**Contents**show

Circuit transfer function is the ratio of output signal to the input signal applied to the LTI (Linear Time Invariant) systems. This transfer function is in the frequency domain instead of time domain. Transfer function is very useful to determine the function of a circuit with an expression.

Unlike circuit analysis where we face a harder time analyzing a more complex circuit, the circuit transfer function can be simpler even if we have a complex circuit. The only tip we can apply is to determine the transfer function of the subcircuits.

Just an advanced word, we will use Laplace Transform, phasor, and complex numbers while doing this thing.

**Circuit Transfer Function**

Transfer function is represented by H(s). The variables we will find the most are voltage or current since our input signals will be either these two. Of course, the output as well will be these two. Like mentioned above, the transfer function is the ratio of the output to the input in frequency domain (s).

Hence, we will denote every variable with (s).

There are four basic transfer function in an electric circuit:

## Transfer Function of a Circuit

In order to fully create a circuit transfer function, we will make the transfer function for each subcircuit to form the bigger circuit. For this, we have to remember what made an electrical circuit? It is the connection, and for this we have known series and parallel connections. And this is no different with the transfer function.

### Transfer Function of a Series Connection

Observe the transfer function diagram below.

There is only one path and it indicates a series connection. Here we have:

- An input, X(s)
- An output, Y(s)
- Two subcircuit transfer functions, H
_{1}(s) and H_{2}(s)

The transfer function is

Series connection will multiply the transfer function.

### Transfer Function of a Parallel Connection

Observe the transfer function diagram below.

There are multiple paths and it indicates a parallel connection. Here we have:

- An input, X(s)
- An output, Y(s)
- Two subcircuit transfer functions, H
_{1}(s) and H_{2}(s)

The transfer function is

Parallel connection will add the transfer function.

### Transfer Function of a Series Parallel Connection

Observe the transfer function diagram below.

There are multiple paths and it indicates a parallel connection. Here we have:

- An input, X(s)
- An output, Y(s)
- Three subcircuit transfer functions, H
_{1}(s), H_{2}(s), and H_{3}(s).

The transfer function is

Series connection will multiply the transfer function H_{1}(s)*H_{2}(s).

Parallel connection will add the transfer function (H_{1}(s)*H_{2}(s)) + H_{3}(s).

### Transfer Function of a Feedback Connection

Observe the transfer function diagram below.

There is a path to the input that indicates we have a feedback system. Here we have:

- An input, X(s)
- An output, Y(s)
- Gain transfer function, G(s)
- Gain input, E(s)
- Feedback transfer function, H(s)
- Feedback output, H(s)Y(s)

The transfer function for the gain input E(s) is

The output transfer function is

If our system is an open loop then the gain is

The closed loop transfer function is

Thus,

**How to do Circuit Transfer Function**

Before anything else, you have to make yourself familiar with phasor and Laplace domain signals. All electrical signals which exist in the time domain with t as an independent variable.

We can transform:

- Time-domain signal into a phasor domain for sinusoidal signals
- Time-domain signal into Laplace domain signal

For example, an impedance in the Laplace domain can be calculated from

An impedance in the phasor domain can be calculated from

Every electrical element has its own Laplace and phasor domain form, as we can see in the table below.

There is a pattern which differentiate between Laplace and phasor and it is the “s” and “jω”.

The Laplace variable “s” in the phasor domain is

Where ω is the radian frequency of a sinusoidal signal.

Looking up a bit, the circuit transfer function H(s) can be expressed as

And a simple diagram as

**Example of Circuit Transfer Function**

For better understanding, let us observe the simple examples below.

1. We have a circuit consisting of a voltage source, a resistor, and a capacitor as shown below.

We call this circuit an RC circuit. Our objective is to determine its transfer function when Vo is measured across the capacitor.

First, looking at the Laplace Transform table above, we convert the resistor and capacitor into the Laplace domain.

The resistor R will stay the same but the capacitor C will be 1/sC. Thus,

With the voltage division rule, the transfer function is

Multiply both top and bottom side with sC we get

We take out the RC

Where

Thus,

We normally express the transfer function where the coefficient of highest power in the denominator is unity (1).

2. Determine the transfer function of a circuit below. Assume that the Op-Amp is ideal.

We refer to the Laplace Transform table and convert the capacitor C into 1/sC while the resistor R is still the same.

The voltages at both input are equal

We will use the voltage division rule again.

Where

We assign node 1 and calculate the node voltage V_{1}.

The voltage V_{1} is

We can simplify the equation above with

Hence,

Where

Thus

The transfer function is

**Frequently Asked Questions**

### What is the transfer function formula?

Transfer function of a circuit is the ratio of output to input in Laplace transform or phasor form. It is often written in H(s) where X(s) is input and Y(s) is output.

### How do you find the transfer function of a RLC circuit?

1. Deriving the circuit into several RLC transfer function subcircuits

2. Determine the output and input function

3. Transform the output and input into Laplace

4. Determine the ratio of the output to input in Laplace transform

### What does the transfer function mean?

In engineering, especially for control systems, the transfer function is known as a mathematical function that represents the system output model. We will find this a lot in electronics and control systems.