Series LCR Circuits and Resonance: Quick Revision

A brief note on Series LCR Circuits and Resonance. Read it before a test. Try each question at the end before you open its answer.

The series circuit

• One current: the same current flows in the resistor, the coil and the capacitor.
• Phases: $V_R$ is in step with the current. $V_L$ is $90^\circ$ ahead. $V_C$ is $90^\circ$ behind.
• Impedance: $Z = \sqrt{R^2 + (X_L - X_C)^2}$, in $\Omega$. Then $V = IZ$ for peak or for rms values.
• Phase angle: $\tan\phi = \dfrac{X_L - X_C}{R}$. Positive: inductive, the current lags. Negative: capacitive, the current leads.
• Reactances: $X_L = \omega L$ and $X_C = \dfrac{1}{\omega C}$, with $\omega = 2\pi f$.

Resonance

• Condition: $X_L = X_C$, so $\omega_0 = \dfrac{1}{\sqrt{LC}}$ and $f_0 = \dfrac{1}{2\pi\sqrt{LC}}$.
• At resonance: $Z = R$, $I_{max} = \dfrac{V}{R}$ and $\phi = 0$.
• Quality factor: $Q = \dfrac{\omega_0L}{R} = \dfrac{1}{R}\sqrt{\dfrac{L}{C}}$. At resonance $V_L = V_C = QV$.
• Bandwidth: $\Delta\omega = \dfrac{R}{L}$ and $\Delta f = \dfrac{f_0}{Q}$. A small $R$ gives a narrow peak.

Power

• Average power: $P = V_{rms}I_{rms}\cos\phi = I_{rms}^2R$, in W.
• Power factor: $\cos\phi = \dfrac{R}{Z}$. It is $1$ at resonance, and $0$ for a pure coil or capacitor.
• Apparent power: $V_{rms}I_{rms}$, in VA.
• A low power factor needs a larger current for the same power, and heats the wires more.

A worked example

A series circuit has $R = 36\ \Omega$, $X_L = 105\ \Omega$ and $X_C = 28\ \Omega$. It is joined to a source of $255$ V (rms). Find the current, the power factor and the average power.

1. Impedance. $Z = \sqrt{(36\ \Omega)^2 + (77\ \Omega)^2} = 85\ \Omega$.
2. Current. $I_{rms} = \dfrac{255\ \text{V}}{85\ \Omega} = 3.0\ \text{A}$.
3. Power factor. $\cos\phi = \dfrac{36\ \Omega}{85\ \Omega} = 0.42$.
4. Power. $P = I^2R = (3.0\ \text{A})^2 \times 36\ \Omega = 324$ W.

Practise with new numbers

Check yourself

1. A series circuit has $R = 9\ \Omega$ and a net reactance of $12\ \Omega$. What is the impedance, and what is the power factor?

2. A coil of $2.0$ mH and a capacitor of $0.50\ \mu\text{F}$ are in series. At what angular frequency is the circuit in resonance?

3. Does a pure capacitor use power from an AC source? Give the reason.