Formula sheet

AC Power Formula Sheet

AC power formulas for RMS values, phasors, impedance, power factor, energy, three-phase and unbalanced circuits, voltage drop, harmonics, fault current, and correction.

Branch: Electrical Engineering
Content: Formula sheet
Updated: Jun 20, 2026
Revision: v1.1.0 · reviewed

This formula sheet collects common relationships used in introductory AC power analysis. It assumes sinusoidal steady-state operation unless stated otherwise. For transients, faults, harmonics, saturation, switching converters, and unbalanced systems, these equations must be extended or replaced by the appropriate time-domain or sequence-domain model.

Design-case selection

Use formulas against a named operating case, not in isolation. A compact design table should include:

\text{case}=\{\text{normal},\ \text{peak},\ \text{starting},\ \text{fault},\ \text{transfer},\ \text{maintenance},\ \text{future}\}

Useful screening quantities are:

\displaystyle \text{loading}=\frac{S_{case}}{S_{rating}}

\text{current margin}=I_{rating}-I_{case}

\text{voltage margin}=V_{limit}-V_{case}

For multiple parallel units where one largest unit may be unavailable, a simple N-1 firm apparent-power capacity is:

\displaystyle S_{firm,N-1}=\sum S_{available}-\max(S_{unit})

This expression checks capacity only. It does not prove protection coordination, equal load sharing, cooling adequacy, physical separation, or maintainability.

Sinusoidal quantities

For a sinusoidal voltage:

v(t)=V_{peak}\sin(\omega t+\phi)

Angular frequency is:

\omega=2\pi f

For an ideal sine wave:

\displaystyle V_{rms}=\frac{V_{peak}}{\sqrt{2}}

\displaystyle I_{rms}=\frac{I_{peak}}{\sqrt{2}}

Peak-to-peak voltage is:

V_{pp}=2V_{peak}

Always state whether a value is RMS, peak, or peak-to-peak. Power-system voltages and currents are normally RMS values.

Phasor and impedance relations

For sinusoidal steady-state analysis:

\tilde{V}=\tilde{I}Z

Impedance is:

Z=R+jX

Admittance is:

\displaystyle Y=\frac{1}{Z}=G+jB

Inductive reactance:

X_L=\omega L

Capacitive reactance:

\displaystyle X_C=-\frac{1}{\omega C}

Impedance magnitude and phase:

|Z|=\sqrt{R^2+X^2}

\theta_Z=\operatorname{atan2}(X,R)

Use the two-argument arctangent so the phase remains in the correct quadrant. These formulas apply to linear components at one frequency. They do not describe nonlinear loads, saturation, or harmonic spectra by themselves.

Single-phase power

For sinusoidal voltage and current with phase angle $\phi$ :

P=V_{rms}I_{rms}\cos\phi

Q=V_{rms}I_{rms}\sin\phi

S=V_{rms}I_{rms}

Complex power:

\tilde{S}=P+jQ

Apparent power magnitude:

S=|\tilde{S}|=\sqrt{P^2+Q^2}

Power factor:

\displaystyle PF=\frac{P}{S}

For purely sinusoidal voltage and current:

PF=\cos\phi

If current is non-sinusoidal, total power factor is still $P/S$ , but it may not equal the displacement power factor of the fundamental component.

Balanced three-phase power

For a balanced three-phase system:

P=\sqrt{3}V_{LL}I_L\cos\phi

Q=\sqrt{3}V_{LL}I_L\sin\phi

S=\sqrt{3}V_{LL}I_L

where $V_{LL}$ is line-to-line RMS voltage and $I_L$ is line current.

For a wye-connected balanced load:

\displaystyle V_{phase}=\frac{V_{LL}}{\sqrt{3}}

I_{phase}=I_L

For a delta-connected balanced load:

V_{phase}=V_{LL}

I_L=\sqrt{3}I_{phase}

Balanced formulas are not valid for unbalanced loads unless each phase is analysed separately or symmetrical components are used.

