# Summing Amplifier

A *summing amplifier* is an adaptation of the inverting amplifier in which the output voltage is the result of *summing* the results due to each separate input signal – hence the name. This is a useful circuit which combines a number of separate signals together, with the option to apply a different scale factor to each input.

The above illustration shows a circuit with three inputs, although a fewer or greater number may be connected, as required. The output voltage for this case is given by:

#### Worked Example 1

Calculate suitable component values to produce the expression V_{out} = − (0.1 V_{1} + 0.2 V_{2} + 0.5 V_{3}).

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Begin by choosing a reasonable value for R_{f}, such as R_{f} = 1 kΩ.

Next, calculate values for R_{1}, R_{2} and R_{3} to give the required ratios of R_{f} / R_{1} = 0.1 (1/10), R_{f} / R_{2} = 0.2 (1/5) and R_{f} / R_{3} = 0.5 (1/2) respectively.

Hence, R_{f} = 1 kΩ, R_{1}, = 10 kΩ, R_{2} = 5 kΩ and R_{3} = 2 kΩ.

### Proof

To understand the circuit operation, begin by considering the case where only a single input V_{1} is applied. This is the same as an inverting amplifier so V_{out} = V_{1} × −R_{f} / R_{1}. Similar expressions are derived if V_{2} or V_{3} are the only inputs.

To analyse the operation of the circuit with multiple active inputs, it is useful to apply a couple of *op-amp assumptions*. Firstly, the voltage on the inverting input (V_{−}) of the op-amp will be 0 V due to the *virtual earth principle*. The input currents through R_{1}, R_{2} and R_{3} may then be calculated as I_{1} = V_{1} / R_{1} · · · (1), I_{2} = V_{2} / R_{2} · · · (2) and I_{3} = V_{3} / R_{3} · · · (3) respectively.

Secondly, we assume that the input impedance of the op-amp is infinite, so the input current to the inverting input of the op-amp will be zero. We can then apply *Kirchhoff's Current Law* to the virtual earth node, showing that the three currents flowing into the junction (I_{1}, I_{2} and I_{3}) will equal the single current flowing out through the feedback resistor, R_{f}. Hence, I_{1} + I_{2} + I_{3} = I_{f} · · · (4).

Based on the assumed direction of I_{f}, V_{out} = 0 − I_{f} × R_{f}, since the left end of R_{f} is at virtual earth potential. Hence I_{f} = −V_{out} / R_{f} · · · (5).

Finally, the overall expression for V_{out} may be found by substituting Equations 1, 2, 3 and 5 into Equation 4, in order to eliminate all currents from the relationship.

## Simplified Summing Amplifier Circuits

If it is not necessary to individually scale input values, then all input resistors may be set to a common value, R_{in}. The expression for V_{out} then simplifies to:

This may be simplified even further if no overall scaling is required, in which case R_{in} = R_{f} and the expression for V_{out} becomes: