Summary of Last Day

Systems with feedback have damped sinusoids in impulse response

Showed how Laplace could be used to identify damped sinusoids and unstable systemsX

Real Axis ()

imaginary axis (j)

0

X

X

X

X

X

XX X

X

Fourier a Subset of Laplace

Fourier and Laplace very similar In fact when = 0, Fourier and

Laplace almost identical

dtethH tj

= )()(

jswheredtethsH st +== ,)()(0

Fourier view of RC circuit

The frequency response H( ) of the series RC circuit is 1/(1+j RC) Can show this using reactance/impedance

methods OR By Taking Fourier Transform of differential equation

relating time-domain inputs and outputs of the RC circuit, given by

dttdvRCtvtv ooi)()()( +=

vi

R

C vo

vR

Frequency Response of RC circuit

10-2 10-1 100 101-30

-25

-20

-15

-10

-5

0

RC=0.7

RC=1

RC=3

Radians/second

20log10|H()|

Frequency Response dependent upon product of R and C

Frequency Response of RC circuitFrequency Response dependent upon product of R and C

0 2 4 6 8 10-30

-25

-20

-15

-10

-5

0

RC=0.7

RC=1

RC=3

Radians/second

20log10(|H()|)

Impulse Response of RC circuit - time domain

It has already been shown the the frequency response of the RC circuit is given by H( ) = 1/(1+j RC)

Using Fourier Transform tables it can then be shown that the impulse response of an RC circuit is h(t)=(1/RC) e-t/RC

The impulse response of the RC circuit can be though of as a damped sinusoid with a frequency of zero. This implies that a pole exists on s-plane.

Where is the pole situated on the s-plane? (Next slide may help)

Pole position relationship with impulse response

X

Real Axis ()

imaginary axis (j)

0

X

X

X

X

X

XX X

X

Pole Position for RC circuit

The impulse response of the RC circuit is h(t)=e-t/RC

There is no oscillating frequency component, so =0. The pole is therefore be positioned on the real axis of the simplified s-plane representation.

Proof: Pole occurs when |H(s)| = , i.e. s = -1/RC No imaginary term Pole always on LHS of s-plane since R and C

always positive (System stable or unstable?)

1/RC)(s1

RC1 H(s)

+=

Pole Position for RC circuit simplified s-plane (pole zero plot)

XReal Axis ()

imaginary axis (j)

-1/RC0

Pole Position for RC circuit 3D view of S-plane

What is the value of RC?

When are Laplace and Fourier effectively the same?

when = 0, Fourier and Laplace almost identical

Where on the S-Plane is = 0?

dtethH tj

= )()(

jswheredtethsH st +== ,)()(0

Pole Position for RC circuit 3D view of S-plane RHS removed

Side View of S-Plane RC = 3RHS of s-plane removed

Shaded Area showsFrequency Responseof System

RC =3 for Slides 10, 12, 13

Compare with Slide 5

Side View of S-Plane RC = 0.7

RC =0.7 in these plotsCompare with Slide 5

Main Points

Pole Positions on s-plane and frequency response of system closely related.

Fourier is effectively a subset of Laplace

Generally a system described in terms of H(s) (aka System Transfer Function) because its the best representation to understand behaviour of the system Why? Shows system stability Its easy enough to get an idea of the frequency

response. With some training it can be easy enough to determine

the step response of a system from H(s)

More examples with 2 poles2 Poles => second order system

Shaded area showsFrequency response of system

S-plane

( ) ( )jsjssH +++= 221)(

Pole ZeroPlot

Real Axis()

imaginaryaxis(j)

0

X

X

5

-5

5

More examples with 2 poles2 Poles => second order system

Pole ZeroPlot

Real Axis()

imaginaryaxis(j)

0

X

X

5

-5

5

S-plane

( ) ( )jsjssH 3.023.021)(

+++=

Shaded area showsFrequency response of system

Summary of Last DayFourier a Subset of LaplaceFourier view of RC circuitFrequency Response of RC circuitSlide 5Impulse Response of RC circuit - time domainPole position relationship with impulse responsePole Position for RC circuitPole Position for RC circuit simplified s-plane (pole zero plot)Pole Position for RC circuit 3D view of S-planeWhen are Laplace and Fourier effectively the same?Pole Position for RC circuit 3D view of S-plane RHS removedSide View of S-Plane RC = 3 RHS of s-plane removedSide View of S-Plane RC = 0.7Main PointsMore examples with 2 poles 2 Poles => second order systemSlide 17