Summing and Difference Amplifiers

The virtual-earth technique behind the inverting amplifier isn't limited to circuits with a single input. Add more input branches to the same virtual-earth node, or compare two signals directly against each other, and the same reasoning extends naturally into two more standard op-amp circuits with real, everyday applications — mixing audio signals and converting digital data back into an analogue voltage.

The summing amplifier

The summing amplifier is a direct extension of the inverting amplifier from the previous lesson: instead of a single input signal arriving through a single input resistor, several separate input signals — $V_1, V_2, V_3, \ldots$ — each arrive through their own input resistor — $R_1, R_2, R_3, \ldots$ — but all of these resistors meet at the same virtual-earth node (the inverting input), which is still fed back to the output through a single shared feedback resistor $R_f$.

Exactly the same virtual-earth reasoning applies as before: the non-inverting input is grounded, so the inverting input sits at a virtual earth of 0 V, and no current flows into the op-amp itself. The only difference is that current continuity at the virtual-earth node must now account for every input branch simultaneously — the total current flowing in from all the inputs combined must equal the current flowing out through $R_f$.

A commonly used special case is where all the input resistors are made equal, $R_1 = R_2 = \cdots = R$. The result then simplifies to $V_{out} = -\dfrac{R_f}{R}(V_1 + V_2 + \cdots)$ — the output becomes proportional simply to the sum of the input voltages, scaled by a single common factor.

The difference amplifier

Where the summing amplifier combines several signals, the difference amplifier does the opposite job: it responds to the difference between two input voltages, ideally rejecting anything the two signals have in common. The standard circuit applies one input to the inverting side through a resistor network and the other input to the non-inverting side through a matched resistor network, so that the output depends only on how the two differ.

For the standard arrangement, where the resistor ratios on both sides are equal, the output is:

This circuit amplifies the difference between the two input voltages by the same ratio $R_f/R_1$ that sets the gain of the basic inverting amplifier, while ideally producing zero output if $V_+$ and $V_-$ happened to be equal — however large either signal is individually.

Applications: audio mixing and digital-to-analogue conversion

These two circuits aren't just abstract exercises — they underpin two directly examinable real-world applications:

• Audio mixing. A summing amplifier is the working core of an audio mixer: several separate audio signals — different microphones or instrument channels, for instance — are each connected through their own input resistor into a single summing amplifier, combining them into one output signal representing the mixed audio. Because each input's contribution to the combined output depends on its own input resistor, adjusting that resistor (in practice, using a variable resistor as a "channel fader") controls how loudly that particular channel is heard in the final mix relative to the others.
• Digital-to-analogue conversion (DAC). A summing amplifier can also convert a digital binary number into a single analogue output voltage, by using binary-weighted input resistors — resistor values in the ratio $R, 2R, 4R, 8R, \ldots$, one per bit of the binary number, each connected (or not) to a fixed reference voltage depending on whether that bit is a 1 or a 0. Because each input's contribution to $V_{out}$ scales inversely with its resistor value, the least-significant bit (through the largest resistor) contributes the smallest share and the most-significant bit (through the smallest resistor) contributes the largest, exactly matching the place-value weighting of a binary number — so the resulting $V_{out}$ ends up proportional to the decimal value the binary digits represent.

This DAC application connects directly back to the earlier "Analogue-to-Digital Conversion" lesson's discussion of pulse-code modulation (PCM): once a signal has been sampled and quantised into a stream of binary numbers for storage or transmission, something has to turn those digital samples back into a smooth, usable analogue signal at the receiving end — for example, to drive a loudspeaker. A binary-weighted summing amplifier performing exactly this conversion is a standard way that final step is carried out.