AC Circuit Fundamentals (Kankon)

eee-1109 EEE-1109 Basic Electrical Engineering

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PART 1: AC Circuit Fundamentals

1. Why AC instead of DC?

While batteries produce DC, the world runs on AC for four practical reasons:

• Generation: It is mechanically simpler to generate AC because the natural rotary motion of turbines produces sinusoidal waves.
• Transmission: AC allows us to use transformers to step up voltage. Transmitting at high voltage and low current significantly reduces power loss ($I^{2}R$) over long distances.
• Cost: Thinner, lighter cables can be used due to lower currents.
• Safety: Voltage can be easily stepped down to safe levels for consumer use.

2. The Sinusoid

A sinusoid is a signal in the form of a sine or cosine function. It is the standard waveform for AC because:

• It is the natural output of rotating generators.
• It changes smoothly with no sudden jumps (unlike square waves), protecting equipment.
• Mathematical Convenience: It is the only waveform that keeps its shape when added, subtracted, differentiated, or integrated.

Key Parameters:

• Amplitude ($V_m$): The peak value.
• Angular Frequency ($\omega$): How fast it oscillates in radians per second.
• Phase ($\varphi$): The horizontal shift of the wave.
• Lag vs. Lead: If a wave peaks before another, it “leads.” If it peaks after, it “lags”,.

Intuition: Think of “Phase” as a head start in a race. If Wave A leads Wave B by $30^{\circ }$, Wave A started the race early and crosses the finish line (peak) first.

3. The Phasor Domain

Solving AC circuits using sine and cosine calculus is difficult. We simplify this using Phasors.

• Definition: A phasor is a complex number (using real and imaginary axes) that represents the amplitude and phase of a sinusoid frozen at $t=0$.
• The Transformation: We convert time-domain functions (e.g., $v(t)=V_m\cos (\omega t+\theta )$) into frequency-domain phasors (e.g., $\mathbf{V}=V_m\angle \theta$).
• Euler’s Identity: The mathematical bridge that connects trigonometry (cos/sin) to complex exponentials ($e^{j\theta }$).

Intuition: A phasor is like taking a snapshot of a spinning wheel. Instead of calculating the spinning motion (calculus), we just look at the angle and length of the spoke in the photo (algebra). This turns difficult differential equations into simple algebra.

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PART 2: Circuit Elements & Impedance

In AC, “Resistance” is replaced by a broader concept called Impedance ($Z$).

1. Impedance ($Z$)

Impedance is the total opposition to current flow in an AC circuit. It has two parts:

• Resistance ($R$): The real part. Friction against electron flow.
• Reactance ($X$): The imaginary part. Energy storage (not burning) by inductors or capacitors.

2. The Passive Elements,:

• Resistor ($R$):
  • Behavior: Voltage and Current are in phase (they peak at the same time).
  • Impedance: Purely real.
• Inductor ($L$):
  • Behavior: Voltage leads Current by $90^{\circ }$.
  • Impedance: Positive imaginary reactance ($j\omega L$).
  • Note: Reactance increases as frequency increases (it blocks high frequencies).
• Capacitor ($C$):
  • Behavior: Voltage lags Current by $90^{\circ }$ (or Current leads Voltage).
  • Impedance: Negative imaginary reactance ($\frac{1}{j\omega C}$).
  • Note: Reactance decreases as frequency increases (it passes high frequencies).

Intuition (The mnemonic “ELI the ICE man”):

• ELI: In an Inductor (L), Voltage (E) leads Current (I).
• ICE: In a Capacitor (C), Current (I) leads Voltage (E).

3. Admittance ($Y$)

This is simply the reciprocal of Impedance ($1/Z$). It measures how easy it is for current to flow.

• Unit: Siemens (S).
• Components: Conductance ($G$) and Susceptance ($B$).

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PART 3: AC Power Analysis

Power in AC is more complex than DC because voltage and current are constantly changing and may not be in phase.

1. RMS Value (Effective Value)

• Definition: The RMS (Root Mean Square) value is the equivalent DC value that would produce the same amount of heat in a resistor.
• For a Sinusoid: The RMS value is the Peak Amplitude divided by $\sqrt{2}$.
• Form Factor: The ratio of RMS value to Average value.

