BM661 - Lecture 3C

Transfer Function

The Laplace or Fourier transform of a system’s impulse response is called the Transfer Function. The Transfer Function is extremely useful in describing the behaviour of a given system.

The convolution property of the Laplace transform mentioned in the previous lecture states that the convolution of two time functions when transformed to the Laplace (or Fourier) domain becomes multiplication. The output of a system is therefore much easier to calculate in the Laplace (or Fourier) domain since only simple multiplication of two functions is involved.

The Laplace transform is also useful in determining the Transfer function and then the impulse response of a system described by a differential equation. The following examples illustrate this.

Example 1:

Consider the simple R-C network shown below. Use the Laplace transform to calculate the current and the voltage across the capacitor.

Now applying the Laplace Transform to both sides of the equation and using the Laplace transform of the integral from the “properties” table:

To get the voltage across the capacitor, we note that:

Now, substituting the expression obtained for the current we get:

If we regard this as a system with V_i as the input and V_c as the output, we can write the transfer function as:

The impulse response of this system can be obtained by taking the inverse Laplace transform of the transfer function. This can be done by simply using the look-up table of common Laplace transforms.

Order of a system:

If a system is described by a first order differential equation or equivalently by a first order polynomial in the transfer function, it is called a first order system. Therefore, the above example is a first order system.

Systems described by an n^th order differential equation or equivalently by a an n^th order polynomial in the transfer function, it is called an n^th order system.

An example of a second order system

Consider the R-L-C network shown below. Use the Laplace transform to calculate the current and the voltage across the capacitor.

Taking the Laplace transform and proceeding as before (using the transforms of both the differential and integral from the Laplace properties table):

The inverse transform can be taken using the tables for specific values of R, L, C. However, we’ll look at the behaviour of second order systems from a broader perspective using an interactive example later.

The Transfer Function, the complex s-plane and Pole-Zero plots

It is often convenient to express a Transfer function as a ratio of two polynomials in ‘s’. The polynomials can be factored into first order terms (the roots of the polynomials may be real or complex).

• The terms in the numerator are called zeros because if s=a₁, or s=a₂, etc., then the function H(s)=0; therefore, the transfer function H(s) is said to have zeros at a₁, a₂, etc.
• The terms in the denominator are called poles because if s=b₁, or s=b₂, etc., then the function H(s)=infinity therefore, the transfer function H(s) is said to have poles at b₁, b₂, etc.

The poles and zeros can be plotted in the complex ‘s’ plane in which the horizontal axis is the real part of ‘s’ and the vertical axis is the imaginary part of ‘s’. The location of the poles can be used to understand the behaviour of the system; this is illustrated in the interactive example below.

Frequency Plots

We have already seen (in the open section of Module 3) how the magnitude of the Fourier transform can be plotted against the frequency to give the Frequency Spectrum. The same idea is used on the Transfer function to obtain the Frequency Response of the system. It is convenient to draw the frequency on a logarithmic scale since it allows a wider range of frequencies to be shown in a single plot. It is conventional to express the magnitude of the transfer function in decibels, which is calculated as :

Use the following interactive example to understand the concepts developed in this module.

Interactive Example: Transfer function plots

Using the transfer function example:

• The screen has four plots. On the left side top is the impulse response plotted against time. On the left bottom, the s-plane is shown with the poles and zeros for the Transfer function of the system. On the right side are the frequency response of the transfer function – magnitude in dB and phase in degrees.
• You can select either a first order system or a second order system.
• The first order system impulse response and transfer function are given here: (as usual u(t) is used to indicate that the response is zero for t<0):

• In a first order system you can select either a low-pass system or a high-pass system. Note that the high-pass system has a zero in the transfer function. Examine the relation between (a) the time constant in the impulse response, (b) the pole location and (c) the frequency response. Note that when the time constant changes, the cutoff frequency changes, but the general shape of the frequency response is the same.
• The second order low-pass transfer function is:
• The impulse response is somewhat complicated and you can see its behaviour in the example. Its exact mathematical form depends on the value of the damping factor.
• In the second order system you can adjust the damping factor and the natural frequency and see the behaviour. In this example only a low-pass system is shown for simplicity. Examine the relation between the system parameters (damping and natural frequency) and (a) the impulse response, (b) the location of the poles (c) and the frequency response.

  Is this system always stable?

• Observe at what values of the damping factor the roots of the second order system are always real. At what values of the damping factor is the system unstable? Look back into module 1 for the meaning of stability. When the system becomes unstable, what is the location of the roots?
• Which is a better low-pass filter, the first order or second order system? Why?

© Suresh Devasahayam