Summing Amplifier

A summing amplifier adds multiple input voltages together with independently set weights, producing one combined output. Digital-to-analog converters, audio mixers, and waveform generators all use this circuit to combine signals from different sources.

Core Concept

The summing amplifier is an extension of the inverting amplifier. Multiple input resistors connect to the same virtual ground node at the inverting input. Because this node is held at 0 V by feedback, each input voltage drives its own independent current through its own resistor, and all those currents add up at the node.

The total current flowing into the virtual ground node must flow through Rf to the output. The output voltage is therefore minus the sum of (Vin/Rn) multiplied by Rf for each input. If all input resistors are equal to Rf, each input contributes equally and the output is the inverted sum.

By choosing different resistor values for each input, you set independent gains. An audio mixer using an LM741 might set R1 = 10 kΩ and Rf = 47 kΩ for the microphone channel (higher gain) while setting R2 = 47 kΩ for a line-level channel (unity gain). The channels do not interact because the virtual ground isolates them.

Key Equations

General output: Vout = -Rf × (V1/R1 + V2/R2 + V3/R3)

Equal-weighted sum (all R equal): Vout = -(Rf/R) × (V1 + V2 + V3)

If Rf = R (all equal), the output is just the inverted sum: Vout = -(V1 + V2 + V3)

Compensating resistor at non-inverting input: Rcomp = R1 || R2 || R3 || Rf to balance bias currents.

Given:
  V1 = 1.0 V, R1 = 10 kΩ
  V2 = 2.0 V, R2 = 20 kΩ
  V3 = 0.5 V, R3 = 5 kΩ
  Rf = 10 kΩ
  Op-amp: LM741
  Find: Vout

Why this formula:
  Virtual ground gives Vout = -Rf × (V1/R1 + V2/R2 + V3/R3)

Formula:
  Vout = -Rf × (V1/R1 + V2/R2 + V3/R3)

Substitution:
  Vout = -10k × (1.0/10k + 2.0/20k + 0.5/5k)

Calculation:
  I1 = 1.0/10,000 = 100 µA
  I2 = 2.0/20,000 = 100 µA
  I3 = 0.5/5,000  = 100 µA
  Total current = 300 µA
  Vout = -10,000 × 300×10^-6
  Vout = -3.0 V

Final Answer:
  Vout = -3.0 V
  Each channel happened to contribute 100 µA despite different V and R.

Exam Tip: GATE problems on summing amplifiers often give unequal resistors and ask for Vout. Use the formula Vout = -Rf(V1/R1 + V2/R2 + ...) and compute each term separately. A common error is treating all inputs as equal-weighted when resistors are unequal. Also, if asked for input resistance seen by each source, the answer is its own input resistor (e.g., R1 for V1), not the parallel combination of all resistors.

Key Properties

• Each input sees an input resistance equal to its own series resistor (R1, R2, R3), because the other end of that resistor is at virtual ground.
• Inputs are isolated from each other by the virtual ground node; a signal on V1 does not appear on V2.
• Weight of each input channel is set by Rf/Rn; reducing Rn increases that channel's contribution.
• An LM741 running on ±15 V supplies can sum signals up to about ±13 V at the output before clipping.
• A compensating resistor at the non-inverting input equal to R1||R2||R3||Rf reduces DC error from bias currents.
• Scaling all resistors by the same factor does not change gain, but it does change input resistance and noise performance.

Quick Revision

• Vout = -Rf × (V1/R1 + V2/R2 + V3/R3).
• If all R equal and Rf = R, then Vout = -(V1 + V2 + V3).
• Virtual ground isolates all input channels from each other.
• Input resistance of each channel = its own input resistor.
• Rcomp = R1||R2||R3||Rf placed at non-inverting input to cancel bias current error.
• Output is inverted; add another inverting stage if non-inverted sum is required.
• Increasing Rf raises all channel gains proportionally.
• Exam trap: students forget the negative sign and write Vout as a positive sum, which is only correct if another inversion stage is added after.