LCR Circuits — Revision Notes

⚡ 30-Second Revision

Inductive Reactance — $X_L = omega L = 2pi f L$
Capacitive Reactance — $X_C = \frac{1}{omega C} = \frac{1}{2pi f C}$
Impedance (Series LCR) — $Z = sqrt{R^2 + (X_L - X_C)^2}$
Phase Angle — $an phi = \frac{X_L - X_C}{R}$

* $X_L > X_C implies phi > 0$ (Inductive, V leads I) * $X_C > X_L implies phi < 0$ (Capacitive, V lags I) * $X_L = X_C implies phi = 0$ (Resonance, V in phase with I)

Resonant Frequency — $omega_0 = \frac{1}{sqrt{LC}}$ or $f_0 = \frac{1}{2pisqrt{LC}}$
At Resonance — $Z = R$ (minimum), $I = V/R$ (maximum), $phi = 0$, $cosphi = 1$
Quality Factor — $Q = \frac{omega_0 L}{R} = \frac{1}{omega_0 C R} = \frac{1}{R}sqrt{\frac{L}{C}}$
Power Factor — $cos phi = \frac{R}{Z}$
Average Power — $P_{avg} = V_{rms} I_{rms} cos phi$

2-Minute Revision

LCR circuits are fundamental AC circuits with a resistor (R), inductor (L), and capacitor (C). Each component offers frequency-dependent opposition: inductive reactance ($X_L = omega L$) increases with frequency, while capacitive reactance ($X_C = 1/omega C$) decreases.

The total opposition, impedance ($Z = sqrt{R^2 + (X_L - X_C)^2}$), is a phasor sum, not arithmetic. The phase angle ($phi$) between voltage and current indicates if the circuit is inductive ($X_L > X_C$, V leads I), capacitive ($X_C > X_L$, V lags I), or resistive ($X_L = X_C$, V in phase with I).

The most critical concept is resonance, occurring when $X_L = X_C$. At the resonant frequency ($f_0 = 1/(2pisqrt{LC})$), impedance is minimum ($Z=R$), current is maximum, and the phase angle is zero. The Q-factor ($Q = (1/R)sqrt{L/C}$) quantifies the sharpness of this resonance, with higher Q meaning greater selectivity. Remember that only the resistor dissipates average power, given by $P_{avg} = V_{rms} I_{rms} cosphi$, where $cosphi$ is the power factor.

5-Minute Revision

An LCR circuit is a series (or parallel) combination of a resistor (R), inductor (L), and capacitor (C) driven by an AC source. The key to understanding it lies in the frequency-dependent reactances: $X_L = omega L$ (inductive reactance, increases with frequency) and $X_C = 1/(omega C)$ (capacitive reactance, decreases with frequency). The resistor's opposition, R, is constant.

The total opposition to current is impedance (Z), calculated as $Z = sqrt{R^2 + (X_L - X_C)^2}$. This is a phasor sum because the voltages across R, L, and C are out of phase. The **phase angle ($phi$)** between the total voltage and current is given by $anphi = (X_L - X_C)/R$. If $X_L > X_C$, the circuit is inductive (voltage leads current, $phi > 0$). If $X_C > X_L$, it's capacitive (voltage lags current, $phi < 0$).

Resonance is a crucial phenomenon where $X_L = X_C$. This occurs at the resonant frequency $f_0 = 1/(2pisqrt{LC})$. At resonance:

1
Impedance is minimum: $Z = R$.
2
Current is maximum: $I_{max} = V/R$.
3
Phase angle is zero: $phi = 0$, meaning voltage and current are in phase.
4
Power factor is maximum: $cosphi = 1$.

The Quality Factor (Q-factor), $Q = (1/R)sqrt{L/C}$, measures the sharpness of resonance. A high Q-factor means the circuit is very selective, responding strongly to a narrow band of frequencies around $f_0$. It's inversely proportional to R and directly proportional to $sqrt{L/C}$.

Power in AC circuits: Only the resistor dissipates average power. The average power is $P_{avg} = V_{rms} I_{rms} cosphi$, where $cosphi = R/Z$ is the power factor. At resonance, $cosphi = 1$, so $P_{avg} = V_{rms} I_{rms}$.

Example: A series LCR circuit has $R=20,Omega$, $L=0.1,\text{H}$, $C=100,mu\text{F}$ and is connected to a $100,\text{V}$, $50,\text{Hz}$ AC source.

