Two golden rules and the virtual earth
When part of the output is fed back to the inverting input (negative feedback), an op-amp follows two simple rules:
- No current flows into V₊ or V₋ (infinite input resistance).
- The output changes until V₊ = V₋ (because the open-loop gain is huge).
If V₊ is connected to 0 V, then V₋ is also held at 0 V even though it is not wired to earth. We call this point a virtual earth.
Inverting amplifier configuration
Input Vin goes through Rin to V₋. Rf joins V₋ to the output. V₊ is at 0 V.
V₋ is a virtual earth, so the current in Rin is Vin/Rin. No current goes into the op-amp, so the same current flows through Rf: −Vout/Rf. Set them equal:
G = Vout / Vin = −Rf / Rin
The minus sign means the output is inverted (180° out of phase). The input resistance of the whole circuit is just Rin.
Non-inverting amplifier configuration
Vin goes straight to V₊. Rf and Rin form a potential divider from the output to earth, with the middle point at V₋.
V₋ = Vout × Rin/(Rin + Rf) and V₋ = V₊ = Vin, so
G = 1 + Rf / Rin
Output is in phase with input. Gain is always at least 1. The input resistance is very high, so it hardly loads a weak sensor. With Rf = 0 (and Rin removed) G = 1: a voltage follower (buffer).
Summing amplifier configuration
Several inputs V1, V2, V3 each go through their own resistor R1, R2, R3 to the virtual earth at V₋. Their currents add and all flow through Rf:
Vout = −Rf (V1/R1 + V2/R2 + V3/R3)
If all resistors are equal to Rf, Vout = −(V1 + V2 + V3). Uses: audio mixers (each channel has its own volume) and digital-to-analogue converters, where resistors R, 2R, 4R, 8R give binary weights 8:4:2:1.
Real operational amplifiers
- Saturation: the output cannot go beyond the supply voltages (in practice about 1–2 V less than ±Vs). If G × Vin is too big, the wave is clipped.
- Finite open-loop gain: about 10⁵, not infinite, and it falls as frequency rises.
- Gain-bandwidth product: gain × bandwidth ≈ constant. An op-amp with GBP = 1 MHz used at a gain of 100 has a bandwidth of only about 10 kHz.
- Finite slew rate: the output can only change so many volts per microsecond.
- Small input currents and offset voltages.
Try it: be the op-amp
Hold a ruler with its middle on a pencil (a see-saw). Push one end down 1 cm at 3 cm from the pencil: the far end, 9 cm away, moves up 3 cm. Like the inverting amp, the output is the opposite way and Rf/Rin = 9/3 = 3 times bigger. Then check it in the 3D with Rin = 10, Rf = 30.
Key formulas and definitions
- Inverting: G = −Rf / Rin
- Non-inverting: G = 1 + Rf / Rin
- Summing: Vout = −Rf (V1/R1 + V2/R2 + …)
- Gain × bandwidth ≈ constant (GBP)
Worked examples
1. Inverting amp: Rin = 10 kΩ, Rf = 50 kΩ, Vin = 0.2 V. Find G and Vout.
G = −50/10 = −5. Vout = −5 × 0.2 = −1.0 V.
2. Non-inverting amp: Rin = 2 kΩ, Rf = 18 kΩ, Vin = 0.3 V. Find Vout.
G = 1 + 18/2 = 10. Vout = 10 × 0.3 = 3.0 V.
3. Design an inverting amp with gain −20 and Rin = 4.7 kΩ.
Rf = 20 × 4.7 = 94 kΩ (use the nearest value, 100 kΩ, if precision is not vital).
4. Summing amp: Rf = 10 kΩ, R1 = 10 kΩ, R2 = 5 kΩ, V1 = 0.5 V, V2 = 0.2 V. Find Vout.
Vout = −10(0.5/10 + 0.2/5) = −10(0.05 + 0.04) = −0.9 V.
5. Inverting amp with gain −15 on a ±12 V supply. Input peak 1.0 V. What does the output look like?
It wants −15 V peaks, but it cannot go past about ±12 V. So the wave is clipped flat at the supply: saturation.
6. An op-amp has a gain-bandwidth product of 1.0 MHz. It is used as a non-inverting amp with Rf = 99 kΩ, Rin = 1 kΩ. Find the bandwidth.
G = 1 + 99/1 = 100. Bandwidth = 1.0 MHz / 100 = 10 kHz.
7. A 3-bit DAC uses a summing amp with Rf = 1 kΩ and input resistors 1 kΩ (MSB), 2 kΩ, 4 kΩ (LSB). Each bit is 0 V or 4 V. Find Vout for input 101.
Vout = −1(4/1 + 0/2 + 4/4) = −(4 + 0 + 1) = −5 V. The size 5 matches binary 101.
Common mistakes
- Forgetting the minus sign for the inverting and summing amplifiers.
- Writing Rf/Rin for the non-inverting gain; it is 1 + Rf/Rin.
- Giving an output bigger than the supply voltage; real op-amps saturate.
- Thinking the virtual earth is wired to 0 V. It is held at 0 V by feedback, with no wire to earth.