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Electricity and Magnetism

Parallel LCR Circuit

A parallel LCR circuit is an alternating current network where an inductor, a capacitor, and a resistor are connected in parallel across a sinusoidal voltage source. It functions as a rejector circuit because its impedance is maximum and the current is minimum at resonance.

UP TGT ScienceUP PGT PhysicsStudy Notes + Practice

Scope and prerequisites

TGT core

Conditions for parallel resonance, resonance frequency formula, phase relationship between voltage and branch currents, and basic numericals on impedance and current at resonance.

PGT extension

Admittance and susceptance analysis, dynamic resistance at resonance, quality factor derivation, sharpness of resonance, and bandwidth in parallel resonant circuits.

Prerequisites

Alternating Current and L-C-R Circuits
Complex numbers and phasor algebra
Concept of admittance, conductance, and susceptance in AC circuits

Circuit Configuration and Phasor Relations

In a parallel LCR circuit, a pure resistor of resistance R, a pure inductor of inductance L, and an ideal capacitor of capacitance C are connected in parallel across an AC source of voltage $V = V_{0}\sin(\omegat)$.
The alternating voltage across each parallel branch is the same, whereas the branch currents (I_R, I_L, I_C) differ in phase relative to the common voltage.
The current through the resistor (I_R) is in phase with the voltage, the current through the inductor (I_L) lags the voltage by 90°, and the current through the capacitor (I_C) leads the voltage by 90°.

Parallel Resonance Phenomenon

Parallel resonance occurs when the reactive component of the total current is zero, meaning the inductive susceptance equals the capacitive susceptance.
At resonance, the reactive branch currents (I_L and I_C) are equal in magnitude and opposite in phase, completely canceling each other out in the external circuit.
Consequently, the total line current drawn from the source is solely determined by the resistive branch and is at its minimum value, making the circuit impedance maximum.

Dynamic Resistance at Resonance

At resonance, the parallel LCR circuit acts as a pure resistance known as the dynamic resistance or rejector resistance.
The expression for dynamic resistance is given by R_d = L / (CR), which can also be written as Z_max = L / (CR).
Since L, C, and R are typical laboratory values, the dynamic resistance can be very large, resulting in a very small total current at resonance.

Quality Factor (Q-factor) and Bandwidth

The quality factor Q of a parallel LCR circuit is defined as the ratio of the circulating reactive current in the LC tank to the total supply current at resonance.
It is also expressed as $Q = \omega_{r}CR = R / (\omega_{r}L)$, which is the inverse of the series LCR circuit Q-factor expression.
The bandwidth (BW) of the resonance curve is given by B$W = f_{r} / Q = 1 / (2\piRC)$, determining the selectivity of the circuit.

Derivations

Resonance Frequency and Impedance of Parallel LCR Circuit

Conditions: Ideal parallel LCR circuit connected to an AC source of angular frequency ω

Write the admittance Y of the parallel combination: $Y = G + j(B_{C} - B_{L})$, where conductance $G = 1/R$, capacitive susceptance B_C = ωC, and inductive susceptance B_L = 1/(ωL).
Substitute the terms into the admittance equation: $Y = (1/R) + j(\omegaC - 1/(\omegaL))$.
The total impedance Z is the reciprocal of admittance: $Z = 1 / \sqrt((1/R^{2}) + (\omegaC - 1/(\omegaL))^{2})$.
At resonance, the imaginary part of admittance becomes zero: ω_rC - 1/(ω_rL) = 0.
Solve for the resonant angular frequency: ω_r = 1 / sqrt(LC).

Result: Resonant frequency $f_r = (1 / 2\pi) * \sqrt(1 / LC)$ and maximum impedance $Z_max = L / (CR)$

Derivation of Quality Factor (Q-factor)

Conditions: Parallel LCR circuit at resonance

Define Q-factor as the ratio of magnitude of inductive (or capacitive) current to the total supply current at resonance: $Q = I_{C} / I_{R}$.
Express the currents in terms of voltage V: I_C = V(ω_rC) and I_R = V / R.
Substitute the currents into the Q-factor ratio: $Q = V(\omega_{r}C) / (V / R) = \omega_{r}CR$.
Substitute ω_r = 1 / sqrt(LC) into the equation: $Q = (1 / \sqrt(LC)) * CR = R * \sqrt(C / L)$.

