Summation Amplifiers - Adding Signals with Operational Amplifiers

Summation amplifiers are circuits that combine several input voltages into a single output voltage, either as a weighted or unweighted sum. They have long been used in analog computers, sensor fusion, and audio mixing, where signals must be combined or averaged. Operational amplifiers make this possible with extremely simple and elegant circuits. The summation amplifier exists in two main configurations:

• Inverting configuration
• Non inverting configuration

The inverting summation amplifier is the most common due to its simplicity and its inherently stable virtual-ground operation.

In the following analysis, the operational amplifier is assumed to be ideal. This means it has infinite open-loop gain, infinite input impedance, zero output impedance, zero input bias currents, and sufficient bandwidth such that frequency-dependent effects can be neglected. Under these assumptions, the amplifier enforces equal voltages at its inverting and non-inverting inputs while drawing no input current, allowing the circuit behavior to be derived using only Kirchhoffs and Ohms laws.

In the end we will take a look at common pitfalls

Inverting Summation Amplifier

The inverting summer connects multiple input voltages $U_{in,1}, U_{in,2}, \ldots, U_{in,N}$ through resistors $R_1, R_2, \ldots, R_N$ to the inverting input of the OpAmp. The non-inverting input is tied to ground, creating a virtual ground at the summing node.

Since the amplifier drives its inverting input to the same potential as the grounded non-inverting input, the node $S_1$ remains at (virtual) ground potential. Applying Kirchhoff’s current law at this node gives:

[ \begin{aligned} \sum I_i &= 0 \\ \to I_1 + I_2 + I_3 + \ldots + I_n + I_f &= 0 \\ - I_f &= \sum_{i = 1}^{N} I_i \end{aligned} ]

The current through each input resistor is determined by the corresponding input voltage:

[ \begin{aligned} U_{in,1} &= R_1 I_1 \\ \to I_1 &= \frac{U_{in,1}}{R_1} \end{aligned} ]

The output voltage in relation to ground is also known and thus determines $I_f$:

[ U_{out} = R_f I_f ]

Assuming current balance in $S_1$ yields:

[ \begin{aligned} U_{out} &= R_f I_f \\ &= - R_f \sum_{i = 1}^{N} I_i \\ &= - R_f \sum_{i = 1}^{N} \frac{1}{R_i} U_{in,i} \end{aligned} ]

In case we of equal input resistors one arrives at a typical (non weighted) sum with a gain term:

[ \begin{aligned} R_1 &= R_2 = \ldots = R_N \\ U_{out} &= - \underbrace{\frac{R_f}{R_{in}}}_{G} \sum_{i = 1}^{N} U_{in,i} \\ U_{out} &= - G \sum_{i = 1}^{N} U_{in,i} \\ \end{aligned} ]

In case one also chooses the feedback resistor to have the same value the gain factor becomes $G=1$

[ \begin{aligned} R_1 &= R_2 = \ldots = R_N = R_f \\ U_{out} &= - \sum_{i = 1}^{N} U_{in,i} \end{aligned} ]

Non Inverting Summation Amplifier

The non inverting summation amplifier works differently. First we can take a look at the inverting input. This provides us a reference potential distinct from ground. This reference potential depends on our output voltage with respect to our system ground (the 0V node):

[ \begin{aligned} I &= \frac{U_\mathrm{out}}{R_f + R_{f,2}} \\ U_\mathrm{inv} &= \frac{R_{f,2}}{R_f + R_{f,2}} U_\mathrm{out} \end{aligned} ]

On the non inverting input we apply Kirchhoffs rule (i.e. the sum over all currents has to be zero due to charge carrier conservation)

[ \begin{aligned} \sum_{n=1}^{N} I_n &= 0 \end{aligned} ]

The current can be expressed by the input voltage. We have to take into account that the operational amplifier drives the output port to compensate the voltage difference between both the inverting and the non inverting input so that

[ \begin{aligned} U_\mathrm{inv} &= U_\mathrm{notinv} \\ I_n &= \frac{U_n - U_\mathrm{inv}}{R_n} \end{aligned} ]

