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ExaminationPaper No: Course Code No. 333 Answer No:

Examiner: D. Pamunuwa Checker: Date Checked:Special requirements, i.e. graph paper etc.

Purpose of question: To test basic understanding of linear system theory and test understanding of the significance of the polesand zeros in a transfer function in shaping a circuits response.

1. Answer the following questions related to analysis of linear time invariant systems.

(a): Using no more than 2-3 sentences, describe the limitations with respect to type of input excitation and outputresponse of the following circuit analysis techniques: phasor analysis, time domain analysis and Laplace transform

based analysis.

ANSWER:Phasor analysis is only valid for sinusoidal inputs and gives the steady-state sinusoidal response.Time domain and Laplace analyses have no restrictions on the type of input signal and provide both the transientand steady-state (forced) responses.

(b): Recall that the Q factor is defined as:

for the quadratic term:

Derive an expression for the poles of H(s) and sketch their locations in the complex plane for the under-dampedcase. Discuss how the poles move and system stability is affected when the Q factor of the quadratic term changes,with special regard to Q=.

ANSWER:The pole locations are:(for the under-damped case)

When the quality factor Q>, the poles are complex and move in a half circle as sketched. For Q=, the poles become real and coincident. The closer the poles are to the j axis, the more oscillatory and the more overshoot in

MarkingScheme

12

2

3

3

2/nQ

22

2

2)(

n

n

s s s H

Real

Imaginary

- n

+j d

+j d

n

d d n j s j s s 2122

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Examiner: Course Code No. : Answer No:

MarkingScheme

the response.

(c): A linear system is represented by the following transfer function:1002)2(50

)( 2 s s s

s H A

Put this transfer function into standard form keeping in mind that the denominator is a quadratic term. Tabulate the break frequencies and intermediate frequencies as necessary in order and list the effect of each on the magnitude(in dB) in order to construct the Bode magnitude plot of the transfer function.

ANSWER:

1002)2(50

)( 2 s s s

s H has one linear term in the numerator and 1 quadratic term in the denominator (complex

roots, cannot be factorised).

In standard form, 22 10102.01

21

10501

21)(

s s s

s s s

s H

The break frequencies and their effect on the Bode asymptotic magnitude and phase plots are given below.

Frequency Effect on Magnitude Asymptotic magnitude at break frequency

Actual magnitude at break frequency

> 10 rad/s Slope remains at -20 dB/decade

The variation at =10 rad/s needs to be calculated as it is a quadratic term.

Calculating the magnitude by substituting s=j and =10 rad/s into the transfer function results in:

222 101010102.0151

10102.01

21

10102.01

21)(

j j

j

j j

j

s s

s s H

1.285.252.0

26)(

2.0

51)( s H

j

j s H dB

(d): Draw the asymptotic Bode magnitude plot, and sketch an approximation to the real curve by using themagnitude at intermediate values as necessary. Use the attached graph sheet and ensure you mark the break frequencies and associated magnitude values on the graph.

ANSWER:

2

For entriesin table or

equivalent:

1

2

2

1

2

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Examiner: Course Code No. : Answer No:

MarkingScheme

4

-40

0

-20

20

40

10 -2 10 -1 100 101 10 2 103

14.1 dB

14 dB

28.1 dB

3 dB

Frequency (rad/s)

M a n i

t u d e d B