Precision Matched Resistors Automatically Improve Differential
Amplifier CMRR – Here’s How
Design Note 1023
Kris Lokere, Tyler Hutchison, Greg Zimmer
INTRODUCTION
Matched resistors are critical to the performance of a large class of differential circuits such as the following:
R1
R2
R1
VCC
R3
VA
R2
R4
R2
INST
AMP
VB
5V
5V
+
VB
VOUT
–
R3
VB
–
OPAMP
VA
VOUT
VA
+
– +
DIFFERENTIAL
AMP
VOUT
+ –
R1
–5V
R3
dn1023 F02
R4
Instrumentation Amp
Difference Amp
dn1023 F07
R4
dn1023 F12
Differential Amp
Ideally, the resistors in these circuits are chosen such that R1/R2 = R4/R3. Any mismatch between these ratios will contribute
a common mode error. The CMRR (common mode rejection ratio) is an important metric in these circuits, as it indicates how
much of the unwanted common mode signal will appear in the output. The CMRR due to the resistors in these circuits can
be calculated using the following formula:
⎛ 1⎞
⎜⎝ ⎟⎠ (G+1)
2
CMRRR _ ONLY ≈
ΔR /R
Where CMRRR_ONLY Is CMRR Due Only to the Resistors
(Ideal Amplifier Case), ΔR/R Is the Resistor Matching
Ratio, G is the Nominal Ratio of R1/R2.
For example, using resistors with 1% tolerance (i.e. matched to 2%) in a differential circuit with G = 1, the CMRR will only
be 34dB.
LT5400 ΔR/RCMRR
In addition to the guaranteed resistor-to-resistor matching over temperature, the LT5400 includes a new metric called “Matching for CMRR.” Only the LT5400 offers a Matching for CMRR, (ΔR/R)CMRR specification. This spec guarantees the contribution
of CMRR error due to the resistors, when the LT5400 is used in a difference configuration using the specific resistor pairs
of R1/R2 and R4/R3.
The definition of Matching for CMRR is the following:
⎛ 1⎞ ⎛ R1⎞ ⎛ R2 R3 ⎞ ⎛ 1⎞ ⎛ R1R3 ⎞
ΔR /RCMRR = ⎜ ⎟ ⎜ ⎟ ⎜ – ⎟ = ⎜ ⎟ ⎜ 1–
⎝ 2 ⎠ ⎝ R2 ⎠ ⎝ R1 R4 ⎠ ⎝ 2 ⎠ ⎝ R2R4 ⎟⎠
L, LT, LTC, LTM, Linear Technology and the Linear logo are registered trademarks of Linear
Technology Corporation. All other trademarks are the property of their respective owners.
03/12/1023
1
Calculating the improved CMRR performance of the LT5400 resistors is easy. Simply replace the resistor matching ratio,
(ΔR/R) with the Matching for CMRR specification (ΔR/RCMRR):
⎛ 1⎞
⎜⎝ ⎟⎠ (G+1)
2
CMRRR _ ONLY ≈
ΔR /RCMRR
Equation 1
Where ΔR/RCMRR Is the Matching for CMRR specification
For example, the LT5400A offers 0.01% resistor matching and Matching for CMRR of 0.005%. With this Matching for CMRR,
the resulting CMRR is 86dB.
CMRR DEFINITION
The amplifier’s common mode rejection ratio (CMRR) is the ratio of the differential mode gain to the common mode gain. For
these calculations, only common mode and differential mode gain is considered for amplifiers. Thus, an amplifier’s output
can be determined as: VOUT = (VCM • ACM) + (VDIFF • ADIFF )
+
AMP
VOUT
–
VDIFF
2
–VDIFF
2
dn1023 F01
VCM
COMBINING AMPLIFIER AND RESISTOR CMRR
In addition to the CMRR due to the resistors, one must also consider the finite CMRR contribution due to the amplifier. The
total CMRR of any of the differential circuits shown above can be calculated using the following formula:
CMRR TOTAL ≈
⎛ 1⎞
⎜⎝ ⎟⎠ (G+1)
2
⎛
⎞ ⎛ 1⎞
1
⎜ CMRR
⎟ ⎜ ⎟ (G+1) + ΔR /RCMRR
⎝
amp ⎠ ⎝ 2 ⎠
Equation 2
Where CMRRamp Is the Amplifier’s CMRR
Specification, and CMRRTOTAL Is the Combined
CMRR Due to the Resistors and Amplifier.
CMRR CALCULATION
The rest of this design note provides the derivation of CMRR for each differential circuit shown above. The results will show
that CMRR can be calculated using Equation 1 and Equation 2 for all three circuits. Examples of calculating CMRRTOTAL
(resistor and amplifier) are also provided at the end of each section.
2
CASE 1: IDEAL INSTRUMENTATION AMP (CMRRamp IS INFINITE)
CMRR Derivation, Step 1: Calculate the Circuit’s Common Mode Gain (ACM)
