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For the RLC circuit shown below, we wish to find v(t) for t>0. a. Derive the governing equation for the voltage v using KCL at the top node using the following definitions: a = 2RC - You should obtain LC a²v 1 đv + 1 v = 0 d²v + 2a + wžv = 0 or dt? RC đt LC at2 dt c. If R=50 2, is the system underdamped, critically damped, or overdamped? What is the equation that describes v(t)? R 1H 1 mF +

For the RLC circuit shown below, we wish to find v(t) for t>0. a. Derive the governing equation for the voltage v using KCL at the top node using the following definitions: a = 2RC - You should obtain LC a²v 1 đv + 1 v = 0 d²v + 2a + wžv = 0 or dt? RC đt LC at2 dt c. If R=50 2, is the system underdamped, critically damped, or overdamped? What is the equation that describes v(t)? R 1H 1 mF +

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
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For the RLC circuit shown below, we wish to find \( v(t) \) for \( t > 0 \). **a. Derive** the governing equation for the voltage \( v \) using KCL at the top node using the following definitions: \( \alpha = \frac{1}{2RC} \), \( \omega_0^2 = \frac{1}{LC} \). You should obtain \[ \frac{d^2v}{dt^2} + \frac{1}{RC} \frac{dv}{dt} + \frac{1}{LC} v = 0 \] or \[ \frac{d^2v}{dt^2} + 2 \alpha \frac{dv}{dt} + \omega_0^2 v = 0 \] **c.** If \( R = 50 \, \Omega \), is the system underdamped, critically damped, or overdamped? What is the equation that describes \( v(t) \)? **Circuit Diagram Explanation:** The circuit diagram consists of a resistor (R), an inductor (1 H), and a capacitor (1 mF), connected in series. The voltage \( v(t) \) is measured across the capacitor. The circuit also includes a current source labeled \( i \). Analyzing such RLC circuits involves setting up and solving differential equations to study the behavior of \( v(t) \) over time.
Transcribed Image Text:For the RLC circuit shown below, we wish to find \( v(t) \) for \( t > 0 \). **a. Derive** the governing equation for the voltage \( v \) using KCL at the top node using the following definitions: \( \alpha = \frac{1}{2RC} \), \( \omega_0^2 = \frac{1}{LC} \). You should obtain \[ \frac{d^2v}{dt^2} + \frac{1}{RC} \frac{dv}{dt} + \frac{1}{LC} v = 0 \] or \[ \frac{d^2v}{dt^2} + 2 \alpha \frac{dv}{dt} + \omega_0^2 v = 0 \] **c.** If \( R = 50 \, \Omega \), is the system underdamped, critically damped, or overdamped? What is the equation that describes \( v(t) \)? **Circuit Diagram Explanation:** The circuit diagram consists of a resistor (R), an inductor (1 H), and a capacitor (1 mF), connected in series. The voltage \( v(t) \) is measured across the capacitor. The circuit also includes a current source labeled \( i \). Analyzing such RLC circuits involves setting up and solving differential equations to study the behavior of \( v(t) \) over time.
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