Parallel Resonance

A parallel resonant circuit, commonly called a tank circuit, is one of the most important building blocks in filter design, radio frequency amplifiers, and oscillators. Unlike series resonance where impedance is minimum at resonance, a parallel RLC circuit presents maximum impedance at its resonant frequency, making it extremely useful for selecting or rejecting specific frequency bands.

Understanding parallel resonance is essential for GATE aspirants because questions frequently test the condition of resonance, the nature of impedance, and the behavior of branch currents at and away from resonance frequency.

Core Concept of Parallel Resonance

In a parallel RLC circuit, the resistor, inductor, and capacitor are all connected across the same voltage source. The total admittance of the circuit is the sum of individual branch admittances. The admittance Y of the parallel combination is given by Y = G + j(BC - BL), where G = 1/R is the conductance, BC = ωC is the susceptance of the capacitor, and BL = 1/(ωL) is the susceptance of the inductor.

At the resonant frequency f₀, the imaginary part of the admittance becomes zero. This means BC = BL, so ωC = 1/(ωL). The total admittance reduces to just G = 1/R, which is the minimum admittance. Since impedance Z = 1/Y, the impedance is maximum at resonance and equals R. This is the defining characteristic that distinguishes parallel resonance from series resonance.

The inductor current IL lags the voltage by 90 degrees, while the capacitor current IC leads the voltage by 90 degrees. At resonance these two currents are equal in magnitude but exactly opposite in phase, so they cancel each other in the external circuit. The source only supplies the resistive current. This is why a parallel resonant circuit draws minimum current from the source at resonance, even though large circulating currents flow between L and C internally.

Mathematical Expression

The resonant frequency for an ideal parallel RLC circuit (with lossless inductor and capacitor) is derived by equating the inductive and capacitive susceptances. Setting ω₀C = 1/(ω₀L) and solving gives the standard result. This is the same expression as series resonance for ideal components.

The resonant angular frequency is ω₀ = 1/√(LC) radians per second. The resonant frequency in hertz is f₀ = 1/(2π√(LC)). At this frequency the impedance of the tank circuit is purely resistive and equal to R (the parallel resistance). For a practical inductor with series resistance r, the resonant frequency is slightly modified and the dynamic resistance of the tank is given by Rd = L/(Cr), which can be very large for high-Q coils.

The dynamic resistance Rd = L/(Cr) is a key result for GATE. It represents the effective impedance of the tank at resonance when the inductor has internal resistance r. The higher the Q of the coil, the higher the dynamic resistance and the sharper the resonance peak.

Practical Understanding

In practical RF circuits, the tank circuit is used as the load of a transistor amplifier to provide high gain only at the resonant frequency. The maximum voltage developed across the tank at resonance is the signal current multiplied by the dynamic resistance. Away from resonance the impedance drops sharply, attenuating unwanted frequencies.

A key practical point is that at resonance the circulating current between L and C can be Q times the supply current. This is called current magnification in a parallel circuit. While this is analogous to voltage magnification in series resonance, engineers must account for it when rating component current capacity in high-Q tank circuits.

Given:
L = 10 mH, C = 10 nF, r (inductor resistance) = 5 Ω

Why this formula applies:
For a practical parallel LC circuit with inductor resistance r, resonant frequency
and dynamic resistance are the key parameters.

Formula:
f₀ = 1 / (2π√(LC))
Rd = L / (C × r)

Substitution:
f₀ = 1 / (2π × √(10×10⁻³ × 10×10⁻⁹))
   = 1 / (2π × √(10⁻¹⁰))
   = 1 / (2π × 10⁻⁵)

Calculation:
f₀ = 1 / (6.2832 × 10⁻⁵)
f₀ ≈ 15,915 Hz ≈ 15.92 kHz

Rd = (10×10⁻³) / (10×10⁻⁹ × 5)
   = 10⁻² / (5×10⁻⁸)
   = 200,000 Ω

Final Answer:
Resonant frequency f₀ ≈ 15.92 kHz
Dynamic resistance Rd = 200 kΩ

Exam Tip: In parallel resonance, impedance is MAXIMUM and current drawn from source is MINIMUM at f₀. For a practical coil with resistance r, dynamic resistance Rd = L/(Cr). Do not confuse this with series resonance where impedance is minimum. GATE frequently tests this distinction.

Key Mechanism Points

• At resonance ω₀ = 1/√(LC), the admittance is purely real (Y = G = 1/R) and impedance is maximum equal to R or Rd for practical circuit.
• The inductor current and capacitor current are equal in magnitude and 180° out of phase at resonance, so they circulate between L and C without loading the source.
• Below f₀ the circuit behaves inductively (net current lags voltage); above f₀ it behaves capacitively (net current leads voltage).
• The source current is minimum at resonance; this is opposite to the series resonant circuit behavior where source current is maximum.
• Dynamic resistance Rd = L/(Cr) increases with lower coil resistance r and higher L/C ratio, giving sharper resonance with higher voltage gain.

Quick Revision

• Resonant frequency: f₀ = 1/(2π√(LC)), same formula as series resonance for ideal components.
• At resonance: impedance is MAXIMUM, current from source is MINIMUM, power factor is unity.
• Dynamic resistance for practical tank: Rd = L/(Cr) where r is the coil resistance.
• Circulating current between L and C = Q × supply current (current magnification).
• Below f₀: inductive; above f₀: capacitive. This is opposite to series resonance.
• Exam trap: parallel resonance gives Z max, NOT Z min. Series resonance gives Z min.
• Tank circuit applications: RF amplifier loads, oscillator frequency selection, bandpass filters.