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Design Note 1023: Precision Matched Resistors Automatically Improve Differential Amplifier CMRR – Here’s How
INTRODUCTION
Matched resistors are critical to the performance of a large class of differential circuits such as the following:
Instrumentation Amp
Difference Amp
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:
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/RCMRRM
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:
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):
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)
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 amplifi er. The total CMRR of any of the differential circuits shown above can be calculated using the following formula:
Equation 2
Where CMRRampTOTAL 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.
CASE 1: IDEAL INSTRUMENTATION AMP (CMRRamp IS INFINITE)
CMRR Derivation, Step 1: Calculate the Circuit’s Common Mode Gain (ACM)
Ideal Instrumentation Amplifier, Gamp = Amplifier Gain
CMRR Derivation, Step 2: Calculate the Circuit’s Differential Mode Gain (ADIFF
Ideal Instrumentation Amplifier, Gamp = Amplifier Gain
CMRR Derivation, Step 3: Calculate the Circuit’s CMRR (ADIFF/ACM)
Equation 3
This Preliminary Result Will Be Referenced in Other Calculations
This equation can be simplified by noting the following:
Thus,
CASE 2: NON-IDEAL INSTRUMENTATION AMP (CMRRamp IS FINITE)
CMRR Derivation, Step 1: Calculate the Circuit’s Common Mode Gain (ACM)
VCM = Common Mode Voltage Seen at Inputs of Amp
VDIFF = Differential Voltage Seen at Inputs of Amp
ACM = Common Mode Gain of the Amp
ADIFF = Differential Mode Gain of the Amp
CMRR Derivation, Step 2: Calculate the Circuit’s Differential Mode Gain (ADIFF)
VCM = Common Mode Voltage Seen at Inputs of Amp
VDIFF = Differential Voltage Seen at Inputs of Amp
ACM = Common Mode Gain of the Amp
ADIFF = Differential Mode Gain of the Amp
This equation can be simplified by noting the following:
Thus,
CMRR Derivation, Step 3: Calculate the Circuit’s CMRR (ADIFF/ACM)
Equation 4
This Preliminary Result Will Be Referenced in Other Calculations
This equation can be simplified by noting the following:
Thus,
Example of CMRR Calculation
Using Linear Technology’s LTC2053 Instrument Amplifier, configured for a gain of 1, and the LT5400A-1
Using Equation 2:
CASE 3: DIFFERENCE AMP USING AN IDEAL OP AMP (CMRRamp IS INFINITE)
CMRR Derivation, Step 1: Calculate the Circuit’s Common Mode Gain (ACM)
Ideal Op Amp
CMRR Derivation, Step 2: Calculate the Circuit’s Differential Mode Gain (ADIFF)
Ideal Op Amp
CMRR Derivation, Step 3: Calculate the Circuit’s CMRR (ADIFF/ACM)
This is Equation 3, and we have already shown that Equation 3 can be reduced to Equation 1.
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)
VCM = Common Mode Voltage Seen at Inputs of Amp VDIFF = Differential Voltage Seen at Inputs of Amp
ACM = Common Mode Gain of the Amp
ADIFF = Differential Mode Gain of the Amp
CMRR Derivation, Step 2: Calculate the Circuit’s Differential Mode Gain (ADIFF)
VCM = Common Mode Voltage Seen at Inputs of Amp VDIFF = Differential Voltage Seen at Inputs of Amp
ACM = Common Mode Gain of the Amp
ADIFF = Differential Mode Gain of the Amp
This equation can be simplified by noting the following:
Thus,
CMRR Derivation, Step 3: Calculate the Circuit’s CMRR (ADIFF/ACM)
This is Equation 4, and we have already shown that Equation 4 can be reduced to Equation 2.
Example of CMRR Calculation
Using Linear Technology’s LT1468 op amp and the LT5400A-3:
Using Equation 2:
CASE 5: IDEAL DIFFERENTIAL AMPLIFIER (CMRRamp IS INFINITE)
CMRR Derivation, Step 1: Calculate the Circuit’s Common Mode Gain (ACM)
Where VOUT Is the Differential Output Voltage Assuming Balanced Outputs
Ideal Amplifier
CMRR Derivation, Step 2: Calculate the Circuit’s Differential Mode Gain (ADIFF)
Where VOUT Is the Differential Output Voltage Assuming Balanced Outputs
Ideal Amplifier
CMRR Derivation, Step 3: Calculate the Circuit’s CMRR (ADIFF/ACM)
This is Equation 3, and we have already shown that Equation 3 can be reduced to Equation 1.
CASE 6: NON-IDEAL DIFFERENTIAL AMPLIFIER (CMRRamp IS FINITE)
CMRR Derivation, Step 1: Calculate the Circuit’s Common Mode Gain (ACM)
Where VOUT Is the Differential Output Voltage
Assuming Balanced Outputs
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
This equation can be simplified by noting the following:
Thus,
CMRR Derivation, Step 2: Calculate the Circuit’s Differential Mode Gain (ADIFF)
Where VOUT Is the Differential Output Voltage
Assuming Balanced Outputs
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
This equation can be simplified by noting the following:
CMRR Derivation, Step 3: Calculate the Circuit’s CMRR (ADIFF/ACM)
This is Equation 4, and we have already shown that Equation 4 can be reduced to Equation 2.
Example of CMRR Calculation
Using Linear Technology’s LTC6362 Differential Amplifier and the LT5400A-6:
Using Equation 2:
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