Pole-Zero Doublets: In this problem we will be looking at the behavior of the circuits shown...

 

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Pole-Zero Doublets: In this problem we will be looking at the behavior of the circuits shown below in order to gain some intuition into the origin and response of pole-zero doublets. Zin Zin L 20 R₂ (1+s/w) (1+s/w)(1+s/02) S C₂ Zin M R₁ R₂ 8L₂ (1) (ii) (iii) a. Without doing any math, how will the magnitude of the impedance of circuits (ii) and (iii) compare with circuit (i)? In other words, will those circuits always have lower, higher, or the same impedance as circuit (i)? b. Derive the impedance Z(s) of circuits (i), (ii), and (iii). c. Now derive and sketch the time-domain voltage response of circuits (i) and (ii) to a current step. Hint: immediately after the current step, what is the voltage across circuit (ii)? What is the final value of the voltage? d. For parts d. and e. of this problem, we will consider the following transfer function (with 0,1, 0p2, and o), all real and positive) : H(s)=7 Sketch || H(jo)|| as a function of @ for (1) @p<0₂ <0p2 and (2) Opi<@p2 <Oz- e. Sketch the time domain step response of H(s) for cases (1) and (2). Will the step response ever overshoot its final value in these two cases? Hint: you may want to find an H₁(s) and H₂(s) such that H(s) = c₁H₁(s) + c₂H₂(s). - i.e., you should use a partial fraction expansion. Pole-Zero Doublets: In this problem we will be looking at the behavior of the circuits shown below in order to gain some intuition into the origin and response of pole-zero doublets. Zin Zin L 20 R₂ (1+s/w) (1+s/w)(1+s/02) S C₂ Zin M R₁ R₂ 8L₂ (1) (ii) (iii) a. Without doing any math, how will the magnitude of the impedance of circuits (ii) and (iii) compare with circuit (i)? In other words, will those circuits always have lower, higher, or the same impedance as circuit (i)? b. Derive the impedance Z(s) of circuits (i), (ii), and (iii). c. Now derive and sketch the time-domain voltage response of circuits (i) and (ii) to a current step. Hint: immediately after the current step, what is the voltage across circuit (ii)? What is the final value of the voltage? d. For parts d. and e. of this problem, we will consider the following transfer function (with 0,1, 0p2, and o), all real and positive) : H(s)=7 Sketch || H(jo)|| as a function of @ for (1) @p<0₂ <0p2 and (2) Opi<@p2 <Oz- e. Sketch the time domain step response of H(s) for cases (1) and (2). Will the step response ever overshoot its final value in these two cases? Hint: you may want to find an H₁(s) and H₂(s) such that H(s) = c₁H₁(s) + c₂H₂(s). - i.e., you should use a partial fraction expansion.

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a Without doing any math how will the magnitude of the impedance of circuits ii and iii compare with ... View the full answer