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14.13 Find the value of the impedance Z, for maximum power transfer in the circuit shown in Figure 14- 29. -j4N j8n 20 2 12 20° V ZL 15 N 4 20° A FIGURE 14-29: Circuit schematic for problem 14.13.

14.13 Find the value of the impedance Z, for maximum power transfer in the circuit shown in Figure 14- 29. -j4N j8n 20 2 12 20° V ZL 15 N 4 20° A FIGURE 14-29: Circuit schematic for problem 14.13.

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
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How do I solve in steps the answer is supposed to be ZL = 21-j4.27ohms

Also shared my work. Is it correct or did I do something wrong? 

**Problem #14.13:** Find the value of the impedance \( Z_L \) for maximum power transfer in the circuit shown in Figure 14-29. **Circuit Description:** - The circuit includes an AC voltage source of \( 120 \angle 0^\circ \) V. - Components include a series resistor of 15Ω, two inductors with impedances of \( j2Ω \) each, and a parallel network. - A current source is provided, \( 4 \angle 0^\circ \) A, in parallel with a 20Ω resistor. - The load impedance is denoted as \( Z_L \). **Solution Steps:** 1. **Determine the Thevenin Impedance (\( Z_{TH} \)):** \[ Z_L = Z_{TH} \] 2. **Calculate the Equivalent Impedance (\( Z_{eq} \)):** - The parallel elements include: - A series combination of \( j2Ω \) and 15Ω. - Another series combination of \( j8Ω \) and 20Ω. - The equivalent impedance calculation: \[ Z_{eq} = \left( \frac{-j(15)}{-j+15} + \frac{j8 \times 20Ω}{j8 + 20Ω} \right) \] - Simplified to: \[ Z_{eq} = 2.1Ω + j4.3Ω \] - Both series and parallel calculations contribute to determining \( Z_{eq} \). 3. **Step 2:** - Calculate \( V_{TH} \): - The calculation for \( V_{TH} \) is not shown but would follow using circuit analysis such as mesh or nodal analysis. **Explanation:** This calculation uses concepts from AC circuit analysis, such as combining impedances in series and parallel, to determine the Thevenin equivalent circuit parameters for the purpose of maximizing power transfer to the load. The given impedances are expressed in complex form, involving real and imaginary components, requiring complex arithmetic for solutions.
Transcribed Image Text:**Problem #14.13:** Find the value of the impedance \( Z_L \) for maximum power transfer in the circuit shown in Figure 14-29. **Circuit Description:** - The circuit includes an AC voltage source of \( 120 \angle 0^\circ \) V. - Components include a series resistor of 15Ω, two inductors with impedances of \( j2Ω \) each, and a parallel network. - A current source is provided, \( 4 \angle 0^\circ \) A, in parallel with a 20Ω resistor. - The load impedance is denoted as \( Z_L \). **Solution Steps:** 1. **Determine the Thevenin Impedance (\( Z_{TH} \)):** \[ Z_L = Z_{TH} \] 2. **Calculate the Equivalent Impedance (\( Z_{eq} \)):** - The parallel elements include: - A series combination of \( j2Ω \) and 15Ω. - Another series combination of \( j8Ω \) and 20Ω. - The equivalent impedance calculation: \[ Z_{eq} = \left( \frac{-j(15)}{-j+15} + \frac{j8 \times 20Ω}{j8 + 20Ω} \right) \] - Simplified to: \[ Z_{eq} = 2.1Ω + j4.3Ω \] - Both series and parallel calculations contribute to determining \( Z_{eq} \). 3. **Step 2:** - Calculate \( V_{TH} \): - The calculation for \( V_{TH} \) is not shown but would follow using circuit analysis such as mesh or nodal analysis. **Explanation:** This calculation uses concepts from AC circuit analysis, such as combining impedances in series and parallel, to determine the Thevenin equivalent circuit parameters for the purpose of maximizing power transfer to the load. The given impedances are expressed in complex form, involving real and imaginary components, requiring complex arithmetic for solutions.
**Problem 14.13** **Objective:** Find the value of the impedance \( Z_L \) for maximum power transfer in the circuit shown in Figure 14.29. **Figure 14.29: Circuit Schematic for Problem 14.13** **Explanation of the Circuit:** - The circuit is composed of multiple components including a voltage source, resistors, an inductor, a capacitor, a current source, and an impedance \( Z_L \). - The voltage source provides \( 12 \angle 0^\circ \) volts. - A resistor with a resistance of \( 15 \, \Omega \) is connected in series with the voltage source. - A capacitor with an impedance of \( -j4 \, \Omega \) is connected in the circuit. - An inductor with an impedance of \( j8 \, \Omega \) is included. - Another resistor with a resistance of \( 20 \, \Omega \) is also present. - The circuit has a current source with a current of \( 4 \angle 0^\circ \) amperes flowing downwards. - The unknown impedance \( Z_L \) is to be determined for the condition of maximum power transfer. --- **Graph/Diagram Details:** The circuit diagram shows a series and parallel combination of components structured as follows: 1. The voltage source (\( 12 \angle 0^\circ \) V) is in series with the \( 15 \, \Omega \) resistor. 2. This combination connects in parallel to the series circuit of a \( -j4 \, \Omega \) capacitor and a \( j8 \, \Omega \) inductor. 3. The \( 20 \, \Omega \) resistor is in series with this combination. 4. The current source (\( 4 \angle 0^\circ \) A) is part of the loop and flows downwards. 5. The load impedance \( Z_L \) is connected after the \( 20 \, \Omega \) resistor. This configuration sets up the conditions to solve for \( Z_L \) using principles such as Thevenin’s theorem and conjugate matching for maximum power transfer. **Note:** For educational purposes, illustrations and detailed calculations are provided to aid in understanding how to solve such circuit problems systematically.
Transcribed Image Text:**Problem 14.13** **Objective:** Find the value of the impedance \( Z_L \) for maximum power transfer in the circuit shown in Figure 14.29. **Figure 14.29: Circuit Schematic for Problem 14.13** **Explanation of the Circuit:** - The circuit is composed of multiple components including a voltage source, resistors, an inductor, a capacitor, a current source, and an impedance \( Z_L \). - The voltage source provides \( 12 \angle 0^\circ \) volts. - A resistor with a resistance of \( 15 \, \Omega \) is connected in series with the voltage source. - A capacitor with an impedance of \( -j4 \, \Omega \) is connected in the circuit. - An inductor with an impedance of \( j8 \, \Omega \) is included. - Another resistor with a resistance of \( 20 \, \Omega \) is also present. - The circuit has a current source with a current of \( 4 \angle 0^\circ \) amperes flowing downwards. - The unknown impedance \( Z_L \) is to be determined for the condition of maximum power transfer. --- **Graph/Diagram Details:** The circuit diagram shows a series and parallel combination of components structured as follows: 1. The voltage source (\( 12 \angle 0^\circ \) V) is in series with the \( 15 \, \Omega \) resistor. 2. This combination connects in parallel to the series circuit of a \( -j4 \, \Omega \) capacitor and a \( j8 \, \Omega \) inductor. 3. The \( 20 \, \Omega \) resistor is in series with this combination. 4. The current source (\( 4 \angle 0^\circ \) A) is part of the loop and flows downwards. 5. The load impedance \( Z_L \) is connected after the \( 20 \, \Omega \) resistor. This configuration sets up the conditions to solve for \( Z_L \) using principles such as Thevenin’s theorem and conjugate matching for maximum power transfer. **Note:** For educational purposes, illustrations and detailed calculations are provided to aid in understanding how to solve such circuit problems systematically.
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