2. The Power Triangle,

• Real Power ($P$): Measured in Watts (W). This is the “useful” power actually consumed by resistance to do work (heat, light).
• Reactive Power ($Q$): Measured in VAR (Volt-Ampere Reactive). This is energy bouncing back and forth between the source and the reactive elements (inductors/capacitors). It does no net work.
• Apparent Power ($S$): Measured in VA. The product of total RMS voltage and current. It represents the capacity of the supply system.

3. Power Factor (pf)

• Definition: The ratio of Real Power to Apparent Power ($\cos \theta$).
• Significance: It tells you how efficiently the current is being converted into useful work.
• Leading vs. Lagging:
  • Lagging pf: Inductive circuit (Current lags Voltage).
  • Leading pf: Capacitive circuit (Current leads Voltage).

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PART 4: Advanced Circuit Concepts

1. Resonance (Series)

Resonance occurs when the Inductive Reactance equals the Capacitive Reactance ($X_L=X_C$).

• Result: The reactances cancel each other out. The total impedance is purely resistive and is at its minimum possible value.
• Effect: Current is at its maximum.

2. Three-Phase Circuits

• Structure: Three AC sources connected together, each out of phase by $120^{\circ }$.
• Line vs. Phase:
  • Phase Voltage: Voltage across one component (source to neutral).
  • Line Voltage: Voltage between two lines (Line A to Line B). In a “Star” (Wye) connection, Line voltage is $\sqrt{3}$ times larger than Phase voltage.
• Advantage: Balanced 3-phase systems deliver constant, non-pulsating instantaneous power, which is better for motors.

3. Coupled Circuits

• Mutual Inductance: When two coils are close, the magnetic field of one induces a voltage in the other (the basis of transformers).
• Coefficient of Coupling ($K$): A measure of how tightly coupled the magnetic flux is between two coils ($0<K<1$).

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PART 5: Magnetic Circuits

Magnetic circuits (like transformer cores or motors) are analyzed using an analogy to Electric circuits, but with different variables.

1. Fundamental Magnetic Quantities

• Magnetic Flux ($\Phi$): The magnetic “current” or flow lines. Unit: Weber (Wb).
• Flux Density ($B$): How tightly packed the flux lines are per unit area. Unit: Tesla (T) or Wb/m².
• Magnetic Field Strength ($H$): The force trying to magnetize a material per unit length. Unit: Ampere-turns/meter (AT/m).
• Permeability ($\mu$): How easily a material allows magnetic flux to pass through it. Iron has high permeability; air has low.

2. The Electric-Magnetic Analogy (Ohm’s Law for Magnetism),

Concept	Electric Circuit	Magnetic Circuit
Driving Force	EMF (Volts)	MMF (Magnetomotive Force, Ampere-Turns)
Flow	Current ($I$)	Flux ($\Phi$)
Opposition	Resistance ($R$)	Reluctance ($S$)
Equation	$I=\frac{V}{R}$	$\Phi =\frac{\text{MMF}}{\text{Reluctance}}$

Intuition:

• MMF: Think of the coil of wire ($N$ turns $\times$ Current $I$) as the “Battery” of the magnetic circuit.
• Reluctance: Think of the iron core as a low-resistance wire for magnetism, and the “Air Gap” as a massive resistor that makes it hard for flux to cross.

3. Key Differences

While the math looks the same, the physics is different:

1. Flow: Current actually flows (electrons move). Flux does not flow; it is just a state established in the core.
2. Energy: Electric circuits require constant energy to maintain current ($I^{2}R$ loss). Magnetic circuits need energy to create the flux, but zero energy to maintain it (excluding small hysteresis losses).
3. Linearity: Resistance is usually constant. Reluctance is not constant—it changes depending on how much flux is already in the material (saturation).

4. Ampere’s Circuital Law

This is the “Kirchhoff’s Voltage Law” for magnetism. It states that the sum of magnetic forces (MMF) around a closed loop equals the current enclosed.

• Practical application: The MMF required to push flux through a composite core (e.g., Iron + Air Gap) is the sum of the MMF needed for the iron part plus the MMF needed for the air gap.