Alternating Current (AC) Theory Basics

• AC (Alternating Current) theory is fundamental for understanding electricity generation, transmission, and diverse applications.
• AC is an electric current that periodically changes direction, unlike direct current (DC), which flows in only one direction.

Basic Concepts

• Alternating Current (AC): An electric current that reverses direction cyclically, typically with a sinusoidal pattern.
• Cycle: A full sequence of positive and negative changes in a waveform.
• Frequency (f): The number of cycles per second, measured in Hertz (Hz).
• Period (T): The duration of one complete cycle, where T = 1/f.
• Amplitude (Peak Value): The highest voltage or current value in a cycle.
• Instantaneous Value: The voltage or current value at a specific moment on the waveform.

Sinusoidal Waveforms

• AC voltage and current often exhibit sinusoidal waveforms, modeled by sine or cosine functions.
• Voltage: Expressed as ( v(t) = V_p \sin(\omega t + \phi) )
• ( V_p ) represents the peak voltage.
• ( \omega = 2\pi f ) defines the angular frequency.
• ( t ) denotes time.
• ( \phi ) is the phase angle measured in radians.
• Current: Represented as ( i(t) = I_p \sin(\omega t + \phi) )
• ( I_p ) indicates the peak current.

Root Mean Square (RMS) Values

• RMS values represent the effective AC voltage or current value.
• The formula for RMS voltage is ( V_{rms} = \frac{V_p}{\sqrt{2}} \approx 0.707 V_p ).
• The formula for RMS current is ( I_{rms} = \frac{I_p}{\sqrt{2}} \approx 0.707 I_p )
• RMS values are equivalent to the DC value providing the same average power.

Average Value

• The average value of a complete sinusoidal AC waveform is zero due to canceling positive and negative halves.
• A non-zero average value is possible with half-wave rectification, calculated over half a cycle.
• Half-wave rectified sine wave voltage: ( V_{avg} = \frac{2V_p}{\pi} \approx 0.637 V_p )
• Half-wave rectified sine wave current: ( I_{avg} = \frac{2I_p}{\pi} \approx 0.637 I_p )

AC Through Resistors

• Voltage and current are in phase within a purely resistive circuit.
• Ohm's Law applies instantaneously: ( v(t) = R \cdot i(t) )
• Power dissipated in the resistor is given by: ( P = V_{rms} I_{rms} = I_{rms}^2 R = \frac{V_{rms}^2}{R} )

AC Through Inductors

• In a purely inductive circuit, current lags voltage by 90 degrees.
• Inductive reactance (( X_L )) arises from the inductor's opposition to current changes.
• Inductive Reactance: ( X_L = \omega L = 2\pi f L )
• ( L ) is the inductance, measured in Henries (H).
• Voltage across the inductor: ( V = I X_L )
• Instantaneous power oscillates, but the average power dissipated by an ideal inductor over a complete cycle is zero.

AC Through Capacitors

• In purely capacitive circuits, current leads voltage by 90 degrees.
• Capacitive reactance (( X_C )) results from the capacitor's opposition to voltage changes.
• Capacitive Reactance: ( X_C = \frac{1}{\omega C} = \frac{1}{2\pi f C} )
• ( C ) represents capacitance, measured in Farads (F).
• Voltage across the capacitor: ( V = I X_C )
• Instantaneous power oscillates, but the average power dissipated by an ideal capacitor over a complete cycle is zero.

Impedance

• Impedance (( Z )) is the total opposition to current flow in an AC circuit, including both resistance and reactance.
• Impedance is a complex value with magnitude and phase.
• ( Z = R + jX )
• ( R ) is resistance.
• ( X = X_L - X_C ) is the net reactance, and ( j ) is the imaginary unit.
• Magnitude of Impedance: ( |Z| = \sqrt{R^2 + X^2} )
• Phase Angle: ( \phi = \arctan\left(\frac{X}{R}\right) )

Series RLC Circuits

• The same current flows through the resistor, inductor, and capacitor in a series RLC circuit.
• Total impedance: ( Z = R + j(X_L - X_C) )
• Current calculation: ( I = \frac{V}{Z} )
• Voltage across each component:
• ( V_R = IR ) (in phase with current)
• ( V_L = IX_L ) (leads current by 90 degrees)
• ( V_C = IX_C ) (lags current by 90 degrees)
• Voltages ( V_L ) and ( V_C ) are 180 degrees out of phase.

