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(a) Derive the second order differential equation for the current across the re- sistor in the 'L+CR' circuit. Hint: The final result can be written as RLCÏR+LIR + RIR = V(t) (b) Derive the second order differential equation for the current across the re- sistor in the 'C+LR' circuit. Hint: The final result can be written as RLC ÏR+LIR + RIR=-LCN²V (t) (c) Prove the following general result: If a current across a resistor with an ex- ternal AC source, V(t) = Vo cos(t + o), satisfies the following second order differential equation: aÏR + bİR+CIR = M cos (nt + o), (2) where a, b, c, M are some parameters that do not depend on time, the average

(a) Derive the second order differential equation for the current across the re- sistor in the 'L+CR' circuit. Hint: The final result can be written as RLCÏR+LIR + RIR = V(t) (b) Derive the second order differential equation for the current across the re- sistor in the 'C+LR' circuit. Hint: The final result can be written as RLC ÏR+LIR + RIR=-LCN²V (t) (c) Prove the following general result: If a current across a resistor with an ex- ternal AC source, V(t) = Vo cos(t + o), satisfies the following second order differential equation: aÏR + bİR+CIR = M cos (nt + o), (2) where a, b, c, M are some parameters that do not depend on time, the average

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
Section: Chapter Questions
Problem 1P: Visit your local library (at school or home) and describe the extent to which it provides literature...
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this is not a graded question! its just an assignment based on completion; but i want to know how to do it; please help me with part c

Consider an AC source represented at any time, t, by V(t) = Vo cos(t + o), where Vo, , are some positive parameters and Vo, o are independent of . In PS3, we considered a circuit called 'R+LC' connected to V(t). There are 2 other independent permutations as shown in figure 1 L+CR 0000 R+LC -WW R R, L, C, N, Vo. d C+LR Figure 1: The 'R+LC', 'L+CR' and 'C+LR' circuits with an external AC source. (a) Derive the second order differential equation for the current across the re- sistor in the 'L+CR' circuit. Hint: The final result can be written as RLCÏR+LIR + RIR = V(t) < PR >= L (b) Derive the second order differential equation for the current across the re- sistor in the 'C+LR' circuit. Hint: The final result can be written as RLCÏR + LIR+RIR = -LCN²V(t) R (c) Prove the following general result: If a current across a resistor with an ex- ternal AC source, V(t) = Vo cos(t + o), satisfies the following second order differential equation: RM² 24 aÏR + bİR+CIR = M cos (nt + o), (2) where a, b, c, M are some parameters that do not depend on time, the average power of the forced oscillations in the resistor over one AC cycle, < PR >, is given by: with A=(c-an²)² + b²N². (3) Notice that in practice, a, b, c, M usually depend on
Transcribed Image Text:Consider an AC source represented at any time, t, by V(t) = Vo cos(t + o), where Vo, , are some positive parameters and Vo, o are independent of . In PS3, we considered a circuit called 'R+LC' connected to V(t). There are 2 other independent permutations as shown in figure 1 L+CR 0000 R+LC -WW R R, L, C, N, Vo. d C+LR Figure 1: The 'R+LC', 'L+CR' and 'C+LR' circuits with an external AC source. (a) Derive the second order differential equation for the current across the re- sistor in the 'L+CR' circuit. Hint: The final result can be written as RLCÏR+LIR + RIR = V(t) < PR >= L (b) Derive the second order differential equation for the current across the re- sistor in the 'C+LR' circuit. Hint: The final result can be written as RLCÏR + LIR+RIR = -LCN²V(t) R (c) Prove the following general result: If a current across a resistor with an ex- ternal AC source, V(t) = Vo cos(t + o), satisfies the following second order differential equation: RM² 24 aÏR + bİR+CIR = M cos (nt + o), (2) where a, b, c, M are some parameters that do not depend on time, the average power of the forced oscillations in the resistor over one AC cycle, < PR >, is given by: with A=(c-an²)² + b²N². (3) Notice that in practice, a, b, c, M usually depend on
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