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

5 Derive CMRR for Discrete Resistor Tolerance

A common assumption in difference amplifier circuits is that the ratio of R₄ and R₃ is equal to the ratio of R₂ and R₁, as described by Equation 23.

Equation 23.

\frac{R_{4}}{R_{3}} = \frac{R_{2}}{R_{1}} = \frac{R_{g}}{R_{i n}}

This assumption is useful because it allows the differential gain equation to be reduced to the form in Equation 24. This is the simplified gain equation for a difference amplifier circuit.

Equation 24.

A_{D} = \frac{R_{g}}{R_{i n}}

Combining Equation 23 with Equation 20 shows that if the resistor ratios are perfectly matched, the common-mode gain (A_CM) is 0 V/V, and therefore the common-mode rejection ratio of the resistor network (CMRR_R) is infinite.

Equation 25.

A_{C M} = \frac{(\frac{R_{g}}{R_{i n} + R_{g}}) - (\frac{R_{g}}{R_{i n} + R_{g}})}{(\frac{R_{i n}}{R_{i n} + R_{g}})} = 0 V / V

In practice, the variation in absolute resistance due to resistor tolerance produces mismatches between the absolute ratios of R₄/R₃ and R₂/R₁. The ratio mismatch presents an asymmetrical resistor divider effect at the amplifier's input terminals. Any common-mode input voltage is attenuated unequally between the two resistor dividers and presents as a small differential voltage which is amplified by the differential gain of the circuit, thus degrading the CMRR performance of the differential stage.

Considering a differential amplifier circuit consisting of four discrete resistors with tolerance t, the worst-case ratio matching occurs when the absolute resistor values differ from the nominal resistor values as shown in Figure 5-1 in which,

Equation 26.

R_{1} = R_{1 N} (1 + t)

Equation 27.

R_{2} = R_{2 N} (1 - t)

Equation 28.

R_{3} = R_{3 N} (1 - t)

Equation 29.

R_{4} = R_{4 N} (1 + t)

Where,

R_XN is the nominal resistance of resistor R_X in Ω
t is the absolute tolerance of the resistor in Ω/Ω

Figure 5-1 Worst-Case Resistor Matching of Difference Amplifier Circuit

The nominal resistor ratios determine the nominal gain of the difference amplifier stage.

Equation 30.

\frac{R_{4 N}}{R_{3 N}} = \frac{R_{2 N}}{R_{1 N}} = G

Where,

R_XN is the nominal resistance of resistor R_X in Ω
G is the nominal differential gain of the amplifier stage in V/V

The contribution of resistor tolerance to the overall common-mode rejection ratio of the difference amplifier can be determined in the worst-case by combining Equation 26 through Equation 29 with Equation 22.

Equation 31.

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

Applying the relationship defined in Equation 30,

Equation 32.

{C M R R}_{R} = \frac{1}{2} \frac{2 {G^{2} R}_{1 N} R_{3 N} (1 - t^{2}) + G R_{1 N} R_{3 N} {(1 + t)}^{2} + G R_{1 N} R_{3 N} {(1 - t)}^{2}}{G R_{1 N} R_{3 N} {(1 + t)}^{2} - G R_{1 N} R_{3 N} {(1 - t)}^{2}}

Which reduces to

Equation 33.

{C M R R}_{R} = \frac{G + 1 + t^{2} (1 - G)}{4 t}

Standard resistor tolerances are very small, typically 1% or less, therefore it is common to further simplify the equation for t << 1. The contribution of discrete resistor tolerance to the overall CMRR of the difference amplifier for t << 1 is defined by Equation 34.

Equation 34.

{C M R R}_{R} \approx \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