# Inverting Summing Amplifier : Circuit, Working, Derivation, Transfer Function & Its Applications

One of the main applications of op-amp is the summing amplifier or adder. When the op-amp’s input impedance is huge, above one input signal is provided to the inverting amplifier to add the given signal at the output, known as the summing amplifier. This is an op-amp circuit wherever different voltage input signals are added to the inverting amplifier into a single output voltage. So, this circuit is classified into two types based on the sign of the output; inverting summing amplifier & non-inverting summing amplifier. This article provides brief information on an inverting summing amplifier, its working, and its applications.

## What is Inverting Summing Amplifier?

An inverting summing amplifier is one of the main op-amp configurations where the input signals are summed & inverted at the output. This amplifier inverts the phase or polarity of the output signal as compared to the input signal. In this amplifier configuration, op-amp’s inverting input gets the input voltage and non-inverting input is connected to GND. Thus, this amplifier’s gain can be controlled through the choice of the feedback resistor & input resistor values.

#### Op-Amp Role in Summing Amplifier:

In summing amplifier circuit, the op-amp or operational amplifier plays a key role. Understanding the op-amp will determine the summing amplifier behavior. An op-amp is a high-gain voltage amplifier including a differential input & single-ended output. The output voltage in op-amp is proportional to the variation within the two input voltages.

The operational amplifier in a summing amplifier is used in two different modes; voltage follower and inverter mode.

• In voltage follower mode, the op-amp output voltage reproduces the input voltage to make the operational amplifier ideal mainly for signal buffering.
• In Inverter mode, the op-amp output voltage can be amplified and inverted to the input voltage.

The summing amplifier’s functioning is extremely dependent on the Op Amp configuration. So the operation of the operational amplifier in the summing amplifier provides precise, amplified & potentially inverted calculation of the input voltages provided to the summing amplifier.

#### Inverting Summing Amplifier Working

This inverting summing amplifier works by inverting the polarity (or) phase of the o/p signal of the amplifier for the i/p signal. So, the input signal of this amplifier is given to the inverting input & the non-inverting input is given to the ground terminal. The amplified output signal that can be generated is always 180° out of phase with the input. A positive input of this amplifier yields a negative output & vice versa. This amplifier’s gain can be controlled by selecting feedback resistor & input resistor values. An inverting summing amplifier output voltage can be expressed as:

Vout = -(Rf/R1)*Vin + -(Rf/R2)*Vin2+…+-(Rf/Rn)*Vinputn

The gain of inverting summing amplifier is Gain (Av) = Vout/Vin = -Rf/Rin

Here it is important to note that, the op-amp summing amplifier can also be designed through the Non-Inverting configuration. But the major distinction between the Inverting & Non-Inverting summing amplifier is the input impedance. An Inverting Summing Amplifier has less input impedance as compared to a Non-Inverting Summing Amplifier because of the feedback network. So the input signals of this amplifier can be amplified based on the resistors connected to the op-amp & the sum of amplified input signals can be inverted & come into view at the op-amp.

### Inverting Summing Amplifier Circuit

The inverting summing amplifier is an extensive version of the inverting amplifier design which means several inputs are provided to the op-amp’s inverting terminal while the non-inverting terminal is connected to GND. The inverting summing amplifier circuit is shown below. This circuit has several input voltages which are connected to the amplifier’s inverting input terminal & the output will be the amount of all the applied input voltages but inverted.

In the above circuit, when the Non-Inverting terminal is connected to GND, the Inverting terminal is at virtual GND. Thus, the inverting input node will become an ideal node mainly for summing the i/p currents.

#### Inverting Summing Amplifier Equation

The inverting summing amplifier using op-amp is shown below. In this circuit, all the added input signals can be given to the inverting input terminal. So, the circuit with two input
In the above circuit, the non-inverting terminal or point B is grounded, because of the virtual GND concept, the node-A can also be at virtual GND potential.

