CHAPTER 15 (2): RC CIRCUITS

Chapter 15: RC Circuits (Part 2) - Parallel RC Circuits and Power Factor

Objectives of the Chapter

  • Analyze a parallel RC circuit.

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

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

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

  • Determine power and power factor of RC circuits.

Admittance and Impedance of Parallel RC Circuits

  • For parallel RC circuits, it is convenient to represent the resistor (R) and capacitor (C) by their admittances:

    • Conductance (G): The measure of how well a circuit conducts current.

    • Susceptance ( B_c): The measure of how a capacitor stores and returns reactive power.

    • Formula: ( Y = G + jB_c )

  • Admittance is the reciprocal of impedance, measured in Siemens (S).

Circuit Analysis Basics

  • The admittance of a parallel circuit can be computed using: ( Y = Y_1 + Y_2 + \ldots + Y_n )

  • Total admittance equation:

    • ( Y_T = Y_R + Y_C )

    • Where:

      • ( Y_R = G )

      • ( Y_C = jB_c )

Voltage and Currents in Parallel RC Circuit

  • The source voltage ( V_S ) is common across the parallel circuit.

  • Phasor relationships:

    • Reference phasor ( V_S = V_{S0} )

    • Supplied current ( I = I_Z )

  • Kirchhoff’s Current Law applies: -( I_T = I_R + I_C )

  • Overall circuit current leads the voltage.

Diagrams in Parallel RC Circuits

  • Admittance Diagram: Represents the relationship between circuit elements using admittance values.

  • Phasor Diagram: Used to visualize the phase relationship, reflecting the same values as the admittance diagram scaled by source voltage.

  • Draw both diagrams for clarity in circuit analysis.

Power in RC Circuits

  • Apparent power (S): Total power transferred; sum of true power (P) and reactive power (Q).

    • Units:

      • Apparent Power: volts-amperes (VA)

      • Reactive Power: volt-amperes reactive (VAR)

      • True Power: watts (W)

  • Power Calculations:

    • True Power: ( P = I^2 R )

    • Reactive Power: ( Q = I^2 X_C )

    • Apparent Power: ( S = V_S I_T )

Power Factor in RC Circuits

  • Power factor (PF) measures efficiency of power conversion:

    • Defined as ( PF = cos(f) ) where 0 ≤ PF ≤ 1.

    • RC circuits exhibit a leading power factor when current leads voltage.

Examples

  • Example calculations include drawing phasor diagrams and admittance diagrams for given circuit conditions, calculating true power, reactive power, apparent power, and establishing the power factor for both series and parallel configurations.

Summary of Chapter 15

  • Understanding parallel RC circuits involves working with admittance instead of impedance.

  • The key components include true power, reactive power, apparent power, and power factor, with the RC circuit demonstrating a leading power factor.