Summing and difference amplifiers

Every input meets a virtual earth

In an inverting amplifier, one input resistor feeds the op-amp's inverting input, and negative feedback holds that input within microvolts of 0 V. That node is the virtual earth. Because the node sits at 0 V, the current arriving through an input resistor is just the input voltage over that resistance, and because the op-amp inputs draw no appreciable current, every bit of that current must leave through the feedback resistor.

Now hang several input resistors on the same node, one per signal. Each input drives its own current V/R$V/R$V/R into the virtual earth without disturbing the others, the currents add at the node, and the whole total is forced through R_f$R_{f}$R_f. The output swings to whatever voltage makes that happen, so it carries an inverted, weighted copy of every input at once:

Describe the drawing

Three input resistors of one hundred, fifty and twenty kilohms carry two, six and five microamps from their sources into one summing point held at zero volts by the op-amp. The thirteen microamp total leaves through the hundred kilohm feedback resistor, so the output sits at minus one point three volts.

Read the equation as a set of independent gains. Input 1 is amplified by R_f/R₁$R_{f}/R_{1}$R_f/R_1, input 2 by R_f/R₂$R_{f}/R_{2}$R_f/R_2, and so on. The circuit does not add the voltages themselves; it adds weighted voltages and inverts the total, and the weights are set by the choice of resistors.

Weighted resistors: from binary to analogue

Choose the weights in powers of two and the mixer becomes a digital-to-analogue converter, a DAC. A 4-bit binary number arrives as four logic levels, each 0 V or 5.0 V. Give the most significant bit the strongest channel and halve the gain at each step down, say R_f$R_{f}$R_f = 10 kΩ with input resistors of 10, 20, 40 and 80 kΩ. The gains run 1, 12$\frac{1}{2}$1/2, 14$\frac{1}{4}$1/4, 18$\frac{1}{8}$1/8, matching the place values 8, 4, 2, 1.

One least significant bit is then worth 5.0 × 10/80 = 0.625 V of output. Feed in the binary count 0000, 0001, 0010 and upward, and the output descends an even staircase, one 0.625 V step per count, sixteen levels in all. Every digital audio player ends in some refinement of this idea.

Describe the drawing

A staircase of sixteen equal treads: the size of the analogue output of a four-bit weighted resistor converter as its input counts from 0000 to 1111. The inverting circuit makes the output itself negative, so the voltage descends as the count rises. Each tread is one least significant bit, nought point six two five volts, and the code 1011 lands eleven steps out at six point eight seven five volts.

The difference amplifier

Swap the geometry and the op-amp subtracts. A difference amplifier feeds one signal towards each input through matched resistor pairs, and its output amplifies only the gap between them:

Watch Difference amplifiers: how the noise cancels on the InkPhysics YouTube channel

Subtraction matters most for small sensor signals. A strain gauge or thermistor usually sits in a bridge, a pair of potential dividers across one supply. In balance both divider midpoints sit at the same voltage. When the sensor shifts, the two midpoints separate by millivolts, and that small difference is superimposed on a large steady level that carries no information. The difference amplifier rejects the shared level and amplifies the difference alone.

The same subtraction is the standard cure for interference. Mains wiring induces the same 50 Hz hum on both leads of a sensor cable, since the leads run side by side. Identical on both inputs, the hum is common-mode, and V₊ - V_-$V_{+} - V_{-}$V_(+) - V_(-) wipes it out before the gain is applied. The wanted signal, applied as a genuine difference between the leads, survives at full gain.

Describe the drawing

Three traces sharing one forty millisecond time axis. At the top the two bridge leads swing together through six tenths of a volt of mains hum, and because they sit only 2.4 mV apart they are drawn one on top of the other. In the middle their difference, on a scale magnified a thousand times, is a flat line at 2.4 mV with no ripple left in it at all, because the hum was identical on both leads and subtracts to nothing. At the foot the output, that difference at a gain of a thousand, is a steady 2.4 V lying well inside the dashed rails at plus and minus thirteen volts, which is where the ideal equation would stop.

Limits: saturation and resistor tolerance

The equations allow any output at all; the hardware does not. An op-amp runs from supply rails, typically ±15 V, and its output cannot pass them. Ask the summing equation for −20 V and the real output runs into saturation a volt or so inside the negative rail and sits there, clipping the waveform flat. Every design should end with the same check, worst-case output against the rails.

The other limit comes from the resistors. Every gain here is a ratio of real resistors, and real resistors come with a tolerance, commonly 5% or 1%. Two 5% resistors can leave a gain almost 10% adrift. A DAC feels this hardest, because its staircase is only trustworthy if each weight is accurate to under half a step. For an 8-bit converter that means resistor ratios good to about 0.2%, and needing eight precision values spanning a 128 to 1 range is exactly what drives real converters away from the weighted-resistor design.