### Analysis of fully differential amplifiers

Amplifiers: Op Amps
Texas Instruments Incorporated
Analysis of fully differential amplifiers
By Jim Karki
Systems Specialist, High-Speed Amplifiers
Introduction
The August issue of Analog Applications Journal
introduced the fully differential amplifiers from Texas
Instruments and illustrated their basic operation (see
by analyzing gain and noise. The fully differential amplifier
has multiple feedback paths, and circuit analysis requires
close attention to detail. Care must be taken to include the
VOCM pin for a complete analysis.
diagram from which specific circuit configurations can be
easily solved. The voltage definitions are required to arrive
at practical solutions.
AF is used to represent the open-loop differential gain of
the amplifier such that (VOUT+) – (VOUT –) = AF(VP – VN).
This assumes that the gains of the two sides of the differential amplifier are well matched and that variations are
insignificant. With negative feedback, this is typically the
case when AF >> 1.
Input voltage definitions:
Circuit analysis
Circuit analysis of fully differential amplifiers follows the
same rules as normal single-ended amplifiers, but subtleties
are present that may not be fully appreciated until a full
analysis is done. The analysis circuit shown in Figure 1 is
used to calculate a generalized circuit formula and block
VID = ( VIN + ) − ( VIN − )
(1)
( VIN + ) + ( VIN − )
2
(2)
VIC =
Output voltage definitions:
Figure 1. Analysis circuit
R2
R1
–
R3
+
VOUT+
–
VOUT –
AF
VP
VIN +
+
VOCM
R4
Figure 2. Block diagram
β2
VIN –
1–β 2
–
+
VIN +
1–β1
–
Σ
(3)
( VOUT + ) + ( VOUT − )
2
(4)
VOC =
VN
VIN –
VOD = ( VOUT + ) − ( VOUT − )
VP– VN
AF
+
β1
( VOUT + ) − ( VOUT − ) = A F ( VP − VN )
(5)
VOC = VOCM
(6)
There are two amplifiers: the main differential amplifier
(from VIN to VOUT) and the VOCM error amplifier. The
operation of the VOCM error amplifier is the simpler of the
two and will be considered first. It may help to review the
simplified schematic shown in Reference 1.
VOUT+ and VOUT – are filtered and summed by an internal
RC network. The VOCM amplifier samples this voltage and
compares it to the voltage applied to the VOCM
pin. An internal feedback loop is used to drive
“error” voltage of the VOCM error amplifier (the
voltage between the input pins) to zero, so that
VOC = VOCM. This is the basis of the voltage definition given in Equation 6.
There is no simple way to analyze the main
differential amplifier except to sit down and
write some node equations, then do the algebra
VOUT+
to massage them into practical form. We will
first derive a solution based solely on nodal
VOUT–
analysis. Then we will make use of the voltage
definitions given in Equations 1–6 to derive solutions for the output voltages, looking at them
single-ended; i.e., VOUT+ and VOUT –. These are
then used to calculate VOD.
48
Analog and Mixed-Signal Products
November 2000
Analog Applications Journal
Amplifiers: Op Amps
Texas Instruments Incorporated
Solving the node equations at VN and VP yields
⎛ R3 ⎞
⎛ R4 ⎞
⎛ R1 ⎞
⎛ R2 ⎞
VN = ( VIN − )⎜
⎟.
⎟ + ( VOUT − )⎜
⎟ and VP = ( VIN + )⎜
⎟ + ( VOUT + )⎜
⎝ R3 + R4 ⎠
⎝ R3 + R4 ⎠
⎝ R1 + R2 ⎠
⎝ R1 + R2 ⎠
By setting
⎛ R1 ⎞
⎛ R3 ⎞
β1 = ⎜
⎟ , VN and VP can be rewritten as
⎟ and β 2 = ⎜
⎝ R1 + R2 ⎠
⎝ R3 + R4 ⎠
VN = ( VIN − )(1 − β 2 ) + ( VOUT + )( β 2 ), and
(7)
VP = ( VIN + )(1 − β 1 ) + ( VOUT − )( β 1 ).
