Library / Circuits, electronics and noise / ELN-27

Circuits, electronics and noise

Matched four-resistor difference amplifier

Output of a difference amplifier with matched resistor ratios, where V₁ feeds the inverting branch and V₂ the non-inverting divider.

Ref: ELN-27
Rev: v1
Field: Circuits, electronics and noise
Result: V_o [V]

V_{o} = k (V_{2} - V_{1})

If the resistor ratios match, a four-resistor difference amplifier amplifies the difference of its inputs by the ratio k = Rf/R₁, which must also equal the ratio of the divider on the other input. Vo = k(V₂ − V₁). Any common voltage on both inputs cancels, but a mismatch of the ratios turns part of the common voltage into an error, so use matched resistor pairs.

Output of a difference amplifier with matched resistor ratios, where V₁ feeds the inverting branch and V₂ the non-inverting divider. The gain k is the feedback-to-input resistance ratio, which must equal the divider ratio.

Calculator

Solve for

V_oOutput voltageComputed

kGain: feedback over input resistance

0.1100

V₁Voltage at the inverting input branch

−55

V₂Voltage at the non-inverting input branch

−55

Input difference, V₂ − V₁0.3 V
Common-mode voltage, (V₁ + V₂)/20.95 V
Output with the two inputs swapped, k(V₁ − V₂)−1.5 V

What drives the result

Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.

Output voltage (V) current value 1.5 V

V₁Voltage at the inverting input branchKey driver

−19.530.5

V₂Voltage at the non-inverting input branch

−2921

kGain: feedback over input resistance

0.0330

The amplifier multiplies a 0.3 V difference by 5 to give 1.5 V, ignoring the 0.95 V that both inputs share.
A common-mode voltage much larger than the difference makes resistor matching critical: 0.1 % resistors limit the rejection to roughly 60 dB at this gain.

Chart data as tables

Ranked variables

Variable	Typical range	Lowest result	Highest result	Swing (V)	Local step	Change in result
V₁ Voltage at the inverting input branch	−5 to 5 V	−19.5	30.5	50	+0.2 V	−66.7 %
V₂ Voltage at the non-inverting input branch	−5 to 5 V	−29	21	50	+0.2 V	+66.7 %
k Gain: feedback over input resistance	0.1 to 100	0.03	30	29.97	+10%	+10 %

Sweep of Voltage at the non-inverting input branch

Voltage at the non-inverting input branch (V)	Output voltage (V)
−5	−29
−3.75	−22.75
−2.5	−16.5
−1.25	−10.25
0	−4
1.25	2.25
2.5	8.5
3.75	14.75
5	21

Sweep one variable

Vary

Output voltage (V)Hover, touch, or use the arrow keys on the chart

Variables and how each one contributes

V_o

Output voltage

Result · V

What the equation gives you. Everything else feeds it.

Gain: feedback over input resistance

· typical 0.1 to 100

The gain: the feedback-to-input resistance ratio. Scales the output in direct proportion. Raising it by 10 % (to 5.5) moves the result by +10 %, from 1.5 to 1.65 V.

V₁

Voltage at the inverting input branch

V · typical −5 to 5

The voltage on the inverting branch, in V. Raises the output by −k per volt, so a rise in V₁ lowers it. Raising it by 10 % (to 0.88) moves the result by −26.7 %, from 1.5 to 1.1 V.

V₂

Voltage at the non-inverting input branch

V · typical −5 to 5

The voltage on the non-inverting branch, in V. Raises the output by k per volt. Raising it by 10 % (to 1.21) moves the result by +36.7 %, from 1.5 to 2.05 V.

Worked example with the values above

Input difference: V₂ − V₁ = 1.1 − 0.8 = 0.3 V.
Gain: k = 5.
Vo = 5 × 0.3 = 1.5 V.

Assumptions and limits

An ideal op amp with infinite gain and no input current, in its linear range with negative feedback.
Output limited only by the supply, and signals within the useful bandwidth.

Common mistakes

Using resistor ratios that do not match, which turns common-mode voltage into error.
Swapping the inputs, which changes the sign of the output.
Forgetting that the input resistance of the two inputs is different.

Statistics and uncertainty

Resistor tolerances set the gain accuracy, so use 1 % parts or better for precision, and a matched network for the difference amplifier, where mismatch converts common-mode voltage into error. Real op amps add offset, bias current and finite gain.

Where it is used

Signal mixing, scaling, integration and current-to-voltage conversion in instrumentation.
Photodiode, sensor and ramp-generator circuits.

Maths you need

Virtual ground: the inverting input sits at the non-inverting voltage
Negative feedback: the output acts to cancel the input difference
Kirchhoff’s current law: currents at a node add to zero

Related formulas

Practise this formula