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Virtual ground and summer design

Short answer

Virtual ground: in an inverting op-amp with negative feedback and non-inverting input grounded, V- ≈ 0 V due to high open-loop gain, acting as ground without physical connection. Design: inverting summer with Rf=100kΩ, R1=100kΩ, R2=10kΩ, R3=4kΩ yields Vo=-(V1+10V2+25V3).

Explain the concept of virtual ground. Design a summer circuit using op-amp to get the output voltage as: V0 = -(V1 + 10V2 + 25V3).

Virtual ground

In an inverting op-amp circuit with negative feedback, the non-inverting input is grounded. Because the open-loop gain is very large (ideally infinite), the differential input voltage is nearly zero, so V− ≈ V+ = 0. The inverting input is therefore at ground potential without being physically connected to ground. This is called a virtual ground. Since the input resistance of the op-amp is infinite, no current flows into that terminal, so all the input current flows through the feedback resistor. This is why the inputs of a summing amplifier do not interact.

Design of V0 = −(V1 + 10V2 + 25V3)

The inverting summing amplifier gives:

V0 = −(Rf/R1 V1 + Rf/R2 V2 + Rf/R3 V3)

 V1 o--[ R1 ]--+
               |
 V2 o--[ R2 ]--+-----[ Rf ]-----+
               |                |
 V3 o--[ R3 ]--+     |\         |
               +-----|- \       |
                     |   >------+----o V0
               +-----|+ /
               |     |/
              GND

Comparing the coefficients: Rf/R1 = 1, Rf/R2 = 10, Rf/R3 = 25.

Choose Rf = 100 kΩ:

  • R1 = Rf/1 = 100 kΩ
  • R2 = Rf/10 = 10 kΩ
  • R3 = Rf/25 = 4 kΩ

Check: V0 = −(100/100 V1 + 100/10 V2 + 100/4 V3) = −(V1 + 10V2 + 25V3).

Answer: Rf = 100 kΩ, R1 = 100 kΩ, R2 = 10 kΩ, R3 = 4 kΩ. A resistor Rcomp = R1 ∥ R2 ∥ R3 ∥ Rf (about 2.7 kΩ) placed at the non-inverting input can reduce the offset error.

✓ Answer checked by Claude Opus 5.5 (high effort)

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