(8)
With Equations 7 and 8, a block diagram of the main differential amplifier can be constructed, like that shown in
Figure 2. Block diagrams are useful tools for understanding circuit operation and investigating other variations.
By using the block diagram, or combining Equations 7 and 8 with Equation 5, we can find the input-to-output
relationship:
(VOUT + )(1 + A F β 2 ) − ( VOUT − )(1 + A F β 1 ) = A F [( VIN + )(1 − β 1 ) − ( VIN − )(1 − β 2 )] .
(9)
Although accurate, Equation 9 is somewhat cumbersome when the feedback paths are not symmetrical. By using
the voltage definitions given in Equations 1–4 and Equation 6, we can derive more practical formulas.
Substituting (VOUT –) = 2VOC – (VOUT+), and VOC = VOCM, we can write
( VOUT + )( 2 + A F β 1 + A F β 2 ) − 2 VOCM (1 + A F β 1 ) = A F [( VIN + )(1 − β 1 ) − ( VIN − )(1 − β 2 )] , or
( VOUT + ) =
1
(β1 + β 2 )
⎛ 1
⎞
+ β1 ⎟
( VIN + )(1 − β 1 ) − ( VIN − )(1 − β 2 ) + 2 VOCM ⎜
⎝ AF
⎠
⎛
⎞
2
⎜1+
⎟
A F β1 + A F β 2 ⎠
⎝
.
(10)
With the “ideal” assumption AFβ1 >> 1 and AFβ2 >> 1, this reduces to
( VOUT + ) =
( VIN + )(1 − β 1 ) − ( VIN − )(1 − β 2 ) + 2 VOCM β 1
.
(β1 + β 2 )
(11)
VOUT – is derived in a similar manner:
( VOUT − ) =
1
(β1 + β 2 )
⎛ 1
⎞
− ( VIN + )(1 − β 1 ) + ( VIN − )(1 − β 2 ) + 2 VOCM ⎜
+ β2 ⎟
A
⎝ F
⎠
⎛
⎞
2
⎜1+
⎟
A F β1 + A F β 2 ⎠
⎝
.
(12)
Again, assuming AFβ1 >> 1 and AFβ2 >> 1, this reduces to
( VOUT − ) =
−( VIN + )(1 − β 1 ) + ( VIN − )(1 − β 2 ) + 2 VOCM ( + β 2 )
.
(β1 + β 2 )
(13)
To calculate VOD = (VOUT+) – (VOUT –), subtract Equation 12 from Equation 10:
VOD =
2[( VIN + )(1 − β 1 ) − ( VIN − )(1 − β 2 )] + 2 VOCM ( β 1 − β 2 )
1
(β1 + β 2 )
⎛
⎞
2
⎜1+
⎟
A F β1 + A F β 2 ⎠
⎝
(14)
Continued on next page
49
Analog Applications Journal
November 2000
Analog and Mixed-Signal Products
Amplifiers: Op Amps
Texas Instruments Incorporated
Continued from previous page
Again, assuming AFβ1 >> 1 and AFβ2 >> 1, this reduces to
VOD =
2[( VIN + )(1 − β 1 ) − ( VIN − )(1 − β 2 )] + 2 VOCM ( β 1 − β 2 )
.
(β1 + β 2 )
It can be seen from Equations 11, 13, and 15 that even
though the obvious use of a fully differential amplifier is
with symmetrical feedback, the gain can be controlled
with only one feedback path.
Using matched resistors R1 = R3 and R2 = R4 in the
analysis circuit of Figure 1 balances the feedback paths so
that β1 = β2 = β, and the transfer function is
( VOUT + ) − ( VOUT − )
AF
1− β
1
.
×
=
=
β
( VIN + ) − ( VIN − )
(1 + A F β )
⎛
1 ⎞
⎜1+
⎟
A Fβ ⎠
⎝
(15)
Figure 3. Single-ended to differential amplifier
R1
R2
–
+
VOUT+
AF
R3
VIN +
–
+
VOUT–
VOCM
The common-mode voltages at the input and output do
not enter into the equation, VIC is rejected, and VOC is set
by the voltage at VOCM. The ideal gain (assuming AFβ >> 1)
is set by the ratio
R4
1 − β R2
=
.
