SBOA582 November 2023 OPA2387 , OPA387 , OPA4387 , RES11A , RES11A-Q1

2 Common-Mode Rejection Ratio in Difference Amplifier Circuits

Difference amplifiers, often referred to as diff-amps, are designed to convert a differential input voltage into a single-ended output voltage as a real world implementation of Equation 1. Figure 2-1 illustrates a typical difference amplifier circuit consisting of four standard resistors and an operational amplifier.

Figure 2-1 Difference Amplifier Circuit

Diff-amp circuits have a differential gain (A_D) which amplifies or attenuates the differential signal voltage, and a common-mode gain (A_CM) which amplifies or attenuates the common-mode voltage. Common-mode rejection ratio (CMRR) is defined as the ratio of differential gain to common-mode gain of the amplifier stage.

Equation 3.

C M R R = \frac{A_{D}}{A_{C M}}

Where,

A_D is the differential gain of the amplifier stage in V/V
A_CM is the common-mode gain of the amplifier stage in V/V

Often, CMRR is expressed in decibels (dB) as defined by Equation 4.

Equation 4.

{C M R R}_{d B} = 20 ∙ {L o g}_{10} (\frac{A_{D}}{A_{C M}})

Diff-amps are designed to have high CMRR to reject noise and other errors from the signal chain. The effective CMRR of the gain stage is determined by non-idealities of the discrete components that make up the difference amplifier circuit. The operational amplifier and the resistor network both have CMRR metrics which contribute to the overall CMRR of the diff-amp stage, as detailed in the following.

Operational amplifiers have a CMRR specification that can be found in the amplifier data sheet. For example, OPA387 is an ultra-high precision, zero-drift amplifier with very high CMRR. The electrical characteristics table of the data sheet specifies a CMRR of 150 dB typical when operating on a 5.5-V supply.

Table 2-1 OPA387 Electrical Characteristics: Common-Mode Rejection Ratio

Parameter		Test Conditions		MIN	TYP	UNIT
CMRR	Common-mode rejection ratio	(V–) – 0.1 V < V_CM < (V+), V_S = 1.7 V		115	138	dB
		(V–) – 0.2 V < V_CM < (V+) + 0.1 V, V_S = 5.5 V	OPA387, OPA2387	140	150
		(V–) – 0.2 V < V_CM < (V+) + 0.1 V, V_S = 5.5 V	OPA4387	130
		(V–) – 0.1 V < V_CM < (V+), T_A = –40°C to +125°C		110	132
		(V–) – 0.2 V < V_CM < (V+) + 0.1, V_S = 5.5 V, T_A = –40°C to +125°C		130

From the typical characteristics section of the OPA387 data sheet, Figure 2-2 shows that the maximum CMRR of the op-amp occurs at DC and low frequencies. As the amplifier's open-loop gain (AOL) decreases over frequency, the CMRR decreases along at a rate of 20 dB per decade. This reduction in CMRR over frequency occurs because the amplifier relies on the high open-loop gain to reject the common-mode error.

Figure 2-2 PSRR and CMRR vs Frequency, OPA387

The resistive components of the diff-amp shown in Figure 2-1 R₁, R₂, R₃, R₄, also contribute to the total CMRR of the diff-amp stage. Practical resistors have a tolerance specification, often described as a percentage, which indicates the maximum deviation between the absolute resistance and the nominal resistance. For example, a discrete resistor with a nominal resistance of 1 kΩ and a tolerance of ±0.5% can have an absolute resistance between 995 Ω and 1005 Ω. This relationship is described by Equation 5.

Equation 5.

R_{a b s o l u t e} = R_{n o m i n a l} (1 + t)

Where t is the absolute tolerance of the resistor in Ω/Ω.

The industry standard is to specify resistor tolerance as a percentage which must be converted into Ω/Ω to be used in the analysis. The percent tolerance t_%, is converted to the absolute tolerance t, by a division of 100 as in Equation 6. The analysis in this document considers all tolerance metrics in Ω/Ω, even when specified as a percentage.

Equation 6.

t = \frac{t_{%}}{100}

Where,

t is the absolute tolerance in Ω/Ω
t_% is the absolute tolerance in %

When the OPA387 is configured as a diff-amp using these 0.5% resistors as shown in Figure 2-3, the resulting CMRR of the diff-amp stage (CMRR_D) is much lower than the 150 dB op-amp specification (CMRR_A), with a worst-case CMRR_D of only 40 dB.

Figure 2-3 Difference Amplifier with 0.5% Tolerance, 1-kΩ Resistors

The degradation of CMRR_D is caused by the deviation in the absolute resistor values due to the resistor tolerance, which produces mismatches between resistor ratios R₂/R₁ and R₄/R₃. The ratiometric mismatch reduces the effective CMRR of the discrete resistor network (CMRR_R) which dominates the common-mode performance of the diff-amp stage. This occurs because the difference between the two resistor ratios causes a portion of the common-mode voltage to present as a differential voltage at the op-amp's input terminals which is amplified by the differential gain of the circuit.

The worst-case CMRR_R of a diff-amp using four discrete resistors with tolerance t, is given by Equation 7. The detailed derivations for Equation 7 can be found in Section 5.

Equation 7.

{C M R R}_{R} = \frac{G + 1}{4 t}

Where,

G is the nominal differential gain in V/V
t is the absolute tolerance of the resistors in Ω/Ω

The total CMRR of the diff-amp stage is the parallel combination of the amplifier CMRR and the resistor CMRR, as defined in Equation 8.

Equation 8.

\frac{1}{{C M R R}_{D}} = \frac{1}{{C M R R}_{A}} + \frac{1}{{C M R R}_{R}}

Where,

CMRR_D is the common-mode rejection ratio of the diff-amp stage in V/V
CMRR_A is the common-mode rejection ratio of the amplifier in V/V
CMRR_R is the common-mode rejection ratio of the resistor network in V/V

As shown in Figure 2-4, the CMRR_D performance is dominated by CMRR_R. As the AOL of the amplifier decreases over frequency, CMRR_A begins to contribute to the overall CMRR. At high frequencies the common-mode performance is dominated by CMRR_A.

Figure 2-4 Resistor and Op-Amp Contributions to Effective CMRR of Difference Amplifier
OPA387 with 0.5% Tolerance Resistors