Understanding Engineering Mathematics

i.e.

x ̃o(s)=

G(s)

1 +F(s)G(s)

x ̃i(s)

x∼o

x∼f

x∼i

x∼∼i − xf x

o

∼

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Control

F(s)

G(s)

Figure 17.8

By such means we can derive the transfer functions for ever more complex control

systems. Many linear control systems have a transfer function that is a rational function.

Show that ifF(s)andG(s)are rational functions then so is the overall transfer function

derived above. Discuss the poles of the overall transfer function in terms of those of

theF(s)andG(s). Essentially, the study of the stability of a control system reduces

to the study of the poles of such transfer functions – this is where all the work we did

on rational functions in Chapter 2 comes in handy.

3.We have already emphasised the use of Fourier series in, for example, breaking up
periodic signals into sinusoidal components that are more easily analysed and then
recombined in a linear system. No doubt you will see more of this in your engineering
subjects, whether it be in the analysis of coupled dynamical systems, heat transfer in
solid bodies, or analysis of electronic and optical signalling systems. Here we look at
a use of Fourier series in providing a succinct expression for the power in a periodic
signal.
The average power in a periodic signal x(t)(assumedtobeofperiod2π)is
defined by

P=

1

2 π

∫π

−π

x^2 (t)dt

If the signal can be expressed as a Fourier series in the form

x(t)=

a 0

2

+

∑∞

n= 1

ancosnt+

∑∞

n= 1

bnsinnt

show that the average power can be written as

P=

a 02

4

+

1

2

∑∞

n= 1

(a^2 n+b^2 n)

This is a particular case of a famous mathematical result calledParseval’s theorem,

and the general principle it expresses is a key principle of the study of any sort of