(a) To determine Find the output voltage v o using convolution integral. Explanation Given data: Refer to given circuit in the textbook. The value of input voltage is, v i ( t ) = 5 [ u ( t ) − u ( t − 0.5 ) ] V (1) Formula used: Write a general expression to calculate the impedance of a resistor in s -domain. Z R = R (2) 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 (3) Here, L is the value of inductance. Write a general expression to calculate the impedance of a capacitor in s -domain. Z C = 1 s C (4) Here, C is the value of capacitance. Write a general expression of convolution integral. y ( t ) = ∫ − ∞ ∞ h ( λ ) x ( t − λ ) d λ (5) Here, h ( λ ) is the unit impulse function, and x ( t − λ ) is the shifted input. Calculation: From the given data the circuit is redrawn as shown in Figure 1. Substitute 2 Ω for R in equation (2) to find Z R . Z R = 2 Substitute 100 mH for L in equation (3) to find Z L . Z L = s ( 100 mH ) = s ( 100 × 10 − 3 H ) = 0.1 s Substitute 100 mF for C in equation (3) to find Z C . Z C = 1 s ( 100 mF ) = 1 s ( 100 × 10 − 3 F ) { ∵ 1 m = 10 − 3 } = 10 s Convert the Figure 1 into s -domain as shown in Figure 2. Use voltage division rule in Figure 2 to find V o . V o = ( ( 10 s ) 2 + 0.1 s + 10 s ) V i = ( ( 10 s ) 0.1 s 2 + 2 s + 10 s ) V i = ( 10 s ) ( s 0.1 ( s 2 + 20 s + 100 ) ) V i = ( 100 s 2 + 20 s + 100 ) V i Rearrange the above equation to find V o V i . V o V i = 100 s 2 + 20 s + 100 On solving the quadratic equation in denominator of above equation it becomes as follows: V o V i = 100 ( s + 10 ) 2 Write a general expression to calculate transfer function. H ( s ) = V o V i Substitute 100 ( s + 10 ) 2 for V o V i in above equation to find H ( s ) . H ( s ) = 100 ( s + 10 ) 2 Apply inverse Laplace transform for above equation to find h ( t ) . h ( t ) = 100 t e − 10 t u ( t ) { ∵ ℒ − 1 ( 1 ( s + a ) n ) = t n − 1 ( n − 1 ) ! e − a t u ( t ) } Substitute λ for t in above equation to find h ( λ ) . h ( λ ) = 100 λ e − 10 λ u ( λ ) Refer to Figure 1, the input is v i and output is v o . Therefore, the equation (5) becomes as follows: v o ( t ) = ∫ − ∞ ∞ h ( λ ) v i ( t − λ ) d λ (6) Substitute t − λ for t in equation (1) to find v i ( t − λ ) . v i ( t ) = 5 [ u ( t − λ ) − u ( t − λ − 0.5 ) ] V (7) The waveform for equation (1) with time λ is drawn as shown in Figure 3. The waveform for inverted function of equation (1) is drawn as shown in Figure 4. The waveform for equation (7) is drawn as shown in Figure 5. Substitute 100 e − 10 λ u ( λ ) for h ( λ ) , and equation (6) in equation (5) to find v o ( t ) . v o ( t ) = ∫ − ∞ ∞ [ 100 t e − 10 λ u ( λ ) ] [ 5 u ( t − λ ) − 5 u ( t − λ − 0.5 ) ] d λ v o ( t ) = ∫ − ∞ ∞ [ [ 500 t e − 10 λ u ( λ ) ] u ( t − λ ) ] − [ [ 500 t e − 10 λ u ( λ ) ] u ( t − λ − 0 (b) To determine Sketch the output voltage for time 0 < t < 1 s .

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ISBN: 8220100801792

Author: Riedel

Publisher: YUZU

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Chapter 13, Problem 70P

(a)

To determine

Find the output voltage vo using convolution integral.

(b)

To determine

Sketch the output voltage for time 0<t<1 s.

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Chapter 13, Problem 71P

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Chapter 13 Solutions

EBK ELECTRIC CIRCUITS

Ch. 13.2 - Prob. 1AP Ch. 13.2 - The parallel circuit in Example 13.1 is placed in...Ch. 13.3 - Prob. 3AP 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) =...Ch. 13.7 - The current source in the circuit shown is...Ch. 13.7 - For the circuit shown, find the steady-state...Ch. 13 - Prob. 1P Ch. 13 - Prob. 2P Ch. 13 - Prob. 3P Ch. 13 - Prob. 4P Ch. 13 - An 8 kΩ resistor, a 25 mH inductor, and a 62.5 pF...Ch. 13 - Prob. 6P Ch. 13 - Find the poles and zeros of the impedance seen...Ch. 13 - Find the poles and zeros of the impedance seen...Ch. 13 - Prob. 9P Ch. 13 - Prob. 10P Ch. 13 - Prob. 13P Ch. 13 - Prob. 15P Ch. 13 - There is no energy stored in the circuit in Fig....Ch. 13 - There is no energy stored in the circuit in Fig....Ch. 13 - Prob. 25P Ch. 13 - Prob. 28P Ch. 13 - The switch in the circuit seen in Fig. P13.32 has...Ch. 13 - Prob. 31P Ch. 13 - Prob. 33P Ch. 13 - Prob. 35P Ch. 13 - Prob. 46P Ch. 13 - Prob. 47P Ch. 13 - Find the transfer function H(s) − Vo/Vi for the...Ch. 13 - Prob. 49P Ch. 13 - Prob. 50P Ch. 13 - Prob. 51P Ch. 13 - Prob. 53P Ch. 13 - Prob. 54P Ch. 13 - The operational amplifier in the circuit in Fig....Ch. 13 - Find the transfer function Io/Ig as a function of...Ch. 13 - Prob. 58P Ch. 13 - Prob. 59P Ch. 13 - Prob. 60P Ch. 13 - Prob. 61P Ch. 13 - Assume the voltage impulse response of a circuit...Ch. 13 - Prob. 68P Ch. 13 - The input voltage in the circuit seen in Fig....Ch. 13 - Find the impulse response of the circuit shown in...Ch. 13 - Prob. 73P Ch. 13 - Prob. 74P Ch. 13 - Prob. 75P Ch. 13 - The op amp in the circuit seen in Fig. P13.81 is...Ch. 13 - Prob. 78P Ch. 13 - The transfer function for a linear time-invariant...Ch. 13 - Prob. 80P Ch. 13 - Prob. 81P Ch. 13 - Prob. 82P Ch. 13 - Prob. 84P Ch. 13 - Prob. 85P Ch. 13 - The parallel combination of R2 and C2 in the...Ch. 13 - Show that if R1C1 = R2C2 in the circuit shown in...Ch. 13 - The switch in the circuit in Fig P13.91 has been...Ch. 13 - Prob. 90P Ch. 13 - Prob. 91P

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