AcademyAlternating Current
Academy
Phasors and AC Signals
Level 1 - Physics topic page in Alternating Current.
Principle
Alternating-current signals vary sinusoidally in time. A phasor represents a sinusoid as a rotating vector, so phase differences become angle differences.
Notation
\(v(t)\)
instantaneous voltage
\(\mathrm{V}\)
\(i(t)\)
instantaneous current
\(\mathrm{A}\)
\(V_0, I_0\)
peak voltage and current
\(\mathrm{V,\;A}\)
\(V_{\mathrm{rms}}, I_{\mathrm{rms}}\)
root-mean-square voltage and current
\(\mathrm{V,\;A}\)
\(\omega\)
angular frequency
\(\mathrm{rad\,s^{-1}}\)
\(f\)
frequency
\(\mathrm{Hz}\)
\(\phi\)
phase angle
\(\mathrm{rad}\)
Method
Derivation 1: Write the sinusoid
An AC source is often modeled by a sine or cosine with fixed amplitude and angular frequency.
Voltage signal
\[v(t)=V_0\cos(\omega t+\phi_v)\]
Current signal
\[i(t)=I_0\cos(\omega t+\phi_i)\]
Frequency relation
\[\omega=2\pi f\]
Derivation 2: Convert peak to rms
RMS values give the equivalent DC value for heating in a resistor. For a sinusoid, the rms value is the peak value divided by \(\sqrt2\).
RMS voltage
\[V_{\mathrm{rms}}=\frac{V_0}{\sqrt2}\]
RMS current
\[I_{\mathrm{rms}}=\frac{I_0}{\sqrt2}\]
Derivation 3: Compare phases
The phase difference is \(\phi_v-\phi_i\). If the voltage phasor is ahead of the current phasor, voltage leads current. If it is behind, voltage lags current.
Rules
Angular frequency
\[\omega=2\pi f\]
Period
\[T=\frac{1}{f}=\frac{2\pi}{\omega}\]
Sinusoidal rms
\[V_{\mathrm{rms}}=\frac{V_0}{\sqrt2},\qquad I_{\mathrm{rms}}=\frac{I_0}{\sqrt2}\]
Phase difference
\[\phi=\phi_v-\phi_i\]
Examples
Question
A sinusoidal voltage has peak value
\[170\,\mathrm V\]
Find its rms value.Answer
\[V_{\mathrm{rms}}=\frac{V_0}{\sqrt2}=\frac{170}{\sqrt2}=120\,\mathrm V\]
Checks
- Use peak values in instantaneous sinusoid equations.
- Use rms values for average power calculations.
- Phase is an angle, so keep radians and degrees consistent.
- Phasors are a steady-state sinusoidal tool; they do not describe transients by themselves.