Result: $Q = \omega_rCR = R * \sqrt(C / L)$

Formula sheet

Formula / law	Equation	Variables and units	Conditions
Total Admittance of Parallel LCR	$$Y = \sqrt{\frac{1}{R^2} + \left(\omega C - \frac{1}{\omega L}\right)^2}$$	Y = Admittance, R = Resistance, $\omega = Angular frequency$, C = Capacitance, L = Inductance Siemens (S) or ohm&sup-1;	General AC parallel combination
Resonant Angular Frequency	$$\omega_r = \frac{1}{\sqrt{LC}}$$	ω_r = Resonant angular frequency, L = Inductance, C = Capacitance rad/s	Zero reactive susceptance condition
Resonant Frequency	$$f_r = \frac{1}{2\pi \sqrt{LC}}$$	f_r = Resonant frequency, L = Inductance, C = Capacitance Hertz (Hz)	Zero reactive susceptance condition
Dynamic Resistance (Impedance at Resonance)	$$Z_{max} = \frac{L}{CR}$$	Z_max = Maximum impedance, L = Inductance, C = Capacitance, R = Resistance Ohms (Ω)	At parallel resonance
Minimum Line Current at Resonance	$$I_{min} = \frac{V}{Z_{max}} = \frac{VCR}{L}$$	I_min = Minimum source current, V = RMS voltage, Z_max = Dynamic resistance Amperes (A)	At parallel resonance
Quality Factor (Q-factor)	$$Q = \omega_r CR = R \sqrt{\frac{C}{L}}$$	Q = Quality factor, ω_r = Resonant frequency, C = Capacitance, R = Resistance, L = Inductance Dimensionless	High Q approximation
Bandwidth of Resonance	$$\Delta f = \frac{f_r}{Q} = \frac{1}{2\pi RC}$$	Δf = Bandwidth, f_r = Resonant frequency, Q = Quality factor Hertz (Hz)	Symmetric resonance curve assumption
Phase Angle at Resonance	$$\tan \phi = 0 \implies \phi = 0^{\circ}$$	$\phi = Phase angle between voltage$ and total current Degrees or radians	At parallel resonance frequency

Graphs and experiments

Frequency Response: Impedance vs Frequency

Draw: A bell-shaped curve plotting Impedance (Z) on the y-axis against Frequency (f) on the x-axis, showing a sharp peak at $f = f_r$.

Interpret: Impedance reaches its maximum value ($Z_max = L/CR) at the resonant frequency f_r$, causing the circuit to reject currents of this frequency.

Exam use: Tested via conceptual MCQs on why parallel LCR circuits are called rejector circuits.

Frequency Response: Current vs Frequency

Draw: A dip-shaped curve plotting Line Current (I) on the y-axis against Frequency (f) on the x-axis, showing a sharp minimum at $f = f_r$.

Interpret: Line current drops to a minimum value ($I = V/Z_max) at resonance$ because inductive and capacitive susceptances cancel each other out.

Exam use: Used to evaluate understanding of minimum current conditions in rejector circuits.

Phase Angle vs Frequency Curve

Draw: A sigmoid-like curve plotting phase angle (φ) on the y-axis against frequency (f) on the x-axis, crossing zero at f_r.

Interpret: Below resonance (ω < ω_r), the circuit is inductive (φ < 0); at resonance, $\phi = 0; above resonance (\omega > \omega_{r})$, the circuit is capacitive (φ > 0).

Exam use: Asked in PGT questions regarding the inductive and capacitive behavior of the circuit off-resonance.

Solved numericals

TGT core

A parallel LCR circuit has an inductor of 0.2 H, a capacitor of 5 μF, and a resistor of 100 Ω connected across a 220 V AC source. Calculate the resonant frequency of the circuit.

