Chapter 11: Problem 24
In a balanced three-phase system, the source has an abc sequence, is Y-connected, and \(\mathbf{V}_{\mathrm{an}}=120 / 20^{\circ} \mathrm{V} .\) The source feeds two loads, both of which are \(Y\) -connected. The impedance of load 1 is \(8+j 6 \Omega / \phi\). The complex power for the a \(\cdot\) phase of load 2 is \(600 / 36^{\circ} \mathrm{VA}\). Find the total complex power supplied by the source.
Short Answer
Step by step solution
Analyze Given Values
Calculate Load 1 Phase Current
Calculate Load 1 Power
Calculate Total Complex Power of Load 1
Calculate Total Complex Power Supplied
Perform Final Complex Addition
Convert to Polar Form
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Complex Power Calculation
To calculate complex power, you use the formula:
- \( S = P + jQ \)
- \( S = VI^* \)
In a three-phase system, particularly when loads are balanced and the sources are symmetric, the complex power per phase can be simplified as proportionate to total power. This simplifies the computation and helps in straightforward analysis.
Y-Connected Loads
Each load in the Y configuration connects to a single neutral point, meaning:
- The voltage across each phase of the load is equal to the line-to-neutral voltage.
- Total power in such configurations is calculated by multiplying the power in one phase by three.
Ohm's Law
- \( V = IZ \)
In the context of our exercise, Ohm's Law was used to find the phase current by dividing the voltage by the impedance of Load 1. This step is crucial to understanding load behavior and necessary for calculating the complex power. By understanding the relationship between voltage, current, and resistance, students can solve for unknown quantities in circuit analysis.
Impedance
- \( Z = R + jX \)
Understanding impedance helps determine how effectively a circuit resists and stores energy. In the exercise, solving impedance values allowed for current phase angles to be calculated, relevant in power calculations. With this understanding, students tackle circuit analysis confidently by calculating the current and subsequently the power values.
Three-Phase Circuits
The advantages of three-phase circuits include:
- More efficient power transmission.
- Constant power delivery, reducing pulsations found in single-phase systems.
- Reduced conductor material requirements.
The exercise reveals key insights into the balanced three-phase systems where calculations for power require symmetry in voltage and phase angles, simplifying analysis. These circuits prove to sustain efficiency and cost-effectiveness over single-phase circuits due to their inherent design and operational benefits.