β
R1
Note that the normal inversion we might expect, given two
balanced inverting amplifiers, is accounted for by the output voltage definitions, resulting in a positive gain.
Many applications require that a single-ended signal be
converted to a differential signal. The circuits in Figures
3–7 show various approaches. Using Equations 11, 13, and
15, we can easily derive circuit solutions.
With a slight variation of Figure 1 as shown in Figure 3,
single-ended signals can be amplified and converted to differential signals. VIN – is now grounded and the signal is
applied to VIN+. Substituting VIN – = 0 in Equations 11, 13,
and 15 results in
( VOUT + ) =
( VIN + )(1 − β 1 ) + 2 VOCM β 1
,
(β1 + β 2 )
( VOUT − ) =
2 VOCM β 2 − ( VIN + )(1 − β 1 )
, and
(β1 + β 2 )
VOD =
Figure 4. β1 = 0
R1
R2
–
+
VOUT+
AF
VIN +
–
+
VOUT–
VOCM
Figure 5. β2 = 0
2( VIN + )(1 − β 1 ) + 2 VOCM ( β 1 − β 2 )
.
(β1 + β 2 )
If the signal is not referenced to ground, the reference
voltage will be amplified along with the desired signal,
reducing the dynamic range of the amplifier. To strip
unwanted dc offsets, use a capacitor to couple the signal
to VIN+. Keeping β1 = β2 will prevent VOCM from causing
an offset in VOD .
The circuits in Figures 4–7 have nonsymmetrical feedback. This causes VOCM to influence VOUT+ and VOUT –
differently, making VOCM show up in VOD . This will change
the operating points between the internal nodes in the
–
VOUT+
AF
R3
VIN +
+
+
–
VOUT–
VOCM
R4
50
Analog and Mixed-Signal Products
November 2000
Analog Applications Journal
Amplifiers: Op Amps
Texas Instruments Incorporated
Figure 7. β1 = 0, and β2 = 1
Figure 6. β2 = 1
–
+
VOUT+
–
AF
R3
–
+
VIN +
VOUT+
+
AF
VOUT–
VIN +
VOCM
+
–
VOUT–
R4
VOCM
differential amplifier, and matching of the open-loop gains
will degrade. CMRR is not a real issue with single-ended
inputs, but the analysis points out that CMRR is severely
compromised when nonsymmetrical feedback is used. In
the discussion of noise analysis that follows, it is shown
that nonsymmetrical feedback also increases noise introduced at the VOCM pin. For these reasons, even though
the circuits shown in Figures 4–7 have been tested to
prove they work in accordance with the equations given,
they are presented mainly for instructional purposes. They
are not recommended without extensive lab testing to
prove their worthiness in your application.
In the circuit shown in Figure 4, VIN – = 0 and β1 = 0.
The output voltages are
( VOUT + ) =
( VIN + )(1 − β 1 )
+ 2 VOCM ,
β1
2( VIN + )(1 − β 1 )
+ 2 VOCM .
β1
( VOUT + ) =
( VIN + )(1 − β 1 ) + 2 VOCM β 1
,
β1 + 1
( VOUT − ) =
2 VOCM − ( VIN + )(1 − β 1 )
, and
β1 + 1
VOD =
With β1 = 0, this circuit is similar to a noninverting amplifier.
In the circuit shown in Figure 5, VIN – = 0 and β2 = 0.
The output voltages are
−( VIN + )(1 − β 1 )
, and
β1
With β2 = 0, the gain is twice that of an inverting amplifier
(without the minus sign).
In the circuit shown in Figure 6, VIN – = 0 and β2 = 1.
The output voltages are
( VIN + )
, and
β2
2( VIN + )
− 2 VOCM .
β2
( VOUT + ) =
VOD =
( VIN + )
,
β2
( VOUT − ) = 2 VOCM −
VOD =
( VOUT − ) =
2( VIN + )(1 − β 1 ) + 2 VOCM ( β 1 − 1)
.