1
$X_L = 2pi f L = 2pi(50)(0.1) = 10pi approx 31.4,Omega$.
2
$X_C = \frac{1}{2pi f C} = \frac{1}{2pi(50)(100 \times 10^{-6})} = \frac{1}{0.01pi} approx 31.8,Omega$.
3
$Z = sqrt{20^2 + (31.4 - 31.8)^2} = sqrt{400 + (-0.4)^2} = sqrt{400 + 0.16} approx sqrt{400.16} approx 20.004,Omega$. (Note: $X_L approx X_C$, so it's near resonance).
4
$anphi = (31.4 - 31.8)/20 = -0.4/20 = -0.02$. $phi approx -1.15^circ$ (slightly capacitive).
5
$f_0 = \frac{1}{2pisqrt{LC}} = \frac{1}{2pisqrt{0.1 \times 100 \times 10^{-6}}} = \frac{1}{2pisqrt{10^{-5}}} approx 50.3,\text{Hz}$. The circuit is indeed very close to resonance.

Prelims Revision Notes

LCR Circuits: Key Facts for NEET UG

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Components & Reactances — A series LCR circuit consists of a Resistor (R), Inductor (L), and Capacitor (C).

* Resistance (R): Constant, independent of frequency. Unit: Ohm ($Omega$). * **Inductive Reactance ($X_L$)**: Opposition by inductor. $X_L = omega L = 2pi f L$. Increases with frequency. Unit: Ohm ($Omega$). * **Capacitive Reactance ($X_C$)**: Opposition by capacitor. $X_C = \frac{1}{omega C} = \frac{1}{2pi f C}$. Decreases with frequency. Unit: Ohm ($Omega$).

1
Impedance (Z) — Total opposition to AC current. $Z = sqrt{R^2 + (X_L - X_C)^2}$. Unit: Ohm ($Omega$).

* Minimum Z occurs at resonance ($X_L = X_C$), where $Z=R$.

1
Phase Angle ($phi$) — Phase difference between total voltage (V) and current (I).

* $an phi = \frac{X_L - X_C}{R}$. * If $X_L > X_C$: Circuit is inductive, V leads I ($phi > 0$). * If $X_C > X_L$: Circuit is capacitive, V lags I ($phi < 0$). * If $X_L = X_C$: Circuit is purely resistive (at resonance), V and I are in phase ($phi = 0$).

1
Resonance — Occurs when $X_L = X_C$.

* Resonant Angular Frequency: $omega_0 = \frac{1}{sqrt{LC}}$ (rad/s). * Resonant Frequency: $f_0 = \frac{1}{2pisqrt{LC}}$ (Hz). * Characteristics at Resonance: $Z=R$ (minimum), $I=V/R$ (maximum), $phi=0$, power factor $cosphi=1$. * Voltage across L and C are equal in magnitude ($V_L = V_C$) but $180^circ$ out of phase, so their vector sum is zero.

1
Quality Factor (Q-factor) — Measures sharpness of resonance.

* $Q = \frac{omega_0 L}{R} = \frac{1}{omega_0 C R} = \frac{1}{R}sqrt{\frac{L}{C}}$. * High Q means sharp resonance, high selectivity (e.g., radio tuning). * $Q propto 1/R$, $Q propto sqrt{L}$, $Q propto 1/sqrt{C}$.

1
Power in AC LCR Circuits

* Power Factor: $cos phi = \frac{R}{Z}$. * Average Power Dissipated: $P_{avg} = V_{rms} I_{rms} cos phi = I_{rms}^2 R$. Only resistor dissipates power. * At resonance, $cosphi = 1$, so $P_{avg} = V_{rms} I_{rms}$.

Common Mistakes to Avoid:

Arithmetic sum of R, $X_L$, $X_C$ for impedance. Use phasor sum.
Arithmetic sum of $V_R, V_L, V_C$ for total voltage. Use phasor sum.
Forgetting to convert units (e.g., $mu\text{F}$ to F, $ext{mH}$ to H).
Confusing $omega$ (angular frequency) with $f$ (linear frequency).
Misinterpreting conditions for voltage leading/lagging current.
Assuming $V_L$ or $V_C$ are zero at resonance (only their sum is zero).

Vyyuha Quick Recall

Leads Current, Resists in Phase

Leads Current: In an inductor (L), voltage leads current by $90^circ$. (Think 'L' for Lead)
Resists in Phase: In a resistor (R), voltage and current are in phase.
Current Leads: In a capacitor (C), current leads voltage by $90^circ$ (or voltage lags current). (Think 'C' for Current leads)

For Resonance: Lovely Cancellation, Really Minimal Zed

Lovely Cancellation: $X_L = X_C$ at resonance.
Really Minimal Zed: Impedance (Z) is minimum (equal to R) at resonance.