Op Amps are among the most widely used components in systems design of electronic circuits. Although functionally simple, they exhibit complex behavior as the Op Amp itself is a carefully crafted sub-circuit consisting of more than a dozen transistors. Idealized models of the Op Amp, namely, infinite values of gain, bandwidth, input impedances and output admittance and zero values of input offset voltage and bias currents, are a good first-order approximation for analyzing Op Amp-based circuits.

Deviation from ideal behavior can be incorporated into analysis depending on the environment in which the Op Amp is operating. One such environment is DC measurement systems. In such applications, the presence of offset voltage cannot be ignored; unlike in a signal processing chain where DC offsets can be easily filtered out with a single capacitor. Offset voltage of an Op Amp results in an error at the output for DC signals. In addition, they can reduce the dynamic range of the output if significant in value. The presence of offset voltage is a well-understood phenomenon and is described in various literature and textbooks.

In this article, a generalized method is proposed to compute offset in the output when an Op Amp with an input offset *e* is used in the circuit.

The transfer function of an ideal Op Amp is described by the equation *y* = *A*(*V*_{+} - *V*_{-}), where *y* is the output; *A* is the gain, with *A* → ∞ , *V*_{+} is the voltage at positive input terminal and *V*_{-} the voltage at the negative input terminal of the Op Amp. The presence of offset can be encapsulated by assuming that the real Op Amp input/output transfer characteristic is *y* = *A*(*V*_{+} - *V*_{-} + *e*) where *e* is the error in the differential input to the ideal Op Amp. This model is consistent with the observation that in a real op amp, the output is zero when there is a difference in the input (*V*_{+} ≠ *V*_{-}) and that a real op-amp produces a nonzero output when *V*_{+} = *V*_{-}.

Assuming a functional model of the Op Amp as *y* = *A*(*V*_{+} - *V*_{-} + *e*) where *e* is the input offset voltage, when used in a negative feedback configuration, we get

or, *V*_{+} - *V*_{-} = - *e* (assuming an infinite gain) for *any* Op Amp used in a negative feedback configuration.

Thus, the “golden rules” of Op Amp widely followed while analyzing ideal Op Amp based circuits is modified to include the presence of input offset voltage of value *e*.

- In a circuit with an op amp used in a negative feedback configuration,
*V*_{-} = *V*_{+} + *e*

- Input current into/from the positive or negative terminals of an Op Amp is zero.

Now, consider a circuit topology shown in Figure 1. The schematic topology is mapped to the circuit shown in Figure 2 with the resistive networks replaced by their Thevenin equivalents (looking away from the terminals) as *V*_{th+}, *R*_{th+} and *V*_{th-}, *R*_{th-} respectively.

**Figure 1**

**A General topology of negative feedback Op Amp based circuit. **

Applying the golden rule,

*V*_{−} = *V*_{+} + *e* = *V*_{th+} + *e*, since *V*_{+} = *V*_{th+} .

Applying Kirchhoff's current law at the negative terminal of the op amp in Figure 2 yields the equation,

Solving for *V*_{out} in terms of *V*_{th+}, *V*_{th-}, and *e*, we get

**Figure 2**

**The circuit topology shown in Figure 1 is mapped to the above circuit**

*F*(*V*_{th+}, *V*_{th-}, 0) is simply the output when the op amp is ideal (i.e. zero input offset voltage) Therefore, the output offset is

Here, we apply the above derived formula for output offset calculations for a variety of widely used Op Amp based circuits.

**1. Difference Amplifier **

See Figure 3. In this case

**Figure 3**

**1a. Noninverting Amplifier:**

A special case of the difference amplifier with *V*_{1} = 0 and *R*_{1} = 0.

**2. Summing Amplifier **

See Figure 4.

**Figure 4**

In this case *R*_{th-} = *R*_{1} || *R*_{2} || *R*_{3} ……….|| *R*_{n}and therefore

**2a. Inverting Amplifier **

Special case of summing amplifier with *n* = 1.

**2b. Binary Weighted Digital to Analog Converter **

Special case of summing amplifier shown above with

**3. R-2R Ladder Digital to Analog Converter **

The equivalent output resistance of a R-2R ladder is well known to be R irrespective of the length of the ladder. Two flavors of implementation are shown in Figures 5 and 6. In Figure 5

**Figure 5**

**R-2R ladder Digital to Analog Converter with buffered output **

**Figure 6**

**R-2R ladder Digital to Analog Converter with inverting output**

In Figure 6,

For low values of *n* the weighted binary DAC produces less offset at the output compared to R-2R DAC in Figure 6.

**Incorporating bias currents **

This section includes output offset calculation if bias currents are a cause of concern. Let bias currents out of positive and negative terminals be *I*_{B+} and *I*_{B-} respectively (+ value for outgoing current). Referring to Figure 2,

Applying Kirchhoff’s current law at negative terminal of op amp yields the equation,

Solving for *V*_{out} in terms of *V*_{th+}, *V*_{th-}, *e*, *I*_{B+}, *I*_{B-}, we get,

F (*V*_{th+}, *V*_{th-},0, 0, 0) is output for an ideal op amp. Thus, the output offset is given by

Having calculated *R*_{th+}, and *R*_{th-} in each of above cases (sections 1-3), the total offset calculation is simply an exercise in substituting the appropriate values in the above equation.