2.2 Impedance and admittance concepts

Impedance and admittance are key concepts in AC circuit analysis. They measure how components oppose or allow current flow, respectively. Understanding these helps us analyze complex circuits and design efficient electrical systems.

Impedance combines resistance and reactance, while admittance includes conductance and susceptance. These concepts are crucial for calculating voltage, current, and power in AC circuits, and they play a vital role in various applications from power systems to audio equipment.

Impedance and Related Concepts

Understanding Impedance and Resistance

• Impedance measures opposition to current flow in AC circuits
• Represented by complex number Z with real and imaginary parts
• Expressed in ohms (Ω) and denoted by symbol Z
• Resistance forms real part of impedance
• Resistance opposes current flow in both AC and DC circuits
• Measured in ohms (Ω) and denoted by symbol R
• Resistance remains constant regardless of frequency

Reactance and Frequency Dependence

• Reactance constitutes imaginary part of impedance
• Opposes changes in voltage or current in AC circuits
• Measured in ohms (Ω) and denoted by symbol X
• Inductive reactance (XL) increases with frequency
• Capacitive reactance (XC) decreases with frequency
• Reactance equation for inductors: $X_L = 2\pi fL$
• Reactance equation for capacitors: $X_C = \frac{1}{2\pi fC}$
• Impedance magnitude calculated using Pythagorean theorem: $|Z| = \sqrt{R^2 + X^2}$
• Impedance phase angle calculated using arctangent: $\theta = \tan^{1}(\frac{X}{R})$

Applications and Examples

• Power systems use impedance for load balancing and fault analysis
• Audio equipment employs impedance matching for optimal sound quality
• Transmission lines rely on impedance calculations for signal integrity
• Biomedical sensors utilize impedance measurements for tissue characterization
• RF circuits depend on impedance matching for maximum power transfer (antenna design)

Admittance and Related Concepts

Admittance Fundamentals

• Admittance measures ease of current flow in AC circuits
• Reciprocal of impedance: $Y = \frac{1}{Z}$
• Expressed in siemens (S) and denoted by symbol Y
• Represented by complex number with real and imaginary parts
• Used in parallel circuit analysis and network calculations
• Admittance magnitude calculated as: $|Y| = \frac{1}{|Z|}$
• Admittance phase angle equals negative of impedance phase angle

Conductance and Susceptance Components

• Conductance forms real part of admittance
• Measures ease of current flow in phase with voltage
• Reciprocal of resistance: $G = \frac{1}{R}$
• Expressed in siemens (S) and denoted by symbol G
• Susceptance constitutes imaginary part of admittance
• Measures ease of current flow out of phase with voltage
• Reciprocal of reactance: $B = \frac{1}{X}$
• Expressed in siemens (S) and denoted by symbol B
• Admittance equation: $Y = G + jB$

Practical Applications

• Power factor correction in electrical systems utilizes admittance calculations
• Microwave circuit design relies on admittance parameters for component selection
• Antenna design incorporates admittance measurements for impedance matching
• Control systems use admittance analysis for stability assessment
• Electronic filter design employs admittance concepts for frequency response shaping

AC Circuit Analysis

Ohm's Law in AC Circuits

• Extends DC Ohm's law to AC circuits using complex numbers
• Expresses relationship between voltage, current, and impedance
• AC Ohm's law equation: $V = IZ$
• Current calculation: $I = \frac{V}{Z}$
• Voltage calculation: $V = IZ$
• Applies to individual circuit elements and entire networks
• Considers both magnitude and phase relationships

Phasor Representation and Complex Algebra

• Phasors represent sinusoidal voltages and currents as rotating vectors
• Simplifies AC circuit analysis by converting time-domain to frequency-domain
• Phasor notation: $V = |V|e^{j\theta}$ or $V = |V|∠\theta$
• Complex algebra used for phasor addition, subtraction, multiplication, and division
• Kirchhoff's laws applied using phasor representations of voltages and currents

Power Calculations in AC Circuits

• Apparent power (S) measured in volt-amperes (VA): $S = VI^*$
• Real power (P) measured in watts (W): $P = |V||I|\cos\theta$
• Reactive power (Q) measured in volt-amperes reactive (VAR): $Q = |V||I|\sin\theta$
• Power triangle relates apparent, real, and reactive power
• Power factor defined as ratio of real power to apparent power: $PF = \frac{P}{S} = \cos\theta$