Unbalanced Phase Screening

Total active power from phase quantities:

P_{total}=\sum_i V_i I_i \cos\phi_i

Total reactive power:

Q_{total}=\sum_i V_i I_i \sin\phi_i

Neutral current from phase-current phasors:

\tilde{I}_N=-(\tilde{I}_A+\tilde{I}_B+\tilde{I}_C)

Voltage unbalance percentage:

\displaystyle VU=\frac{\max|V_i-\bar{V}|}{\bar{V}}\times100\%

Unbalanced checks should identify phase loading, neutral current, grounding path, single-phase loads, harmonic triplen currents, and protection assumptions.

Current from power

Single-phase current from active power:

\displaystyle I=\frac{P}{VPF}

Balanced three-phase line current:

\displaystyle I_L=\frac{P}{\sqrt{3}V_{LL}PF}

Current from apparent power:

\displaystyle I=\frac{S}{V}

for single-phase systems, and:

\displaystyle I_L=\frac{S}{\sqrt{3}V_{LL}}

for balanced three-phase systems.

These equations explain why low power factor increases current for the same delivered active power.

Energy and Demand

Energy over time:

E=Pt

Average power:

\displaystyle P_{avg}=\frac{E}{t}

Demand factor:

\displaystyle DF=\frac{P_{max,demand}}{P_{connected}}

Load factor:

\displaystyle LF=\frac{P_{avg}}{P_{peak}}

Energy and demand checks should state billing interval, duty cycle, metering location, diversity assumptions, power factor, and whether demand is peak, average, or contractual.

Losses and voltage drop

Resistive loss:

P_{loss}=I^2R

Approximate single-phase voltage drop:

\Delta V \approx I(R\cos\phi+X\sin\phi)

Approximate balanced three-phase line-to-line voltage drop:

\Delta V_{LL} \approx \sqrt{3}I_L(R\cos\phi+X\sin\phi)

The signs of the reactive term depend on load power factor and the reference convention. For accurate design, use the governing standard, conductor temperature, actual route length, grouping factors, harmonic loading, and supply impedance.

Power factor correction

Reactive compensation needed to move from initial power factor $PF_1=\cos\phi_1$ to target power factor $PF_2=\cos\phi_2$ at active power $P$ :

Q_C=P(\tan\phi_1-\tan\phi_2)

For a three-phase capacitor bank with line-to-line RMS voltage $V_{LL}$ :

\displaystyle C_{Y,phase}=\frac{Q_C}{\omega V_{LL}^2}

for a wye-connected bank, and:

\displaystyle C_{\Delta,phase}=\frac{Q_C}{3\omega V_{LL}^2}

for a delta-connected bank.

These formulas assume balanced sinusoidal operation. Check switching steps, overcorrection, resonance, harmonic distortion, discharge resistors, voltage rating, and protection before specifying a capacitor bank.

Transformer and equipment loading

Apparent power rating:

S=VI

for single phase, and:

S=\sqrt{3}V_{LL}I_L

for balanced three phase.

Active power available at a given power factor:

P=S \cdot PF

A transformer loaded to 100 kVA can deliver 100 kW at unity power factor but only 80 kW at power factor 0.8, while carrying the same apparent loading.

Equipment utilization for a design case:

\displaystyle u=\frac{S_{case}}{S_{rating}}

Remaining apparent-power margin:

S_{margin}=S_{rating}-S_{case}

For load growth from present apparent power $S_0$ to future apparent power $S_f$ :

\displaystyle \text{growth margin}=\frac{S_{rating}-S_f}{S_{rating}}

These equations should be evaluated for normal operation, emergency loading, ambient-temperature derating, harmonic heating, and maintenance states. A positive margin in normal operation may disappear when one transformer, feeder, or cooling path is unavailable.

Per-Unit and Base Quantities

Three-phase base current:

\displaystyle I_{base}=\frac{S_{base}}{\sqrt{3}V_{LL,base}}

Single-phase base current:

\displaystyle I_{base}=\frac{S_{base}}{V_{base}}

Three-phase base impedance:

\displaystyle Z_{base}=\frac{V_{LL,base}^2}{S_{base}}

Per-unit impedance:

\displaystyle Z_{pu}=\frac{Z}{Z_{base}}

Percent impedance:

Z_{\%}=100Z_{pu}

Per-unit values simplify transformer, cable, motor, and fault studies, but bases must be stated clearly whenever voltage level or apparent-power base changes.

