Explanation Given data: Refer to given circuit and the waveform in the textbook. Formula used: Write a general expression to calculate the impedance of a resistor in s -domain. Z R = R (1) Here, R is the value of resistance. Write a general expression to calculate the impedance of an inductor in s -domain. Z L = s L (2) Here, L is the value of inductance. Write a general expression of convolution integral. y ( t ) = ∫ − ∞ ∞ x ( λ ) h ( t − λ ) d λ (3) Here, x ( λ ) is the input function, and h ( t − λ ) is the shifted impulse unit function. Calculation: The given circuit is redrawn as shown in Figure 1. Substitute 5 Ω for R in equation (1) to find Z R . Z R = 5 Substitute 2.5 H for L in equation (2) to find Z L . Z L = s ( 2.5 H ) = 2.5 s Convert the Figure 1 into s -domain as shown in Figure 2. Apply voltage division rule in Figure 2 to find V o . V o = ( 5 5 + 2.5 s ) V i = ( 5 2.5 ( s + 2 ) ) V i = ( 2 s + 2 ) V i Rearrange the above equation to find H ( s ) = V o V i . H ( s ) = V o V i = 2 s + 2 Apply inverse Laplace transform for above equation to find h ( t ) . h ( t ) = 2 e − 2 t u ( t ) { ∵ L − 1 ( 1 s + a ) = e − a t u ( t ) } Substitute t − λ for t in above equation to find h ( λ ) . h ( t − λ ) = 2 e − 2 ( t − λ ) u ( t − λ ) = 2 e − 2 t + 2 λ u ( t − λ ) = 2 e − 2 t e 2 λ u ( t − λ ) The given waveform is drawn as shown in Figure 3. Write a general expression to calculate ω . ω = 2 π T (4) Here, T is the total time period. Refer to Figure 3, the half cycle of sine wave is given. Therefore, T 2 = π 2 Simplify the above equation to find T . T = ( π 2 ) ( 2 ) = π Substitute π for T in equation (4) to find ω . ω = 2 π π = 2 Write a general expression of sinusoidal waveform. v i ( t ) = A sin ω t (5) Refer to Figure 3, the amplitude of the waveform is 20 . Substitute 20 for A , and 2 for ω in equation (5). v i ( t ) = 20 sin 2 t With the unit step function, the above equation is written as follows: v i ( t ) = { 20 sin 2 t u ( t ) 0 < t < π 2 0 t > π 2 Substitute λ for t in above equation to find v i ( λ ) . v i ( λ ) = { 20 sin 2 λ u ( λ ) 0 < λ < π 2 0 λ > π 2 Substitute v o ( t ) for y ( t ) , and v i ( λ ) for x ( λ ) in equation (3) to find v o ( t )

ELECTRIC CIRCUITS-W/MASTERINGENGINEERING

11th Edition

ISBN: 9780134894300

Author: NILSSON

Publisher: PEARSON

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Current Division Method

The generalized current divider formula is given by the following relation below,

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Chapter 13, Problem 75P

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Find the value of vo at time t=2.2 s.

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Chapter 13, Problem 73P

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ELECTRIC CIRCUITS-W/MASTERINGENGINEERING

Ch. 13.2 - The parallel circuit in Example 13.1 is placed in...Ch. 13.3 - Prob. 2AP Ch. 13.3 - The energy stored in the circuit shown is zero at...Ch. 13.3 - The dc current and dc voltage sources are applied...Ch. 13.3 - Prob. 6AP Ch. 13.3 - Using the results from Example 13.7 for the...Ch. 13.3 - The energy stored in the circuit shown is zero at...Ch. 13.4 - Derive the numerical expression for the transfer...Ch. 13.5 - Find (a) the unit step and (b) the unit impulse...Ch. 13.5 - The unit impulse response of a circuit is υo(t) =...

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