8.2 Impedance and admittance

Impedance and admittance are key concepts in AC circuit analysis. They describe how components resist or allow current flow in sinusoidal circuits. Understanding these properties helps engineers design and analyze complex electrical systems.

Impedance combines resistance and reactance, while admittance is its inverse. These concepts are crucial for calculating voltage, current, and power in AC circuits. They form the foundation for more advanced topics in steady-state sinusoidal analysis.

Impedance and Reactance

Impedance and its Components

• Impedance ($Z$) represents the total opposition to current flow in an AC circuit
  • Measured in ohms ($\Omega$)
  • Consists of resistance ($R$) and reactance ($X$)
    • $Z = R + jX$, where $j$ is the imaginary unit
• Reactance ($X$) is the opposition to current flow due to capacitance or inductance
  • Measured in ohms ($\Omega$)
  • Two types: capacitive reactance ($X_C$) and inductive reactance ($X_L$)
    • $X = X_L - X_C$

Capacitive and Inductive Reactance

• Capacitive reactance ($X_C$) is the opposition to current flow due to capacitance
  • Inversely proportional to frequency ($f$) and capacitance ($C$)
    • $X_C = \frac{1}{2\pi fC}$
  • Decreases with increasing frequency or capacitance
• Inductive reactance ($X_L$) is the opposition to current flow due to inductance
  • Directly proportional to frequency ($f$) and inductance ($L$)
    • $X_L = 2\pi fL$
  • Increases with increasing frequency or inductance

Impedance Triangle and Phasor Representation

• Impedance triangle graphically represents the relationship between resistance, reactance, and impedance
  • Resistance ($R$) on the horizontal axis, reactance ($X$) on the vertical axis, and impedance ($Z$) as the hypotenuse
  • $|Z| = \sqrt{R^2 + X^2}$ and $\theta = \tan^{-1}(\frac{X}{R})$, where $\theta$ is the phase angle
• Impedance can be represented as a complex number or phasor
  • $Z = |Z|\angle\theta$ or $Z = R + jX$
  • Phasors allow for easy analysis of AC circuits using complex algebra

Admittance

Admittance and its Components

• Admittance ($Y$) is the reciprocal of impedance and represents the ease of current flow in an AC circuit
  • Measured in siemens ($S$) or mhos
  • $Y = \frac{1}{Z} = G + jB$, where $G$ is conductance and $B$ is susceptance
• Conductance ($G$) is the reciprocal of resistance and represents the ease of current flow due to resistance
  • Measured in siemens ($S$) or mhos
  • $G = \frac{1}{R}$
• Susceptance ($B$) is the reciprocal of reactance and represents the ease of current flow due to capacitance or inductance
  • Measured in siemens ($S$) or mhos
  • $B = \frac{1}{X} = B_C - B_L$, where $B_C$ is capacitive susceptance and $B_L$ is inductive susceptance

Relationship between Admittance and Impedance

• Admittance and impedance are reciprocals of each other
  • $Y = \frac{1}{Z}$ and $Z = \frac{1}{Y}$
• Conductance and resistance are reciprocals, while susceptance and reactance are reciprocals with opposite signs
  • $G = \frac{1}{R}$, $R = \frac{1}{G}$, $B = -\frac{1}{X}$, and $X = -\frac{1}{B}$
• Admittance can be used to simplify parallel circuit analysis, while impedance simplifies series circuit analysis

AC Circuit Analysis

Ohm's Law for AC Circuits

• Ohm's law relates voltage, current, and impedance in AC circuits
  • $V = IZ$, where $V$ is the voltage phasor, $I$ is the current phasor, and $Z$ is the impedance
• Voltage and current phasors have magnitude and phase angle
  • $V = |V|\angle\theta_V$ and $I = |I|\angle\theta_I$
  • Phase angle difference between voltage and current depends on the circuit's impedance
• Ohm's law can be applied to series and parallel AC circuits
  • Series: $V_{total} = V_1 + V_2 + ... + V_n$, $I_{total} = I_1 = I_2 = ... = I_n$, $Z_{total} = Z_1 + Z_2 + ... + Z_n$
  • Parallel: $I_{total} = I_1 + I_2 + ... + I_n$, $V_{total} = V_1 = V_2 = ... = V_n$, $\frac{1}{Z_{total}} = \frac{1}{Z_1} + \frac{1}{Z_2} + ... + \frac{1}{Z_n}$

Power in AC Circuits

• Power in AC circuits consists of real power ($P$), reactive power ($Q$), and apparent power ($S$)
  • Real power: $P = |V||I|\cos\theta$, measured in watts ($W$)
  • Reactive power: $Q = |V||I|\sin\theta$, measured in volt-amperes reactive ($VAR$)
  • Apparent power: $S = |V||I|$, measured in volt-amperes ($VA$)
• Power factor ($PF$) is the ratio of real power to apparent power
  • $PF = \frac{P}{S} = \cos\theta$
  • Ranges from 0 to 1, with 1 being the ideal case (purely resistive load)
• Power triangle graphically represents the relationship between real, reactive, and apparent power
  • Real power ($P$) on the horizontal axis, reactive power ($Q$) on the vertical axis, and apparent power ($S$) as the hypotenuse
  • $S = \sqrt{P^2 + Q^2}$ and $\theta = \tan^{-1}(\frac{Q}{P})$