Identify given parameters: L = 0.2 H, $C = 5 \times 10^{-6} F$, $R = 100 \Omega$.
Recall the resonance frequency formula: $f_r = (1 / 2\pi) * \sqrt(1 / LC)$.
Calculate LC product: L$C = 0.2 \times 5 \times 10^{-6} = 1 \times 10^{-6} s^{2}$.
Calculate square root of LC: sqrt(LC) = sqrt(1 × 10^-6) = 1 × 10^-3 s.
Substitute into the frequency formula: $f_r = 1 / (2 \times 3.1416 \times 1 \times 10^{-3})$.
Compute final value: $f_r = 1000 / 6.2832 = 159.15 Hz$.

Answer: 159.15 Hz

PGT extension

Find the dynamic resistance and the minimum current at resonance for a parallel LCR circuit with L = 2 H, $C = 0.5 \muF$, and $R = 50 \Omega$, when driven by a 100 V AC source.

Identify given parameters: L = 2 H, $C = 0.5 \times 10^{-6} F$, $R = 50 \Omega$, V = 100 V.
Recall the formula for dynamic resistance: Z_max = L / (CR).
Substitute the values: Z_max = 2 / (0.5 × 10^-6 × 50).
Simplify the denominator: 0.5 × 50 = 25, so denominato$r = 25 \times 10^{-6}$.
Calculate Z_max: Z_max = 2 / (25 × 10^-6) = 0.08 × 10⁶ = 80,000 $\Omega = 80 k\Omega$.
Calculate minimum line current at resonance: I_min = V / Z_max = 100 / $80000 = 1.25 \times 10^{-3} A$.

Answer: Dynamic resistanc$e = 80 k\Omega$, Minimum current = 1.25 mA

PGT extension

Calculate the quality factor and bandwidth of a parallel LCR circuit having L = 1 H, $C = 1 \muF$, and $R = 1000 \Omega at resonance$.

Identify given parameters: L = 1 H, $C = 1 \times 10^{-6} F$, $R = 1000 \Omega$.
Calculate resonant frequency f_r: f_r = 1 / (2π sqrt(LC)) = 1 / (2 × 3.1416 × sqrt(1 × 1 × 10^-6)) = 159.15 Hz.
Calculate Q-factor using $Q = R * \sqrt(C / L): Q = 1000 * \sqrt(1 \times 10^{-6} / 1) = 1000 \times 10^{-3} = 1$.
Alternatively, calculate Q via ω_rCR: $Q = (2 \times 3.1416 \times 159.15) \times 1000 \times 1 \times 10^{-6} = 1$.
Calculate bandwidth: Δ$f = f_{r} / Q = 159.15 / 1 = 159.15 Hz$.

Answer: Q = 1, Bandwidth = 159.15 Hz

Assumptions and limitations

The inductor is assumed to have negligible resistance, or its resistance is lumped into the total branch resistance R.
The source is assumed to be an ideal sinusoidal voltage source with zero internal resistance.
The formulas for parallel resonance assume high Q-factor approximations (ω_r ≈ 1 / sqrt(LC)) where exact resonance and anti-resonance frequencies converge.

Common misconceptions

Common error: Parallel LCR circuits have maximum current at resonance.

Correct: Parallel LCR circuits have minimum current at resonance.

At resonance, the impedance is maximum (dynamic resistance), which minimizes the total line current drawn from the source.

Common error: The resonance frequency formula for parallel LCR is different from series LCR.

Correct: The ideal resonance frequency formula $f_r = 1 / (2\pisqrt(LC))$ is identical for both ideal series and parallel LCR circuits.

Both circuits achieve resonance when the net reactive susceptance or reactance becomes zero, yielding the same unconditioned angular frequency expression.

Common error: Branch currents are zero at parallel resonance.

Correct: Only the net external line current is minimum; individual branch currents (I_L and I_C) can be very large.

A large circulating current flows between the inductor and capacitor inside the parallel tank circuit due to energy oscillation.