VA = VB = 0 —— (I)

From the input side of this circuit;

I1 = V1-VA/R1 = V1/R1 —— (ii)

I2 = V2-VA/R2 = V2/ R2 —— (iii)

Applying at node-A & current at input op-amp is zero.

I = I1 + I2—— (iv)

From the output of the amplifier,

I = VA-Vo/Rf = -Vo/Rf————– (v)

Substitute ii, iii equations in iv.

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

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

Vo = – ((Rf /R1) V1 + (Rf /R2) V2).

If the three R1, R2 & Rf resistances are equal then R1= R2 = Rf, so the above equation will become as;

Vo = – (V1 + V2)………(Vi)

By selecting R1, R2 & Rf properly, we can get weighted addition of the input signals such as; aV1 + bV2 which is indicated by the Vi equation. Actually in such a manner, ‘n’ input voltages are added.

Therefore, the output voltage’s magnitude is the amount of the input voltages and thus this circuit is known as an adder or summer circuit. At the output, because of the negative indication of the sum it is known as inverting summing amplifier.

### How to Derive the Inverting Summing Amplifier Transfer Function

This amplifier adds the input signals & inverts the output. The input signals in this amplifier are added with their gain. The following circuit shows the inverting summing amplifier including two inputs. The transfer function of this amplifier is shown below.

Vout = -[V1(Rf/R1)+V2(Rf/R2)]

Using the superposition theorem, let’s begin by making V2 input zero as shown in the following figure. Here the main point is to understand that the level of voltage at the inverting input of the op-amp is zero volts because the non-inverting input is connected to GND.

This operational amplifier will set the o/p level at a voltage that brings its inverting input to a similar range to the noninverting input. So this is because of the very high differential gain of this op-amp like 100,000. If the o/p is a few volts (5V), the differential voltage at the input of the operational amplifier has to be

Vd = 5V/100,000 = 50uV.

The inverting & non-inverting input is considered at a similar potential with few microvolts in between the inputs of the op-amp. The virtual GND within the inverting input assists in determining the voltage drop on the ‘Rf’ feedback resistor. Since the inverting input is at 0V, the voltage drop above Rf is similar to Vout. Thus, the current throughout Rf, If can be written as;

If = Vout/Rf

The flow of current throughout the R1 resistor is current ‘I1’ & can be written like the following equation.

I1= V1/R1

#### Operational Amplifier is Ideal

The operational amplifier can be considered ideal, so the input bias current ‘Ib’ is very close to zero. In addition, resistor ‘R2’ is connected with a single leg to GND whereas the other leg is connected to a virtual GND node. The flow of current throughout resistor ‘R2’ is very close to zero. Here Kirchoff current law says that the sum of all currents within a node is zero, thus we can write that,

If + I1 + I2 + Ib = 0

After replacing ‘If’ & I1,

Vout/Rf = -V1/R1 or -V1 (Rf/R1)

The above equation looks similar to the op amp transfer function in an inverting configuration. The amplifier including V1 in its i/p is a regular inverter since the flow of current throughout ‘R2’ is zero.
In the following superposition theorem conditions, we store ‘V2’ & make ‘V1’zero. The following similar ideas as for ‘V1’, the o/p voltage Vout2 whenever there is just ‘V2’ within the input amplifier is;

Vout2 = -V2 (Rf/R1)

#### Transfer Function:

By adding the two o/p voltages, the T.F of inverting summing amplifier

Vout = Vout1 + Vout2

Vout = – [V1 (Rf/R1) + V2 (Rf/R2)]

The transfer function of this amplifier with ‘n’ input signals is

Vout = – [V1 (Rf/R1) + V2 (Rf/R2) +…+ Vn (Rf/Rn)]

#### Example1:

Let’s assume the values of resistors for inverting summing amplifier Rf = 100KOhms, R1=10KOhms & R2=10KOhms. The input audio signals of this amplifier are’ Vinput1 = 1V and Vinput2 = 2V, so calculate Vout for this amplifier.

