Using two port theory (and replacing the transformer by its leakage T network mo
ID: 2248829 • Letter: U
Question
Using two port theory (and replacing the transformer by its leakage T network model (L M, M, L - M) find the circuit transfer function Vo/Ii shown in Fig. 2. Note that excitation is a current source For the further parts of this problem: Since magnitude and frequency scaling can be applied take R as 1 and C as 1 F with L and M as unknowns in the Fig. If you have done the transfer function right, then now the denominator of your transfer function is factorizable as Name: two quadratic polynomials with (L-M) and (L+M) as coefficients of s2, s suitably. With these values for R, C, factorize the fourth order denominator polynomial into quadratic polynomials Use Maxima to verify your factorization. Now answer the following: 1) What is the resonant frequency for each of these quadratic polynomials? 2) What is the damping factor in each of these quadratic polynomials? 3) What will happen to the damping factors as R increases from the value of 1 chosen leaving C, L, M fixed? 4) When the circuit is underdamped in both quadratic polynomials in the denominator and L-M and L+M are quite different numerically (with C = 1), what sort of frequency response will the overall transfer function have - explain clearly. Figure 2: Circuit Diagram for Q-3Explanation / Answer
Answer:
Solution: The system has four poles and no zeros. The two real poles correspond to decaying exponential terms C1e3t and C2e0.1t , and the complex conjugate pole pair introduce an oscillatory component Aet sin (2t + ), so that the total homogeneous response is yh(t) = C1e3t + C2e0.1t + Aet sin (2t + ) (12) Although the relative strengths of these components in any given situation is determined by the set of initial conditions, the following general observations may be made: 1. The term e3t , with a time-constant of 0.33 seconds, decays rapidly and is significant only for approximately 4 or 1.33seconds. 2. The response has an oscillatory component Aet sin(2t + ) defined by the complex conjugate pair, and exhibits some overshoot. The oscillation will decay in approximately four seconds because of the et damping term. 3. The term e0.1t , with a time-constant = 10 seconds, persists for approximately 40 seconds. It is therefore the dominant long term response component in the overall homogeneous response.
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