🔦Electrical Circuits and Systems II Unit 2 – Phasors and Complex Impedance in Circuits

Key Concepts and Definitions

• Phasors represent sinusoidal signals using complex numbers, capturing both magnitude and phase information
• Complex numbers consist of a real part and an imaginary part, expressed as $a + jb$ where $j = \sqrt{-1}$
• Impedance is the complex resistance of a circuit element, defined as $Z = R + jX$ where $R$ is resistance and $X$ is reactance
  • Reactance can be capacitive ($X_C = -\frac{1}{\omega C}$) or inductive ($X_L = \omega L$)
• Admittance is the reciprocal of impedance, defined as $Y = \frac{1}{Z} = G + jB$ where $G$ is conductance and $B$ is susceptance
• Kirchhoff's laws (KVL and KCL) still apply in AC circuits, but voltages and currents are represented as phasors
• AC power consists of real power (P), reactive power (Q), and apparent power (S), related by the power triangle ($S^2 = P^2 + Q^2$)
• Resonance occurs when the inductive and capacitive reactances cancel each other, resulting in a purely resistive impedance

Phasor Representation Basics

• Phasors are complex numbers that represent the amplitude and phase of sinusoidal signals
• The magnitude of a phasor represents the peak amplitude of the sinusoidal signal
• The angle of a phasor represents the phase shift of the sinusoidal signal relative to a reference
• Phasors can be expressed in rectangular form ($a + jb$) or polar form ($A\angle\theta$)
  • To convert from rectangular to polar: $A = \sqrt{a^2 + b^2}$ and $\theta = \tan^{-1}(\frac{b}{a})$
  • To convert from polar to rectangular: $a = A\cos\theta$ and $b = A\sin\theta$
• Phasors simplify the analysis of AC circuits by eliminating the need to work with time-varying sinusoidal functions
• Phasor addition and subtraction follow the rules of complex number arithmetic

Complex Numbers in Circuit Analysis

• Complex numbers are used to represent impedances, admittances, voltages, and currents in AC circuits
• The real part of a complex number represents the resistive component, while the imaginary part represents the reactive component
• Complex impedances can be combined using series and parallel combinations, similar to resistors in DC circuits
  • Series: $Z_{total} = Z_1 + Z_2 + ... + Z_n$
  • Parallel: $\frac{1}{Z_{total}} = \frac{1}{Z_1} + \frac{1}{Z_2} + ... + \frac{1}{Z_n}$
• Ohm's law can be applied using complex numbers: $V = IZ$ or $I = \frac{V}{Z}$
• Complex power is calculated using $S = VI^*$, where $I^*$ is the complex conjugate of the current phasor

Impedance and Admittance

• Impedance is the complex resistance of a circuit element, representing the opposition to current flow in an AC circuit
• Resistance (R) is the real part of impedance and is frequency-independent
• Reactance (X) is the imaginary part of impedance and is frequency-dependent
  • Capacitive reactance: $X_C = -\frac{1}{\omega C}$, where $\omega = 2\pi f$ is the angular frequency and $C$ is the capacitance
  • Inductive reactance: $X_L = \omega L$, where $L$ is the inductance
• Admittance is the reciprocal of impedance and represents the ease with which current flows in an AC circuit
• Conductance (G) is the real part of admittance and is frequency-independent
• Susceptance (B) is the imaginary part of admittance and is frequency-dependent
  • Capacitive susceptance: $B_C = \omega C$
  • Inductive susceptance: $B_L = -\frac{1}{\omega L}$

AC Circuit Analysis with Phasors

• Kirchhoff's voltage law (KVL) and Kirchhoff's current law (KCL) are used to analyze AC circuits using phasors
• KVL states that the sum of phasor voltages around a closed loop is zero: $\sum V = 0$
• KCL states that the sum of phasor currents entering a node is equal to the sum of phasor currents leaving the node: $\sum I_{in} = \sum I_{out}$
• Nodal analysis and mesh analysis techniques can be applied to AC circuits using phasors
  • Nodal analysis: Write KCL equations for each node and solve for node voltages
  • Mesh analysis: Write KVL equations for each mesh and solve for mesh currents
• Superposition theorem, Thévenin's theorem, and Norton's theorem can be used with phasors to simplify AC circuit analysis
• Phasor diagrams are graphical representations of phasor voltages and currents in an AC circuit, helping to visualize phase relationships

Power Calculations in AC Circuits

• AC power consists of three components: real power (P), reactive power (Q), and apparent power (S)
• Real power (P) is the average power consumed by resistive elements, measured in watts (W)
  • $P = VI\cos\theta$, where $\theta$ is the phase angle between voltage and current
• Reactive power (Q) is the power exchanged between inductive and capacitive elements, measured in volt-amperes reactive (VAR)
  • $Q = VI\sin\theta$
• Apparent power (S) is the total power supplied to the circuit, measured in volt-amperes (VA)
  • $S = VI$
• The power triangle relates the three power components: $S^2 = P^2 + Q^2$
• Power factor (PF) is the ratio of real power to apparent power: $PF = \frac{P}{S} = \cos\theta$
  • A power factor of 1 indicates a purely resistive load, while a power factor less than 1 indicates the presence of reactive elements

Frequency Response and Resonance

• Frequency response describes how a circuit's impedance, current, or voltage changes with frequency
• Resonance occurs when the inductive and capacitive reactances are equal in magnitude, resulting in a purely resistive impedance
• Series resonance occurs when the total impedance of a series RLC circuit is minimized
  • At series resonance: $X_L = X_C$, $f_r = \frac{1}{2\pi\sqrt{LC}}$, and $Z = R$
• Parallel resonance occurs when the total impedance of a parallel RLC circuit is maximized
  • At parallel resonance: $B_L = B_C$, $f_r = \frac{1}{2\pi\sqrt{LC}}$, and $Z = \frac{R}{1 + Q^2}$, where $Q = \frac{R}{\omega L}$ is the quality factor
• Bandwidth (BW) is the range of frequencies over which the power output is at least half the maximum value
  • $BW = \frac{f_r}{Q}$ for series and parallel resonant circuits
• Quality factor (Q) is a measure of the sharpness of the resonance peak and the selectivity of the circuit
  • $Q = \frac{f_r}{BW}$ for series and parallel resonant circuits

Practical Applications and Examples

• Phasors and complex impedance are used in the analysis and design of various electrical systems and components
• Power systems: Phasor analysis is used to study the flow of power in transmission lines, transformers, and generators
  • Example: A 500 kV transmission line with a 100 MW load and a 0.8 lagging power factor
• Filters: RLC circuits are used to design filters that attenuate or amplify specific frequency ranges
  • Example: A bandpass filter with a center frequency of 1 MHz and a bandwidth of 100 kHz
• Impedance matching: Complex impedance is used to match the impedance of a source to a load for maximum power transfer
  • Example: Matching a 50 Ω antenna to a 75 Ω transmission line using a quarter-wave transformer
• Oscillators: Resonant circuits are used to generate sinusoidal signals at a specific frequency
  • Example: A Colpitts oscillator using an LC tank circuit to generate a 10 MHz signal
• AC motors: Phasor analysis is used to study the performance and efficiency of AC motors under various load conditions
  • Example: A 3-phase induction motor with a rated power of 50 HP and a power factor of 0.85 at full load