Parallel RLC Circuits

• In a parallel RLC circuit, the resistor, inductor, and capacitor experience the same voltage.
• Total admittance (( Y )), the reciprocal of impedance, simplifies analysis: ( Y = \frac{1}{Z} = \frac{1}{R} + j\left(\frac{1}{X_C} - \frac{1}{X_L}\right) )
• Current through each component:
• ( I_R = \frac{V}{R} ) (in phase with voltage)
• ( I_L = \frac{V}{X_L} ) (lags voltage by 90 degrees)
• ( I_C = \frac{V}{X_C} ) (leads voltage by 90 degrees)
• Currents ( I_L ) and ( I_C ) are 180 degrees out of phase.

Resonance

• Series Resonance: In a series RLC circuit, it happens when ( X_L = X_C ).
• Impedance is minimal (equal to R), and current is maximal.
• The circuit acts purely resistive.
• Resonant frequency: ( f_r = \frac{1}{2\pi\sqrt{LC}} )
• Parallel Resonance: In a parallel RLC circuit, it occurs when ( X_L = X_C ).
• Impedance is maximal, and current from the source is minimal.
• The circuit is purely resistive at resonance.
• Resonant frequency: ( f_r = \frac{1}{2\pi\sqrt{LC}} )

Power in AC Circuits

• Instantaneous Power: ( p(t) = v(t) \cdot i(t) )
• Average (Real) Power (P): The power consumed by the circuit, dissipated in resistors.
• ( P = V_{rms} I_{rms} \cos(\phi) )
• ( \phi ) is the phase angle between voltage and current.
• Reactive Power (Q): Power exchanged between the source and reactive components (inductors, capacitors).
• ( Q = V_{rms} I_{rms} \sin(\phi) )
• Reactive power circulates in the circuit, not consumed.
• Apparent Power (S): Product of ( V_{rms} ) and ( I_{rms} ).
• ( S = V_{rms} I_{rms} )
• ( S = \sqrt{P^2 + Q^2} )
• Power Factor (PF): Real power to apparent power ratio.
• ( PF = \cos(\phi) = \frac{P}{S} )
• Indicates power usage efficiency; 1 is ideal, 0 means no real power consumption.

Power Factor Correction

• Power factor improvement usually involves adding capacitors to inductive circuits.
• Reduced current flow for the same real power minimizes transmission losses.
• Utilities may impose extra charges for low power factors.

Transformers

• Used to increase or decrease AC voltages.
• Constructed with two or more coils around a shared core.
• Turns Ratio: ( a = \frac{N_p}{N_s} )
• ( N_p ) is the number of turns in the primary coil.
• ( N_s ) is the number of turns in the secondary coil.
• Voltage Transformation: ( \frac{V_p}{V_s} = a )
• Current Transformation: ( \frac{I_p}{I_s} = \frac{1}{a} )
• Power (Ideal Transformer): ( V_p I_p = V_s I_s )

AC Circuit Analysis Techniques

• Ohm's Law and Kirchhoff's Laws can be applied in AC circuit analysis using complex impedance.
• Nodal Analysis: Summing currents at nodes to analyze circuits.
• Mesh Analysis: Summing voltages around loops for circuit analysis.
• Superposition Theorem: Analyzing linear circuits by assessing each independent source's impact separately.
• Thevenin's and Norton's Theorems: Simplifying circuits to a voltage source and series impedance (Thevenin) or a current source and parallel impedance (Norton).

Polyphase Systems

• AC voltages, out of phase with each other, are used.
• Three-phase systems are typical for power distribution.
• Benefits include smoother power delivery, smaller conductors, and efficient motors.

AC Measurements

• Voltmeters and ammeters measure AC voltage and current, generally showing RMS values.
• Oscilloscopes display AC waveforms to measure frequency, amplitude, and phase.
• Power analyzers measure real power, reactive power, apparent power, and power factor.