 if VD D = VSS = 10 V, IREF = 250 A, K n = 25 A/V2 , VT N = 0.75 V, K p = 10 A/V2 , and VT P = −0.75 V. (b) What is the approximate value of the W/L ratio for M6 of the CMOS op amp in order for the offset voltage to be zero? What is the differential-mode voltage gain of the op amp if λ = 0.017 V−1 for both transistor types? ∗ 16.112. (a) Simulate the amplifier in Prob. 16.111 and compare its differential-mode voltage gain to the hand 20 1 –VSS MGG 5 1 vO –VSS I1 10 M10 1 +VDD I2 M11 20 1 M6 v1 20 1 IREF 10 1 25 1 M7 M12 20 1 Figure 16.96 –VSS Problems 16.114. ∗ 16.115. ∗ 16.116. 16.117. 16.118. 16.119. 16.120. 16.121. IREF = 100 A, K n = 25 A/V2 , VT ON = 0.75 V, K p = 10 A/V2 , VT OP = −0.75 V, γn = 0, and γ p = 0. Use λ = 0.017 V−1 for both transistor types. (a) Use SPICE to find the Q-points of the transistors of the amplifier in Prob. 16.113. (b) Repeat with 2φ F = 0.8 V, γn = 0.60 V0.5 , and γ p = 0.75 V0.5 , and compare the results to (a). Find the Q-points of the transistors in Fig. 16.96 if VD D = VSS = 7.5 V, IREF = 250 A, (W/L)12 = 40/1, K n = 25 A/V2 , VT N = 0.75 V, K p = 10 A/V2 , and VT P = −0.75 V. What is the differential-mode voltage gain of the op amp if λ = 0.017 V−1 for both transistor types? (a) Estimate the minimum values of VD D and VSS needed for proper operation of the amplifier in Prob. 16.113. Use K n = 25 A/V2 , VT N = 0.75 V, K p = 10 A/V2 , and VT P = −0.75 V. (b) What are the minimum values of VD D and VSS needed to have at least a ±5-V common-mode input range in the amplifier? (a) Find the Q-points of the transistors in the CMOS op amp in Fig. 16.48 if VD D = VSS = 5 V, IREF = 250 A, K n = 25 A/V2 , VT N = 0.75 V, K p = 10 A/V2 , and VT P = −0.75 V. (b) What is the voltage gain of the op amp assuming the output stage has unity gain and λ = 0.017 V−1 for both transistor types? (c) What is the voltage gain if IREF is changed to 500 A? Based on the example calculations and your knowledge of MOSFET characteristics, what will be the gain of the op amp in Ex. 16.8 if the IREF is set to (a) 250 A? (b) 20 A? (Note: These should be short calculations.) Based on the exercise answers and your knowledge of BJT characteristics, what will be the gain of the op amp in Fig. 16.51 if the IREF is set to (a) 250 A? (b) 50 A? (Note: These should be short calculations.) Draw the amplifier that represents the mirror image of Fig. 16.48 by interchanging NMOS and PMOS transistors. Choose the W/L ratios of the NMOS and PMOS transistors so the voltage gain of the new amplifier is the same as the gain of the amplifier in Fig. 16.48. Maintain the operating currents the same and use the device parameter values from Ex. 16.8. Draw the amplifier that represents the mirror image of Fig. 16.50 by interchanging npn and pnp 1271 transistors. If βon = 150, βop = 60, and V AN = V A P = 60 V, which of the two amplifiers will have the highest voltage gain? Why? ∗ 16.122. What is the approximate emitter area of Q 16 needed to achieve zero offset voltage in the amplifier in Fig. 16.97 if I B = 250 A and VCC = VE E = 5 V? What is the value of R B B needed to set the quiescent current in the output stage to 75 A? What are the voltage gain and input resistance of this amplifier? Assume βon = 150, βop = 60, V AN = V A P = 60 V, and I S Onpn = I S O pnp = 15 fA. +VCC A Q5 A Q6 A Q3 A Q4 4A Q8 A Q7 v1 Q1 Q2 A A v2 Q6 A –VEE RBB vO A Q10 IB Q12 A Q14 A Q16 –VEE AE16 Figure 16.97 16.123. Use SPICE to simulate the characteristics of the amplifier in Prob. 16.122. Determine the offset voltage, voltage gain, input resistance, output resistance, and CMRR of the amplifier. 16.124. (a) What are the minimum values of VCC and VE E needed for proper operation of the amplifier in Fig. 16.97? (b) What are the minimum values of VCC and VE E needed to have at least a ±1-V common-mode input range in the amplifier? 