VCC
R3
VA
R2
VB
+
INST
AMP
–
VCM
R4
VOUT
R1
dn1023 F03
⎛ R4 ⎞
VA = VCM ⎜
⎝ R3+R4 ⎟⎠
⎛ R1 ⎞
VB = VCM ⎜
⎝ R1+R2 ⎟⎠
VOUT = ( VA − VB ) •Gamp
Ideal Instrumentation Amplifier, Gamp = Amplifier Gain
⎡
⎛ R4 ⎞
⎛ R1 ⎞ ⎤
VOUT = ⎢ VCM ⎜
⎟⎠ − VCM ⎜⎝
⎟ G
⎝
R3+R4
R1+R2 ⎠ ⎥⎦ amp
⎣
⎡⎛ R4 ⎞ ⎛ R1 ⎞ ⎤
VOUT = VCMGamp ⎢⎜
⎟ −⎜
⎟⎥
⎣⎝ R3+R4 ⎠ ⎝ R1+R2 ⎠ ⎦
VOUT
R1 ⎞
⎛ R4
= Gamp ⎜
−
⎝ R3+R4 R1+R2 ⎟⎠
VCM
CMRR Derivation, Step 2: Calculate the Circuit’s Differential Mode Gain (ADIFF)
VCC
VDIFF
2
–VDIFF
2
R3
VA
R2
VB
R4
+
INST
AMP
VOUT
–
R1
dn1023 F04
VA =
VDIFF ⎛ R4 ⎞
⎜
⎟
2 ⎝ R3+R4 ⎠
VB = −
VDIFF ⎛ R1 ⎞
⎜
⎟
2 ⎝ R1+R2 ⎠
VOUT = ( VA − VB ) •Gamp
Ideal Instrumentation Amplifier, Gamp = Amplifier Gain
3
⎡⎛ V
⎞ ⎛ R4 ⎞ ⎛ VDIFF ⎞ ⎛ R1 ⎞ ⎤
VOUT = ⎢⎜ DIFF ⎟ ⎜
⎟ −⎜−
⎟⎜
⎟ Gamp
2 ⎠ ⎝ R1+R2 ⎠ ⎥⎦
⎣⎝ 2 ⎠ ⎝ R3+R4 ⎠ ⎝
R1 ⎞
⎛ 1⎞ ⎛ R4
VOUT = VDIFF Gamp ⎜ ⎟ ⎜
+
⎝ 2 ⎠ ⎝ R3+R4 R1+R2 ⎟⎠
VOUT
R4 ⎞
⎛ 1⎞ ⎛ R1
= Gamp ⎜ ⎟ ⎜
+
⎝ 2 ⎠ ⎝ R1+R2 R3+R4 ⎟⎠
VDIFF
CMRR Derivation, Step 3: Calculate the Circuit’s CMRR (ADIFF/ACM)
R4 ⎞
⎛ 1⎞ ⎛ R1
Gamp ⎜ ⎟ ⎜
+
⎝ 2 ⎠ ⎝ R1+R2 R3+R4 ⎟⎠
CMRRR _ ONLY =
R1 ⎞
⎛ R4
Gamp ⎜
−
⎝ R3+R4 R1+R2 ⎟⎠
R4 ⎞
⎛ 1⎞ ⎛ R1
+
⎜⎝ ⎟⎠ ⎜⎝
⎟
2 R1+R2 R3+R4 ⎠
=
R1 ⎞
⎛ R4
−
⎜⎝
⎟
R3+R4 R1+R2 ⎠
⎛ 1⎞
⎜⎝ ⎟⎠ (R1R3+R1R4+R1R4+R2R4)
2
=
(R1R4+R2R4−R1R3−R1R4)
⎛ 1⎞
⎜⎝ ⎟⎠ ( 2R1R4+R2R4+R1R3)
2
=
(R2R4−R1R3)
⎛ 1⎞ ⎛ 1 ⎞
⎜⎝ ⎟⎠ ⎜⎝
⎟
2 R2R4 ⎠
=
⎛ 1⎞ ⎛ 1 ⎞
⎜⎝ ⎟⎠ ⎜⎝
⎟
2 R2R4 ⎠
⎛ 1⎞
⎜⎝ ⎟⎠ ( 2R1R4+R2R4+R1R3)
2
(R2R4−R1R3)
⎛ 1⎞ ⎛ 1⎞ ⎛ 2R1R4 R2R4 R1R3 ⎞
+
+
⎜⎝ ⎟⎠ ⎜⎝ ⎟⎠ ⎜⎝
⎟
2 2 R2R4 R2R4 R2R4 ⎠
=
⎛ 1⎞ ⎛ R2R4 R1R3 ⎞
−
⎜⎝ ⎟⎠ ⎜⎝
⎟
2 R2R4 R2R4 ⎠
R1R3 ⎞
⎛ 1⎞ ⎛ 1⎞ ⎛ 2R1
+1+
⎜⎝ ⎟⎠ ⎜⎝ ⎟⎠ ⎜⎝
⎟
2 2 R2
R2R4 ⎠
CMRRR _ ONLY =
⎛ 1⎞ ⎛ R1R3 ⎞
⎜⎝ ⎟⎠ ⎜⎝ 1−
⎟
2
R2R4 ⎠
4
Equation 3
This Preliminary Result Will Be Referenced
in Other Calculations
This equation can be simplified by noting the following:
R1 R4
⎛ 1⎞ ⎛ R1R3 ⎞
⎛ R1R3 ⎞
and
≈1
ΔR /RCMRR = ⎜ ⎟ ⎜ 1−
≈
≈ G and ⎜
⎝ 2 ⎠ ⎝ R2R4 ⎟⎠
⎝ R2R4 ⎟⎠
R2 R3
Thus,
⎛ 1⎞ ⎛ 1⎞
⎜⎝ ⎟⎠ ⎜⎝ ⎟⎠ ( 2G+ 2)
2 2
CMRRR _ ONLY ≈
⎛ 1⎞ ⎛ R1R3 ⎞
⎜⎝ ⎟⎠ ⎜⎝ 1−
⎟
2
R2R4 ⎠
⎛ 1⎞
⎜⎝ ⎟⎠ (G+1)
2
CMRRR _ ONLY ≈
ΔR /RCMRR
CASE 2: NON-IDEAL INSTRUMENTATION AMP (CMRRamp IS FINITE)
CMRR Derivation, Step 1: Calculate the Circuit’s Common Mode Gain (ACM)
VCC
R3
VA
R2
VB
+
INST
AMP
–
VCM
R4
VOUT
R1
dn1023 F05
VCM ´= (VA + VB ) / 2
VDIFF ´= (VA – VB )
VCM´ = Common Mode Voltage Seen at Inputs of Amp
VDIFF´ = Differential Voltage Seen at Inputs of Amp
⎛ R4 ⎞
VA = VCM • ⎜
⎝ R3+R4 ⎟⎠
⎛ R1 ⎞
VB = VCM • ⎜
⎝ R1+R2 ⎟⎠
VOUT = (VCM ´•A CM ´)+(VDIFF ´•ADIFF ´)
ACM´ = Common Mode Gain of the Amp
ADIFF´ = Differential Mode Gain of the Amp
1
⎡
⎡
⎛ R4 ⎞
⎛ R1 ⎞ ⎤
⎛ R4 ⎞
⎛ R1 ⎞ ⎤
+ VCM ⎜
− VCM ⎜
VOUT = A CM ´ ⎢ VCM ⎜
+ ADIFF ´ ⎢ VCM ⎜
⎟
⎟
⎟
⎥
⎝ R3+R4 ⎠
⎝ R1+R2 ⎠ ⎦
⎝ R3+R4 ⎠
⎝ R1+R2 ⎟⎠ ⎥⎦
2
⎣
⎣
⎡1
⎡⎛ R4 ⎞ ⎛ R1 ⎞ ⎤
⎡⎛ R4 ⎞ ⎛ R1 ⎞ ⎤ ⎤
+⎜
VOUT = VCM ⎢ A CM ´ ⎢⎜
+ ADIFF ´ ⎢⎜
⎟
⎟
⎟ −⎜
⎟ ⎥⎥
⎥
⎣⎝ R3+R4 ⎠ ⎝ R1+R2 ⎠ ⎦
⎣⎝ R3+R4 ⎠ ⎝ R1+R2 ⎠ ⎦ ⎦
⎣2
VOUT ⎡ 1
⎡⎛ R4 ⎞ ⎛ R1 ⎞ ⎤
⎡⎛ R4 ⎞ ⎛ R1 ⎞ ⎤ ⎤
+⎜
+ ADIFF ´ ⎢⎜
= ⎢ A CM ´ ⎢⎜
⎟
⎟
⎟ −⎜
⎟ ⎥⎥
⎥
VCM ⎣ 2
⎣⎝ R3+R4 ⎠ ⎝ R1+R2 ⎠ ⎦
⎣⎝ R3+R4 ⎠ ⎝ R1+R2 ⎠ ⎦ ⎦
5
CMRR Derivation, Step 2: Calculate the Circuit’s Differential Mode Gain (ADIFF)
VCC
VDIFF
2
R3
VA
R2
VB
–VDIFF
2
R4
+
INST
AMP
VOUT
–
R1
dn1023 F06
VCM ' = (VA + VB ) / 2
VDIFF ' = (VA − VB )
VA =
VDIFF
2
VB = −
VCM´ = Common Mode Voltage Seen at Inputs of Amp
VDIFF´ = Differential Voltage Seen at Inputs of Amp
⎛ R4 ⎞
•⎜
⎝ R3+R4 ⎟⎠
VDIFF
2
⎛ R1 ⎞
•⎜
⎝ R1+R2 ⎟⎠
VOUT = (VCM ´•A CM ´)+(VDIFF ´•ADIFF ´)
ACM´ = Common Mode Gain of the Amp
ADIFF´ = Differential Mode Gain of the Amp
1
⎡V
⎛ R4 ⎞ ⎛ VDIFF ⎞ ⎛ R1 ⎞ ⎤
VOUT = A CM ´ ⎢ DIFF ⎜
⎟ +⎜−
⎟⎜
⎟
2
2 ⎠ ⎝ R1+R2 ⎠ ⎥⎦
⎣ 2 ⎝ R3+R4 ⎠ ⎝
⎡V
⎛ R4 ⎞ ⎛ VDIFF ⎞ ⎛ R1 ⎞ ⎤
− −
+ADIFF ´ ⎢ DIFF ⎜
⎟⎜
⎟
⎝ R3+R4 ⎟⎠ ⎜⎝
2
2 ⎠ ⎝ R1+R2 ⎠ ⎥⎦
⎣
VOUT =
VDIFF
2
⎡1
⎡⎛ R4 ⎞ ⎛ R1 ⎞ ⎤
⎡⎛ R4 ⎞ ⎛ R1 ⎞ ⎤ ⎤
