Alternating Current (AC) Theory

• AC (Alternating Current) theory studies time-varying voltage/current circuits, unlike DC.
• AC signals periodically reverse direction, whereas DC signals maintain a single direction.
• Sine waves represent the typical form for AC voltage and current.

Sinusoidal Waveforms

• Sinusoidal voltage is ( v(t) = V_m \sin(\omega t + \phi) ), with ( V_m ) as peak voltage, ( \omega ) as angular frequency, and ( \phi ) as phase angle.
• Sinusoidal current is ( i(t) = I_m \sin(\omega t + \phi) ), with ( I_m ) denoting peak current.
• ( T ) represents the time to complete one full cycle.
• Frequency ( f ) indicates cycles per second, measured in Hertz (Hz), with ( f = 1/T ).
• Angular frequency ( \omega ) and frequency are related by ( \omega = 2\pi f ).
• The initial angle of a sinusoidal waveform at ( t = 0 ) is the phase angle ( \phi ), expressed in degrees or radians.
• Waveforms shift left with a positive ( \phi ) (leading) and right with a negative ( \phi ) (lagging).

AC Values: RMS, Average, and Peak

• RMS value equates AC voltage/current to a DC voltage/current's heating effect.
• For sinusoidal waveforms, ( V_{rms} = V_m / \sqrt{2} ) and ( I_{rms} = I_m / \sqrt{2} ).
• The average sinusoidal voltage/current over a full cycle equals zero.
• Over a half cycle, sinusoidal voltage averages to ( V_{avg} = 2V_m / \pi ), and current to ( I_{avg} = 2I_m / \pi ).
• Peak value ( V_m ) denotes the waveform's maximum voltage relative to zero.
• Peak-to-peak value ( V_{pp} = 2V_m ) signifies the difference between the maximum positive and negative voltage values.
• Crest factor ( C = V_m / V_{rms} ) represents the ratio of peak to RMS value.
• Form factor ( F = V_{rms} / V_{avg} ) represents the ratio of RMS to average value.

AC Circuit Elements

• In AC circuits, resistors behave like they do in DC circuits, with voltage and current in phase.
• Resistor voltage: ( v(t) = R \cdot i(t) ); instantaneous power: ( p(t) = v(t) \cdot i(t) = i^2(t) \cdot R ).
• Average power dissipated by a resistor in AC: ( P = I_{rms}^2 \cdot R = V_{rms}^2 / R ).
• Inductors oppose current changes, making voltage lead current by 90 degrees.
• Inductor voltage is ( v(t) = L \cdot di(t)/dt ), where L is the inductance.
• Inductive reactance, or ( X_L = \omega L = 2\pi f L ), is the opposition to AC current caused by inductors.
• Capacitors oppose voltage changes in AC circuits, so current leads voltage by 90 degrees.
• Capacitor current: ( i(t) = C \cdot dv(t)/dt ), where C is the capacitance.
• Capacitive reactance, or ( X_C = 1 / (\omega C) = 1 / (2\pi f C) ), is the opposition to AC current caused by capacitors.

Impedance

• Impedance ( Z ) signifies total AC current opposition, uniting resistance, inductive/capacitive reactance.
• Impedance is a complex value: ( Z = R + jX ) (( R ) = resistance, ( X = X_L - X_C ) = net reactance, ( j ) = imaginary unit).
• Impedance magnitude: ( |Z| = \sqrt{R^2 + X^2} ).
• Impedance phase angle: ( \theta = \arctan(X/R) ).
• In series RLC circuits: ( Z = R + j(X_L - X_C) ).
• In parallel RLC circuits, ( Y = 1/Z = 1/R + j(1/X_C - 1/X_L) ) (admittances are added).

