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Part II Physics

May 2004

EXAMPLES CLASS ­ SYSTEMS
1. The input function u(t) of a linear system is related to its output y(t) by the differential equations. (x(t) being an internal variable.)

y=u­x x = 5y + 6y ­ u
Derive the transfer function of the system H(s) = Y(s) / U(s) and deduce (from a partial fraction expansion) the response of the system to (a) a unit impulse (0) and (b) a unit step [U(s)) = 1/s]. Verify that your answer to (b) is the convolution of (a) with a unit step. 2. Consider a simple spring and mass system as shown in the diagram. The support of the spring is subject to an external drive x(t), which can be regarded as the "input", while the "output" is the displacement of the mass, y(t). Include damping proportional to y (t ) . Deduce a transfer function and sketch how the poles move in the s plane as the damping varies. Hence, identify the pole positions in the low, critical and high damping regimes and explain why the resonant frequency is little affected by damping in the low damping case.

k

x(t)

y(t)

3. Determine the transfer function and impulse response of the following linear system.
1 U(s) 1/s -2 1/s -1 Y(s)

How can such a system be realised using operational amplifiers? 4. Which of the following have roots with no positive real parts? (i) (ii) (iii) s3 + s2 + 4s + 10 s3 + 10s2 + 4s ­ 10 s3 + 5s2 + 4s + 15

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Part II Physics

May 2004

5. Find the values of K for which the system shown is stable. What is the impulse response of the system when K=8.

+ u -

s s - 2s + 5
2

y

K
6. On an audio CD the signal is recorded digitally, with a sampling rate of 44.1 kHz. Assume that, during playback, the data are converted instantaneously to an analogue voltage which is then constant until the next sample ("sample and hold"). Describe this process by a suitable convolution, and show that, relative to low frequencies, a signal component at 10 kHz is reduced in amplitude by about 8%. Is any further filtering of the analogue signal desirable?

7. (Non-linear pendulum) Find and classify the fixed points of the damped pendulum, for which + b + sin = 0 , for all b > 0 , and plot the phase portraits for the qualitatively different cases. For the undamped pendulum driven by a constant torque we have + sin = . a) Find all equilibrium points and classify them as varies. b) Sketch the vector field: i.e. vectors representing the trajectory and speed in the , phase-plane. Also sketch the lines where the vector field is purely vertical and where it is purely horizontal (these are called nullclines). c) Sketch the phase-portrait on the plane as varies. d) Find the approximate frequency of small oscillations about any centres in the phaseportrait.

()

8. A system with transfer function
H (s) =

Y (s) s -1 =2 X ( s ) s + 3s + 2

is excited by white noise with power spectral density N0. Determine the auto-correlation function, the mean and variance of the output variable y(t). 9. A radio-astronomical source of white noise with power density P is observed through a rectangular bandpass filter of width B Hz. The output power is measured by detecting (i.e. squaring) the signal and averaging for a time . Show that the uncertainty in the detected power, P , is given by P = P B . (Hint: Find the autocorrelation function of the filter output, and hence show that samples of the power separated by times t = m / B , for m a non-zero integer, must be independent.)

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Part II Physics

May 2004

10. The impulse responses of an analogue filter and its digital equivalent are shown.

Compare the responses of each filter (relative to that for = 0) at frequencies, , of /6T, /3T and 2/3T.

­3T

0

T

2T

3T

t

What is the reason for the differences in responses?
Rachael Padman May 10, 2004

­2T

0

T

2T

t

3