⎢ 2 A CM ´ ⎢⎜⎝ R3+R4 ⎟⎠ − ⎜⎝ R1+R2 ⎟⎠ ⎥ + ADIFF ´ ⎢⎜⎝ R3+R4 ⎟⎠ + ⎜⎝ R1+R2 ⎟⎠ ⎥ ⎥
⎣
⎦
⎣
⎦⎦
⎣
VOUT 1 ⎡ 1
⎡⎛ R4 ⎞ ⎛ R1 ⎞ ⎤
⎡⎛ R4 ⎞ ⎛ R1 ⎞ ⎤ ⎤
= ⎢ A CM ´ ⎢⎜
⎟⎠ − ⎜⎝
⎟⎠ ⎥ + ADIFF ´ ⎢⎜⎝
⎟ +⎜
⎟ ⎥⎥
⎝
R1+R2 ⎦
VDIFF 2 ⎣ 2
⎣ R3+R4
⎣ R3+R4 ⎠ ⎝ R1+R2 ⎠ ⎦ ⎦
This equation can be simplified by noting the following:
1
⎡⎛ R4 ⎞ ⎛ R1 ⎞ ⎤
⎡⎛ R4 ⎞ ⎛ R1 ⎞ ⎤
+⎜
ADIFF ´ ⎢⎜
>> A CM ´ ⎢⎜
⎟
⎟
⎟ −⎜
⎟⎥
⎥
⎣⎝ R3+R4 ⎠ ⎝ R1+R2 ⎠ ⎦ 2
⎣⎝ R3+R4 ⎠ ⎝ R1+R2 ⎠ ⎦
Thus,
VOUT 1 ⎡
⎡⎛ R4 ⎞ ⎛ R1 ⎞ ⎤ ⎤
≈ ⎢ ADIFF ´ ⎢⎜
⎟ +⎜
⎟ ⎥⎥
VDIFF 2 ⎣
⎣⎝ R3+R4 ⎠ ⎝ R1+R2 ⎠ ⎦ ⎦
6
CMRR Derivation, Step 3: Calculate the Circuit’s CMRR (ADIFF/ACM)
⎡⎛ R4 ⎞ ⎛ R1 ⎞ ⎤
⎛ 1⎞
⎜⎝ ⎟⎠ ADIFF ´ ⎢⎜⎝
⎟ +⎜
⎟
R3+R4 ⎠ ⎝ R1+R2 ⎠ ⎥⎦
2
⎣
CMRR TOTAL ≈
1
⎡⎛ R4 ⎞ ⎛ R1 ⎞ ⎤
⎡⎛ R4 ⎞ ⎛ R1 ⎞ ⎤
A CM ´ ⎢⎜
⎟⎠ + ⎜⎝
⎟⎠ ⎥ + ADIFF ´ ⎢⎜⎝
⎟ −⎜
⎟⎥
⎝
R1+R2 ⎦
2
⎣ R3+R4
⎣ R3+R4 ⎠ ⎝ R1+R2 ⎠ ⎦
≈
Equation 4
This Preliminary Result Will Be
Referenced in Other Calculations
⎛ 1⎞
⎜⎝ ⎟⎠ ⎡⎣(R1R3+R1R4+R1R4+R2R4) ⎤⎦
2
⎞
1
⎛ 1⎞ ⎛
(R1R3+R1R4+R1R4+R2R4) + (R1R4+R2R4−R1R3−R1R4)
⎜⎝ ⎟⎠ ⎜
2 ⎝ CMRR amp ⎟⎠
CMRRamp =
A DIFF ´
A CM´
⎛ 1⎞ ⎛ 1 ⎞ ⎛ 1⎞
⎜⎝ ⎟⎠ ⎜⎝
⎟ ⎜ ⎟ (R1R3+ 2R1R4+R2R4)
2 R2R4 ⎠ ⎝ 2 ⎠
≈
⎤
⎞
1
⎛ 1⎞ ⎛ 1 ⎞ ⎡⎛ 1⎞ ⎛
(R1R3+ 2R1R4+R2R4) + (R2R4−R1R3)⎥
⎜⎝ ⎟⎠ ⎜⎝
⎟⎠ ⎢⎜⎝ ⎟⎠ ⎜
⎟
2 R2R4 ⎣⎢ 2 ⎝ CMRR amp ⎠
⎦⎥
⎛ 1⎞ ⎛ 1⎞ ⎛ R1R3 2R1R4 R2R4 ⎞
+
+
⎜⎝ ⎟⎠ ⎜⎝ ⎟⎠ ⎜⎝
⎟
2 2 R2R4 R2R4 R2R4 ⎠
≈
⎞ ⎛ R1R3 2R1R4 R2R4 ⎞ ⎛ 1⎞ ⎛ R2R4 R1R3 ⎞
1
⎛ 1⎞ ⎛ 1⎞ ⎛
+
+
−
⎜⎝ ⎟⎠ ⎜⎝ ⎟⎠ ⎜
⎜
⎟ +⎜ ⎟ ⎜
⎟
2 2 ⎝ CMRR amp ⎟⎠ ⎝ R2R4 R2R4 R2R4 ⎠ ⎝ 2 ⎠ ⎝ R2R4 R2R4 ⎠
⎛ 1⎞ ⎛ 1⎞ ⎛ R1R3 2R1 ⎞
+
+1⎟
⎜⎝ ⎟⎠ ⎜⎝ ⎟⎠ ⎜⎝
2 2 R2R4 R2 ⎠
≈
⎞ ⎛ R1R3 2R1 ⎞ ⎛ 1⎞ ⎛ R1R3 ⎞
1
⎛ 1⎞ ⎛ 1⎞ ⎛
+
+1⎟ + ⎜ ⎟ ⎜ 1−
⎜
⎜⎝ ⎟⎠ ⎜⎝ ⎟⎠ ⎜
⎟
2 2 ⎝ CMRR amp ⎟⎠ ⎝ R2R4 R2 ⎠ ⎝ 2 ⎠ ⎝ R2R4 ⎠
This equation can be simplified by noting the following:
R1 R4
⎛ 1⎞ ⎛ R1R3 ⎞
⎛ R1R3 ⎞
and
≈1
ΔR /RCMRR = ⎜ ⎟ ⎜ 1−
≈
≈ G and ⎜
⎟
⎝ 2 ⎠ ⎝ R2R4 ⎠
⎝ R2R4 ⎟⎠
R2 R3
Thus,
CMRR TOTAL ≈
CMRR TOTAL ≈
⎛ 1⎞ ⎛ R1⎞
⎜⎝ ⎟⎠ ⎜⎝ 1+ ⎟⎠
2
R2
⎛
⎞ ⎛ 1⎞ ⎛ R1⎞ ⎛ 1⎞ ⎛ R1R3 ⎞
1
⎜ CMRR
⎟ ⎜ ⎟ ⎜ 1+ R2 ⎟⎠ + ⎜⎝ 2 ⎟⎠ ⎜⎝ 1− R2R4 ⎟⎠
⎝
amp ⎠ ⎝ 2 ⎠ ⎝
⎛ 1⎞
⎜⎝ ⎟⎠ (1+G)
2
⎛
⎞ ⎛ 1⎞
1
⎜ CMRR
⎟ ⎜ ⎟ (1+G) + ΔR /RCMRR
⎝
amp ⎠ ⎝ 2 ⎠
7
Example of CMRR Calculation
Using Linear Technology’s LTC2053 Instrument Amplifier, configured for a gain of 1, and the LT5400A-1:
CMRR amp(2053) = 100dB
G≈
R1 R4 10k
≈
≈
≈1
R2 R3 10k
100dB ➔ (1/CMRRamp(2053)) = 0.001%
ΔR/RCMRR = 0.005%
Using Equation 2:
CMRR TOTAL ≈
(1)
⎛
⎞
1
⎜ CMRR
⎟ (1) + ( ΔR /RCMRR )
⎝
amp ⎠
1
0.00001+ 0.00005
≈ 16667
≈
CMRR TOTAL (dB) ≈ 20log (16667 )
≈ 84.44dB
CASE 3: DIFFERENCE AMP USING AN IDEAL OP AMP (CMRRamp IS INFINITE)
CMRR Derivation, Step 1: Calculate the Circuit’s Common Mode Gain (ACM)
R1
5V
R2
VB
R3
VA
VCM
–
OPAMP
VOUT
+
–5V
R4
dn1023 F08
⎛ R1 ⎞
⎛ R2 ⎞
+V
VB = VCM ⎜
⎝ R1+R2 ⎟⎠ OUT ⎜⎝ R1+R2 ⎟⎠
⎛ R4 ⎞
VA = VCM ⎜
⎝ R3+R4 ⎟⎠
VA = VB
⎛ R1 ⎞
⎛ R2 ⎞
⎛ R4 ⎞
+V
=V
VCM ⎜
⎝ R1+R2 ⎟⎠ OUT ⎜⎝ R1+R2 ⎟⎠ CM ⎜⎝ R3+R4 ⎟⎠
R1 ⎞
⎛ R2 ⎞
⎛ R4
=V
VOUT ⎜
−
⎝ R1+R2 ⎟⎠ CM ⎜⎝ R3+R4 R1+R2 ⎟⎠
8
Ideal Op Amp
VOUT
VCM
R1 ⎞
⎛ R4
−
⎜⎝
⎟
R3+R4 R1+R2 ⎠
=
⎛ R2 ⎞
⎜⎝
⎟
R1+R2 ⎠
CMRR Derivation, Step 2: Calculate the Circuit’s Differential Mode Gain (ADIFF)
R1
5V
–VDIFF
2
R2
VB
R3
VA
VDIFF
2
–
OPAMP
–5V
R4
VA =
dn1023 F09
VDIFF ⎛ R4 ⎞
⎜
⎟
2 ⎝ R3+R4 ⎠
VB = –
VDIFF ⎛ R1 ⎞
⎛ R2 ⎞
⎜⎝
⎟⎠ + VOUT ⎜⎝
⎟
R1+R2 ⎠
2 R1+R2
VA = VB
–
VOUT
+
Ideal Op Amp
VDIFF ⎛ R1 ⎞
⎛ R2 ⎞ VDIFF ⎛ R4 ⎞
⎜
⎟ +V ⎜
⎟=
⎜
⎟
2 ⎝ R1+R2 ⎠ OUT ⎝ R1+R2 ⎠
2 ⎝ R3+R4 ⎠
R4 ⎞
⎛ R2 ⎞ ⎛ VDIFF ⎞ ⎛ R1
=⎜
VOUT ⎜
+
⎟
⎟
⎜
⎝ R1+R2 ⎠ ⎝ 2 ⎠ ⎝ R1+R2 R3+R4 ⎟⎠
R4 ⎞
⎛ 1⎞ ⎛ R1
+
VOUT ⎜⎝ 2 ⎟⎠ ⎜⎝ R1+R2 R3+R4 ⎟⎠
=
⎛ R2 ⎞
VDIFF
⎜⎝
⎟
R1+R2 ⎠
9
CMRR Derivation, Step 3: Calculate the Circuit’s CMRR (ADIFF/ACM)
R4 ⎞
⎛ 1⎞ ⎛ R1
+
⎜⎝ ⎟⎠ ⎜⎝
⎟
2 R1+R2 R3+R4 ⎠
⎛ R2 ⎞
⎜⎝
⎟
R1+R2 ⎠
CMRRR _ ONLY =
R1 ⎞
⎛ R4
–
⎜⎝
⎟
R3+R4 R1+R2 ⎠
⎛ R2 ⎞
⎜⎝
⎟
R1+R2 ⎠
R4 ⎞
⎛ 1⎞ ⎛ R1
+
⎜⎝ ⎟⎠ ⎜⎝
⎟
2 R1+R2 R3+R4 ⎠
=
R1 ⎞
⎛ R4
−
⎜⎝
⎟
R3+R4 R1+R2 ⎠
This is Equation 3, and we have already shown that Equation 3 can be reduced to Equation 1.