AC Circuit Analysis

• Ohm's Law in AC circuits: ( V = I \cdot Z ) (( V ) and ( I ) are RMS values, ( Z ) is impedance).
• Kirchhoff's Laws also apply to AC circuits, utilizing phasors and complex impedances.
• Mesh and nodal analysis can be adapted for AC circuits using impedances instead of resistances.
• Superposition theorem involves separately summing the results of each independent source.
• Thevenin's/Norton's theorems simplify AC circuits into equivalent sources with impedances.

Power in AC Circuits

• Instantaneous power varies over time: ( p(t) = v(t) \cdot i(t) ).
• Average (real) power ( P ) measures actual dissipated watts in the circuit.
• In AC circuits, ( P = V_{rms} \cdot I_{rms} \cdot \cos(\theta) ), where ( \theta ) is the voltage/current phase angle.
• Reactive power ( Q ) reflects power exchange with reactive components, in VAR (Volt-Ampere Reactive).
• ( Q = V_{rms} \cdot I_{rms} \cdot \sin(\theta) ).
• Apparent power ( S = V_{rms} \cdot I_{rms} ) is the product of RMS voltage and current in VA (Volt-Amperes).
• ( S = \sqrt{P^2 + Q^2} )
• Power factor ( \pf = P/S = \cos(\theta) ) is the ratio of real to apparent power.
• Power factors near 1 signify efficient power use; low power factors indicate much reactive power.
• Power factor correction adds capacitors to inductive circuits, reducing the phase angle and improving the power factor.

Resonance

• Resonance occurs in AC circuits when ( X_L = X_C ).
• Series RLC circuit resonance frequency: ( f_r = 1 / (2\pi \sqrt{LC}) ).
• At resonance in a series RLC circuit, impedance is minimal (equal to ( R )), and current is maximal.
• Parallel RLC resonance also occurs when ( X_L = X_C ); impedance is maximal and the current is minimal.
• Bandwidth (( BW )) is the frequency range where current is at least ( 1/\sqrt{2} ) of its peak.
• Quality factor ( Q = f_r / BW ) indicates resonance peak sharpness.
• Series RLC circuit ( Q = \omega_r L / R = 1 / (\omega_r C R) ).
• Parallel RLC circuit ( Q = R / \omega_r L = \omega_r C R ).

Transformers

• Transformers use electromagnetic induction to transfer electrical energy between circuits.
• They have magnetically linked, electrically isolated coils.
• The primary winding connects to the source.
• The secondary winding connects to the load.
• Voltage ratio: ( V_1/V_2 = N_1/N_2 ) (( N_1 ) and ( N_2 ) are primary/secondary winding turns).
• Current ratio: ( I_1/I_2 = N_2/N_1 ).
• Ideal transformers: ( V_1 I_1 = V_2 I_2 ) (primary power = secondary power).
• Real transformers experience winding resistance losses, core losses (hysteresis/eddy currents), and leakage flux.
• Transformer efficiency ( \eta = (P_{out} / P_{in}) \times 100% ).
• Step-up transformers increase voltage (( N_2 > N_1 )), step-down transformers decrease voltage (( N_2 < N_1 )).

Three-Phase AC Systems

• Three-phase AC features three AC voltages, each 120 degrees out of phase.
• They are more efficient than single-phase for power transmission/distribution.
• Common configurations are wye (Y) and delta ((\Delta)).
• Wye connection: ( V_L = \sqrt{3} \cdot V_{ph} ), ( I_L = I_{ph} ) (( V_{ph} ) and ( I_{ph} ) are phase voltage/current).
• Delta connection: ( V_L = V_{ph} ), ( I_L = \sqrt{3} \cdot I_{ph} ).
• Total apparent power ( S ) in a three-phase: ( S = \sqrt{3} \cdot V_L \cdot I_L ).
• Total real power ( P ) is ( P = \sqrt{3} \cdot V_L \cdot I_L \cdot \cos(\theta) ) (( \theta ) is phase angle).
• Total reactive power ( Q ) is ( Q = \sqrt{3} \cdot V_L \cdot I_L \cdot \sin(\theta) ).