Draw the circuit of a Summing Amplifier and obtain an expression for the output.

Summing Amplifier Circuit

A summing amplifier is a type of operational amplifier (op-amp) circuit that can combine several input signals into a single output signal. It is essentially a weighted sum of the input signals. The circuit for a summing amplifier using an op-amp is shown below:

      V1      R1
       ├─────┬───┐
       │     │   │
      ┌┴┐   ┌┴┐  │
      │ │   │ │  │
      │ │   │ │  │
      └┬┘   └┬┘  │
       │     │   │
       ├─────┼───┼───┐
       │     │   │   │
      V2     R2  │   │
       ├─────┬───┘   │
       │     │       │
      ┌┴┐   ┌┴┐      │
      │ │   │ │      │
      │ │   │ │      │
      └┬┘   └┬┘      │
       │     │       │
       ├─────┼───────┼───┐
       │     │       │   │
      V3     R3      │   │
       ├─────┬───────┘   │
       │     │           │
      ┌┴┐   ┌┴┐          │
      │ │   │ │          │
      │ │   │ │          │
      └┬┘   └┬┘          │
       │     │           │
       ├─────┼───────────┼───┐
       │     │           │   │
       │     Rf          │   │
       │     │           │   │
       │     ├───────────┤   │
       │     │           │   │
       └─────┤           ├───┘
             │           │
             ├───────────┤
             │           │
             └───────────┘

In this circuit, V1, V2, and V3 are the input voltages, R1, R2, and R3 are the input resistors, and Rf is the feedback resistor. The op-amp is represented by the triangle, and it is assumed to be ideal, meaning it has infinite input impedance, zero output impedance, and infinite gain.

Expression for the Output

To obtain an expression for the output voltage Vout, we will use the properties of an ideal op-amp and the concept of virtual ground.

1. Virtual Ground: Since the op-amp is in a negative feedback configuration, the voltage at the inverting input (−) is virtually the same as the non-inverting input (+), which is grounded. Therefore, the voltage at the inverting input is 0V.

2. Current through the Feedback Resistor (If): The current through the feedback resistor Rf is the same as the sum of the currents through R1, R2, and R3 because of Kirchhoff's current law (KCL). This can be expressed as:

[ If = I1 + I2 + I3 ]

where I1, I2, and I3 are the currents through resistors R1, R2, and R3, respectively.

1. Currents through the Input Resistors (I1, I2, I3): Using Ohm's law, the currents through the input resistors are given by:

[ I1 = \frac{V1}{R1} ] [ I2 = \frac{V2}{R2} ] [ I3 = \frac{V3}{R3} ]

1. Current through the Feedback Resistor (If) using Ohm's Law: The current through the feedback resistor is also given by Ohm's law:

[ If = \frac{Vout - 0}{Rf} = \frac{Vout}{Rf} ]

1. Combining the Equations: By substituting the expressions for I1, I2, I3, and If, we get:

[ \frac{Vout}{Rf} = \frac{V1}{R1} + \frac{V2}{R2} + \frac{V3}{R3} ]

1. Solving for Vout: Multiplying through by Rf gives us the output voltage Vout:

[ Vout = -\left( \frac{Rf}{R1}V1 + \frac{Rf}{R2}V2 + \frac{Rf}{R3}V3 \right) ]

The negative sign indicates that the output is inverted with respect to the input signals.

Summary Table

Parameter	Description	Formula
V1, V2, V3	Input Voltages	-
R1, R2, R3	Input Resistors	-
Rf	Feedback Resistor	-
If	Current through Rf	If = I1 + I2 + I3
I1, I2, I3	Currents through R1, R2, R3	I1 = V1/R1, I2 = V2/R2, I3 = V3/R3
Vout	Output Voltage	Vout = -(Rf/R1)V1 - (Rf/R2)V2 - (Rf/R3)V3

Example

Let's say we have a summing amplifier with the following values:

• V1 = 1V, R1 = 10kΩ
• V2 = 2V, R2 = 10kΩ
• V3 = 3V, R3 = 10kΩ
• Rf = 10kΩ

Using the formula for Vout, we get:

[ Vout = -\left( \frac{10kΩ}{10kΩ} \cdot 1V + \frac{10kΩ}{10kΩ} \cdot 2V + \frac{10kΩ}{10kΩ} \cdot 3V \right) ] [ Vout = -(1 + 2 + 3)V ] [ Vout = -6V ]

The output voltage Vout is -6V, which is the inverted sum of the input voltages.