⎛ 1⎞
⎜⎝ ⎟⎠ (G+1)
2
CMRRR _ ONLY ≈
ΔR /RCMRR
CASE 4: DIFFERENCE AMP USING A NON-IDEAL OP AMP (CMRRamp IS FINITE)
CMRR Derivation, Step 1: Calculate the Circuit’s Common Mode Gain (ACM)
R1
5V
VCM
R2
VB
R3
VA
–
OPAMP
VOUT
+
–5V
R4
VCM ´= (VA + VB ) / 2
VDIFF ´= (VA − VB )
dn1023 F10
VCM´ = Common Mode Voltage Seen at Inputs of Amp
VDIFF´ = Differential Voltage Seen at Inputs of Amp
⎛ R4 ⎞
VA = VCM • ⎜
⎝ R3+R4 ⎟⎠
⎛ R1 ⎞
⎛ R2 ⎞
+V
VB = VCM • ⎜
•
⎝ R1+R2 ⎟⎠ OUT ⎜⎝ R1+R2 ⎟⎠
VOUT = (VCM ´•A CM ´)+(VDIFF ´•ADIFF ´)
10
ACM´ = Common Mode Gain of the Amp
ADIFF´ = Differential Mode Gain of the Amp
1
⎡
⎛ R4 ⎞
⎛ R1 ⎞
⎛ R2 ⎞ ⎤
+ VCM ⎜
+ VOUT ⎜
VOUT = A CM ´ ⎢ VCM ⎜
⎟
⎟
⎝ R3+R4 ⎠
⎝ R1+R2 ⎠
⎝ R1+R2 ⎟⎠ ⎥⎦
2
⎣
⎡
⎛ R4 ⎞
⎛ R1 ⎞
⎛ R2 ⎞ ⎤
+ADIFF ´ ⎢ VCM ⎜
⎟⎠ − VCM ⎜⎝
⎟⎠ − VOUT ⎜⎝
⎟
⎝
R3+R4
R1+R2
R1+R2 ⎠ ⎥⎦
⎣
⎡1
⎛ R2 ⎞
⎛ R2 ⎞ ⎤
VOUT = VOUT ⎢ A CM ´⎜
⎟⎠ − ADIFF ´⎜⎝
⎟
⎝
R1+R2
R1+R2 ⎠ ⎥⎦
⎣2
⎡1
⎡⎛ R1 ⎞ ⎛ R4 ⎞ ⎤
⎡⎛ R4 ⎞ ⎛ R1 ⎞ ⎤ ⎤
+⎜
+VCM ⎢ A CM ´ ⎢⎜
+ ADIFF ´ ⎢⎜
⎟
⎟
⎟ −⎜
⎟ ⎥⎥
⎥
⎣⎝ R1+R2 ⎠ ⎝ R3+R4 ⎠ ⎦
⎣⎝ R3+R4 ⎠ ⎝ R1+R2 ⎠ ⎦ ⎦
⎣2
⎡1
⎛ R2 ⎞
⎛ R2 ⎞ ⎤
VOUT − VOUT ⎢ A CM ´⎜
⎟⎠ − ADIFF ´⎜⎝
⎟
⎝
R1+R2
R1+R2 ⎠ ⎥⎦
⎣2
⎡1
⎡⎛ R1 ⎞ ⎛ R4 ⎞ ⎤
⎡⎛ R4 ⎞ ⎛ R1 ⎞ ⎤ ⎤
+⎜
= VCM ⎢ A CM ´ ⎢⎜
+ ADIFF ´ ⎢⎜
⎟
⎟
⎟ −⎜
⎟ ⎥⎥
⎥
⎣⎝ R1+R2 ⎠ ⎝ R3+R4 ⎠ ⎦
⎣⎝ R3+R4 ⎠ ⎝ R1+R2 ⎠ ⎦ ⎦
⎣2
⎡ ⎡1
⎛ R2 ⎞
⎛ R1 ⎞ ⎤ ⎤
− ADIFF ´⎜
VOUT ⎢1− ⎢ A CM ´⎜
⎟
⎝ R1+R2 ⎠
⎝ R1+R2 ⎟⎠ ⎥⎦ ⎥⎦
⎣ ⎣2
⎡1
⎡⎛ R1 ⎞ ⎛ R4 ⎞ ⎤
⎡⎛ R4 ⎞ ⎛ R1 ⎞ ⎤ ⎤
+⎜
= VCM ⎢ A CM ´ ⎢⎜
+ ADIFF ´ ⎢⎜
⎟
⎟
⎟ −⎜
⎟ ⎥⎥
⎥
⎣⎝ R1+R2 ⎠ ⎝ R3+R4 ⎠ ⎦
⎣⎝ R3+R4 ⎠ ⎝ R1+R2 ⎠ ⎦ ⎦
⎣2
VOUT
VCM
1
⎡⎛ R1 ⎞ ⎛ R4 ⎞ ⎤
⎡⎛ R4 ⎞ ⎛ R1 ⎞ ⎤
A CM ´ ⎢⎜
⎟⎠ + ⎜⎝
⎟⎠ ⎥ + ADIFF ´ ⎢⎜⎝
⎟ −⎜
⎟⎥
⎝
2
R3+R4 ⎦
⎣ R1+R2
⎣ R3+R4 ⎠ ⎝ R1+R2 ⎠ ⎦
=
⎡1
⎛ R2 ⎞
⎛ R1 ⎞ ⎤
− ( ADIFF ´ ) ⎜
1− ⎢ A CM ´⎜
⎟
⎝ R1+R2 ⎠
⎝ R1+R2 ⎟⎠ ⎥⎦
⎣2
CMRR Derivation, Step 2: Calculate the Circuit’s Differential Mode Gain (ADIFF)
R1
5V
–VDIFF
2
VDIFF
2
R2
VB
R3
VA
–
OPAMP
–5V
R4
VCM ´= (VA + VB ) / 2
VDIFF ´= (VA − VB )
VOUT
+
dn1023 F11
VCM´ = Common Mode Voltage Seen at Inputs of Amp
VDIFF´ = Differential Voltage Seen at Inputs of Amp
11
VA =
VDIFF
2
VB = −
⎛ R4 ⎞
•⎜
⎝ R3+R4 ⎟⎠
VDIFF ⎛ R1 ⎞
⎛ R2 ⎞
•⎜
⎟⎠ + VOUT • ⎜⎝
⎟
⎝
2
R1+R2
R1+R2 ⎠
ACM´ = Common Mode Gain of the Amp
ADIFF´ = Differential Mode Gain of the Amp
VOUT = (VCM ´•A CM ´)+(VDIFF ´•ADIFF ´)
⎛1
⎞ ⎡V
⎛ R4 ⎞ ⎛ VDIFF ⎞ ⎛ R1 ⎞
⎛ R2 ⎞ ⎤
+⎜−
+ VOUT ⎜
VOUT = ⎜ A CM ´ ⎟ ⎢ DIFF ⎜
⎟
⎟
⎜
⎟
⎝2
⎠ ⎣ 2 ⎝ R3+R4 ⎠ ⎝
⎝ R1+R2 ⎟⎠ ⎥⎦
2 ⎠ ⎝ R1+R2 ⎠
⎡V
⎛ R4 ⎞ ⎛ VDIFF ⎞ ⎛ R1 ⎞
⎛ R2 ⎞ ⎤
+ADIFF ´ ⎢ DIFF ⎜
⎟⎠ − ⎜⎝ −
⎟⎠ ⎜⎝
⎟⎠ − VOUT ⎜⎝
⎟
⎝
R1+R2
R1+R2 ⎠ ⎥⎦
2
⎣ 2 R3+R4
⎡1
⎛ R2 ⎞
⎛ R2 ⎞ ⎤
VOUT = VOUT ⎢ A CM ´⎜
⎟⎠ − ADIFF ´⎜⎝
⎟
⎝
R1+R2
R1+R2 ⎠ ⎥⎦
⎣2
+
VDIFF
2
⎡1
⎡⎛ R4 ⎞ ⎛ R1 ⎞ ⎤
⎡⎛ R4 ⎞ ⎛ R1 ⎞ ⎤ ⎤
⎢ 2 A CM ´ ⎢⎜⎝ R3+R4 ⎟⎠ − ⎜⎝ R1+R2 ⎟⎠ ⎥ + ADIFF ´ ⎢⎜⎝ R3+R4 ⎟⎠ + ⎜⎝ R1+R2 ⎟⎠ ⎥ ⎥
⎣
⎦
⎣
⎦⎦
⎣
⎡1
⎛ R2 ⎞
⎛ R2 ⎞ ⎤
VOUT − VOUT ⎢ A CM ´⎜
⎟⎠ − ADIFF ´⎜⎝
⎟
⎝
R1+R2
R1+R2 ⎠ ⎥⎦
⎣2
=
VOUT
=
VDIFF
VDIFF ⎡ 1
⎡⎛ R4 ⎞ ⎛ R1 ⎞ ⎤