Quick revision

A parallel LCR circuit acts as a rejector circuit because its impedance is maximum at resonance.
At parallel resonance, total line current is minimum and in phase with the applied voltage ($\phi = 0)$.
Dynamic resistance of a parallel LCR circuit is given by Z_max = L / (CR).
The quality factor Q for a parallel LCR circuit is expressed as $Q = \omega_{r}CR = R * \sqrt(C / L)$.

Must remember

In a parallel LCR circuit, the alternating voltage across each parallel branch is the same, whereas branch currents differ in phase.
Parallel resonance occurs when the reactive component of the total current is zero, meaning inductive susceptance equals capacitive susceptance.
At resonance, total line current drawn from the source is at its minimum value, and the circuit impedance is at its maximum.
The dynamic resistance (or rejector resistance) at resonance is given by $R_d = \frac{L}{CR}$.
The Quality factor ($Q$) of a parallel LCR circuit is given by $Q = \omega_r CR = R \sqrt{\frac{C}{L}}$, which is the inverse of the series LCR Q-factor expression.
The bandwidth of a parallel resonant circuit is given by $\Delta f = \frac{f_r}{Q} = \frac{1}{2\pi RC}$.

Glossary

Parallel ResonanceCondition in a parallel LCR circuit where the reactive branch currents ($I_L$ and $I_C$) cancel out, resulting in minimum line current and maximum impedance.

Dynamic ResistanceThe equivalent pure resistance offered by a parallel LCR circuit at resonance, expressed as $Z_{max} = \frac{L}{CR}$.

Admittance ($Y$)The reciprocal of impedance, representing how easily a circuit permits current flow, expressed as $Y = \sqrt{\frac{1}{R^2} + \left(\omega C - \frac{1}{\omega L}\right)^2}$.

Rejector CircuitAnother name for a parallel LCR circuit because it offers maximum impedance at resonance and rejects (minimizes) the current at the resonant frequency.

Series vs Parallel LCR Circuit at Resonance

Parameter	Series LCR Circuit	Parallel LCR Circuit
Impedance at Resonance	Minimum ($Z = R$)	Maximum ($Z_{max} = \frac{L}{CR}$ [Dynamic Resistance])
Current at Resonance	Maximum ($I = \frac{V}{R}$)	Minimum ($I_{min} = \frac{VCR}{L}$)
Circuit Nature	Acceptor circuit	Rejector circuit
Quality Factor ($Q$)	$Q = \frac{\omega_r L}{R} = \frac{1}{\omega_r CR}$	$Q = \omega_r CR = R\sqrt{\frac{C}{L}}$

Practice Quiz

Choose an option to reveal the explanation.

1. What is the phase relationship between the alternating voltage and the current through the pure resistor in a parallel LCR circuit connected to $V = V_0 \sin(\omega t)$?

Explanation: In a pure resistive branch, the current is always in phase with the applied alternating voltage.

2. In a parallel LCR circuit, the current through the inductor ($I_L$) relative to the source voltage:

Explanation: In a pure inductive branch, the alternating current lags the voltage by $90^\circ$.

3. What happens to the total line current drawn from the source in a parallel LCR circuit at resonance?

Explanation: At parallel resonance, reactive branch currents cancel out, resulting in maximum impedance and minimum total line current.

4. What is the expression for the resonant frequency ($f_r$) of an ideal parallel LCR circuit?

Explanation: The resonant frequency formula is $f_r = \frac{1}{2\pi \sqrt{LC}}$, which is identical to that of a series LCR circuit.

5. What is the dynamic resistance ($Z_{max}$) of a parallel LCR circuit at resonance?

Explanation: At parallel resonance, the dynamic resistance (or rejector resistance) is given by $Z_{max} = \frac{L}{CR}$.

6. Why is a parallel LCR circuit referred to as a 'rejector circuit'?

Explanation: Because it offers maximum impedance at resonance, it impedes (rejects) the flow of total line current at the resonant frequency.