We know that Rf = 100KOhms, R1=10KOhms & R2=10KOhms.

Vinput1 = 1V & Vinput2 = 2V

If we substitute these values in the summing amplifier equation, we can get;

Vout = – (Rf/R1) * Vinput1 – (Rf/R2) * Vinput2

= – (100/10) * 1 – (100/10) * 2

= – (10) * 1 – (10) * 2 = – 10 * – 20 = -30V.

The output voltage is -30Volts, which is an amplified & summate of input signals after resistance values adjustment. Different factors change the output of an amplifier like; gain bandwidth product, voltage supply & loading effects. However, the above example of a summing amplifier provides insight into the fundamental arithmetic & interaction of components that drive this amplifier. The summing & amplifying signals process can be scaled up to include various signals jointly.

#### Example2:

What will be the output voltage for the following summing amplifier circuit if three audio signals drive this amplifier?

For every channel in the above circuit, the closed-loop voltage gains can be measured as;

ACL1 = – (Rf / R1) => – (100 Kilo Ohms / 20 Kilo Ohms) => – 5 Kilo Ohms.

ACL2 = – (Rf / R2) => – (100 Kilo Ohms / 10 Kilo Ohms) => ACL2 = – 10 Kilo Ohms.

ACL3 = – (Rf / R3) => – (100 Kilo Ohms / 50 Kilo Ohms) => ACL3 = – 2 Kilo Ohms.

The o/p voltage for this summing amplifier can be given as;

VOUT => (ACL1 V1 + ACL2 V1 + ACL3 V1)

= – [(5 * 100 mVolts) + (10 * 200 mVolts) + (2 * 300 mVolts)]

= – (0.5 Volts + 2 Volts + 0.6 Volts) => – 3.1 Volts.

The advantages of inverting a summing amplifier include the following.

• The summing point in this amplifier is at earth potential virtually and thus the settings as well as signals from every different channel do not influence each other. Like this, every channel is mixed or summed apart from the signal level, etc.
• This amplifier permits audio experts to merge signals from different channels & reproduce them into an only track. Each single audio input is configured separately without disturbing the output.
This kind of amplifier gives isolation between the individual inputs & output because of its virtual GND at the node.

The disadvantages of inverting a summing amplifier include the following.

• The main disadvantage of an inverting summing amplifier is that it has a fairly lower gain as compared to the non-inverting type.
• This amplifier is sensitive to noise so it degrades the S/N ratio & decreases the precision of the output signal.
• The calculation of this amplifier becomes complex when the number of inputs increases.
• The inversion of the sum in this amplifier might not be desirable in some cases.

### Applications

The inverting summing amplifier applications include the following.

• Inverting the summing amplifier helps in inverting the polarity (or) phase of the o/p signal of the amplifier with the input signal.
• This is a very specialized amplifier configuration wherever the input signals are summed & inverted at the output.
• This type of summing amplifier is used for adding the signals.
• This amplifier is used for adding different signals with equivalent gains in the audio mixer.
• This summing amplifier is utilized to apply a DC offset voltage through an AC signal voltage.
• It can also work as a subtractor by simply providing an o/p voltage which is equivalent to the variation of two voltages.

Thus, this is an overview of an inverting amplifier, circuits, working, derivation, advantages, disadvantages, and applications. The main function of this amplifier is to invert the phase of the o/p signal. These amplifiers have low output impedance, high input impedance & very flexible circuit values which can be easily adjusted to handle each input signal’s gain.

The operational amplifier in the summing amplifier circuit determines its behavior. The op-amp in this amplifier operates in voltage follower or inverter mode. The equation of this amplifier simply indicates the o/p voltage which is relative to input voltages as well as the resistors within the circuit. These summing amplifiers are used in different practical applications like; audio mixers, wherever different input signals are merged into a single output. Here is a question for you, what is a non-inverting summing amplifier?