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

6 Derive CMRR for Matched Ratio Tolerance

As derived in Section 4, the CMRR of a difference amplifier stage is dominated by the mismatch between the two resistor divider ratios. The absolute accuracy of the four resistor values does not directly contribute to the CMRR performance. The RES11A-Q1 consists of two precision thin-film resistor dividers: R_G1/R_IN1 and R_G2/R_IN2. The ratio tolerance t_D of each resistor divider is specified in the RES11A-Q1 data sheet and is defined by the following relationships.

Equation 35.

\frac{R_{G 1}}{R_{I N 1}} = G (1 + t_{D 1})

Equation 36.

\frac{R_{G 2}}{R_{I N 2}} = G (1 + t_{D 2})

Where,

G is the nominal gain ratio in V/V
t_D1 is the ratio tolerance of divider 1 in Ω/Ω
t_D2 is the ratio tolerance of divider 2 in Ω/Ω

From the previous equations, it is also shown that:

Equation 37.

R_{G 1} = G (1 + t_{D 1}) R_{I N 1}

Equation 38.

R_{G 2} = G (1 + t_{D 2}) R_{I N 2}

Figure 6-1 shows the RES11A-Q1 in a difference amplifier configuration. From the analysis in Section 4, the common-mode rejection ratio of the resistor network is defined by Equation 39.

Equation 39.

{C M R R}_{R} = \frac{1}{2} \frac{{{2 R}_{G 1} R}_{G 2} + R_{I N 1} R_{G 2} + R_{G 1} R_{I N 2}}{{R_{I N 1} R}_{G 2} - R_{G 1} R_{I N 2}}

Figure 6-1 RES11A-Q1 Difference Amplifier Circuit

The effect of the ratio tolerance is considered by substituting the relationships from Equation 37 and Equation 38 into Equation 39.

Equation 40.

{C M R R}_{R} = \frac{1}{2} \frac{2 G^{2} {R_{I N 1} R}_{I N 2} (1 + t_{D 1}) (1 + t_{D 2}) + G {R_{I N 1} R}_{I N 2} (1 + t_{D 2}) + G {R_{I N 1} R}_{I N 2} (1 + t_{D 1})}{G {R_{I N 1} R}_{I N 2} (1 + t_{D 2}) - G {R_{I N 1} R}_{I N 2} (1 + t_{D 1})}

Which reduces to Equation 41.

Equation 41.

{C M R R}_{R} = \frac{G + 1 + G (t_{D 1} + t_{D 2} + t_{D 1} t_{D 2}) + t_{D 1} + t_{D 2}}{t_{D 2} - t_{D 1}}

The RES11A-Q1 tolerances t_D1 and t_D2 are very small, with a maximum tolerance of ±0.05%. Therefore, it is consistent with Equation 34 for standard resistors to further simplify the equation for t_D << 1, resulting in Equation 42.

Equation 42.

{CMRR}_{R} \approx \frac{G + 1}{t_{D 2} - t_{D 1}}

The RES11A-Q1 data sheet specifies the matched ratio tolerance t_m, as the difference between the absolute ratio tolerances of the two resistor dividers, as defined in Equation 43. The matched ratio tolerance specification describes the maximum and typical mismatch between the two resistor divider ratios.

Equation 43.

t_{m} = t_{D 2} - t_{D 1}

Therefore, the simplified equation for the common-mode rejection ratio of the RES11A-Q1 is expressed by Equation 44. This equation can be used with data sheet specifications to directly calculate the minimum and typical CMRR_R of the RES11A-Q1 precision matched resistor divider pair.

Equation 44.

{C M R R}_{R} = \frac{G + 1}{t_{m}}

Where,

G is the nominal differential gain in V/V
t_m is the matched ratio tolerance between resistor dividers R_G1/R_IN1 and R_G2/R_IN2 in Ω/Ω