TECHNICAL NOTE 31 Practical Analysis of the Pierce Oscillator Introduction To achieve optimum performance from a Pierce crystal oscillator, e.g. good frequency stability and low long term aging, the crystal parameters, crystal drive current, and oscillator gain requirements must be carefully considered. Over many years of experience, it has been found that excessive crystal drive current is one of the main causes of oscillator malfunction. Overdriving the crystal causes frequency instability over time, and for tuning-fork crystals the excessive motional displacement can break the crystal tines. (a) The loop gain must be greater than unity. (b) The phase shift around the loop must be an integer multiple of 2π. The CMOS amplifier provides the amplification while the two capacitors CD and CG, the resistor RA, and the crystal work as the feedback network. The resistor RA stabilizes the output voltage of the amplifier and is used to reduce the crystal drive level. This technical note describes a practical approach to measuring the key parameters of a Pierce oscillator. This allows the designer to check the oscillator design against the actual oscillator performance and ensure that the oscillator design rules are met. (For analysis, see Statek Technical Note 30.) This note covers the following topics: 1. A practical procedure for determining the crystal drive current. 2. A practical procedure for measuring the amplifier’s transconductance gm and output resistance RO. 3. A practical procedure for determining the total load capacitance CL of the Pierce oscillator circuit. Basic Crystal Oscillator The basic quartz crystal CMOS Pierce oscillator circuit configuration is shown in Figure 1. The crystal oscillator circuit consists of an amplifying section and a feedback network. For oscillation to occur, the Barkhausen criteria must be met: Figure 1: Basic Pierce Oscillator Circuit The crystal drive current is given by the equation ib = 2 X e Re 1 + + X 0′ X 0′ 2 2 ′ Re + RA 1 − X e + X e′ + RA Re X ′ X D 0 2 V , where X 0′ = 1 1 . = ωC0′ ω (C0 + Cs ) The gain equation is REV D STATEK Corporation, 512 N. Main St., Orange, CA 92868 714-639-7810 FAX: 714-997-1256 www.statek.com TN-31 2 2 Page 2 of 4 [ gm > 4π f CG (CD + Cd) Re + + CG ( ) RA CD 1+ R +Cd O ( 2 2 4π f CdCDRA + 1 RO ( Cd + )( 1+ Re C RO d ) ] RA ) 2 2 RA + Re -4π f CdCDRA Re RO The operating frequency is given by scope probe to avoid changing the DC bias voltage. 6. Calculate the peak-to-peak currents: (IG)p-p = (VG)p-p/XG and (ID)p-p = (VD)p-p/XD. 7. Calculate the RMS currents: (ID)RMS = (ID)p-p/ 2 2 and (IG)RMS = (IG)p-p/ 2 2 . ( ) ( ) C1 , f = fs 1 + 2(C0 + CL ) 8. The AC current through the crystal will be between (IG)RMS and (ID)RMS, being closer to (IG)RMS. where CL is the load capacitance of oscillation. The equations for the oscillation frequency, gain and crystal drive are derived using a closed loop and phase diagram analysis of a CMOS quartz crystal oscillator. For more details see Statek’s Technical Note 30. 9. The measured current should not exceed the maximum recommended value Measuring the Crystal Drive Level Measuring the Transconductance and Output Impedance of the Amplifier One of the most important parameters for a good oscillator design is the crystal drive current. With increasing demand for ultra miniature quartz crystal resonators, the crystal parameters and the oscillator components must be carefully considered. The maximum recommended crystal drive level should not be exceeded. IRMS ≤ (maximum allowed power ) . R1 For the oscillator to start up, the transconductance of the amplifier must be greater than the value given by the gain equation, and as a general rule it is best that it at least 2-3 times this minimum. The main factors affecting the drive level are the supply voltage (VDD), R1, RA, CD, CG, and the stray capacitance, Cs. Measurement Procedure 1. Measure CG: Remove the crystal and neutralize node (d). With no power applied, measure CG using a capacitance meter. 2. Measure CD: Remove the crystal and disconnect RA. With no power applied, measure CD and Cd using a capacitance meter. 