Resonance in AC Circuits — Revision Notes

⚡ 30-Second Revision

Resonant Frequency: — $f_0 = \frac{1}{2\pi\sqrt{LC}}$ or $\omega_0 = \frac{1}{\sqrt{LC}}$
Series RLC Resonance:

- $X_L = X_C$ - Impedance $Z_{min} = R$ - Current $I_{max} = V/R$ - Phase angle $\phi = 0^\circ$, Power Factor $\cos\phi = 1$ - Voltage magnification: $V_L = V_C = Q \times V_{source}$ - Q-factor: $Q = \frac{\omega_0 L}{R} = \frac{1}{\omega_0 CR} = \frac{1}{R}\sqrt{\frac{L}{C}}$

Parallel RLC Resonance:

- $X_L = X_C$ - Impedance $Z_{max}$ (ideally infinite) - Current $I_{min} = V/Z_{max}$ - Phase angle $\phi = 0^\circ$, Power Factor $\cos\phi = 1$ - Current magnification: $I_L = I_C = Q \times I_{source}$ - Q-factor: $Q = \frac{R}{\omega_0 L} = \omega_0 CR = R\sqrt{\frac{C}{L}}$

Bandwidth: — $BW = f_0/Q$

2-Minute Revision

Resonance in AC circuits is a critical phenomenon where the inductive reactance ($X_L = 2\pi fL$) and capacitive reactance ($X_C = 1/(2\pi fC)$) become equal in magnitude. This occurs at a specific resonant frequency, $f_0 = 1/(2\pi\sqrt{LC})$.

For a series RLC circuit at resonance: The total impedance is minimum and equal to the resistance ($Z=R$). This leads to maximum current flow ($I_{max} = V/R$). The circuit behaves purely resistively, so the phase angle between voltage and current is $0^\circ$, and the power factor is unity.

A key feature is voltage magnification, where voltages across L and C can be much larger than the source voltage, with $V_L = Q \times V_{source}$. The Q-factor for series resonance is $Q = (\omega_0 L)/R$.

For a parallel RLC circuit at resonance: The total impedance is maximum (ideally infinite), leading to minimum current drawn from the source. Again, the phase angle is $0^\circ$ and the power factor is unity. Here, current magnification occurs, meaning currents circulating between L and C can be much larger than the source current, with $I_L = Q \times I_{source}$. The Q-factor for parallel resonance is $Q = R/(\omega_0 L)$.

The Quality Factor (Q) determines the sharpness of resonance, and bandwidth ($BW = f_0/Q$) indicates the range of frequencies around $f_0$ for which the circuit responds significantly. High Q means narrow BW and high selectivity.

5-Minute Revision

Resonance in AC circuits is a state where the inductive reactance ($X_L$) and capacitive reactance ($X_C$) cancel each other out, leading to unique circuit behaviors. This condition, $X_L = X_C$, defines the resonant frequency $f_0 = 1/(2\pi\sqrt{LC})$.

Series RLC Resonance:

Condition: — $X_L = X_C \implies 2\pi f_0 L = 1/(2\pi f_0 C)$.
Impedance: — Minimum, $Z = R$. This is because the reactive components cancel out, leaving only resistance.
Current: — Maximum, $I_{max} = V/R$. Due to minimum impedance.
Phase Angle & Power Factor: — $\phi = 0^\circ$, $\cos\phi = 1$. The circuit behaves purely resistively.
Voltage Magnification: — The voltage across the inductor ($V_L = IX_L$) and capacitor ($V_C = IX_C$) can be significantly larger than the source voltage ($V_{source}$). $V_L = V_C = Q \times V_{source}$.
Q-factor: — $Q = (\omega_0 L)/R = 1/(\omega_0 CR) = (1/R)\sqrt{L/C}$. A higher Q means a sharper resonance.
Application: — Acts as an 'acceptor' circuit, used in band-pass filters and radio tuning.

