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1 Exercises EC&S CHAPTER 2 Draw pole-zero maps of the following functions :

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1 1 Exercises EC&S CHAPTER 2 Draw pole-zero maps of the following functions :

2 2 Exercises (ctd.) EC&S CHAPTER 2 Find functions of s corresponding to the following pole-zero diagrams : multiplier = 100 s-plane j6 j6 j4 j4 j2 j2 -j6 -j2 -j4 -8 -10 -12 1 -2 -6 Im Re -4 s-plane j3 j3 j2 j2 j1 j1 -j3 -j1 -j2 -5 -6 1 -3 2 Im Re -2 -4 multiplier = 20

3 3 Sketching time functions EC&S CHAPTER 2 Illustrative function and its pole-zero map : Im s-plane j3 j3 j2 j2 j1 j1 -j3 -j1 -j2 -5 -6 1 -3 2 Re -2 -4 j4 j4 -j4 If we used partial fractions, function would decompose into following form :

4 4 Sketching time functions EC&S CHAPTER 2 Re-write as : Im s-plane j3 j3 j2 j2 j1 j1 -j3 -j1 -j2 -5 -6 1 -3 2 Re -2 -4 j4 j4 -j4 Corresponding time function is of form : Thus response has three components : 1 st is due to pole at s = 0 2 nd is due to pole at s = - 6 3rd is due to conjugate pair at s = - 1 ± j4

5 5 Sketching time functions EC&S CHAPTER 2 Im s-plane j3 j3 j2 j2 j1 j1 -j3 -j1 -j2 -5 -6 1 -3 2 Re -2 -4 j4 j4 -j4 Pole @ origin gives constant term Pole @ s = - 6 gives term decaying as e - 6t Pole pair gives oscillatory term ( frequency 4 rad/s decaying as e - 1t ) Can find (constant) FV without inverting LT : Also

6 6 Sketching time functions EC&S CHAPTER 2 2 Im j3 j3 j2 j2 j1 j1 -j3 -j1 -j2 -5 -6 1 -3 Re -2 -4 j4 j4 -j4 Dominant term FV theorem takes care of pole @ origin Fast decay Slow decay FV thm. Dominant poles Decaying oscillation Period of osc n = T = 2  /4 = 1.6 sec Decay time const.  = 1/1 = 1 sec 1 4

7 7 Sketching time functions EC&S CHAPTER 2 2 Im j3 j3 j2 j2 j1 j1 -j3 -j1 -j2 -5 -6 1 -3 Re -2 -4 j4 j4 -j4 -1 0V FV = 2V 23 45 (sec) 0 63% 4V  =1 T = 1.6 sec T/2 3T/2 upper boundary lower boundary ignore Approximating sketch

8 8 Guidelines for sketching EC&S CHAPTER 2 1. Write down LT in factored form 2. Draw pole-zero map 3. Draw time axis 4. Determine IV (dot on vertical axis)  FV time IV 0 63% 55 95% 3  5. Determine FV 6. Calculate  7. Place dot 63% between IV and FV 8. Sketch exponential curve

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