( β 1 + 1)
The gain is 1 with β1 = 0.333; or, with β1 = 0.6, the gain
is 1/2.
In the circuit shown in Figure 6, VIN – = 0, β1 = 0, and
β2 = 1. The output voltages are
( VOUT + ) = ( VIN + ), ( VOUT − ) = 2 VOCM − ( VIN + ),
and VOD = 2[( VIN + ) − VOCM ] .
This circuit realizes a gain of 2 with no resistor.
Continued on next page
51
Analog Applications Journal
November 2000
Analog and Mixed-Signal Products
Amplifiers: Op Amps
Texas Instruments Incorporated
Continued from previous page
Figure 8. Noise analysis circuit
Noise analysis
The noise sources are identified in Figure 8, which
will be used for analysis with the following definitions.
EIN is the input-referred RMS noise voltage of the
amplifier: EIN ≈ eIN x √ENB (assuming the 1/f noise is
negligible), where eIN is the input white noise spectral
density in volts per square root of the frequency in
Hertz, and ENB is the effective noise bandwidth. EIN
is modeled as a differential voltage at the input.
IIN+ and IIN – are the input-referred RMS noise
currents that flow into each input. They are taken as
equal and called IIN. IIN ≈ iIN x √ENB (assuming the
1/f noise is negligible), where iIN is the input white
noise spectral density in amps per square root of the
frequency in Hertz, and ENB is the effective noise
bandwidth. IIN develops a voltage in proportion to the
equivalent input impedance seen from the input
nodes. Assume the equivalent input impedance is
dominated by the parallel combination of the gain
setting resistors:
R EQ1 =
R1R2
R1 + R2
and R EQ2 =
R3R4
.
R3 + R4
ECM is the RMS noise at the VOCM pin, taking into account
the spectral density and bandwidth as with the inputreferred noise sources.
Noise current into the VOCM pin will develop a noise
voltage across the impedance seen from the node. It is
assumed that proper bypassing of the VOCM pin is done to
reduce the effective bandwidth, so this voltage is negligible.
If this is not the case, the added noise should be added to
ECM in a similar manner, as shown below.
ER1 through ER4 are the RMS noise voltages from the
resistors, calculated by ERn = √4kTR x ENB, where n is
the resistor number, k is Boltzmann’s constant (1.38 x
10–23j/K), T is the absolute temperature in Kelvin (K), R is
the resistance in ohms (Ω), and ENB is the effective noise
bandwidth.
E R1
R1
E R2
R2
IIN –
–
E IN
+
VOUT+
AF
IIN +
+
–
VOUT–
VOCM
E R3
R4
R3
E R4
ECM
EOD is the differential RMS output noise voltage.
EOD = A(EID), where EID is the input noise source, and A
is the gain from the source to the output. Half of EOD is
attributed to the positive output (+EOD/2), and half is
attributed to the negative output (–EOD/2). Therefore,
(+EOD/2) and (–EOD/2) are correlated to one another and
to the input source, and can be directly added together; i.e.,
⎛ + EOD ⎞ ⎛ − EOD ⎞
⎜
⎟ −⎜
⎟ = EOD = A( EID ).
⎝ 2 ⎠ ⎝ 2 ⎠
Independent noise sources typically are not correlated.
To combine noncorrelated noise voltages, a sum-of-squares
technique is used. The total RMS voltage squared is equal to
the square of the individual RMS voltages added together.
The output noise voltages from the individual noise
sources are calculated one at a time and then combined
in this fashion.
The block diagram shown in Figure 9 helps in analyzing
the amplifier’s noise sources.
Considering only EIN, from the block diagram we
can write:
( − EOD ) β 1 ( + EOD ) β 2
⎡
−
EOD = A F ⎢ EIN +
2
2
⎣
Figure 9. Block diagram of the amplifier’s
input-referred noise
⎤
⎥.
⎦
Solving yields
β2
I IN x REQ1
–
E IN
+
–
Σ
+
VP – VN
+ EOD
2
AF
+
I IN x REQ2
β1
– EOD
2
E CM
EOD
⎛
⎞
⎜
⎟
⎛ 2EIN ⎞ ⎜
1
⎟.
