Resistor tolerances will cause amplification to depart from the ideal as derived in part 1 of this series. This is of significance because, in many seemingly undemanding designs, 1% resistors are not adequate, usually because of excessive CM gain, which ideally is ACM = 0. If a resistor of value R is used in the amplifier and has a tolerance of +/- ε, its range of values is R •(1 +/- ε)
Resistors in series have a resistance of
For equal-tolerance resistors, ┓ = └ = ε. Then, for the worst case, their errors have the same sign, and
The tolerance of resistors in series is the same as the resistor tolerance.
For resistors in parallel,
Again, the combination of resistors, in series or parallel, results in a combined tolerance equal to the resistor tolerance.
For a divider, the result is not as beneficial. A ratio of resistors of equal tolerance has a worst-case error when one has +ε and the other has −ε
Thus, the tolerance range for resistor ratios is +/-2∙ε, ε << 1.
These resistor error formulas can be applied to the one-op-amp diff-amp. The worst case (maximum error) occurs for maximum mismatch in the dividers T i + and T i – . Defining the ideal diff-amp voltage gain as
and for which ε << 1, Av0 >> 1. Thus, for small ε and Av >> 1, the error with resistor tolerance of ε in the differential gain is about +/-2•ε. The case for +2•ε is derived by exchanging 1 + 2•ε and 1 – 2•ε for the resistance ratios in the above derivation.
The CM gain is
Applying error tolerances,
Applying these derivations to a design example, the requirements are that Av = 1, and that the CM input voltage range ≥ 30 V. The differential input range is ±2.5 V, input resistance is 2 MΩ, and VOS = 2.5 V, half the ADC reference voltage.
The value of Av0 = Rf /Ri results in Rf = Ri . To make input resistance meet spec, let Rf = Ri = 1.00 MΩ +/-1%. Then, for the CM range requirement using a rail-rail-input (RRI) op-amp,
Solving for ROS+ , ROS+ ≤ 111.1 kΩ. Let ROS+ = 100 kΩ, +/-1%. Then the CM range is 32.5 V > 30 V. Resistor mismatch causes a worst-case gain error of about
The maximum output voltage error of the diff-amp is thus about (+/-2.5 V)•(0.01) = +/-25 mV. For the CM gain,
Over a 30 V CM range, the maximum output voltage error is about 30 V/50 = 600 mV. For a 10-bit A/D converter with a 5V input range and 1,000 counts of range (after 24 counts are allotted for calibration), this is 120 counts of error, or about 12% of the input range. The CM error dominates. It usually does unless low-tolerance or closely matched resistors are used.