Single-Phase & Three-Phase Circuits Notes

Last reviewed 16 Sept 2026 · 7 min read

Single-phase series AC circuits

Z = R + j (X_{L} - X_{C}), ∣ Z ∣ = R^{2} + (X_{L} - X_{C})^{2}, ϕ = tan^{- 1} \frac{X _{L} - X _{C}}{R}

Circuit	Impedance	Phase of current	Power factor
R–L series	$R^{2} + X_{L}^{2}$	Lags voltage by φ	Lagging, $cos ϕ = R / Z$
R–C series	$R^{2} + X_{C}^{2}$	Leads voltage by φ	Leading
R–L–C series	$R^{2} + (X_{L} - X_{C})^{2}$	Lags if $X_{L} > X_{C}$ , leads if $X_{C} > X_{L}$ , in phase at resonance	—

Impedance triangle — sides R (base), X (perpendicular), Z (hypotenuse), angle φ.

FormulaPower triangle

Active (real/true) power $P = V I cos ϕ$ — watts (W); converted to useful work/heat; $= I^{2} R$
Reactive power $Q = V I sin ϕ$ — volt-ampere reactive (VAR); oscillates between source and L/C; $= I^{2} X$
Apparent power $S = V I$ — volt-amperes (VA)

S^{2} = P^{2} + Q^{2}, Power factor = cos ϕ = \frac{P}{S} = \frac{R}{Z}

Inductive loads absorb reactive power (lagging pf); capacitive loads supply reactive power (leading pf).
Power factor ranges from 0 to 1 (unity for pure resistance).

At resonance, $X_{L} = X_{C}$ :

FormulaSeries resonance

f_{0} = \frac{1}{2 π L C}

Impedance minimum $= R$ ; current maximum $= V / R$ ; power factor unity
Quality factor $Q = \frac{ω _{0} L}{R} = \frac{1}{ω _{0} C R} = \frac{1}{R} \frac{L}{C}$
Voltage across L or C $= Q \times V$ — voltage magnification
Bandwidth $B W = f_{2} - f_{1} = \frac{R}{2 π L} = \frac{f _{0}}{Q}$ (half-power frequencies)

Series resonant circuits are acceptor circuits (tuning in radio receivers).

Admittance $Y = 1/ Z = G + j B$ (siemens); conductance $G$ , susceptance $B$ .
Parallel branch currents add as phasors; solve using admittances or phasor diagrams.

For a coil (R, L) in parallel with C:

f_{0} = \frac{1}{2 π} \frac{1}{L C} - \frac{R ^{2}}{L ^{2}} (\approx \frac{1}{2 π L C} for small R)

Impedance maximum (dynamic impedance $Z_{D} = \frac{L}{C R}$ ); line current minimum; unity power factor.
Current magnification — circulating current in the tank is Q times the line current.
Rejector circuit (filters, oscillators).

Three EMFs of equal magnitude and frequency, displaced by 120° electrically, generated in three windings of an alternator.

Phase sequence (RYB or ABC) — order in which phases reach maximum; determines direction of rotation of three-phase motors.

More power for the same conductor material — transmission needs less copper than single-phase for the same power.
Three-phase motors are self-starting and produce uniform (constant) torque; single-phase motors are not self-starting.
Constant total instantaneous power in balanced systems (less vibration).
Three-phase machines are smaller and more efficient for the same rating.
Single-phase loads can be supplied from three-phase systems.

Quantity	Star (Y)	Delta (Δ)
Line voltage vs phase voltage	$V_{L} = 3 V_{p h}$ (line voltage leads phase voltage by 30°)	$V_{L} = V_{p h}$
Line current vs phase current	$I_{L} = I_{p h}$	$I_{L} = 3 I_{p h}$ (line current lags phase current by 30°)
Neutral	Available (3-phase 4-wire)	Not available
Use	Distribution (LV 400/230 V supply), transmission	Loads, motors, transformer windings

FormulaThree-phase power (balanced)

P = 3 V_{p h} I_{p h} cos ϕ = 3 V_{L} I_{L} cos ϕ

Q = 3 V_{L} I_{L} sin ϕ, S = 3 V_{L} I_{L}

φ is the angle between phase voltage and phase current.

For the same impedance per phase and same line voltage, a delta-connected load draws three times the power of a star-connected load.
In a balanced star system, neutral current is zero.

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