Sample 3114 Final a Solutions

Non-worked out solutions to ELEC3114 Sample Final Exam Paper A

1. (a) e  = 0 for step and ramp inputs, and e  = 2/K a  = 18 for r (t) = t 2 . ∞

∞

01, 0.1, 1, 3. (b) 0.01,

(c)  For the normalised second order component in the numerator, s2 + 0.2s  + 1, the the corr correc ecti tion on is 20 log log 0.2 = 13. 13.98 dB. For the normalised second order 2 component in the denominator, s denominator, s /9+1 9+ 1.2s/9+1, s/9+1, the correction is 20log0. 20log0.4 = 7.96 dB.

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(d)  The asymptotic linear equations: (a) In the interval ω (0, (0, 0.01] : 20 log H ( jω)  jω ) 20log9 40log ω = 19. 19.085 40log ω dB. (b) In the interval ω [0. [0.01, 01, 0.1] : 20 log H ( jω)  jω ) 19. 19.085 40log0. 40log0.01 20(log ω log0. log0.01) = 60. 60.915 20(log ω log0. log0.01) dB. (c) In the interval ω [0. [0.1, 1] : 20 lo l og H ( jω)  jω ) 20. 20.9151  20log  20log 0.1 40(log ω log0. log0.1) = 40. 40.9150 40(log ω log0. log0.1) dB. (d) In the interval ω [1, [1, 3] : 20 log H ( jω) 40. 40.9150 40(lo 40(logg 1 log0. log0.1) +  jω ) 0 (log ω log0. log0.1) = 0. 0.9150 dB. (e) In the interv interval al ω  ω [3, [3, ) : 20log H ( jω)  jω ) 0.9150 40(log ω log log 3) dB. dB.

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(e)  The slope table for the asymptotic magnitude response between break points is given given in Table 1. The slope table table for the asympto asymptotic tic phase response between between break points is given in Table 2. (f) (f )  The approximate magnitude and phase Bode plots are shown in Figs. 1 and 2, respectively.

1

Figure 1: Bode magnitude plot for Q1(f).

Figure 2: Asymptotic Bode phase plot for Q1(f).

2

Table 1: Asymptotic slope table for magnitude 0.001 0.01 0.1 1 3 30 Double pole at 0 -40 -40 -40 -40 -40 -40 Zero at -0.01 20 20 20 20 20 Pole at -0.1 -20 -20 -20 -20 Second order numerator 40 40 40 break point at 1 rad/s Second order denominator -40 -40 break point at 3 rad/s Total slope

-40

-20

-40 0

-40

Table 2: Asymptotic slope table for phase 10 4 10 3 10 2 0.1 0.3 1 Zero at -0.01 45 45 Pole at -0.1 -45 -45 -45 Second order numerator 90 90 90 break point at 1 rad/s Second order denominator -90 -90 break point at 3 rad/s −

Total slope

0

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−

45

0

45

-45

0

-40

3

10

30

90 -90

-90

0

-90

0

2. (a)  Open-loop system is not stable. In this case, it is only marginally stable due to the (non-repeating) pole at the origin.

(b) ω = 1 and ω =

−1.

(c)  See Figure 3 for Nyquist plot. (d) 0 < K