Inserting this into Kirchhoffs law yields

[ \begin{aligned} \sum_{n=1}^{N} \frac{U_n - U_\mathrm{inv}}{R_n} &= 0 \\ \to \sum_{n=1}^{N} \frac{U_n}{R_n} &= \sum_{n=1}^{N} \frac{U_\mathrm{inv}}{R_n} \\ &= \left( \sum_{n=1}^{N} \frac{1}{R_n} \right) U_\mathrm{inv} \\ &= \left( \sum_{n=1}^{N} \frac{1}{R_n} \right) \frac{R_{f,2}}{R_f + R_{f,2}} U_\mathrm{out} \end{aligned} ]

Reorganizing to express $U_\mathrm{out}$:

[ \begin{aligned} U_\mathrm{out} &= \frac{R_f + R_{f,2}}{R_{f,2}} \frac{1}{\sum_{n=1}^{N} \frac{1}{R_n}} \sum_{n=1}^{N} \frac{U_n}{R_n} \\ &= \underbrace{\left(\frac{R_f}{R_{f,2}} + 1 \right)}_{G} \frac{1}{\sum_{n=1}^{N} \frac{1}{R_n}} \sum_{n=1}^{N} \frac{U_n}{R_n} \\ \end{aligned} ]

The first part $G$ expresses the gain of the circuit. The second part is the expression of a weighted average of the input voltages. If we want to further simplify the expression we can assume that all input resistors share the same value $R_\mathrm{in}$

[ \begin{aligned} U_\mathrm{out} &= \underbrace{\left(\frac{R_f}{R_{f,2}} + 1\right)}_{G} \frac{1}{N \frac{1}{R_\mathrm{in}}} \frac{1}{R_\mathrm{in}} \sum_{n=1}^{N} U_n \\ &= \underbrace{\left(\frac{R_f}{R_{f,2}} + 1\right)}_{G} \frac{1}{N} \sum_{n=1}^{N} U_n \\ \end{aligned} ]

In this case the second term collapses from the weighted average to the average of all input voltages. The gain of the system in both cases is given by

[ \begin{aligned} G &= \frac{R_f}{R_{f,2}} + 1 \end{aligned} ]

A non-inverting summation amplifier is primarily used when the input signals must be combined without significantly loading their sources. Because the summation takes place at the non-inverting input, each signal sees a high input impedance and is largely decoupled from the output, which can be advantageous when working with sensors, high-impedance voltage sources, or preceding stages that should not be disturbed. The output voltage remains non-inverted and its overall gain can be adjusted independently using the feedback network, while the input resistor network determines the relative weighting or averaging of the signals.

The main tradeoff of this configuration is increased sensitivity and complexity compared to the inverting summation amplifier. The summing node is not held at virtual ground, so resistor tolerances, input bias currents, and noise have a more direct influence on the result. For these reasons, non-inverting summation amplifiers are typically chosen for impedance-matching or buffering considerations rather than for simplicity or maximum precision.

Pitfalls

• The summing resistors draw current from the voltage sources - the circuit loads the signal sources. If any source has high output impedance (for example a voltage divider), this can lead to drifts in those circuits. Use a buffer amplifier (voltage follower) before the summing stage in such cases.
• Differences in resistor tolerances cause gain errors and unwanted weighting.
• Input bias currents flowing through large resistors create offset voltages. Adding a balancing resistor to the non-inverting input can compensate for this.
• For low-level signals, minimize noise by keeping resistor values moderate ($10 k\Omega$ to $100 k\Omega$ range).
• In audio and sensor-fusion circuits, equal resistors typically provide good linear superposition.

For most applications taking a digital approach is - in my opinion - better, more flexible and more resilient. First providing buffers and preamplifications and then performing digital to analog conversion before continuing processing in a microcontroller or FPGA avoids all pitfalls with respect to loading, cross-influence of upstream circuits due to loading, etc. - and it’s also cheap and flexible since you can change the algorithms even after building the circuits. In addition you can always tap into the raw signals at all stages for debugging in a very simple fashion.

This article is tagged: Tutorial, DIY, Electronics, Physics, Basics, Measurements, OpAmp