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Problem 5- Find ia(t) and vs(t) in the circuit below, given that the voltage of the 50mH inductor is V50mH (t) = 40Cos (200t - 20°) V Dg(1) 200 µF H 30 92 1092 in 50 mH

Problem 5- Find ia(t) and vs(t) in the circuit below, given that the voltage of the 50mH inductor is V50mH (t) = 40Cos (200t - 20°) V Dg(1) 200 µF H 30 92 1092 in 50 mH

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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### Problem 5: Electrical Circuit Analysis **Objective**: Find \( i_d(t) \) and \( v_s(t) \) in the circuit below, given that the voltage of the 50 mH inductor is \( v_{50mH}(t) = 40\cos(200t - 20^\circ) \) V. #### Given: - Voltage across the 50 mH inductor: \( v_{50mH}(t) = 40\cos(200t - 20^\circ) \) V #### Circuit Description: The circuit consists of the following components connected in series and parallel: - A voltage source \( v_s(t) \) - A 200 µF capacitor - A 30 Ω resistor - A 10 Ω resistor - A 50 mH inductor ##### Circuit Diagram: 1. **Voltage Source \( v_s(t) \):** Represented by a circle with a plus and minus sign indicating its polarity. 2. **200 µF Capacitor:** Connected in parallel with the 30 Ω resistor. 3. **30 Ω Resistor:** A standard resistor symbol connected parallel to the capacitor. 4. **10 Ω Resistor:** A series resistor symbol connected in series with the 50 mH inductor. 5. **50 mH Inductor:** An inductor symbol with \( i_d(t) \) represented as the current through it. ```plaintext v_s(t) | | ( + )--------| |----- 200 µF -------| ( - ) | | 30 Ω | | | |----- 10 Ω --------- .----| | | . | 50 mH L------> i_d(t) ``` #### Steps to Solve: 1. **Determine the impedance** \( Z_X(f) \) **of each component**: - Capacitor (C = 200 µF): \( Z_C = \frac{1}{j\omega C} = \frac{1}{j200 \cdot 200 \mu F} \) - Resistors (10 Ω, 30 Ω): \( Z_R = R \) (no frequency dependence) - Inductor (L = 50 mH): \( Z_L = j\omega L = j 200 \cdot 50 mH \
Transcribed Image Text:### Problem 5: Electrical Circuit Analysis **Objective**: Find \( i_d(t) \) and \( v_s(t) \) in the circuit below, given that the voltage of the 50 mH inductor is \( v_{50mH}(t) = 40\cos(200t - 20^\circ) \) V. #### Given: - Voltage across the 50 mH inductor: \( v_{50mH}(t) = 40\cos(200t - 20^\circ) \) V #### Circuit Description: The circuit consists of the following components connected in series and parallel: - A voltage source \( v_s(t) \) - A 200 µF capacitor - A 30 Ω resistor - A 10 Ω resistor - A 50 mH inductor ##### Circuit Diagram: 1. **Voltage Source \( v_s(t) \):** Represented by a circle with a plus and minus sign indicating its polarity. 2. **200 µF Capacitor:** Connected in parallel with the 30 Ω resistor. 3. **30 Ω Resistor:** A standard resistor symbol connected parallel to the capacitor. 4. **10 Ω Resistor:** A series resistor symbol connected in series with the 50 mH inductor. 5. **50 mH Inductor:** An inductor symbol with \( i_d(t) \) represented as the current through it. ```plaintext v_s(t) | | ( + )--------| |----- 200 µF -------| ( - ) | | 30 Ω | | | |----- 10 Ω --------- .----| | | . | 50 mH L------> i_d(t) ``` #### Steps to Solve: 1. **Determine the impedance** \( Z_X(f) \) **of each component**: - Capacitor (C = 200 µF): \( Z_C = \frac{1}{j\omega C} = \frac{1}{j200 \cdot 200 \mu F} \) - Resistors (10 Ω, 30 Ω): \( Z_R = R \) (no frequency dependence) - Inductor (L = 50 mH): \( Z_L = j\omega L = j 200 \cdot 50 mH \
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