16.125. Find the output voltage of the bandgap reference circuit in Fig. 16.98 if R1 = 537 , R2 = 2.43 k, R L = 5 k, and A E1 = 8A E2 . What are the values of the five collector currents? Assume infinite current gains, VCC = 5 V, and I S2 = 10−16 A. 1272 Chapter 16 Analog Integrated Circuits ∗ VCC Q3 Q4 Q5 Q1 VO Q2 RL R1 R2 16.133. Create a small-signal SPICE model for the circuit in Fig. 16.61 and verify the values of y11 , y21 , and y22 . What is the value of y12 ? Figure 16.98 16.8 The A741 Operational Amplifier ∗∗ 16.126. (a) What are the three bias currents in the source in Fig. 16.99 if R1 = 100 k, R2 = 4 k, and VCC = VE E = 3 V. (b) Repeat for VCC = VE E = 22 V. (c) Why is it important that I1 in the A741 be independent of power supply voltage but it does not matter as much for I2 and I3 ? +VCC A Q22 3A 16.129. (a) Based on the schematic in Fig. 16.53, what are the minimum values of VCC and VE E needed for proper operation of A741 amplifier? (b) What are the minimum values of VCC and VE E needed to have at least a ±1-V common-mode input range in the amplifier? 16.130. What are the values of the elements in the Norton equivalent circuit in Fig. 16.60 if I1 in Fig. 16.53 is increased to 50 A? 16.131. Suppose Q 23 in Fig. 16.54 is replaced by a cascode current source. (a) What is the new value of output resistance R2 ? (b) What are the new values of the y-parameters of Fig. 16.61? (c) What is the new value of Adm for the op amp? 16.132. Draw a schematic for the cascode current source in Prob. 16.131. Q23 I2 A Q24 16.134. Figure 16.100 represents an op amp input stage that was developed following the introduction of the A741. (a) Find the Q-points for all the transistors in the differential amplifier in Fig. 16.100 if VCC = VE E = 15 V and IREF = 100 A. (b) Discuss how this bias network operates to establish the Q-points. (c) Label the inverting and noninverting input terminals. (d) What are the transconductance and output resistance of this amplifier? Use V A = 60 V. I3 VCC R1 R I1 Q20 A A v1 Q1 v2 Q2 Q21 IREF Q3 Q4 Q5 Q6 R2 VEE VCC vO Figure 16.99 Q9 16.127. Choose the values of R1 and R2 in Fig. 16.99 to set I2 = 250 A and I1 = 50 A if VCC = VE E = 12 V. What is I3 ? 16.128. Choose the values of R1 and R2 in Fig. 16.99 to set I3 = 300 A and I1 = 75 A if VCC = VE E = 15 V. What is I2 ? Q10 Q11 Q7 –VEE Figure 16.100 Q8 Problems ∗∗ 16.135. Figure 16.101 represents an op amp input stage that was developed following the introduction of the A741. Find the Q-points for all the transistors in the differential amplifier in Fig. 16.97 if VCC = VE E = 15 V and IREF = 100 A. (b) Discuss how this bias network operates to establish the Q-points. (c) Label the inverting and noninverting input terminals. (d) What are the transconductance and output resistance of this amplifier? Use V A = 60 V. 16.137. VCC R v1 Q1 v2 Q2 16.138. IREF Q3 Q4 VCC 16.139. vO Q8 Q7 Q9 Q10 Q6 Q5 –VEE Figure 16.101 16.9 The Gilbert Analog Multiplier 16.136. Find the Q-points of the six transistors in Fig. 16.67 if VCC = −VE E = 5 V, I B B = 100 A, R1 = 10 k, and R = 50 k. Draw the circuit assuming 16.140. 1273 the bases of Q 1 and Q 2 are biased at a commonmode voltage of −2.5 V with v1 = 0. Assume the bases of Q 3 through Q 6 are biased at a commonmode voltage of 0 V with v2 = 0. (a) Find the collector currents of the six transistors in Fig. 16.67 if VCC = −VE E = 7.5 V, I B B = 200 A, R1 = 10 k, and R = 50 k. Draw the circuit assuming the bases of Q 1 and Q 2 are biased at a common-mode voltage of −3 V with v1 = 0.5 V. Assume the bases of Q 3 through Q 6 are biased at a common-mode voltage of 0 V with v2 = 0. (b) Repeat with v2 = 1 V. (c) Repeat with v2 = −1 V. Write an expression for the output voltage for the circuit in Fig. 16.67 if v1 = 0.5 sin 2000π t, and v2 is generated by the circuit in Fig. 16.68 with v3 = 0.5 sin 10,000π t? Assume VCC = −VE E = 10 V, I E E = 500 A, R1 = R3 = 2 k, and R = 10 k. (a) Write expressions for the total collector currents i C1 and i C2 in Fig. 16.67 if I B B = 1 mA, R1 = 2 k, and v1 = 0.4 sin 5000π t V. Assume the transistors are operating in the active region. (b) What is the transconductance G m of the voltage-to-current converter formed by Q 1 and Q 2 ? [G m = (i C1 − i C2 )/v1 ] Use SPICE to plot the VTC for the circuit in Fig. 16.68 with VB B = 3 V, −VE E = −5 V, I E E = 300 A, and R3 = 3.3 k.