AC Electrical Circuit Analysis Study Notes

AC SINUSOIDAL WAVEFORMS AND FUNDAMENTALS

Introduction to Alternating Current (AC)

• Definition of AC: Alternating current is current that alternates or flips back and forth between two polarities through time. This differs from Direct Current (DC), which is fixed in polarity and constant over time.
• Waveform Varieties: AC can range from simple laboratory standards (sine, triangle, square waves) to complex biological or musical signals.
• The Sine Wave: This is the simplest waveform, representing a simple vector rotating at a constant speed.
  • Cycle: One full repeat of the waveform.
  • Period ($T$): The time required to complete one cycle.
  • Frequency ($f$): The number of cycles per second, measured in Hertz ($Hz$). Formula: $f = \frac{1}{T}$.
  • Amplitude: Can be expressed as Peak ($V<em>p$) or Peak-to-Peak ($V</em>{pp}$). For a sine wave, $V<em>{pp} = 2V</em>p$.

Mathematical Description of Sine Waves

• General Format: The general expression for a voltage sine wave is: $v(t) = V<em>{DC} + V</em>P \sin(2\pi f t + \theta)$
  • $v(t)$: Voltage at time $t$.
  • $V_{DC}$: DC offset (vertical shift).
  • $V_P$: Peak value.
  • $f$: Frequency.
  • $\theta$: Phase shift in degrees ($+$ for leading/left-shifted, $-$ for lagging/right-shifted).
• Phase Shift Calculation: To find the shift in degrees: $\theta = 360^{\circ} \frac{\Delta t}{T}$
  • $\Delta t$: Time differential between the wave and a reference.
• Sines and Cosines:
  • A cosine wave is a sine wave leading by $90^{\circ}$: $\sin(2\pi f t + 90^{\circ}) = \cos(2\pi f t)$.
  • The cosine wave represents the derivative (slope) of the sine wave ($\frac{dv}{dt}$).

Effective Value and RMS

• Root Mean Square (RMS): Used to determine the equivalent DC power (effective heating value).
• Process: Square the waveform, find the average (mean) of the squared wave, and take the square root of that mean.
• Sine Wave RMS Conversion:
  • $V<em>{RMS} = V</em>P \times 0.707$
  • Alternately: $V<em>{RMS} = \frac{V</em>P}{\sqrt{2}}$
• Crest Ratio: The ratio of peak value to RMS value (approx. $1.414$ for sine waves).

Fourier Analysis

• Fourier Theorem: States that any repetitive waveform can be represented as a collection of sine and cosine waves of proper amplitude and frequency.
• Fundamental: The lowest frequency component.
• Harmonics: Integer multiples of the fundamental frequency.
  • Square Wave: Comprised of an infinite series of odd harmonics ($f, 3f, 5f…$) with inverse amplitude ($1, 1/3, 1/5…$).
  • Triangle Wave: Comprised of an infinite series of odd harmonics with an inverse square amplitude characteristic ($1, 1/9, 1/25…$).

COMPLEX NUMBERS, REACTANCE, AND IMPEDANCE

Complex Numbers

• Definition: AC quantities are vectors (magnitude and direction/phase) sitting on a complex plane with real ($Re$) and imaginary ($j$) axes.
• Forms:
  • Rectangular: $Real + jImaginary$
  • Polar: $Magnitude \angle \theta$
• Conversions:
  • $Magnitude = \sqrt{Real^2 + Imaginary^2}$
  • $\theta = \tan^{-1}(\frac{Imaginary}{Real})$
  • $Real = Magnitude \cos \theta$
  • $jImaginary = Magnitude \sin \theta$
• Operations:
  • Addition/Subtraction: Use Rectangular form.
  • Multiplication/Division: Use Polar form (multiply magnitudes and add angles; divide magnitudes and subtract angles).

Reactance and Impedance

• Reactance ($X$): The opposition to current flow in ideal capacitors and inductors, where voltage and current are $90^{\circ}$ out of phase.
• Capacitive Reactance ($X_C$):
  • Inversely proportional to frequency.
  • $X_C = -j \frac{1}{2\pi f C}$
• Inductive Reactance ($X_L$):
  • Directly proportional to frequency.
  • $X_L = +j 2\pi f L$
• Impedance ($Z$): The combination of resistance and reactance.
  • $Z = R + jX$
• Admittance Systems:
  • Admittance ($Y$): Reciprocal of impedance ($Y = 1/Z$), measured in Siemens.
  • Susceptance ($B$): Reciprocal of reactance ($B = 1/X$).
  • Conductance ($G$): Reciprocal of resistance ($G = 1/R$).