Parallel RLC Resonance:

Condition: — Same as series: $X_L = X_C$.
Impedance: — Maximum, $Z_{max}$. This is because the reactive currents in the L and C branches cancel out in the main line, drawing minimal current from the source.
Current: — Minimum from source, $I_{min} = V/Z_{max}$.
Phase Angle & Power Factor: — $\phi = 0^\circ$, $\cos\phi = 1$. The circuit behaves purely resistively from the source's perspective.
Current Magnification: — The currents circulating between the inductor and capacitor ($I_L$ and $I_C$) can be much larger than the source current. $I_L = I_C = Q \times I_{source}$.
Q-factor: — $Q = R/(\omega_0 L) = \omega_0 CR = R\sqrt{C/L}$.
Application: — Acts as a 'rejector' circuit, used in band-stop filters and oscillators (tank circuits).

Bandwidth (BW): The range of frequencies over which the circuit's response is significant. $BW = f_0/Q$. A high Q-factor implies a narrow bandwidth, indicating high selectivity.

Example: A series RLC circuit has $L = 200,\text{mH}$, $C = 50,mu\text{F}$, and $R = 5,Omega$. Find $f_0$ and Q.

1
Convert units: $L = 0.2,\text{H}$, $C = 50 \times 10^{-6},\text{F}$.
2
$f_0 = \frac{1}{2\pi\sqrt{LC}} = \frac{1}{2\pi\sqrt{0.2 \times 50 \times 10^{-6}}} = \frac{1}{2\pi\sqrt{10^{-5}}} \approx 50.3,\text{Hz}$.
3
$\omega_0 = 2\pi f_0 \approx 316.2,\text{rad/s}$.
4
$Q = \frac{\omega_0 L}{R} = \frac{316.2 \times 0.2}{5} = \frac{63.24}{5} \approx 12.65$.

Prelims Revision Notes

Resonance in AC circuits is a key concept for NEET, focusing on the interplay of inductive and capacitive reactances. The fundamental condition for resonance is $X_L = X_C$, which leads to the resonant frequency $f_0 = \frac{1}{2\pi\sqrt{LC}}$. Remember that $f_0$ depends only on L and C, not R.

Series RLC Circuit at Resonance:

Impedance: — Minimum, $Z = R$. This is the smallest opposition to current.
Current: — Maximum, $I_{max} = V/R$. This is why it's called an 'acceptor' circuit.
Phase Angle: — $\phi = 0^\circ$. Voltage and current are in phase.
Power Factor: — $\cos\phi = 1$ (unity).
Voltage Magnification: — $V_L$ and $V_C$ can be much greater than $V_{source}$. $V_L = V_C = Q \times V_{source}$.
Q-factor: — $Q = \frac{\omega_0 L}{R} = \frac{1}{\omega_0 CR} = \frac{1}{R}\sqrt{\frac{L}{C}}$. Higher Q means sharper resonance.
Bandwidth: — $BW = f_0/Q$. Narrower bandwidth for higher Q.

Parallel RLC Circuit at Resonance:

Impedance: — Maximum, $Z_{max}$ (ideally infinite). This is the largest opposition to current from the source.
Current from Source: — Minimum, $I_{min} = V/Z_{max}$. This is why it's called a 'rejector' circuit.
Phase Angle: — $\phi = 0^\circ$. Voltage and current are in phase.
Power Factor: — $\cos\phi = 1$ (unity).
Current Magnification: — Currents circulating between L and C ($I_L, I_C$) can be much greater than $I_{source}$. $I_L = I_C = Q \times I_{source}$.
Q-factor: — $Q = \frac{R}{\omega_0 L} = \omega_0 CR = R\sqrt{\frac{C}{L}}$.

Key Formulas to Memorize:

$f_0 = \frac{1}{2\pi\sqrt{LC}}$
$X_L = 2\pi fL$, $X_C = \frac{1}{2\pi fC}$
Series RLC Impedance: $Z = \sqrt{R^2 + (X_L - X_C)^2}$
Series Q-factor: $Q = \frac{\omega_0 L}{R}$
Parallel Q-factor: $Q = \frac{R}{\omega_0 L}$
Bandwidth: $BW = \frac{f_0}{Q}$

Common Traps: Confusing series vs. parallel characteristics (e.g., minimum impedance for parallel), unit conversion errors, and miscalculating powers of 10. Always ensure units are in SI before calculation.

Vyyuha Quick Recall

Reactances Equal, Series Minimum Impedance, Parallel Maximum Impedance. (Resonance: $X_L=X_C$. Series: Min Z, Max I. Parallel: Max Z, Min I).