⎡⎛ R4 ⎞ ⎛ R1 ⎞ ⎤ ⎤
A CM ´ ⎢⎜
⎟⎠ − ⎜⎝
⎟⎠ ⎥ + ADIFF ´ ⎢⎜⎝
⎟ +⎜
⎟ ⎥⎥
⎢
⎝
R1+R2 ⎦
2 ⎣2
⎣ R3+R4
⎣ R3+R4 ⎠ ⎝ R1+R2 ⎠ ⎦ ⎦
⎡⎛ R4 ⎞ ⎛ R1 ⎞ ⎤
⎡⎛ R4 ⎞ ⎛ R1 ⎞ ⎤ ⎤
⎛ 1⎞ ⎡ 1
⎜⎝ ⎟⎠ ⎢ A CM ´ ⎢⎜⎝
⎟⎠ − ⎜⎝
⎟⎠ ⎥ + ADIFF ´ ⎢⎜⎝
⎟ +⎜
⎟ ⎥⎥
R1+R2 ⎦
2 ⎣2
⎣ R3+R4
⎣ R3+R4 ⎠ ⎝ R1+R2 ⎠ ⎦ ⎦
⎡⎛ 1
⎞ ⎛ R2 ⎞
⎛ R2 ⎞ ⎤
− ADIFF ´⎜
1− ⎢⎜ A CM ´ ⎟ ⎜
⎟
⎠ ⎝ R1+R2 ⎠
⎝ R1+R2 ⎟⎠ ⎥⎦
⎣⎝ 2
This equation can be simplified by noting the following:
R4 ⎞ ⎛ R1 ⎞ ⎤ ⎛ 1
⎞ ⎡⎛ R4 ⎞ ⎛ R1 ⎞ ⎤
( ADIFF ´ ) ⎡⎢⎛⎜⎝ R3+R4
⎟⎠ + ⎜⎝
⎟⎠ ⎥ >> ⎜⎝ A CM ´ ⎟⎠ ⎢⎜⎝
⎟ −⎜
⎟
R3+R4 ⎠ ⎝ R1+R2 ⎠ ⎥
R1+R2
2
⎣
⎦
⎣
Thus,
VOUT
VDIFF
12
⎡⎛ R4 ⎞ ⎛ R1 ⎞ ⎤
⎛ 1⎞
⎜⎝ ⎟⎠ ADIFF ´ ⎢⎜⎝
⎟ +⎜
⎟
2
R3+R4 ⎠ ⎝ R1+R2 ⎠ ⎥⎦
⎣
≈
⎡⎛ 1
⎞ ⎛ R2 ⎞
⎛ R2 ⎞ ⎤
1− ⎢⎜ A CM ´ ⎟ ⎜
⎟⎠ − ADIFF ´⎜⎝
⎟
⎝
⎠
⎝
R1+R2
R1+R2 ⎠ ⎥⎦
2
⎣
⎦
CMRR Derivation, Step 3: Calculate the Circuit’s CMRR (ADIFF/ACM)
⎡⎛ R4 ⎞ ⎛ R1 ⎞ ⎤
⎛ 1⎞
⎜⎝ ⎟⎠ ADIFF ´ ⎢⎜⎝
⎟ +⎜
⎟⎥
2
⎣ R3+R4 ⎠ ⎝ R1+R2 ⎠ ⎦
⎡⎛ 1
⎞ ⎛ R2 ⎞
⎛ R2 ⎞ ⎤
1− ⎢⎜ A CM ´ ⎟ ⎜
⎟⎠ − ADIFF ´⎜⎝
⎟
⎝
⎠
⎝
2
R1+R2
R1+R2 ⎠ ⎥⎦
⎣
CMRR TOTAL ≈
1
⎡⎛ R1 ⎞ ⎛ R4 ⎞ ⎤
⎡⎛ R4 ⎞ ⎛ R1 ⎞ ⎤
A CM ´ ⎢⎜
⎟⎠ + ⎜⎝
⎟⎠ ⎥ + ADIFF ´ ⎢⎜⎝
⎟ −⎜
⎟⎥
⎝
2
R3+R4 ⎦
⎣ R1+R2
⎣ R3+R4 ⎠ ⎝ R1+R2 ⎠ ⎦
⎡⎛ 1
⎞ ⎛ R2 ⎞
⎛ R1 ⎞ ⎤
1− ⎢⎜ A CM ´ ⎟ ⎜
− ( ADIFF ´ ) ⎜
⎟
⎠ ⎝ R1+R2 ⎠
⎝ R1+R2 ⎟⎠ ⎥⎦
⎣⎝ 2
⎡⎛ R4 ⎞ ⎛ R1 ⎞ ⎤
⎛ 1⎞
⎜⎝ ⎟⎠ ADIFF ´ ⎢⎜⎝
⎟ +⎜
⎟
2
R3+R4 ⎠ ⎝ R1+R2 ⎠ ⎥⎦
⎣
≈
1
⎡⎛ R1 ⎞ ⎛ R4 ⎞ ⎤
⎡⎛ R4 ⎞ ⎛ R1 ⎞ ⎤
A CM ´ ⎢⎜
+⎜
+ ADIFF ´ ⎢⎜
⎟
⎟
⎟ −⎜
⎟⎥
⎥
2
⎣⎝ R1+R2 ⎠ ⎝ R3+R4 ⎠ ⎦
⎣⎝ R3+R4 ⎠ ⎝ R1+R2 ⎠ ⎦
This is Equation 4, and we have already shown that Equation 4 can be reduced to Equation 2.
CMRR TOTAL ≈
⎛ 1⎞
⎜⎝ ⎟⎠ (1+G)
2
⎛
⎞ ⎛ 1⎞
1
⎜ CMRR
⎟ ⎜ ⎟ (1+G) + ΔR /RCMRR
⎝
amp ⎠ ⎝ 2 ⎠
Example of CMRR Calculation
Using Linear Technology’s LT1468 op amp and the LT5400A-3:
CMRR amp(1468) = 96dB
G≈
R1 R4 100k
≈
≈
≈ 10
R2 R3 10k
96dB ➔ (1/CMRRamp(1468)) = 0.00158%
ΔR/RCMRR = 0.005%
Using Equation 2:
CMRR TOTAL ≈
≈
(5.5)
⎛
⎞
1
⎜ CMRR
⎟ ( 5.5) + ( ΔR /RCMRR )
⎝
amp ⎠
5.5
(0.00005) + (0.0000869)
≈ 40175
CMRR TOTAL (dB) ≈ 20log(40175)
≈ 92.08dB
13
CASE 5: IDEAL DIFFERENTIAL AMPLIFIER (CMRRamp IS INFINITE)
CMRR Derivation, Step 1: Calculate the Circuit’s Common Mode Gain (ACM)
R2
R1
5V
VB
VA
VCM
– +
DIFFERENTIAL
AMP
+ –
VC
VD
VOUT
R3
R4
dn1023 F13
VOUT = VC − VD
⎛ 1⎞
⎜⎝ ⎟⎠ VOUT = VC = −VD
2
Where VOUT Is the Differential Output Voltage
Assuming Balanced Outputs
⎛ R4 ⎞
⎛ R3 ⎞
+ VD ⎜
VA = VCM ⎜
⎟
⎝ R3+R4 ⎠
⎝ R3+R4 ⎟⎠
⎛ R1 ⎞
⎛ R2 ⎞
+ VC ⎜
VB = VCM ⎜
⎟
⎝ R1+R2 ⎠
⎝ R1+R2 ⎟⎠
VA = VB
⎛ R4 ⎞
⎛ R3 ⎞
⎛ R1 ⎞
⎛ R2 ⎞
+ VD ⎜
= VCM ⎜
+ VC ⎜
VCM ⎜
⎟
⎟
⎟
⎝ R3+R4 ⎠
⎝ R3+R4 ⎠
⎝ R1+R2 ⎠
⎝ R1+R2 ⎟⎠
R1
⎛ R4 ⎞
⎛ 1⎞ R3
⎛ 1⎞ R2
+ ( −VOUT ) ⎜ ⎟
VCM ⎜
= VCM
+ VOUT ⎜ ⎟
⎟
⎝ R3+R4 ⎠
⎝ 2 ⎠ R3+R4
⎝ 2 ⎠ R1+R2
R1+R2
⎛ 1⎞ ⎛ R2 ⎞
⎛ 1⎞ ⎛ R3 ⎞
⎛ R4 ⎞
⎛ R1 ⎞
+V
=V
−V
VOUT ⎜ ⎟ ⎜
⎝ 2 ⎠ ⎝ R1+R2 ⎟⎠ OUT ⎜⎝ 2 ⎟⎠ ⎜⎝ R3+R4 ⎟⎠ CM ⎜⎝ R3+R4 ⎟⎠ CM ⎜⎝ R1+R2 ⎟⎠
R3 ⎞
R1 ⎞
⎛ 1⎞ ⎛ R2
⎛ R4
=V
VOUT ⎜ ⎟ ⎜
+
−
⎝ 2 ⎠ ⎝ R1+R2 R3+R4 ⎟⎠ CM ⎜⎝ R3+R4 R1+R2 ⎟⎠
R1 ⎞
⎛ R4
−
