SLPS755 Data sheet

SLPS755B October 2023 – October 2025 RES11A-Q1

PRODUCTION DATA

8.1.3.2 Difference-Amplifier Gain Error Analysis

The transfer function Equation 41 assumes that R_G1 = R_G2 and R_IN1 = R_IN2. Without this assumption, the transfer function of the difference amplifier is better described by the following:

Equation 50.

V_{OUT} = V_{IN+} \times (\frac{R_{G1}}{R_{G1} + R_{IN1}}) (\frac{R_{G2} + R_{IN2}}{R_{IN2}}) - V_{IN–} \times (\frac{R_{G2}}{R_{IN2}}) + V_{REF}

If the end-to-end values of R_G2 + R_IN2 and R_G1 + R_IN1 are sufficiently matched, the correspond terms cancel out in the above equation. The end-to-end mismatch specification of the RES11A-Q1 describes the typical error of this in ratiometric terms; for brevity, this error term is denoted as t_E2E.

Equation 51.

\frac{R_{G2} + R_{IN2}}{R_{G1} + R_{IN1}} = 1 + t_{E2E}

Equation 52.

V_{OUT} = V_{IN+} \times (\frac{R_{G1}}{R_{IN2}}) (1 + t_{E2E}) - V_{IN–} \times (\frac{R_{G2}}{R_{IN2}}) + V_{REF}

The ratio error of R_G2 / R_IN2 is described by t_D2. The ratio error of R_G1 / R_IN2 is described by the R_G mismatch between dividers, ratiometric specification; for brevity, this error term is denoted as t_D2D.

Equation 53.

\frac{R_{G2}}{R_{IN2}} = (1 + t_{D2}) \times G_{nom}

Equation 54.

\frac{R_{G2}}{R_{IN1}} = (1 + t_{D2D}) \times G_{nom}

The effective transfer function is thus

Equation 55.

V_{OUT} = V_{IN+} \times G_{nom} \times (1 + t_{E2E}) (1 + t_{D2D}) - V_{IN–} \times G_{nom} \times (1 + t_{D2}) + V_{REF}

For further analysis, the input voltages V_IN+ and V_IN– are first expressed as a common-mode input voltage (V_CM) and a differential input voltage (V_DIFF).

Equation 56.

V_{CM} = \frac{(V_{IN+} + V_{IN–})}{2}

Equation 57.

V_{DIFF} = V_{IN+} - V_{IN–}

Equation 55 is expressed in terms of V_CM and V_DIFF as

Equation 58.

V_{OUT} = V_{CM} \times (\frac{\frac{R_{G1}}{R_{IN1} + R_{G1}} - \frac{R_{G2}}{R_{IN2} + R_{G2}}}{\frac{R_{IN2}}{R_{IN2} + R_{G2}}}) + V_{DIFF} \times (\frac{\frac{R_{G1}}{R_{IN1} + R_{G1}} + \frac{R_{G2}}{R_{IN2} + R_{G2}}}{2 \times \frac{R_{IN2}}{R_{IN2} + R_{G2}}})

Equation 59.

V_{OUT} = V_{CM} \times (\frac{R_{G1}}{R_{IN2}} \times \frac{R_{IN2} + R_{G2}}{R_{IN1} + R_{G1}} - \frac{R_{G2}}{R_{IN2}}) + \frac{V_{DIFF}}{2} \times (\frac{R_{G1}}{R_{IN2}} \times \frac{R_{IN2} + R_{G2}}{R_{IN1} + R_{G1}} + \frac{R_{G2}}{R_{IN2}})

Equation 60.

V_{OUT} = V_{CM} \times G_{nom} \times ((1 + t_{D2D}) \times (1 + t_{E2E}) - (1 + t_{D2})) + \frac{V_{DIFF}}{2} \times G_{nom} \times ((1 + t_{D2D}) \times (1 + t_{E2E}) + (1 + t_{D2}))

The gain error with respect to V_CM or to V_DIFF is calculating by taking a partial derivative of Equation 60 with respect to the given variable.

Equation 61.

\frac{{∂V}_{OUT}}{{∂V}_{CM}} = G_{nom} \times ((1 + t_{D2D}) \times (1 + t_{E2E}) - (1 + t_{D2}))

Equation 62.

\frac{{∂V}_{OUT}}{{∂V}_{DIFF}} = \frac{G_{nom}}{2} \times ((1 + t_{D2D}) \times (1 + t_{E2E}) + (1 + t_{D2}))

Because the error tolerance terms (1 + t_D2D) and (1 + t_E2E) are multiplicative, and t_D2D and t_E2E are both are zero-mean with a standard deviation in the sub-200ppm range, the error contribution of t_D2D × t_E2E is less than 0.01ppm and is assumed to be negligible. The result is an algebraic sum of three terms, all considered as independent zero-mean Gaussian values, such that:

Equation 63.

\frac{t_{{E R R}_{effective}}}{1} = \sqrt{{(\frac{t_{D2D}}{1})}^{2} + {(\frac{t_{E2E}}{1})}^{2} + {(\frac{t_{D2}}{1})}^{2}}

By substituting the typical values of t_D2D, t_E2E, and t_D2, root sum of squares error analysis is performed on the resulting terms to describe a typical error for the transfer function.

Consider an example where a RES11A50-Q1 is used, such that G_nom = 5. Assume t_D2 = 81ppm, t_E2E = 18ppm, and t_D2D = 86ppm. Using Equation 63, t_{ERR_effective} is calculated as ±120ppm, and is used to calculate ∂V_OUT with respect to V_CM and to V_DIFF. The former is the common-mode gain error, while the latter is composed of the desired nominal gain term (G_nom) and an undesired gain error.

Equation 64.

\frac{{∂V}_{OUT}}{{∂V}_{CM}} = G_{nom} \times {t_{ERR}}_{effective} = G_{nom} \times ±120 ppm = ±600 ppm

Equation 65.

\frac{{∂V}_{OUT}}{{∂V}_{DIFF}} = \frac{G_{nom}}{2} \times (2 + {t_{ERR}}_{effective}) = G_{nom} \pm 300 ppm

Multiplication of the t_{ERR_effective} error by the desired process control value, such as × 6 for a six-sigma approach, gives conservative maximum bounds. Because the ±1σ values reported in Electrical Characteristics already include guardbanding and account for mean shifts, in many cases a lower process control value (such as five-sigma) is sufficient. For example, solving the previous expressions for CMRR yields only 78.5dB, whereas the actual typical CMRR for the RES11A50-Q1 is 102.1dB. The discrepancy arises because the measurement resolution of t_D1, t_D2, t_M, and CMRR is higher than that of t_D2D and t_E2E, and therefore the reported values of the latter parameters include additional guardbanding. Additionally, the conservative modeling approach assumes t_D2D, t_E2E, and t_D2 are uncorrelated, whereas for many devices there are weak correlations (such as t_D2D and t_E2E having different polarities) that cause the actual observed error to be lower than the modeled error.