ANALYSIS TECHNIQUES FOR RLC CIRCUITS

Series RLC Circuits

• Rule: Current is the same everywhere in a series loop ($i<em>{total} = i</em>R = i<em>L = i</em>C$).
• Series Impedance: The vector sum of resistance and reactants.
  • $Z<em>{total} = R + jX</em>L - jX_C$
• Kirchhoff's Voltage Law (KVL): The vector sum of voltage drops must equal the source voltage. Simply summing magnitudes will produce incorrect results due to phase differences.
• Voltage Divider Rule (VDR):
  • $v<em>A = e \frac{Z</em>A}{Z_{Total}}$
• Inductor Quality Factor ($Q_{coil}$): Ratio of inductive reactance to internal coil resistance.
  • $Q<em>{coil} = \frac{X</em>L}{R_{coil}}$

Parallel RLC Circuits

• Rule: Voltage is the same across all components in a parallel connection.
• Parallel Impedance:
  • $Z<em>{total} = \frac{1}{\frac{1}{Z</em>1} + \frac{1}{Z_2} + …}$
  • Product-Sum Rule for two components: $Z<em>{total} = \frac{Z</em>1 Z<em>2}{Z</em>1 + Z_2}$
• Kirchhoff's Current Law (KCL): Sum of entering currents equals the sum of exiting currents (vector summation).
• Current Divider Rule (CDR):
  • For two branches: $i<em>1 = i</em>{Total} \frac{Z<em>2}{Z</em>1 + Z_2}$

Series-Parallel RLC Circuits

• These require identifying sub-groups that are strictly series or parallel.
• Simplify the circuit step-by-step from the farthest point of the source until a single equivalent impedance is found.
• Use Ohm's Law and divider rules to expand back and find specific branch values.

ANALYSIS THEOREMS

Source Conversions

• A voltage source ($E$) with series impedance ($Z<em>s$) is equivalent to a current source ($I$) with parallel impedance ($Z</em>p$).
• $Z<em>s = Z</em>p$
• $I = \frac{E}{Z_s}$
• $E = I \times Z_p$

Superposition Theorem

• In a multi-source linear bilateral network, any voltage or current can be found by summing the contributions of each source acting alone.
• Replace other voltage sources with shorts and current sources with opens.
• Crucial for circuits with sources of different frequencies; each frequency must be analyzed in its own sub-circuit.

Thévenin's and Norton's Theorems

• Thévenin: Any single-port network can be reduced to one voltage source ($E<em>{th}$) in series with an impedance ($Z</em>{th}$).
• Norton: Any single-port network can be reduced to one current source ($I<em>n$) in parallel with an impedance ($Z</em>n$).
• $Z<em>{th} = Z</em>n$ is the impedance looking into the port with all sources replaced by their internal impedances.

Maximum Power Transfer

• To achieve maximum power in a load from a source with internal impedance ($Z<em>i = R</em>i + jX_i$), the load impedance must be the complex conjugate of the internal impedance.
• $Z<em>{load} = R</em>i - jX_i$
• This results in the reactances cancelling, producing a purely resistive circuit.
• Efficiency at maximum power transfer is only $50\%$.

RESONANCE

Series Resonance

• Occurs when magnitudes of $X<em>L$ and $X</em>C$ are equal.
• Resonant Frequency ($f<em>0$): $f</em>0 = \frac{1}{2\pi \sqrt{LC}}$.
• At resonance, impedance is at a minimum ($Z = R$) and current is maximum.
• Circuit Q: $Q<em>{series} = \frac{X</em>0}{R_{total}}$. High Q results in high voltages across L and C (scaled by factor Q).
• Bandwidth (BW): The range of frequencies for which power is at least half the maximum. $BW = f<em>2 - f</em>1 = \frac{f_0}{Q}$.

Parallel Resonance

• At resonance, impedance is at a maximum.
• Phase Resonant Frequency: Frequency where phase angle is $0^{\circ}$.
• For low Q parallel circuits, the resonant frequency shifts downward:
  $f<em>0 = \frac{1}{2\pi \sqrt{LC}} \sqrt{1 - \frac{C R</em>{coil}^2}{L}}$
• Circuit Q: $Q<em>{parallel} = \frac{R</em>{parallel}}{X_L}$.

POLYPHASE POWER

Three-Phase Systems

• Consists of three sub-generators separated by $120^{\circ}$ phase angles.
• Configurations: Delta ($\Delta$) or Y (Wye).
• Y-Connection:
  • Line current ($I<em>L$) = Phase current ($I</em>{ph}$).
  • Line voltage ($V<em>L$) = $\sqrt{3} \times V</em>{ph}$.
• Delta-Connection:
  • Line voltage ($V<em>L$) = Phase voltage ($V</em>{ph}$).
  • Line current ($I<em>L$) = $\sqrt{3} \times I</em>{ph}$.
• Total Power: $P<em>{total} = 3 \times V</em>{ph} \times I_{ph} \times \cos \theta$

DECIBELS AND BODE PLOTS

The Decibel (dB)

• Power Gain: $G' = 10 \log_{10} G$
• Voltage Gain: $A'<em>v = 20 \log</em>{10} A_v$
• Common references:
  • dBW: Relative to 1 Watt.
  • dBm: Relative to 1 milliwatt.
  • dBV: Relative to 1 Volt.

Bode Plots

• A graphical representation of gain magnitude and phase vs. frequency.
• Lead Network: High-pass response (slope $+6 dB/octave$ or $+20 dB/decade$ below critical frequency).
• Lag Network: Low-pass response (slope $-6 dB/octave$ or $-20 dB/decade$ above critical frequency).
• Critical Frequency ($f_c$): The frequency where the gain is down $3 dB$ and the phase shift is $45^{\circ}$.