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9. Kirchhoff's voltage law states that the sum of the voltage drops across a resistor, R, an inductor, L, and a capacitor, C, in an electrical circuit must be the same as the voltage source, E (t), applied to that RLC circuit. Applying the additional fact that the current, I, is related to the charge, q, by the relationship I = a, the resulting ODE model for the charge dt' in a circuit is: d²q dq 1 +R dt² dt +=q= E (t).

9. Kirchhoff's voltage law states that the sum of the voltage drops across a resistor, R, an inductor, L, and a capacitor, C, in an electrical circuit must be the same as the voltage source, E (t), applied to that RLC circuit. Applying the additional fact that the current, I, is related to the charge, q, by the relationship I = a, the resulting ODE model for the charge dt' in a circuit is: d²q dq 1 +R dt² dt +=q= E (t).

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
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### Kirchhoff's Voltage Law in RLC Circuits Kirchhoff’s voltage law states that the sum of the voltage drops across a resistor (R), an inductor (L), and a capacitor (C) in an electrical circuit must be equal to the voltage source \( E(t) \) applied to that RLC circuit. By incorporating that the current \( I \) is related to the charge \( q \) by the relationship \( I = \frac{dq}{dt} \), the resulting ordinary differential equation (ODE) model for the charge in a circuit is given by: \[ L\frac{d^2q}{dt^2} + R\frac{dq}{dt} + \frac{1}{C}q = E(t). \] #### Example Problem If a \( 100 \sin(60t) \) source is connected to an RLC circuit with: - An inductor \( L \) of \( \frac{1}{20} \) henry, - A resistor \( R \) of 1 ohm, and - A capacitor \( C \) of \( \frac{1}{130} \) farad, Find the charge \( q(t) \), given that \( q(0) = 0 \) and \( I(0) = q'(0) = 0 \).
Transcribed Image Text:### Kirchhoff's Voltage Law in RLC Circuits Kirchhoff’s voltage law states that the sum of the voltage drops across a resistor (R), an inductor (L), and a capacitor (C) in an electrical circuit must be equal to the voltage source \( E(t) \) applied to that RLC circuit. By incorporating that the current \( I \) is related to the charge \( q \) by the relationship \( I = \frac{dq}{dt} \), the resulting ordinary differential equation (ODE) model for the charge in a circuit is given by: \[ L\frac{d^2q}{dt^2} + R\frac{dq}{dt} + \frac{1}{C}q = E(t). \] #### Example Problem If a \( 100 \sin(60t) \) source is connected to an RLC circuit with: - An inductor \( L \) of \( \frac{1}{20} \) henry, - A resistor \( R \) of 1 ohm, and - A capacitor \( C \) of \( \frac{1}{130} \) farad, Find the charge \( q(t) \), given that \( q(0) = 0 \) and \( I(0) = q'(0) = 0 \).
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