CHAPTER 17 (2): RLC CIRCUITS

Chapter 17: RLC Circuits (Part 2)

Parallel RLC Circuits and Power

Objectives

  • After completing Part 2 of this chapter, you will be able to:

    • Determine AC impedance and admittance of parallel RLC circuits.

    • Analyze a parallel RLC circuit.

    • Understand the phasor relationship between applied voltage and circuit currents in parallel RLC circuits.

    • Draw admittance and phasor diagrams of parallel RLC circuits.

    • Determine power and power factor of RLC circuits.

Admittance and Impedance of Parallel RLC Circuits

  • In parallel RLC circuits, it's more convenient to represent resistor (R), inductor (L), and capacitance (C) by their admittances.

    • Admittance (Y) is the reciprocal of impedance (Z).

Key Equations:

  • Admittance of the circuit:Y = Y1 + Y2 + Y3 + ... + Yn

  • Components of Admittance:

    • G (conductance)

    • B (susceptance)

    • Y = G + jB

Components Relationships:

  • For Resistor:

    • Y_R = G

  • For Inductor:

    • Y_L = jB_L = -jX_L

  • For Capacitor:

    • Y_C = jB_C = j1/X_C

Analyzing Circuits

  • The total circuit current (I) and voltages are described using Kirchhoff’s Current Law and Ohm’s Law.

  • If B_C > B_L:

    • I total leads the voltage (V_s).

  • Voltage is a reference phasor (V = V_s / 0°).

Power in Parallel RLC Circuits

Power Components:

  • Apparent Power (S): Total power transferred from the source to RLC circuit, in volt-amperes (VA).

  • Reactive Power (Q): Power exchanged between the source and reactive components (inductor and capacitor); measured in volt-amperes reactive (VAR).

  • True Power (P): Power dissipated as heat in resistive components; measured in watts (W).

Power Analysis:

  • Varies for inductive and capacitive loads:

    • Inductive loads contribute to a lagging power factor.

    • Capacitive loads contribute to a leading power factor.

Power Calculations:

  • True Power: P = I²R

  • Apparent Power: S = I²Z_T

  • Reactive Power: Q = I²X_tot

Power Factor

  • Measures conversion efficiency of apparent power to true power.

  • Defined as:

    • 0 ≤ cos φ ≤ 1

  • Lagging when I(t) lags V_s(t), leading when I(t) leads V_s(t).

Specific Conditions:

  • For parallel RLC circuit:

    • Lagging power factor if B_C < B_L

    • Leading power factor if B_C > B_L

Example Problems

  • Example 17-2: Determine currents in a given circuit; draw phasor diagrams.

  • Example 17-3: Given values, calculate power factor, true power, reactive power, and apparent power in an RLC circuit.

Summary

  • In a parallel RLC circuit:

    • Total current leads the source voltage if B_C > B_L.

    • Total current lags the source voltage if B_C < B_L.

  • Power analysis of inductive RLC circuits is similar to that of RL or RC circuits depending on inductive or capacitive nature.