AcademyAlternating Current
Academy
LRC Series Circuits in AC
Level 1 - Physics topic page in Alternating Current.
Principle
In a series LRC circuit, the same current passes through every component, but the component voltages are not generally in phase. Impedance combines resistance and net reactance.
Notation
\(R\)
series resistance
\(\mathrm{\Omega}\)
\(L\)
series inductance
\(\mathrm{H}\)
\(C\)
series capacitance
\(\mathrm{F}\)
\(X_L\)
inductive reactance
\(\mathrm{\Omega}\)
\(X_C\)
capacitive reactance
\(\mathrm{\Omega}\)
\(Z\)
series impedance magnitude
\(\mathrm{\Omega}\)
\(\phi\)
phase angle of source voltage relative to current
\(\mathrm{rad}\)
Method
Derivation 1: Find the reactances
Convert frequency to angular frequency, then compute the capacitor and inductor reactances.
Angular frequency
\[\omega=2\pi f\]
Reactances
\[X_L=\omega L,\qquad X_C=\frac{1}{\omega C}\]
Derivation 2: Build the impedance triangle
In a series circuit, the resistive voltage is in phase with current, the inductor voltage leads by \(90^\\circ\), and the capacitor voltage lags by \(90^\\circ\). The net reactive part is \(X_L-X_C\).
Impedance magnitude
\[Z=\sqrt{R^2+(X_L-X_C)^2}\]
Current
\[I_{\mathrm{rms}}=\frac{V_{\mathrm{rms}}}{Z}\]
Phase angle
\[\tan\phi=\frac{X_L-X_C}{R}\]
Derivation 3: Interpret the sign
If \(X_L>X_C\), the circuit is inductive and source voltage leads current. If \(X_C>X_L\), the circuit is capacitive and source voltage lags current.
Rules
Series LRC impedance
\[Z=\sqrt{R^2+(X_L-X_C)^2}\]
Series current
\[I_{\mathrm{rms}}=\frac{V_{\mathrm{rms}}}{Z}\]
Phase
\[\tan\phi=\frac{X_L-X_C}{R}\]
Component voltages
\[V_R=IR,\qquad V_L=IX_L,\qquad V_C=IX_C\]
Examples
Question
A series circuit has
\[R=30\,\Omega\]
\[X_L=80\,\Omega\]
and \[X_C=40\,\Omega\]
Find \(Z\).Answer
\[Z=\sqrt{30^2+(80-40)^2}=50\,\Omega\]
Checks
- Series elements share the same current.
- Reactances subtract because inductor and capacitor voltages are opposite in phase.
- The source voltage is the phasor sum of component voltages.
- \(Z\) is never less than \(R\) in a series LRC circuit.