AcademyAlternating Current
Academy
Power in AC Circuits
Level 1 - Physics topic page in Alternating Current.
Principle
Average power in a sinusoidal AC circuit depends on rms voltage, rms current, and the phase angle between them.
Notation
\(P_{\mathrm{avg}}\)
average real power
\(\mathrm{W}\)
\(V_{\mathrm{rms}}\)
rms source voltage
\(\mathrm{V}\)
\(I_{\mathrm{rms}}\)
rms current
\(\mathrm{A}\)
\(\phi\)
phase angle between voltage and current
\(\mathrm{rad}\)
\(\cos\phi\)
power factor
\(R\)
resistance where energy is dissipated
\(\mathrm{\Omega}\)
\(Z\)
impedance magnitude
\(\mathrm{\Omega}\)
Method
Derivation 1: Use rms quantities
For sinusoidal steady state, average power is the rms voltage-current product multiplied by the power factor.
Average power
\[P_{\mathrm{avg}}=V_{\mathrm{rms}}I_{\mathrm{rms}}\cos\phi\]
Power factor
\[\cos\phi=\frac{R}{Z}\]
Derivation 2: Use the resistor power
Only resistance dissipates net energy over a full cycle in an ideal LRC circuit.
Resistive power
\[P_{\mathrm{avg}}=I_{\mathrm{rms}}^2R\]
Equivalent voltage form
\[P_{\mathrm{avg}}=\frac{V_{\mathrm{rms}}^2R}{Z^2}\]
Derivation 3: Interpret reactive components
Ideal capacitors and inductors store energy during part of a cycle and return it during another part. Their average power over a cycle is zero.
Rules
Average AC power
\[P_{\mathrm{avg}}=V_{\mathrm{rms}}I_{\mathrm{rms}}\cos\phi\]
Power factor
\[\cos\phi=\frac{R}{Z}\]
Series LRC power
\[P_{\mathrm{avg}}=I_{\mathrm{rms}}^2R\]
Apparent power
\[S=V_{\mathrm{rms}}I_{\mathrm{rms}}\]
Examples
Question
An AC circuit has
\[V_{\mathrm{rms}}=120\,\mathrm V\]
\[I_{\mathrm{rms}}=2.0\,\mathrm A\]
and power factor \[0.80\]
Find average power.Answer
\[P_{\mathrm{avg}}=V_{\mathrm{rms}}I_{\mathrm{rms}}\cos\phi=(120)(2.0)(0.80)=192\,\mathrm W\]
Checks
- Use rms values in average power formulas.
- Ideal capacitors and inductors have zero average power over a cycle.
- The power factor is between \(0\) and \(1\) for passive series circuits.
- Apparent power is measured in volt-amperes; real average power is measured in watts.