Fault Current Screening

Approximate single-phase short-circuit current:

\displaystyle I_{sc}\approx \frac{V}{|Z_{source}+Z_{line}|}

Approximate balanced three-phase short-circuit current:

\displaystyle I_{sc,3\phi}\approx \frac{V_{LL}}{\sqrt{3}|Z_{eq}|}

Short-circuit apparent power:

S_{sc}=\sqrt{3}V_{LL}I_{sc}

Transformer-limited fault current from percent impedance:

\displaystyle I_{sc}\approx I_{rated}\frac{100}{Z_{\%}}

These screening equations do not replace protection coordination. Check available fault current, X/R ratio, breaker interrupting rating, let-through energy, grounding method, arc-flash assumptions, and upstream/downstream selectivity.

Mini example: transformer-limited fault current

A 10 MVA transformer feeds a 13.8 kV bus and has 5.75% impedance. Rated line current is:

\displaystyle I_{rated}=\frac{10\times10^6}{\sqrt{3}(13.8\times10^3)}=418\ \text{A}

Transformer-limited short-circuit current is approximately:

\displaystyle I_{sc}\approx418\frac{100}{5.75}=7270\ \text{A}

This is a first screen. A complete study also checks upstream source strength, cable impedance, motor contribution, parallel transformers, grounding, X/R ratio, and source modes with generators, UPS systems, or battery converters.

Harmonic checks

Total harmonic distortion of current is commonly expressed as:

\displaystyle THD_I=\frac{\sqrt{I_2^2+I_3^2+I_4^2+\cdots}}{I_1}

where $I_1$ is fundamental RMS current and $I_2, I_3, \ldots$ are harmonic RMS currents.

Total RMS current is:

I_{rms,total}=\sqrt{I_1^2+I_2^2+I_3^2+\cdots}

Harmonic current increases heating even when it does not increase useful power. Neutral conductors, transformers, capacitor banks, and protective devices may need special evaluation when harmonic content is significant.

Mini example: three-phase load current

A balanced three-phase motor load requires:

P=75\ \text{kW}

at:

V_{LL}=400\ \text{V}

with:

PF=0.86

The line current is:

\displaystyle I_L=\frac{P}{\sqrt{3}V_{LL}PF}

Substitute:

\displaystyle I_L=\frac{75\,000}{\sqrt{3}(400)(0.86)}

I_L=126\ \text{A}

If the power factor were improved to 0.96 for the same active power and voltage:

\displaystyle I_L=\frac{75\,000}{\sqrt{3}(400)(0.96)}=113\ \text{A}

The active power is unchanged, but current and associated $I^2R$ losses are lower. A full design would still check motor starting, cable rating, voltage drop, protection coordination, harmonics, and duty cycle.

Validation notes

For calculation records, state:

whether values are RMS, peak, phase, line-to-line, or line-to-neutral;
whether the model assumes balanced, sinusoidal, steady-state operation;
voltage level, frequency, grounding method, and source mode;
power factor convention and sign convention for reactive power;
conductor temperature, cable route length, grouping, and installation assumptions;
transformer impedance, source impedance, and fault-current contribution assumptions;
metering location and whether demand is instantaneous, interval-averaged, or contractual;
acceptance criteria for current, voltage drop, harmonic distortion, equipment loading, and protection operation.

The calculation is not complete until the result is linked to a rating, limit, measurement, or operating decision.

Common cautions

Do not use balanced three-phase equations for unbalanced systems without checking phase quantities. Do not size equipment from kW alone when power factor or harmonic current is important. Do not add power factor correction capacitors without checking resonance and light-load overvoltage. Do not treat RMS, peak, and peak-to-peak values as interchangeable. Do not assume that a breaker can interrupt a fault simply because its continuous current rating is high enough.

REF

Disciplines