Summing Amplifier

The summing amplifier circuit is shown on the Picture 1. It is inverting amplifier with multiple inputs connected together in one terminal, thus the resulting input signal is a sum of all inputs. If we apply the Kirchoff's current law (KCL) at the inverting input of this circuit, we got:

V1/R1 + V2/R2 + V3/R3 = -Vo/Rf

The summing amplifier shown on Picture 1 has three inputs. However, in general, the summing amplifier can have n inputs (n > 0), thus the applied KCL for the general summing amplifier will look like:

V1/R1 + V2/R2 + ... + Vn/Rn = -Vo/Rf

Picture 1: Summing Amplifier with 3 inputs

According to this, for the output voltage of the our circuit example shown on Picture 1, we can write:


Vo = -Rf(V1/R1 + V2/R2 + V3/R3)


or, for the general summing amplifier with n inputs, we have:

Vo = -Rf(V1/R1 + V2/R2 + ... + Vn/Rn)


So, as we can see from the last relations, the output voltage of this amplifier is actually a sum of all inputs multiplied with constants Rf/Rn. If we choose values for Rn and Rf as: R1 = R2 = ... = Rn = R, and Rf = R, => Rn = Rf = R, then we will got pure sum of the inputs of the circuit as output voltage: Vo = -(V1 + V2 + ... + Vn).

Picture 2: Output voltage of the Summing Amplifier (all inputs at same frequency)


On the Picture 2 is shown the output voltage of our summing amplifier with three inputs in case when all three inputs have same frequency and different amplitudes. In this case, the frequency of input signals are f1 = f2 = f3 = 500 Hz, and the amplitudes are V1 = +/- 100 mV, V2 = +/- 300 mV and V3 = +/- 200 mV. The values of the input resistors are R1 = R2 = R3 = 1K and the value of the output resistor is Rf = 5k ohms. So, if we write now the relation for the output voltage, we will have:

Vo = -Rf(V1/R1 + V2/R2 + V3/R3) = -5K/1K(V1 + V2 + V3)

=> Vo = -5(V1 + V2 + V3) = -5(0.1 + 0.3 + 0.2) = -3 [V]

Picture 3: Output voltage of the Summing Amplifier (inputs at different frequency)


If we have input signals at different frequencies one from each other, then the output signal wave form will not have a pure sinusoidal form. As example, if we set different frequencies for all three input signals and equal amplitude, we will got a waveform as shown on Picture 3. In this case, we have equal amplitude of V1 = V2 = V3 = +/- 100 mV and frequencies of the signals are f1 = 500 Hz, f2 = 1000 Hz and f3 = 2500 Hz. In this case, the relation for the output voltage is not the same as it was for the previous situation. Namely, we can't just sum the input signals multiplied by factor 5, because they have different frequencies and the frequency take a role here, so the relation will be correct if for every Vi input signal we write its time-domain relation, which is Vi = Vimax * sin(wt + fi), where w is the circular frequency w = 2*Pi*f, and Fi is the phase of the input signal. Anyway, here we will not going that deep into math, since we have a waveform of the output voltage, shown on Picture 3, from which we can see the non-sinusoidal shape of it, and that the max values of the signal are about +/- 1 V.