POWER

Angular frequency

ω = 2πf

Peak voltage from RMS

Vm = √2 Vrms

Peak current from RMS

Im = √2 Irms

Voltage time-domain form

v(t) = √2 Vrms sin(ωt + θ)

Current time-domain form

i(t) = √2 Irms sin(ωt + θ)

Ohm's law for AC circuits

I = V/Z

General impedance

Z = R + jX

Inductor impedance

ZL = jωL

Capacitor impedance

ZC = -j/(ωC)

Single-phase complex power

S = VI* = P + jQ

Real power from resistance

P = |I|²R

Reactive power from reactance

Q = |I|²X

Power factor from angles

PF = cos(θV - θI)

Power factor from power

PF = P/|S|

Power-factor angle

θ = cos⁻¹(PF)

Inductive load

Current lags voltage; Q is positive; power factor is lagging

Capacitive load

Current leads voltage; Q is negative; power factor is leading

Resistive load

Current and voltage are in phase; Q = 0; PF = 1

Y-connected voltage relationship

VLL = √3 Vphase

Y-connected phase voltage

Vphase = VLL/√3

Y-connected current relationship

IL = Iphase

Line-to-line voltage phasor

Vab = √3 Van ∠30°

Three-phase real power using line values

P3φ = √3 VLL IL cos(θ)

Three-phase reactive power using line values

Q3φ = √3 VLL IL sin(θ)

Three-phase apparent power using line values

|S3φ| = √3 VLL IL

Total current for parallel loads

IT = I1 + I2 + I3 + … using phasor addition

Total real power

PT = P1 + P2 + P3 + …

Total reactive power

QT = Q1 + Q2 + Q3 + …

Total complex power

ST = S1 + S2 + S3 + …

Balanced delta to Y shortcut

ZY = ZΔ/3

Balanced Y to delta shortcut

ZΔ = 3ZY

Single-phase instantaneous power frequency

Instantaneous power oscillates at twice the source frequency: 2f

Balanced three-phase instantaneous power

Total instantaneous power is constant

Positive reactive power

Inductive load

Negative reactive power

Capacitive load

Leading power factor

Capacitive load; current leads voltage

Lagging power factor

Inductive load; current lags voltage

Unity power factor

PF = 1 and Q = 0

Phasor division

Divide magnitudes and subtract angles

Phasor multiplication

Multiply magnitudes and add angles

Adding phasors

Convert to rectangular form, add real and imaginary parts, then convert back

Complex conjugate

Keep the magnitude and negate the angle