CHAPTER 16 (2): RL CIRCUITS

Chapter 16: RC Circuits (Part 2)

Parallel RL Circuits and Power Factor

  1. Overview

    • The chapter covers analysis of parallel RL circuits, power factor, and the resulting implications in AC circuits.

Objectives

  • Be able to analyze a parallel RL circuit.

  • Determine AC impedance and admittance of a parallel RL circuit.

  • Understand the phase relationship between applied voltage and currents in a parallel RL circuit.

  • Draw impedance and phasor diagrams of a parallel RL circuit.

  • Determine power and power factor of RL circuits.

Admittance and Impedance of Parallel RL Circuits

Key Definitions and Concepts

  • Admittance (Y): Measure of how easily a circuit allows current to flow, inversely related to impedance, expressed in Siemens (S).

  • Impedance (Z): Total opposition that a circuit offers to the flow of alternating current (AC) at a given frequency.

Analysis Techniques

  1. When analyzing parallel RL circuits, it is often more convenient to work with admittance:

    • Admittance Formula:[ Y_{T} = Y_{R} + Y_{L} = G - jB ]

    • Where:

      • G = conductance

      • B = susceptance

      • j = imaginary unit

      • Z = impedance

Extra Components of Analysis

  • Relationship of Voltages and Currents:

    • Kirchhoff’s Current Law applies, meaning the sum of currents entering a junction equals the sum of currents leaving.

  • Use of phasor diagrams assists in visualizing the relationships between these quantities.

Step-by-Step Analysis Procedure

  1. Calculate circuit admittance:[ Y = G - jB ]

  2. Determine the current through the components in the RL circuit using the calculated admittance.

  3. Draw phasor and admittance diagrams if necessary.

Power in RL Circuits

Categories of Power

  1. True Power (P): Power dissipated as heat, useful energy (Watts).

  2. Reactive Power (Q): Power that oscillates between the source and the inductor, does no useful work (VAR).

  3. Apparent Power (S): Total power in the circuit accounting for both true and reactive power (VA). (S = P+Q)

Power Factor

  • Defined as the ratio of True Power to Apparent Power (cos φ).

  • For RL circuits, power factor is generally lagging.

Power Calculation Formulas

  • True Power (P): [ P = I^2 R = (V_R)^2 / R ]

  • Reactive Power (Q): [ Q = I^2 X_L = (V_L)^2 / X_L ]

  • Apparent Power (S): [ S = I^2 Z_T = (V_S)^2 / Z_T = V_S I ]

Example Analysis

Example 16-3: Parallel RL Circuit Analysis

  • Given parameters: V = 10 V, R = 1 kΩ, X_L = 2 kΩ

  • Find: Power factor, true power, reactive power, and apparent power using definitions and formulas established.

  • Solution Summary:

    • Power factor = cos φ = 0.448 (lagging)

    • True Power (P) = 20 mW

    • Reactive Power (Q) = 40 mVAR

    • Apparent Power (S) = 44.6 mVA

Summary

  • Analysis of parallel RL circuits relies heavily on concepts of admittance, true power, reactive power, and power factor.

  • Important to derive results from phasor diagrams and component relationships in the circuit.