=⎜
⎟
⎟
2
⎝ β1 + β 2 ⎠ ⎜
⎜ 1+
⎟
A F ( β1 + β 2 ) ⎠
⎝
Assuming AFβ1 >> 1 and AFβ2 >> 1,
EOD =
2EIN
.
(β1 + β 2 )
52
Analog and Mixed-Signal Products
November 2000
Analog Applications Journal
Amplifiers: Op Amps
Texas Instruments Incorporated
Given β1 = β2 = β (symmetrical feedback),
EOUT =
EIN
,
β
the same as a standard single-ended voltage feedback op amp.
Similarly, the noise contributions from IIN x REQ1 and IIN x REQ2 will be
2I IN × R EQ1
(β1 + β 2 )
and
2I IN × R EQ2
(β1 + β 2 )
, respectively.
The VOCM error amplifier will produce a common-mode noise voltage at the output equal to ECM. Due to the feedback
paths, β1 and β2, a noise voltage is seen at the input that is equal to ECM(β1 – β2 ). This is amplified, just as an input, and
seen at the output as a differential noise voltage equal to
2ECM ( β 1 − β 2 )
.
(β1 + β 2 )
Noise gain from the VOCM pin ranges from 0 (given β1 = β2) to a maximum absolute value of 2 (given β1 = 1 and β2 = 0, or β1
= 0 and β2 = 1).
Noise from resistors R1 and R3 appears like signals at VIN+ and VIN – in Figure 1. From the circuit analysis presented
earlier, the differential output noise contribution is
2( E R1 )(1 − β 2 )
and
(β1 + β 2 )
2( E R3 )(1 − β 1 )
(β1 + β 2 )
for each resistor respectively.
Noise from resistors R2 and R4 (ER2 and ER4, respectively) is imposed directly on the output with no amplification.
Adding the individual noise sources yields the total output differential noise:
EOD =
( 2EIN ) 2 + ( 2I IN × R EQ1 ) 2 + ( 2I IN × R EQ2 ) 2 + [ 2ECM ( β 1 − β 2 )] 2 + [ 2( E R1 )(1 − β 2 )] 2 + [ 2( E R3 )(1 − β 1 )] 2
(β1 + β 2 ) 2
The individual noise sources are added in sum-of-squares
fashion. Input-referred terms are amplified by the noise
gain of the circuit:
Gn =
2
.
β1 + β 2
If symmetrical feedback is used where β1 = β2 = β, the
noise gain is
Gn =
R
1
= 1+ F ,
β
RG
+ E R2 2 + E R4 2 .
Reference
litnumber and replace “litnumber” with the TI Lit. # for
the materials listed below.
Document Title
TI Lit. #
1. Jim Karki, “Fully differential amplifiers,”
Analog Applications Journal (August
2000), pp. 38-41 . . . . . . . . . . . . . . . . . . . . . . .slyt165
Related Web site
amplifier.ti.com
where RF is the feedback resistor and RG is the input
resistor, the same as a standard single-ended voltage
feedback amplifier.
53
Analog Applications Journal
November 2000
Analog and Mixed-Signal Products
IMPORTANT NOTICE
Texas Instruments Incorporated and its subsidiaries (TI) reserve
the right to make corrections, modifications, enhancements,
improvements, and other changes to its products and services at
any time and to discontinue any product or service without notice.
Customers should obtain the latest relevant information before
placing orders and should verify that such information is current
conditions of sale supplied at the time of order acknowledgment.
TI warrants performance of its hardware products to the
specifications applicable at the time of sale in accordance with TI's
standard warranty. Testing and other quality control techniques are
used to the extent TI deems necessary to support this warranty.
Except where mandated by government requirements, testing of
all parameters of each product is not necessarily performed.
TI assumes no liability for applications assistance or customer
product design. Customers are responsible for their products and
applications using TI components. To minimize the risks
associated with customer products and applications, customers
should provide adequate design and operating safeguards.
TI does not warrant or represent that any license, either express or
right, or other TI intellectual property right relating to any
combination, machine, or process in which TI products or services
products or services does not constitute a license from TI to use
such products or services or a warranty or endorsement thereof.