⎜⎝
⎟
VOUT
R3+R4 R1+R2 ⎠
=
R3 ⎞
VCM ⎛ 1⎞ ⎛ R2
+
⎜⎝ ⎟⎠ ⎜⎝
⎟
2 R1+R2 R3+R4 ⎠
14
Ideal Amplifier
CMRR Derivation, Step 2: Calculate the Circuit’s Differential Mode Gain (ADIFF)
R2
R1
5V
VB
VA
– +
DIFFERENTIAL
AMP
+ –
VC
VD
VOUT
R3
R4
–VDIFF
2
VOUT = VC − VD
⎛ 1⎞
⎜⎝ ⎟⎠ VOUT = VC = −VD
2
VA =
VDIFF
2
dn1023 F14
Where VOUT Is the Differential Output Voltage
Assuming Balanced Outputs
VDIFF ⎛ R4 ⎞
⎛ R3 ⎞
⎜⎝
⎟⎠ + VD ⎜⎝
⎟
R3+R4 ⎠
2 R3+R4
VB = −
VDIFF ⎛ R1 ⎞
⎛ R2 ⎞
⎜
⎟ +V ⎜
⎟
2 ⎝ R1+R2 ⎠ C ⎝ R1+R2 ⎠
VA = VB
Ideal Amplifier
VDIFF ⎛ R1 ⎞
VDIFF ⎛ R4 ⎞
⎛ 1⎞ ⎛ R3 ⎞
⎛ 1⎞ ⎛ R2 ⎞
⎜⎝
⎟⎠ + ( −VOUT ) ⎜⎝ ⎟⎠ ⎜⎝
⎟⎠ = −
⎜⎝
⎟⎠ + VOUT ⎜⎝ ⎟⎠ ⎜⎝
⎟
2 R3+R4
2 R1+R2 ⎠
2 R3+R4
2 R1+R2
⎛ 1⎞ ⎛ R2 ⎞
⎛ 1⎞ ⎛ R3 ⎞ VDIFF ⎛ R1 ⎞ VDIFF ⎛ R4 ⎞
+V
=
VOUT ⎜ ⎟ ⎜
⎜
⎟+
⎜
⎟
⎝ 2 ⎠ ⎝ R1+R2 ⎟⎠ OUT ⎜⎝ 2 ⎟⎠ ⎜⎝ R3+R4 ⎟⎠
2 ⎝ R1+R2 ⎠
2 ⎝ R3+R4 ⎠
R4 ⎞
⎛ 1⎞ ⎛ R1
+
VOUT ⎜⎝ 2 ⎟⎠ ⎜⎝ R1+R2 R3+R4 ⎟⎠
=
R3 ⎞
VDIFF ⎛ 1⎞ ⎛ R2
+
⎜⎝ ⎟⎠ ⎜⎝
⎟
2 R1+R2 R3+R4 ⎠
15
CMRR Derivation, Step 3: Calculate the Circuit’s CMRR (ADIFF/ACM)
R4 ⎞
⎛ 1⎞ ⎛ R1
+
⎜⎝ ⎟⎠ ⎜⎝
⎟
2 R1+R2 R3+R4 ⎠
R3 ⎞
⎛ 1⎞ ⎛ R2
+
⎜⎝ ⎟⎠ ⎜⎝
⎟
2 R1+R2 R3+R4 ⎠
CMRRR _ ONLY =
R1 ⎞
⎛ R4
−
⎜⎝
⎟
R3+R4 R1+R2 ⎠
R3 ⎞
⎛ 1⎞ ⎛ R2
+
⎜⎝ ⎟⎠ ⎜⎝
⎟
2 R1+R2 R3+R4 ⎠
R4 ⎞
⎛ 1⎞ ⎛ R1
+
⎜⎝ ⎟⎠ ⎜⎝
⎟
2 R1+R2 R3+R4 ⎠
CMRRR _ ONLY =
R1 ⎞
⎛ R4
−
⎜⎝
⎟
R3+R4 R1+R2 ⎠
This is Equation 3, and we have already shown that Equation 3 can be reduced to Equation 1.
⎛ 1⎞
⎜⎝ ⎟⎠ (G+1)
2
CMRRR _ ONLY ≈
ΔR /RCMRR
CASE 6: NON-IDEAL DIFFERENTIAL AMPLIFIER (CMRRamp IS FINITE)
CMRR Derivation, Step 1: Calculate the Circuit’s Common Mode Gain (ACM)
R2
R1
5V
VB
VA
VCM
VC
– +
DIFFERENTIAL
AMP
+ –
VD
VOUT
R3
R4
dn1023 F15
VOUT = VC − VD
Where VOUT Is the Differential Output Voltage
⎛ 1⎞
⎜⎝ ⎟⎠ VOUT = VC = −VD
2
Assuming Balanced Outputs
VCM ´= (VA + VB ) / 2
VDIFF ´= (VA − VB )
⎛ R4 ⎞
⎛ R3 ⎞
+V
VA = VCM ⎜
⎝ R3+R4 ⎟⎠ D ⎜⎝ R3+R4 ⎟⎠
⎛ R1 ⎞
⎛ R2 ⎞
+V
VB = VCM ⎜
⎝ R1+R2 ⎟⎠ C ⎜⎝ R1+R2 ⎟⎠
VOUT = (VCM ´•A CM ´)+(VDIFF ´•ADIFF ´)
16
VCM´ = Common Mode Voltage Seen at Inputs of Amp
VDIFF´ = Differential Voltage Seen at Inputs of Amp
Where ACM Is the Common Mode Input to Differential
Output Gain and ADM is the Differential Mode Input
to Differential Output Gain
ACM´ = Common Mode Gain of the Amp
ADIFF´ = Differential Mode Gain of the Amp
1
⎡
⎛ R4 ⎞ ⎛ VOUT ⎞ ⎛ R3 ⎞
⎛ R1 ⎞ VOUT ⎛ R2 ⎞ ⎤
+⎜−
+ VCM ⎜
+
VOUT = A CM ´ ⎢ VCM ⎜
⎟
⎜
⎟
⎜
⎟
⎟
⎝ R3+R4 ⎠ ⎝
⎝ R1+R2 ⎟⎠
2
2 ⎠ ⎝ R3+R4 ⎠
2 ⎝ R1+R2 ⎠ ⎥⎦
⎣
⎡
⎛ R4 ⎞ ⎛ VOUT ⎞ ⎛ R3 ⎞
⎛ R1 ⎞ VOUT ⎛ R2 ⎞ ⎤
+ADIFF ´ ⎢ VCM ⎜
⎟⎠ + ⎜⎝ −
⎟⎠ − VCM ⎜⎝
⎟−
⎜
⎟
⎟⎠ ⎜⎝
⎝
R3+R4
R1+R2 ⎠
R3+R4
2
2 ⎝ R1+R2 ⎠ ⎥⎦
⎣
⎡1
⎡⎛ R4 ⎞ ⎛ R1 ⎞ ⎤
⎡⎛ R4 ⎞ ⎛ R1 ⎞ ⎤ ⎤
+⎜
VOUT = VCM ⎢ A CM ´ ⎢⎜
+ ADIFF ´ ⎢⎜
⎟
⎟
⎟ −⎜
⎟ ⎥⎥
⎥
⎣⎝ R3+R4 ⎠ ⎝ R1+R2 ⎠ ⎦
⎣⎝ R3+R4 ⎠ ⎝ R1+R2 ⎠ ⎦ ⎦
⎣2
−
VOUT +
VOUT
2
⎡1
⎡⎛ R3 ⎞ ⎛ R2 ⎞ ⎤
⎡⎛ R3 ⎞ ⎛ R2 ⎞ ⎤ ⎤
⎢ 2 A CM ´ ⎢⎜⎝ R3+R4 ⎟⎠ − ⎜⎝ R1+R2 ⎟⎠ ⎥ + ADIFF ´ ⎢⎜⎝ R3+R4 ⎟⎠ + ⎜⎝ R1+R2 ⎟⎠ ⎥ ⎥
⎣
⎣
⎦
⎦⎦
⎣
