**Title: Analysis of a Balanced Three-Phase Delta-Wye System** **Problem Statement:** In the balanced three-phase delta-wye system shown below: - \( V_{ab} = 440 \angle 0^\circ \) V rms - \( V_{bc} = 440 \angle -120^\circ \) V rms - \( V_{ca} = 440 \angle -240^\circ \) V rms - \( Z_Y = 4.5 + j10.5 \, \Omega \) - \( Z_F = 0.5 + j1.5 \, \Omega \) Find the (a) magnitude of line current and (b) the average power delivered to the load. **Diagram Explanation:** 1. The diagram depicts a balanced delta-wye (Δ-Y) electrical network. 2. Points a, b, and c represent the connection points in the delta configuration. 3. \( Z_F \) represents the impedance in the delta configuration. 4. Points A, B, and C represent the connection points in the wye (Y) configuration. 5. \( Z_Y \) represents the impedance in the wye configuration. 6. The line current \( I_a \) flows from point a to A. **Solution Steps:** 1. **Convert Delta Impedances to Wye:** - Use the transformation formulas to convert \( Z_F \) into equivalent wye impedances. This enables the analysis of line and phase currents. 2. **Calculate Line Current Magnitude:** - Use the phase voltages \( V_{ab}, V_{bc}, \) and \( V_{ca} \) to determine the phase currents through \( Z_Y \). - Apply Ohm’s Law in the wye connection. 3. **Determine Average Power Delivered:** - Use the formula for power in a three-phase system: \[ P_{\text{avg total}} = \sqrt{3} \times V_L \times I_L \times \cos(\theta) \] - Calculate the power factor using the impedance angles. By following these steps, you can solve for the line current and average power delivered to the load in a balanced delta-wye system.
**Title: Analysis of a Balanced Three-Phase Delta-Wye System** **Problem Statement:** In the balanced three-phase delta-wye system shown below: - \( V_{ab} = 440 \angle 0^\circ \) V rms - \( V_{bc} = 440 \angle -120^\circ \) V rms - \( V_{ca} = 440 \angle -240^\circ \) V rms - \( Z_Y = 4.5 + j10.5 \, \Omega \) - \( Z_F = 0.5 + j1.5 \, \Omega \) Find the (a) magnitude of line current and (b) the average power delivered to the load. **Diagram Explanation:** 1. The diagram depicts a balanced delta-wye (Δ-Y) electrical network. 2. Points a, b, and c represent the connection points in the delta configuration. 3. \( Z_F \) represents the impedance in the delta configuration. 4. Points A, B, and C represent the connection points in the wye (Y) configuration. 5. \( Z_Y \) represents the impedance in the wye configuration. 6. The line current \( I_a \) flows from point a to A. **Solution Steps:** 1. **Convert Delta Impedances to Wye:** - Use the transformation formulas to convert \( Z_F \) into equivalent wye impedances. This enables the analysis of line and phase currents. 2. **Calculate Line Current Magnitude:** - Use the phase voltages \( V_{ab}, V_{bc}, \) and \( V_{ca} \) to determine the phase currents through \( Z_Y \). - Apply Ohm’s Law in the wye connection. 3. **Determine Average Power Delivered:** - Use the formula for power in a three-phase system: \[ P_{\text{avg total}} = \sqrt{3} \times V_L \times I_L \times \cos(\theta) \] - Calculate the power factor using the impedance angles. By following these steps, you can solve for the line current and average power delivered to the load in a balanced delta-wye system.
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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