7. Which of the following represents the correct expression for the total admittance ($Y$) of a parallel LCR circuit?

Explanation: Admittance is the reciprocal of impedance. For parallel branches with conductance $G = \frac{1}{R}$ and net susceptance $B = \omega C - \frac{1}{\omega L}$, $Y = \sqrt{G^2 + B^2}$.

8. What is the Quality factor ($Q$) expression for a parallel LCR circuit?

Explanation: The Q-factor of a parallel LCR circuit is given by $Q = \omega_r CR = R\sqrt{\frac{C}{L}}$, which is the inverse of the series LCR Q-factor expression.

9. How is the bandwidth ($\Delta f$) of a parallel LCR resonance curve related to its resonant frequency and Q-factor?

Explanation: The bandwidth of resonance is given by $\Delta f = \frac{f_r}{Q} = \frac{1}{2\pi RC}$.

10. What is the phase angle ($\phi$) between the voltage and total current in a parallel LCR circuit at resonance?

Explanation: At resonance, the reactive components cancel out entirely, making the circuit purely resistive, so the phase angle $\phi = 0^\circ$.

11. What is the minimum line current ($I_{min}$) drawn from the source at parallel resonance for a voltage $V$?

Explanation: The minimum line current is given by $I_{min} = \frac{V}{Z_{max}} = \frac{VCR}{L}$.

12. Which of the following branch currents is in quadrature (phase difference of $90^\circ$) leading the voltage in a parallel LCR circuit?

Explanation: The current through an ideal capacitor ($I_C$) leads the alternating voltage by $90^\circ$.

13. If the inductance $L$ and capacitance $C$ of a parallel LCR circuit are both doubled, the resonant frequency $f_r$ will:

Explanation: Since $f_r = \frac{1}{2\pi \sqrt{LC}}$, if both $L$ and $C$ are multiplied by 2, the product $LC$ increases by a factor of 4, meaning the frequency becomes half ($\frac{1}{2}$).

14. What physical phenomenon occurs inside the parallel tank circuit (LC combination) at resonance, despite the external line current being minimal?

Explanation: Large reactive branch currents ($I_L$ and $I_C$) circulate within the LC tank due to energy oscillation, even though they cancel each other out in the external line feed.

15. What are the SI units of electrical admittance ($Y$)?

Explanation: Admittance is the inverse of impedance, measured in Siemens (S) or ohm$^{-1}$.

PYQ Patterns

These are pattern-based practice questions. Authentic year labels are only used when verified from a source.

1. In a parallel LCR circuit, the condition for resonance is that the reactive component of the total current is zero. This implies:

Explanation: Parallel resonance occurs when inductive susceptance equals capacitive susceptance, i.e., $\omega C = \frac{1}{\omega L}$ (or equivalently, inductive reactance equals capacitive reactance $\omega L = \frac{1}{\omega C}$).

2. A parallel LCR circuit has $L = 2\text{ H}$, $C = 32\mu\text{F}$, and $R = 10\,\Omega$. What is the dynamic resistance of the circuit at resonance?

Explanation: Dynamic resistance $Z_{max} = \frac{L}{CR} = \frac{2}{(32 \times 10^{-6}) \times 10} = \frac{2}{3.2 \times 10^{-4}} = 6.25 \times 10^3\,\Omega = 6.25\, k\Omega$.

3. Which of the following statements is INCORRECT regarding a parallel LCR resonant circuit?

Explanation: At resonance, the circuit is purely resistive ($\phi = 0^\circ$), so the power factor is $\cos(0^\circ) = 1$ (unity), not zero.

4. The Q-factor of a parallel LCR circuit is given by $Q = 50$. If the resonant frequency is $10\text{ kHz}$, what is the bandwidth of the circuit?

Explanation: Bandwidth $\Delta f = \frac{f_r}{Q} = \frac{10000}{50} = 200\text{ Hz}$.

5. How does the resonant frequency of an ideal parallel LCR circuit change if the resistance $R$ of the resistor branch is increased?

Explanation: The ideal resonant frequency formula is $f_r = \frac{1}{2\pi \sqrt{LC}}$, which is independent of resistance $R$ in the idealized circuit model.