VOUT ⎡ 1
⎡⎛ R3 ⎞ ⎛ R2 ⎞ ⎤
⎡⎛ R3 ⎞ ⎛ R2 ⎞ ⎤ ⎤
A CM ´ ⎢⎜
⎟⎠ − ⎜⎝
⎟⎠ ⎥ + ADIFF ´ ⎢⎜⎝
⎟ +⎜
⎟ ⎥⎥
⎢
⎝
R1+R2 ⎦
2 ⎣2
⎣ R3+R4
⎣ R3+R4 ⎠ ⎝ R1+R2 ⎠ ⎦ ⎦
⎡1
⎡⎛ R4 ⎞ ⎛ R1 ⎞ ⎤
⎡⎛ R4 ⎞ ⎛ R1 ⎞ ⎤ ⎤
= VCM ⎢ A CM ´ ⎢⎜
⎟⎠ + ⎜⎝
⎟⎠ ⎥ + ADIFF ´ ⎢⎜⎝
⎟ −⎜
⎟ ⎥⎥
⎝
R1+R2 ⎦
⎣ R3+R4
⎣ R3+R4 ⎠ ⎝ R1+R2 ⎠ ⎦ ⎦
⎣2
⎛ ⎛ 1⎞ ⎡ 1
⎡⎛ R3 ⎞ ⎛ R2 ⎞ ⎤
⎡⎛ R3 ⎞ ⎛ R2 ⎞ ⎤ ⎤⎞
−⎜
VOUT ⎜ 1+ ⎜ ⎟ ⎢ A CM ´ ⎢⎜
+ ADIFF ´ ⎢⎜
⎟
⎟
⎟ +⎜
⎟ ⎥ ⎥⎟
⎥
⎝ ⎝ 2⎠ ⎣ 2
⎣⎝ R3+R4 ⎠ ⎝ R1+R2 ⎠ ⎦
⎣⎝ R3+R4 ⎠ ⎝ R1+R2 ⎠ ⎦ ⎦⎠
⎡1
⎡⎛ R4 ⎞ ⎛ R1 ⎞ ⎤
⎡⎛ R4 ⎞ ⎛ R1 ⎞ ⎤ ⎤
= VCM ⎢ A CM ´ ⎢⎜
⎟⎠ + ⎜⎝
⎟⎠ ⎥ + ADIFF ´ ⎢⎜⎝
⎟ −⎜
⎟ ⎥⎥
⎝
R1+R2 ⎦
⎣ R3+R4
⎣ R3+R4 ⎠ ⎝ R1+R2 ⎠ ⎦ ⎦
⎣2
VOUT
VCM
1
⎡⎛ R4 ⎞ ⎛ R1 ⎞ ⎤
⎡⎛ R4 ⎞ ⎛ R1 ⎞ ⎤
A CM ´ ⎢⎜
+⎜
−
+ ADIFF ´ ⎢⎜
⎟
⎟
⎥
⎝ R3+R4 ⎠ ⎝ R1+R2 ⎠ ⎦
⎝ R3+R4 ⎟⎠ ⎜⎝ R1+R2 ⎟⎠ ⎥⎦
2
⎣
⎣
=
⎡⎛ R3 ⎞ ⎛ R2 ⎞ ⎤
⎡⎛ R3 ⎞ ⎛ R2 ⎞ ⎤ ⎤
⎛ 1⎞ ⎡ 1
1+ ⎜ ⎟ ⎢ A CM ´ ⎢⎜
⎟⎠ − ⎜⎝
⎟⎠ ⎥ + ADIFF ´ ⎢⎜⎝
⎟ +⎜
⎟ ⎥⎥
⎝ 2⎠ ⎣ 2
⎝
R1+R2 ⎦
⎣ R3+R4
⎣ R3+R4 ⎠ ⎝ R1+R2 ⎠ ⎦ ⎦
This equation can be simplified by noting the following:
R4 ⎞ ⎛ R1 ⎞ ⎤ ⎛ 1
⎞ ⎡⎛ R4 ⎞ ⎛ R1 ⎞ ⎤
( ADIFF ´ ) ⎡⎢⎛⎜⎝ R3+R4
⎟⎠ + ⎜⎝
⎟⎠ ⎥ >> ⎜⎝ A CM ´ ⎟⎠ ⎢⎜⎝
⎟ −⎜
⎟
R3+R4 ⎠ ⎝ R1+R2 ⎠ ⎥
R1+R2
2
⎣
⎦
⎣
⎦
Thus,
VOUT
VCM
1
⎡⎛ R4 ⎞ ⎛ R1 ⎞ ⎤
⎡⎛ R4 ⎞ ⎛ R1 ⎞ ⎤
A CM ´ ⎢⎜
+⎜
−
+ ADIFF ´ ⎢⎜
⎟
⎟
⎥
⎝ R3+R4 ⎠ ⎝ R1+R2 ⎠ ⎦
⎝ R3+R4 ⎟⎠ ⎜⎝ R1+R2 ⎟⎠ ⎥⎦
2
⎣
⎣
≈
⎡⎛ R3 ⎞ ⎛ R2 ⎞ ⎤
⎛ 1⎞
1+ ⎜ ⎟ ADIFF ´ ⎢⎜
⎟ +⎜
⎟⎥
⎝ 2⎠
⎣⎝ R3+R4 ⎠ ⎝ R1+R2 ⎠ ⎦
17
CMRR Derivation, Step 2: Calculate the Circuit’s Differential Mode Gain (ADIFF)
R2
R1
5V
VB
VA
VC
– +
DIFFERENTIAL
AMP
+ –
VD
VOUT
R3
R4
–VDIFF
2
VDIFF
2
dn1023 F16
VOUT = VC − VD
Where VOUT Is the Differential Output Voltage
⎛ 1⎞
⎜⎝ ⎟⎠ VOUT = VC = −VD
2
Assuming Balanced Outputs
VCM ´= (VA + VB ) / 2
VDIFF ´= (VA − VB )
VA =
VDIFF ⎛ R4 ⎞
⎛ R3 ⎞
⎜⎝
⎟⎠ + VD ⎜⎝
⎟
R3+R4 ⎠
2 R3+R4
VB = −
VDIFF ⎛ R1 ⎞
⎛ R2 ⎞
⎜⎝
⎟⎠ + VC ⎜⎝
⎟
R1+R2 ⎠
2 R1+R2
VOUT = (VCM ´•A CM ´)+(VDIFF ´•ADIFF ´)
VCM´ = Common Mode Voltage Seen at Inputs of Amp
VDIFF´ = Differential Voltage Seen at Inputs of Amp
Where ACM Is the Common Mode Input to Differential
Output Gain and ADM is the Differential Mode Input
to Differential Output Gain
ACM´ = Common Mode Gain of the Amp
ADIFF´ = Differential Mode Gain of the Amp
1
⎡V
⎛ R4 ⎞
⎛ 1⎞ ⎛ R3 ⎞ ⎛ VDIFF ⎞ ⎛ R1 ⎞
⎛ 1⎞ ⎛ R2 ⎞ ⎤
+ ( −VOUT ) ⎜ ⎟ ⎜
+⎜−
+ VOUT ⎜ ⎟ ⎜
VOUT = A CM ´ ⎢ DIFF ⎜
⎟
⎟
⎟
⎜
⎟
⎝ 2 ⎠ ⎝ R3+R4 ⎠ ⎝
⎝ 2 ⎠ ⎝ R1+R2 ⎟⎠ ⎥⎦
2
2 ⎠ ⎝ R1+R2 ⎠
⎣ 2 ⎝ R3+R4 ⎠
⎡V
⎛ R4 ⎞
⎛ 1⎞ ⎛ R3 ⎞ ⎛ VDIFF ⎞ ⎛ R1 ⎞
⎛ 1⎞ ⎛ R2 ⎞ ⎤
+ ( −VOUT ) ⎜ ⎟ ⎜
−⎜−
− VOUT ⎜ ⎟ ⎜
+ADIFF ´ ⎢ DIFF ⎜
⎟
⎟
⎟
⎜
⎟
⎝ 2 ⎠ ⎝ R3+R4 ⎠ ⎝
⎝ 2 ⎠ ⎝ R1+R2 ⎟⎠ ⎥⎦
2 ⎠ ⎝ R1+R2 ⎠
⎣ 2 ⎝ R3+R4 ⎠
VOUT =
VDIFF
2
⎡1
⎡⎛ R4 ⎞ ⎛ R1 ⎞ ⎤
⎡⎛ R4 ⎞ ⎛ R1 ⎞ ⎤ ⎤
⎢ 2 A CM ´ ⎢⎜⎝ R3+R4 ⎟⎠ − ⎜⎝ R1+R2 ⎟⎠ ⎥ + ADIFF ´ ⎢⎜⎝ R3+R4 ⎟⎠ + ⎜⎝ R1+R2 ⎟⎠ ⎥ ⎥
⎣
⎣
⎦
⎦⎦
⎣
⎡⎛ R3 ⎞ ⎛ R2 ⎞ ⎤
⎡⎛ R3 ⎞ ⎛ R2 ⎞ ⎤ ⎤
⎛ 1⎞ ⎡ 1
−⎜
+ ADIFF ´ ⎢⎜
−VOUT ⎜ ⎟ ⎢ A CM ´ ⎢⎜
⎟
⎟
⎟ +⎜
⎟ ⎥⎥
⎥
⎝ 2⎠ ⎣ 2
⎣⎝ R3+R4 ⎠ ⎝ R1+R2 ⎠ ⎦
⎣⎝ R3+R4 ⎠ ⎝ R1+R2 ⎠ ⎦ ⎦
⎡1
⎡⎛ R3 ⎞ ⎛ R2 ⎞ ⎤
⎡⎛ R3 ⎞ ⎛ R2 ⎞ ⎤ ⎤
