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:

    \begin{align*}&\mbox{H(s)=Voltage gain}=\frac{V_o(s)}{V_i(s)}\\&\mbox{H(s)=Current gain}=\frac{I_o(s)}{I_i(s)}\\&\mbox{H(s)=Impedance}=\frac{V(s)}{I(s)}\\&\mbox{H(s)=Admittance}=\frac{I(s)}{V(s)}\\\end{align*}

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.

circuit transfer function 1

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, H1(s) and H2(s)

The transfer function is

    \begin{align*}H(s)=\frac{Y(s)}{X(s)}=H_1(s)\cdot H_2(s)\end{align*}

Series connection will multiply the transfer function.

Transfer Function of a Parallel Connection

Observe the transfer function diagram below.

circuit transfer function 2

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, H1(s) and H2(s)

The transfer function is

    \begin{align*}H(s)=\frac{Y(s)}{X(s)}=H_1(s)+H_2(s)\end{align*}

Parallel connection will add the transfer function.

Transfer Function of a Series Parallel Connection

Observe the transfer function diagram below.

circuit transfer function 3

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, H1(s), H2(s), and H3(s).

The transfer function is

    \begin{align*}H(s)=\frac{Y(s)}{X(s)}=[H_1(s)\cdot H_2(s)]+H_3(s)\end{align*}

Series connection will multiply the transfer function H1(s)*H2(s).

Parallel connection will add the transfer function (H1(s)*H2(s)) + H3(s).

Transfer Function of a Feedback Connection

Observe the transfer function diagram below.

circuit transfer function 4

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

    \begin{align*}E(s)=X(s)-[H(s)\cdot Y(s)]\end{align*}

The output transfer function is

    \begin{align*}Y(s)=G(s)\cdot E(s)\end{align*}

If our system is an open loop then the gain is

    \begin{align*}G(s)\cdot H(s)\end{align*}

The closed loop transfer function is

    \begin{align*}\frac{\mbox{gain}}{1+(\mbox{open loop gain})}\end{align*}

Thus,

    \begin{align*}\frac{Y(s)}{X(s)}=\frac{G(s)}{1+G(s)\cdot H(s)}\end{align*}

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

    \begin{align*}\frac{\mbox{Voltage - Laplace transform}}{\mbox{Current - Laplace transform}}\end{align*}

An impedance in the phasor domain can be calculated from

    \begin{align*}\frac{\mbox{Voltage - Phasor form}}{\mbox{Current - Phasor form}}\end{align*}

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

circuit transfer function 5

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

    \begin{align*}s=j\omega\end{align*}

Where ω is the radian frequency of a sinusoidal signal.

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

    \begin{align*}H(s)&=\mbox{Circuit transfer function}\\&=\frac{\mbox{Output in Laplace}}{\mbox{Input in Laplace}}\\&=\frac{\mbox{Output in phasor}}{\mbox{Input in phasor}}\\\end{align*}

And a simple diagram as

circuit transfer function 6

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.

circuit transfer function 7

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,

circuit transfer function 8

With the voltage division rule, the transfer function is

    \begin{align*}H(s)&=\frac{V_o}{V_{in}}\\&=\frac{\frac{1}{sC}}{R+\frac{1}{sC}}\end{align*}

Multiply both top and bottom side with sC we get

    \begin{align*}H(s)&=\frac{\frac{1}{sC}}{R+\frac{1}{sC}}\cdot \frac{sC}{sC}\\&=\frac{1}{1+sRC}\end{align*}

We take out the RC

    \begin{align*}H(s)&=\frac{1}{1+sRC}\\&=\frac{1}{RC}\frac{1}{s+\frac{1}{RC}}\end{align*}

Where

    \begin{align*}\alpha=\frac{1}{RC}\end{align*}

Thus,

    \begin{align*}H(s)&=\frac{1}{RC}\frac{1}{s+\frac{1}{RC}}\\&=\frac{\alpha}{s+\alpha}\end{align*}

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.

circuit transfer function 9

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

    \begin{align*}V_1=V_2\end{align*}

We will use the voltage division rule again.

    \begin{align*}V_1&=V_2=\frac{\frac{1}{sC}}{R+\frac{1}{sC}}V_{in}\\&=\frac{1}{1+sRC}V_{in}\\&=\frac{1}{RC}\frac{1}{s+\frac{1}{RC}}V_{in}\\&=\frac{\alpha}{s+\alpha}V_{in}\end{align*}

Where

    \begin{align*}\alpha=\frac{1}{RC}\end{align*}

We assign node 1 and calculate the node voltage V1.

circuit transfer function 10

The voltage V1 is

    \begin{align*}(V_1-V_o)sC_1+\frac{V_1}{R_1}=0\end{align*}

We can simplify the equation above with

    \begin{align*}&V_1-V_o+\frac{V_1}{sC_1R_1}=0\\V_o&=V_1+\frac{V_1}{sC_1R_1}\\V_o&=V_1[1+\frac{1}{sC_1R_1}]\\V_o&=V1\frac{1+sC_1R_1}{sC1R_1}\end{align*}

Hence,

    \begin{align*}V_o&=V_1\frac{1+sC_1R_1}{sC_1R_1}\\&=V_1\frac{s+\frac{1}{C_1R_1}}{s}\end{align*}

Where

    \begin{align*}\beta=\frac{1}{C_1R_1}\end{align*}

Thus

    \begin{align*}V_o&=V_1\frac{s+\frac{1}{C_1R_1}}{s}\\&=V_1\frac{s+\beta}{s}\end{align*}

The transfer function is

    \begin{align*}H(s)&=\frac{V_o}{V_{in}}=\frac{V_o}{V_1}\frac{V_1}{V_{in}}\\&=\frac{s+\beta}{s}\frac{\alpha}{s+\alpha}\\&=\frac{\alpha(s+\beta)}{s(s+\alpha)}\end{align*}

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.

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