Use of such information may require a license from a third party
under the patents or other intellectual property of the third party, or a
license from TI under the patents or other intellectual property of TI.
Reproduction of information in TI data books or data sheets is
permissible only if reproduction is without alteration and is
accompanied by all associated warranties, conditions, limitations,
and notices. Reproduction of this information with alteration is an
unfair and deceptive business practice. TI is not responsible or
liable for such altered documentation.
Resale of TI products or services with statements different from or
beyond the parameters stated by TI for that product or service
voids all express and any implied warranties for the associated TI
product or service and is an unfair and deceptive business
practice. TI is not responsible or liable for any such statements.
Following are URLs where you can obtain information on other
Texas Instruments products and application solutions:
Products
Amplifiers
Data Converters
DSP
Interface
Logic
Power Mgmt
Microcontrollers
amplifier.ti.com
dataconverter.ti.com
dsp.ti.com
interface.ti.com
logic.ti.com
power.ti.com
microcontroller.ti.com
Applications
Audio
Automotive
Digital control
Military
Optical Networking
Security
Telephony
Video & Imaging
Wireless
www.ti.com/audio
www.ti.com/automotive
www.ti.com/digitalcontrol
www.ti.com/military
www.ti.com/opticalnetwork
www.ti.com/security
www.ti.com/telephony
www.ti.com/video
www.ti.com/wireless
TI Worldwide Technical Support
Internet
support.ti.com
support.ti.com/sc/knowledgebase
Product Information Centers
Americas
Phone
Internet/Email
+1(972) 644-5580
Fax
support.ti.com/sc/pic/americas.htm
+1(972) 927-6377
Europe, Middle East, and Africa
Phone
Belgium (English) +32 (0) 27 45 54 32
Netherlands (English) +31 (0) 546 87 95 45
Finland (English) +358 (0) 9 25173948
Russia
+7 (0) 95 7850415
France
+33 (0) 1 30 70 11 64
Spain
+34 902 35 40 28
Germany
+49 (0) 8161 80 33 11
Sweden (English)
+46 (0) 8587 555 22
Israel (English)
1800 949 0107
United Kingdom
+44 (0) 1604 66 33 99
Italy
800 79 11 37
Fax
+(49) (0) 8161 80 2045
Internet
support.ti.com/sc/pic/euro.htm
Japan
Fax
International
Internet/Email
International
Domestic
Asia
Phone
International
Domestic
Australia
China
Hong Kong
Indonesia
Korea
Malaysia
Fax
Internet
+81-3-3344-5317
Domestic
0120-81-0036
support.ti.com/sc/pic/japan.htm
www.tij.co.jp/pic
+886-2-23786800
Toll-Free Number
1-800-999-084
800-820-8682
800-96-5941
001-803-8861-1006
080-551-2804
1-800-80-3973
886-2-2378-6808
support.ti.com/sc/pic/asia.htm
New Zealand
Philippines
Singapore
Taiwan
Thailand
Email
Toll-Free Number
0800-446-934
1-800-765-7404
800-886-1028
0800-006800
001-800-886-0010
[email protected]
[email protected]
C011905
Safe Harbor Statement: This publication may contain forwardlooking statements that involve a number of risks and
uncertainties. These “forward-looking statements” are intended
to qualify for the safe harbor from liability established by the
Private Securities Litigation Reform Act of 1995. These forwardlooking statements generally can be identified by phrases such
as TI or its management “believes,” “expects,” “anticipates,”
“foresees,” “forecasts,” “estimates” or other words or phrases
of similar import. Similarly, such statements herein that describe
the company's products, business strategy, outlook, objectives,
plans, intentions or goals also are forward-looking statements.
All such forward-looking statements are subject to certain risks
and uncertainties that could cause actual results to differ
materially from those in forward-looking statements. Please
refer to TI's most recent Form 10-K for more information on the
risks and uncertainties that could materially affect future results
of operations. We disclaim any intention or obligation to update
any forward-looking statements as a result of developments
occurring after the date of this publication.