⎛ 1⎞
VOUT + ⎜ ⎟ VOUT ⎢ A CM ´ ⎢⎜
⎟⎠ − ⎜⎝
⎟⎠ ⎥ + ADIFF ´ ⎢⎜⎝
⎟ +⎜
⎟ ⎥⎥
⎝ 2⎠
⎝
R1+R2 ⎦
⎣ R3+R4
⎣ R3+R4 ⎠ ⎝ R1+R2 ⎠ ⎦ ⎦
⎣2
=
18
VDIFF ⎡ 1
⎡⎛ R4 ⎞ ⎛ R1 ⎞ ⎤
⎡⎛ R4 ⎞ ⎛ R1 ⎞ ⎤ ⎤
A CM ´ ⎢⎜
⎟⎠ − ⎜⎝
⎟⎠ ⎥ + ADIFF ´ ⎢⎜⎝
⎟ +⎜
⎟ ⎥⎥
⎢
⎝
R1+R2 ⎦
2 ⎣2
⎣ R3+R4
⎣ R3+R4 ⎠ ⎝ R1+R2 ⎠ ⎦ ⎦
⎛ ⎛ 1⎞ ⎡ 1
⎡⎛ R3 ⎞ ⎛ R2 ⎞ ⎤
⎡⎛ R3 ⎞ ⎛ R2 ⎞ ⎤ ⎤⎞
−⎜
VOUT ⎜ 1+ ⎜ ⎟ ⎢ A CM ´ ⎢⎜
´
+
A
⎟
⎟
DIFF
⎢⎜⎝ R3+R4 ⎟⎠ + ⎜⎝ R1+R2 ⎟⎠ ⎥ ⎥⎟
⎥
⎝ ⎝ 2⎠ ⎣ 2
⎣⎝ R3+R4 ⎠ ⎝ R1+R2 ⎠ ⎦
⎣
⎦ ⎦⎠
=
VOUT
=
VDIFF
VDIFF
2
⎡1
⎡⎛ R4 ⎞ ⎛ R1 ⎞ ⎤
⎡⎛ R4 ⎞ ⎛ R1 ⎞ ⎤ ⎤
⎢ 2 A CM ´ ⎢⎜⎝ R3+R4 ⎟⎠ − ⎜⎝ R1+R2 ⎟⎠ ⎥ + ADIFF ´ ⎢⎜⎝ R3+R4 ⎟⎠ + ⎜⎝ R1+R2 ⎟⎠ ⎥ ⎥
⎣
⎣
⎦
⎦⎦
⎣
⎡⎛ R4 ⎞ ⎛ R1 ⎞ ⎤
⎡⎛ R4 ⎞ ⎛ R1 ⎞ ⎤ ⎤
⎛ 1⎞ ⎡ 1
⎜⎝ ⎟⎠ ⎢ A CM ´ ⎢⎜⎝
⎟⎠ − ⎜⎝
⎟⎠ ⎥ + ADIFF ´ ⎢⎜⎝
⎟ +⎜
⎟ ⎥⎥
R3+R4
R1+R2
2 ⎣2
⎣
⎣ R3+R4 ⎠ ⎝ R1+R2 ⎠ ⎦ ⎦
⎦
⎡⎛ R3 ⎞ ⎛ R2 ⎞ ⎤
⎡⎛ R3 ⎞ ⎛ R2 ⎞ ⎤ ⎤
⎛ 1⎞ ⎡ 1
1+ ⎜ ⎟ ⎢ A CM ´ ⎢⎜
⎟⎠ − ⎜⎝
⎟⎠ ⎥ + ADIFF ´ ⎢⎜⎝
⎟ +⎜
⎟ ⎥⎥
⎝ 2⎠ ⎣ 2
⎝
R3+R4
R1+R2
⎣
⎣ R3+R4 ⎠ ⎝ R1+R2 ⎠ ⎦ ⎦
⎦
This equation can be simplified by noting the following:
1
⎡⎛ R4 ⎞ ⎛ R1 ⎞ ⎤
⎡⎛ R4 ⎞ ⎛ R1 ⎞ ⎤
ADIFF ´ ⎢⎜
⎟⎠ + ⎜⎝
⎟⎠ ⎥ >> A CM ´ ⎢⎜⎝
⎟ −⎜
⎟⎥
⎝
R1+R2
R3+R4
2
⎣
⎣ R3+R4 ⎠ ⎝ R1+R2 ⎠ ⎦
⎦
and
1
⎡⎛ R3 ⎞ ⎛ R2 ⎞ ⎤
⎡⎛ R3 ⎞ ⎛ R2 ⎞ ⎤
+⎜
>> A CM ´ ⎢⎜
ADIFF ´ ⎢⎜
⎟
⎟
⎟ −⎜
⎟⎥
⎥
⎣⎝ R3+R4 ⎠ ⎝ R1+R2 ⎠ ⎦ 2
⎣⎝ R3+R4 ⎠ ⎝ R1+R2 ⎠ ⎦
VOUT
≈
VDIFF
⎡⎛ R4 ⎞ ⎛ R1 ⎞ ⎤ ⎤
⎛ 1⎞ ⎡
⎜⎝ ⎟⎠ ⎢ ADIFF ´ ⎢⎜⎝
⎟ +⎜
⎟ ⎥⎥
2 ⎣
⎣ R3+R4 ⎠ ⎝ R1+R2 ⎠ ⎦ ⎦
⎡⎛ R3 ⎞ ⎛ R2 ⎞ ⎤
⎛ 1⎞
1+ ⎜ ⎟ ADIFF ´ ⎢⎜
⎟ +⎜
⎟⎥
⎝ 2⎠
⎣⎝ R3+R4 ⎠ ⎝ R1+R2 ⎠ ⎦
CMRR Derivation, Step 3: Calculate the Circuit’s CMRR (ADIFF/ACM)
⎡⎛ R4 ⎞ ⎛ R1 ⎞ ⎤ ⎤
⎛ 1⎞ ⎡
⎜⎝ ⎟⎠ ⎢ ADIFF ´ ⎢⎜⎝
⎟ +⎜
⎟ ⎥⎥
2 ⎣
⎣ R3+R4 ⎠ ⎝ R1+R2 ⎠ ⎦ ⎦
⎡⎛ R3 ⎞ ⎛ R2 ⎞ ⎤
⎛ 1⎞
+
1+ ⎜ ⎟ ADIFF ´ ⎢⎜
⎝ 2⎠
⎝ R3+R4 ⎟⎠ ⎜⎝ R1+R2 ⎟⎠ ⎥⎦
⎣
CMRR TOTAL ≈
1
⎡⎛ R4 ⎞ ⎛ R1 ⎞ ⎤
⎡⎛ R4 ⎞ ⎛ R1 ⎞ ⎤
A CM ´ ⎢⎜
+⎜
+ ADIFF ´ ⎢⎜
⎟
⎟
⎟ −⎜
⎟⎥
⎥
2
⎣⎝ R3+R4 ⎠ ⎝ R1+R2 ⎠ ⎦
⎣⎝ R3+R4 ⎠ ⎝ R1+R2 ⎠ ⎦
⎡⎛ R3 ⎞ ⎛ R2 ⎞ ⎤
⎛ 1⎞
1+ ⎜ ⎟ ADIFF ´ ⎢⎜
⎟ +⎜
⎟⎥
⎝ 2⎠
⎣⎝ R3+R4 ⎠ ⎝ R1+R2 ⎠ ⎦
≈
⎡⎛ R4 ⎞ ⎛ R1 ⎞ ⎤ ⎤
⎛ 1⎞ ⎡
⎜⎝ ⎟⎠ ⎢ ADIFF ´ ⎢⎜⎝
⎟ +⎜
⎟ ⎥⎥
2 ⎣
⎣ R3+R4 ⎠ ⎝ R1+R2 ⎠ ⎦ ⎦
1
⎡⎛ R1 ⎞ ⎛ R4 ⎞ ⎤
⎡⎛ R4 ⎞ ⎛ R1 ⎞ ⎤
A CM ´ ⎢⎜
⎟⎠ + ⎜⎝
⎟⎠ ⎥ + ADIFF ´ ⎢⎜⎝
⎟ −⎜
⎟⎥
⎝
R3+R4 ⎦
2
⎣ R1+R2
⎣ R3+R4 ⎠ ⎝ R1+R2 ⎠ ⎦
This is Equation 4, and we have already shown that Equation 4 can be reduced to Equation 2.
19
CMRR TOTAL ≈
⎛ 1⎞
⎜⎝ ⎟⎠ (1+G)
2
⎛
⎞ ⎛ 1⎞
1
⎜ CMRR
⎟ ⎜ ⎟ (1+G) + ΔR /RCMRR
⎝
amp ⎠ ⎝ 2 ⎠
Example of CMRR Calculation
Using Linear Technology’s LTC6362 Differential Amplifier and the LT5400A-6:
CMRR amp(6362) = 70dB
G≈
R1 R4 5k
≈
≈ ≈5
R2 R3 1k
70dB ➔ (1/CMRRamp(6362))=0.0316%
ΔR/RCMRR = 0.005%
Using Equation 2:
CMRR TOTAL ≈
≈
( 3)
⎛
⎞
1
⎜ CMRR
⎟ ( 3) + ( ΔR /RCMRR )
⎝
amp ⎠
3
(0.00005) + (3 • 0.000316)
≈ 3006
CMRR TOTAL (dB) ≈ 20log(3006)
≈ 69.56dB
Data Sheet Download
For applications help,
call (408) 432-1900, Ext. 2020
www.linear.com
20
Linear Technology Corporation
dn1023f LT 0312 • PRINTED IN THE USA
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