SLPS785 Data sheet

SLPS785 December 2023 RES11A

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8.1.1.1 Difference-Amplifier Common-Mode Rejection Analysis

In this simple difference amplifier configuration, the nominal CMRR is calculated as:

Equation 14.

CMRR = 20 \times \log_{10} (|\frac{A_{D}}{A_{CM}}|)

The term A_D is the differential gain of the circuit, and the term A_CM is the common-mode gain of the circuit. These are defined as the following:

Equation 15.

A_{D} \times = \frac{V_{OUT}}{V_{D}} = 0.5 \times \frac{(\frac{R_{G1}}{R_{G1} + R_{IN1}}) + (\frac{R_{G2}}{R_{G2} + R_{IN2}})}{(\frac{R_{IN2}}{R_{G2} + R_{IN2}})}

Equation 16.

A_{C M} = \frac{V_{OUT}}{V_{CM}} = \frac{(\frac{R_{G1}}{R_{G1} + R_{IN1}}) - (\frac{R_{G2}}{R_{G2} + R_{IN2}})}{(\frac{R_{IN2}}{R_{G2} + R_{IN2}})}

Therefore,

Equation 17.

CMRR = 20 \times \log_{10} (|2\times \frac{R_{G1} \times (R_{IN2} + R_{G2}) - R_{G2} \times (R_{IN1} + R_{G1})}{R_{G1} \times (R_{IN2} + R_{G2}) + R_{G2} \times (R_{IN1} + R_{G1})}|)

When this expression is evaluated with the definitions given in Section 7.3.1, assuming the worst-case scenario of the most unbalanced divider matching possible,

Equation 18.

CMRR = 20 \times \log_{10} (|\frac{G_{nom} +1+ {t_{Rx}}^{2} (1- G_{nom})}{4\times t_{Rx}}|)

Because t_Rx² << 1, the worst-case CMRR is approximated as

Equation 19.

CMRR = 20 \times \log_{10} (|\frac{G_{nom} +1}{4\times t_{Rx}}|)

By definition, the parameter t_M describes the effective error that is otherwise equivalent to 4 × t_x for an unmatched divider network, and so the maximum value of t_M can be used to calculate the same worst-case result. Likewise, the typical value of t_M can be used to approximate the typical CMRR.

Equation 20.

CMRR = 20 \times \log_{10} (|\frac{G_{nom} +1}{t_{M}}|)

For example, the worst-case CMRR for a RES11A40 device with G = 4 is approximately 74.0 dB, with a typical CMRR of approximately 95.4 dB. In comparison, implementation of a comparable G = 4 difference amplifier with unmatched 0.1%-tolerance resistors results in a worst-case CMRR of approximately 62 dB.

In a difference amplifier configuration, the CMRR of the op amp contributes error as well. The op-amp CMRR is considered in parallel with the CMRR of the resistor network, as per the following equation:

Equation 21.

\frac{1}{{CMRR}_{TOTAL}} = \frac{1}{{CMRR}_{AMP}} + \frac{1}{{CMRR}_{RESISTORS}}

Additional mismatches in the divider end-to-end resistances reduce the effective CMRR of a difference amplifier. While the low absolute tolerance span of the RES11A (235 ppm typical) helps reduce these concerns, parasitic trace resistances can lead to additional mismatches that impact the CMRR specs. Bench results from a difference amplifier implementation of the RES11A40 and the OPA210 are presented for various deliberate input-impedance mismatches.

Figure 8-2 Effect of Input Impedance Mismatch on Common-mode Rejection Ratio