Impedance and admittance are key concepts in understanding how circuit elements behave with alternating current. They combine resistance, inductance, and capacitance effects into complex numbers, allowing us to analyze AC circuits using familiar DC techniques.

These concepts are crucial for sinusoidal steady-state analysis, as they help us calculate voltages, currents, and power in AC circuits. By representing circuit elements with impedances, we can apply methods like mesh analysis and node voltage to solve complex AC networks.

Impedance and Admittance

Fundamental Concepts

• Impedance (Z) represents total opposition to alternating current flow in sinusoidal steady-state analysis measured in ohms (Ω)
• Admittance (Y) represents ease of alternating current flow measured in siemens (S)
• Both quantities incorporate frequency-dependent effects of resistance, inductance, and capacitance
• Impedance expressed as Z = R + jX with real part (resistance) and imaginary part (reactance)
• Admittance expressed as Y = G + jB with real part (conductance) and imaginary part (susceptance)
• Relationship between impedance and admittance given by Y = 1/Z allows conversion between representations

Complex Number Representation

• Rectangular form (a + jb) used to represent real and imaginary components
• Magnitude calculated using Pythagorean theorem $|Z| = \sqrt{R^2 + X^2}$ or $|Y| = \sqrt{G^2 + B^2}$
• Phase angle determined using arctangent function $\theta = \tan^{-1}(X/R)$ for impedance or $\theta = \tan^{-1}(B/G)$ for admittance
• Conversion between rectangular (a + jb) and polar (r∠θ) forms simplifies calculations
• Euler's formula $e^{j\theta} = \cos \theta + j \sin \theta$ relates exponential and trigonometric representations
• j-operator notation (j = √(-1)) distinguishes resistive (real) and reactive (imaginary) components
• Phasor diagrams visualize magnitude and phase relationships between voltages and currents

Impedance of Circuit Elements

Resistors

• Impedance purely real and frequency-independent $Z_R = R$
• Admittance $Y_R = 1/R$
• Phase angle between voltage and current 0°
• Power dissipation in resistors calculated using $P = I^2R$ or $P = V^2/R$

Inductors

• Impedance $Z_L = j\omega L$ where ω represents angular frequency
• Admittance $Y_L = -j/(\omega L)$
• Impedance increases linearly with frequency
• Phase angle between voltage and current +90°
• Energy storage in magnetic field $E = \frac{1}{2}LI^2$

Capacitors

• Impedance $Z_C = 1/(j\omega C)$
• Admittance $Y_C = j\omega C$
• Impedance decreases inversely with frequency
• Phase angle between voltage and current -90°
• Energy storage in electric field $E = \frac{1}{2}CV^2$

Non-Ideal Components

• Real-world inductors include parasitic resistance (RL) modeled as series combination
• Real-world capacitors include parasitic resistance (RC) modeled as parallel combination
• Skin effect in conductors increases effective resistance at high frequencies
• Self-resonance in inductors and capacitors limits usable frequency range

Series and Parallel Combinations

Series Connections

• Total impedance sum of individual impedances $Z_{total} = Z_1 + Z_2 + ... + Z_n$
• Voltage division across series elements $V_k = \frac{Z_k}{Z_{total}} V_{source}$
• Series resonance occurs when inductive and capacitive reactances cancel
• Quality factor (Q) for series RLC circuit $Q = \frac{\omega L}{R} = \frac{1}{\omega RC}$

Parallel Connections

• Total admittance sum of individual admittances $Y_{total} = Y_1 + Y_2 + ... + Y_n$
• Equivalent impedance calculated using reciprocal of sum of reciprocals $\frac{1}{Z_{total}} = \frac{1}{Z_1} + \frac{1}{Z_2} + ... + \frac{1}{Z_n}$
• Current division in parallel branches $I_k = \frac{Y_k}{Y_{total}} I_{source}$
• Parallel resonance occurs when admittances of inductor and capacitor are equal and opposite
• Quality factor (Q) for parallel RLC circuit $Q = R\sqrt{\frac{C}{L}}$

Complex Circuit Analysis

• Convert between series and parallel representations of R-L, R-C, or R-L-C combinations
• Use delta-wye (Δ-Y) transformations for simplifying complex networks
• Apply superposition principle for circuits with multiple AC sources
• Utilize Thévenin and Norton equivalent circuits for AC network analysis
• Employ mesh current and node voltage methods for solving AC circuits