Understanding Active Power in Parallel Circuits: A Comprehensive Guide

Parallel circuits are fundamental in electrical engineering and form the backbone of many power distribution systems. When a resistor in parallel with an inductive circuit is supplied by an AC source, understanding the behavior and calculation of active power in the inductive circuit becomes crucial. In this article, we will explore the concept of parallel circuits, the parameters involved, and how to calculate the active power in an inductive circuit when a 220V, 50Hz supply is distributing 7A total current between a 55 ohm resistor and the inductive circuit.

Introduction to Parallel Circuits

A parallel circuit consists of two or more paths through which current can flow, with the same voltage across each path. In such a configuration, components can be arranged in parallel to distribute the load or to add additional devices in a circuit without affecting the existing ones.

Components and Parameters

In our scenario, we have a 55 ohm pure resistor and an inductive circuit both connected in parallel to a 220V, 50Hz AC supply. The supply distributes a total current of 7A. To solve the problem, we need to understand the current split between the resistor and the inductive circuit and then determine the active power in the inductive circuit.

Active Power and Power Factor

Active power (P) is the actual power consumed by a circuit to perform work. It is a measure of the real energy used, as opposed to apparent power which includes both active power and reactive power.

Power Factor and Reactive Power

The power factor (PF) is the ratio of active power to apparent power, and is crucial in understanding the efficiency of the circuit. Reactive power (Q), which is the power stored and released but not consumed, is not included in the active power calculation. It is commonly expressed as Q V * I * sin(θ), where θ is the phase angle between the voltage and current.

Calculating Active Power in the Inductive Circuit

Given:

Total current, I 7A Current through the resistor, IR 5A Supply voltage, V 220V Resistor value, RL 55 ohms

First, we need to find the current through the inductive circuit, IL.

IL I - IR 7A - 5A 2A

The active power (P) in the inductive circuit can be calculated using the formula:

P VI cos(θ)

Since the circuit is an inductive circuit, we need to determine the phase angle between the voltage and current. We can use Ohm's law and the fact that the current in the pure resistor is 5A to find the voltage drop across the resistor.

VR IR * RL 5A * 55 ohms 275V

Now, using the total voltage and the voltage drop across the resistor, we can find the equivalent impedance (Z) of the circuit.

Z V / I 220V / 3.46A 63.63 ohms (assuming the circuit is purely inductive for this example)

The phase angle (?) can be calculated using the formula:

tan(?) X L / R

Giving:

X L √(Z^2 - R^2) √(34.22^2 - 55^2) 34.22 ohms (hypotenuse of the impedance triangle)

tan(?) 34.22 / 55 0.623

? tan -1 (0.623) 32.2°

Now, we can calculate the active power:

P VI cos(?) 220V * 2A * cos(32.2°) 377.6 W

Conclusion

Understanding active power in parallel circuits is crucial for electrical engineers and technicians. The example problem demonstrates the importance of calculating the phase angle and using the power factor to determine the active power in the inductive circuit. By breaking down the problem step by step, we can accurately determine the power dissipated in the inductive circuit, even without a full detailed analysis of the circuit components.

To recap:

Parallel circuits distribute current in multiple paths Active power is the real energy consumed Power factor and phase angle are key in understanding the circuit's performance The active power in an inductive circuit can be calculated using the given current and voltage

Key Takeaways: Parallel circuits and their properties Calculating the current through inductive and resistive circuits Understanding power factor and reactive power Determining active power using voltage, current, and phase angle

References:

Stephen D. Brown, "Principles of Electronic Instrumentation", 4th Edition Richard M. Feller, "Basic Electricity", 6th Edition A. B. Dickson, "Electrical and Electronic Science for Technicians", 6th Edition