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

4 Derive Differential and Common-Mode Gain, Difference Amplifier

This section details the step-by-step derivations for differential and common-mode gain of a difference amplifier circuit. These gain equations are used to determine the CMRR of a difference amplifier as a function of the absolute resistance of the resistor network. The resulting relationships are used in the following sections to derive the simplified CMRR equations for both discrete resistor tolerance t, and matched ratio tolerance t_m.

Figure 4-1 illustrates a typical difference amplifier circuit. Assuming an ideal op-amp, Kirchoff's Current Law (KCL) and Kirchoff's Voltage Law (KVL) can be applied to determine the transfer function. The ideal op-amp assumptions are that the voltage at the inverting input (V_N) is equal to the voltage at the non-inverting input (V_P), and there is zero current flowing through the input terminals.

Figure 4-1 Difference Amplifier Circuit with Ideal Op-Amp

Analysis of Figure 4-1 using KVL, KCL, and ideal op-amp assumptions produces the following equations.

Equation 10.

V_{N} = V_{P}

Equation 11.

V_{P} = (\frac{R_{4}}{R_{3} + R_{4}}) V_{S i g+}

Equation 12.

V_{O U T} = V_{N} - I_{N} R_{2}

Equation 13.

I_{N} = \frac{V_{S i g-} - V_{N}}{R_{1}}

Combining the previous equations results in Equation 14.

Equation 14.

V_{O U T} = (\frac{R_{4}}{R_{3} + R_{4}}) (1 + \frac{R_{2}}{R_{1}}) V_{S i g+} - (\frac{R_{2}}{R_{1}}) V_{S i g-}

Some trivial algebra rewrites Equation 14 in a more intuitive form, Equation 15. Note that in this form, Equation 15 is represented as a combination of resistor dividers.

Equation 15.

V_{O U T} = \frac{(\frac{R_{4}}{R_{3} + R_{4}}) V_{S i g+} - (\frac{R_{2}}{R_{1} + R_{2}}) V_{S i g-}}{(\frac{R_{1}}{R_{1} + R_{2}})}

At this point in the analysis, it is useful to consider the differential and common-mode voltage components of the input signal. In Figure 4-2, the difference amplifier circuit is redrawn to show the input voltage as a combination of differential and common-mode voltage sources. This shows that V_Sig+ and V_Sig- each consist of half of the input differential voltage, of opposite polarity, referenced to the input common-mode voltage, as expressed by Equation 16 and Equation 17.

Equation 16.

V_{S i g+} = V_{C M} + \frac{V_{D i f f}}{2}

Equation 17.

V_{S i g-} = V_{C M} - \frac{V_{D i f f}}{2}

Figure 4-2 Difference Amplifier With Differential and Common-Mode Input Signals

Rewriting Equation 15 considering the differential and common-mode components of the input signal produces Equation 18.

Equation 18.

V_{O U T} = \frac{(\frac{R_{4}}{R_{3} + R_{4}}) (V_{C M} + \frac{V_{D i f f}}{2}) - (\frac{R_{2}}{R_{1} + R_{2}}) (V_{C M} - \frac{V_{D i f f}}{2})}{(\frac{R_{1}}{R_{1} + R_{2}})}

Superposition allows the differential and common-mode voltage components to be considered independently, as shown in Figure 4-3 and Figure 4-4. Applying superposition to Equation 18 produces the differential-mode transfer function as defined by Equation 19 and the common-mode transfer function as defined by Equation 20.

Equation 19.

A_{D} = \frac{V_{O U T}}{V_{D i f f}} = \frac{1}{2} \frac{(\frac{R_{4}}{R_{3} + R_{4}}) + (\frac{R_{2}}{R_{1} + R_{2}})}{(\frac{R_{1}}{R_{1} + R_{2}})}

Equation 20.

A_{C M} = \frac{V_{O U T}}{V_{C M}} = \frac{(\frac{R_{4}}{R_{3} + R_{4}}) - (\frac{R_{2}}{R_{1} + R_{2}})}{(\frac{R_{1}}{R_{1} + R_{2}})}

Figure 4-3 Superposition: Differential-Mode

Figure 4-4 Superposition: Common-Mode

Combining the differential and common-mode gain equations with the definition of CMRR from Equation 3, results in Equation 21.

Equation 21.

C M R R = \frac{1}{2} \frac{(\frac{R_{4}}{R_{3} + R_{4}}) + (\frac{R_{2}}{R_{1} + R_{2}})}{(\frac{R_{4}}{R_{3} + R_{4}}) - (\frac{R_{2}}{R_{1} + R_{2}})}

In this form, it is clear that the CMRR of the resistor network is determined by the difference between the two resistor dividers R₂/R₁ and R₄/R₃. The CMRR equation can also be expressed in the form below, which is used for the analysis in Section 5 and Section 6.

Equation 22.

C M R R = \frac{1}{2} \frac{2 R_{2} R_{4} + R_{1} R_{4} + R_{2} R_{3}}{R_{1} R_{4} - R_{2} R_{3}}