6. In a parallel LCR circuit at resonance, the branch currents $I_L$ and $I_C$ are:

Explanation: At parallel resonance, inductive and capacitive susceptances are equal, meaning $I_L$ and $I_C$ are equal in magnitude and $180^\circ$ out of phase, canceling each other out.

Timed Topic Test

Attempt these questions under a ten-minute limit using a separate timer.

1. What is the primary operational characteristic that distinguishes a parallel LCR circuit from a series LCR circuit?

Explanation: A parallel LCR circuit acts as a rejector circuit with maximum impedance at resonance, whereas a series LCR circuit acts as an acceptor circuit with minimum impedance.

2. If a parallel LCR circuit has $L = 0.5\text{ H}$ and $C = 2\,\mu\text{F}$, what is its resonant angular frequency ($\omega_r$)?

Explanation: $\omega_r = \frac{1}{\sqrt{LC}} = \frac{1}{\sqrt{0.5 \times 2 \times 10^{-6}}} = \frac{1}{\sqrt{10^{-6}}} = 10^3 = 1000\text{ rad/s}$.

3. The sharpness of resonance in a parallel LCR circuit is directly determined by:

Explanation: The Q-factor determines the sharpness of the resonance peak and selectivity of the circuit.

4. In terms of susceptance $B_L$ and $B_C$, the parallel resonance condition can be written as:

Explanation: Parallel resonance requires that the inductive susceptance ($B_L = \frac{1}{\omega L}$) equals the capacitive susceptance ($B_C = \omega C$), so $B_L = B_C$.

5. If the capacitance of a parallel resonant circuit is quadrupled (multiplied by 4), how does the dynamic resistance $Z_{max}$ change, assuming $L$ and $R$ are constant?

Explanation: Since $Z_{max} = \frac{L}{CR}$, if $C$ is multiplied by 4, the dynamic resistance becomes $\frac{1}{4}$ of its initial value.

6. What is the power factor of a parallel LCR circuit when operating at frequencies higher than the resonant frequency $f_r$?

Explanation: For $f > f_r$, capacitive susceptance exceeds inductive susceptance ($\omega C > \frac{1}{\omega L}$), making the circuit capacitive, so the total current leads the voltage (leading power factor).

7. Calculate the bandwidth of a parallel LCR circuit having $R = 1\,\text{k}\Omega$ and $C = 100\,\text{pF}$ (where $\pi \approx 3.14$).

Explanation: Bandwidth $\Delta f = \frac{1}{2\pi RC} = \frac{1}{2 \times 3.1416 \times (10^3\,\Omega) \times (100 \times 10^{-12}\text{ F})} = \frac{1}{6.283 \times 10^{-7}} \approx 1.59 \times 10^6\text{ Hz}$? Wait: $10^3 \times 10^{-10} = 10^{-7}$. $\frac{1}{6.28 \times 10^{-7}} = 1.59 \times 10^6\text{ Hz} = 1590\text{ kHz} = 1.59\text{ MHz}$. Let's check options. Option A is $1.59\text{ MHz}$, Option B is $1.59\text{ kHz}$. Correct is $1.59\text{ MHz}$. Let's select correct index 0.

8. Which component in a parallel LCR circuit introduces electrical conductance ($G = \frac{1}{R}$)?

Explanation: The pure resistor branch provides conductance $G = \frac{1}{R}$ in parallel admittance calculations.

9. At a frequency below parallel resonance, the parallel LCR circuit behaves predominantly as:

Explanation: At frequencies below resonance, inductive susceptance dominates over capacitive susceptance ($\frac{1}{\omega L} > \omega C$), making the circuit inductive (lagging current).

10. If an AC source of constant RMS voltage $V$ supplies a parallel LCR circuit at resonance, the minimum line current depends directly on:

Explanation: Since $I_{min} = \frac{VCR}{L}$, the minimum current depends on the voltage and the circuit parameter ratio $\frac{CR}{L}$.