3. Calculate the impedances XG =1/(ωCG), XD = 1/(ωCD), and Xd = 1/(ωCd). 4. Mount the crystal, reconnect RA and turn on the operating supply voltage. 5. Using a scope probe with no more than 2 pF capacitance, measure the peak-to-peak voltages across CG and CD. Note: AC-couple the signal to the Figure 2: Measuring the Transconductance and Output Impedance of the Amplifier Measurement Procedure 1. Apply a sinusoidal voltage VI to the input of the amplifier, coupled through a 1 µF capacitor (with the voltage sufficiently low so that the output does not saturate). (See Figure 2.) 2. Measure the output voltage VO through a 1 µF capacitor. REV D STATEK Corporation, 512 N. Main St., Orange, CA 92868 714-639-7810 FAX: 714-997-1256 www.statek.com TN-31 Page 3 of 4 3. Measure the output voltage with various load resistances to ground. For example consider the sequence of measurements in Table 1. Table 1: Measuring RO VO [V] VI [V] RL [kΩ] 0.0168 OPEN 1.068 0.0168 30 0.636 0.0168 18 0.520 0.0168 15 0.464 The output resistance RO of the amplifier is approximately equal to that load resistance RL such that the output voltage VO is one half of the output voltage when the load resistance is infinite (open). In the above example, at a load resistance RL of 18 kΩ, the output voltage VO is approximately equal to one-half of VO when RL is an open circuit. Therefore, RO is equal to approximately 18 kΩ. The amplifier’s transconductance gm is the equal to the voltage gain divided by RO. g m = (VO / VI ) / RO = (1.068 / 0.0168) /(18 kΩ) = 3.53 mS. Table 3: Crystal Parameters 2. Measure stray capacitance Cs across the crystal: Remove the crystal, neutralize the positive (Vdd) and ground, then measure the capacitance across the crystal termination. Table 2: Oscillator Parameters Gate capacitance Drain capacitance Stray capacitance Amp. Output capacitance Amp. Output resistance Limiting resistance Symbol CG CD Cs Cd RO RA Value 7.4 5.4 1.4 7.0 70 30 Unit pF pF pF pF kΩ kΩ Value 1.0 3.0 1.2 Unit MHz KΩ pF Using the gain equation, we find for the oscillator and crystal described in Tables 2 and 3 g m ≥ 0.119 mS . The ratio of the amplifier’s transconductance to the minimum transconductance required for oscillation is (3.53/0.119) = 29.7. Therefore, this circuit meets both the required minimum transconductance and the 2-3 times the minimum rule. Measuring the Load Capacitance of the Oscillator Circuit Properly specifying the load capacitance of the oscillator circuit allows the crystal manufacturer to tune the crystal frequency to the operating frequency of the oscillator. Given a crystal of known fs, C1, and C0, operating at a frequency f in a circuit, the load capacitance of the circuit is found from the frequency equation The required minimum transconductance of the oscillator is calculated as follows: 1. Measure CG and CD, as described above. Symbol fs R1 C0 Frequency Motional resistance Shunt capacitance CL = C1 fs − C0 . 2 f − fs Measurement Procedure 1. Measure the crystal parameters C1, C0, and fs with the use of a CI meter or an impedance analyzer. 2. Install the measured crystal in the oscillator circuit and measure the oscillation frequency f. 3. Then calculate the load capacitance CL. For example with fs C1 C0 f = = = = 32.7644 kHz 2.3 fF 1.5 pF 32.768 kHz = 9.0 pF. we find CL REV D STATEK Corporation, 512 N. Main St., Orange, CA 92868 714-639-7810 FAX: 714-997-1256 www.statek.com TN-31 Page 4 of 4 The calculated load capacitance includes the stray capacitance across the crystal (Cs). Glossary Table 4: Glossary of Symbols L1 C1 C0 R1 RA fs f VI VO CL Cs CD CG ib gm RO Cd Crystal Motional Inductance Crystal Motional Capacitance Crystal Shunt Capacitance Crystal Motional Resistance Limiting Resistance Series Resonant Frequency of the Crystal Operating Frequency Input Voltage Output Voltage Total Load Capacitance of the Oscillator Total Stray Capacitance Across the Crystal Drain Capacitance Gate Capacitance Crystal Drive Current Transconductance Amplifier Output Resistance Amplifier Output Capacitance REV D STATEK Corporation, 512 N. Main St., Orange, CA 92868 714-639-7810 FAX: 714-997-1256 www.statek.com