RLC Parallel Response

The parallel RLC circuit is the dual of the series RLC circuit. Here, the resistor, inductor, and capacitor are all connected across the same two terminals with a common voltage. The governing differential equation is again second order, and the same three types of damped responses appear. Understanding the parallel RLC circuit is essential for analyzing resonant circuits, bandpass filters, and antenna matching networks.

Core Concept Explanation

In a parallel RLC circuit excited by a current source Is, all three elements share the same terminal voltage V(t). The total current from the source splits among R, L, and C according to their individual current-voltage relationships. Applying KCL at the top node: Is = V/R + (1/L)∫V dt + C·dV/dt. Differentiating and rearranging gives the second-order differential equation: C·d²V/dt² + (1/R)·dV/dt + (1/L)·V = dIs/dt.

The natural response of the parallel RLC circuit is the response when the source is set to zero (open circuit for current source) and only initial conditions drive the circuit. This occurs after a switch disconnects the source while some initial voltage exists across the capacitor or initial current through the inductor.

A key difference from the series RLC is the damping coefficient. For parallel RLC, α = 1/(2RC), while for series RLC, α = R/(2L). This duality is important: increasing R in a series circuit increases damping, but increasing R in a parallel circuit decreases damping (since 1/R decreases). The natural frequency ω₀ = 1/√(LC) remains the same for both configurations.

The resonant frequency ω₀ is where the inductive and capacitive susceptances cancel: ωL = 1/(ωC), giving ω₀ = 1/√(LC). At resonance, the parallel LC combination presents infinite impedance — the circuit acts as an open circuit to the source at that frequency.

Mathematical Expression

The characteristic equation is: s² + (1/RC)·s + 1/(LC) = 0, giving roots s = -α ± √(α² - ω₀²) with α = 1/(2RC). The complete response is the sum of the natural response (determined by initial conditions and roots) and the forced response (determined by the source). For the natural response with initial voltage V0 across the capacitor and initial inductor current I0: V(t) = A₁·e^(s₁t) + A₂·e^(s₂t) for overdamped, or V(t) = e^(-αt)·[A·cos(ωd·t) + B·sin(ωd·t)] for underdamped.

Practical Understanding

Parallel LC resonant circuits (tank circuits) are the core of radio frequency (RF) amplifiers and oscillators. The high impedance at resonance means the circuit selectively amplifies signals near ω₀ while rejecting other frequencies. A small resistance (from component losses) limits the peak impedance and determines the bandwidth.

In power factor correction, capacitor banks are placed in parallel with inductive loads. This creates a parallel LC combination that can resonate with the supply frequency if not carefully designed, potentially causing large circulating currents. Proper damping through resistance is essential to prevent such resonance issues.

Given:
R = 1/(0.2) = 5 Ω,  L = 1 H,  C = 0.1 F
Initial conditions: V(0) = 10 V, iL(0) = 0 A, source open-circuited

Why this formula applies:
Natural response with initial voltage on capacitor. Determine damping type first.

Formula:
α = 1/(2RC) = 1/(2 × 5 × 0.1) = 1 rad/s
ω₀ = 1/√(LC) = 1/√(1×0.1) = 1/0.316 ≈ 3.162 rad/s

Substitution:
α = 1  <  ω₀ = 3.162  →  Underdamped
ωd = √(ω₀² - α²) = √(10 - 1) = √9 = 3 rad/s

Calculation:
V(t) = e^(-t)·[A·cos(3t) + B·sin(3t)]
At t=0: V(0) = A = 10
dV/dt|₀ = -αA + ωd·B = -(V(0)/R + iL(0))/C = -(10/5 + 0)/0.1 = -20
-1×10 + 3B = -20  →  B = -10/3 ≈ -3.33

Final Answer: V(t) = e^(-t)·[10·cos(3t) - 3.33·sin(3t)] V. The voltage oscillates at 3 rad/s with exponentially decaying amplitude. The circuit is underdamped.

Exam Tip: In parallel RLC, α = 1/(2RC) — increasing R REDUCES damping. This is dual to series RLC where α = R/(2L) and increasing R INCREASES damping. Resonant frequency ω₀ = 1/√(LC) is the same for both. GATE often tests this duality directly.

Quick Revision

• Parallel RLC: α = 1/(2RC). Series RLC: α = R/(2L). Duality — R has opposite effect on damping.
• ω₀ = 1/√(LC) for both series and parallel RLC. Resonant frequency is determined only by L and C.
• Natural response types: overdamped (α > ω₀), underdamped (α < ω₀), critically damped (α = ω₀).
• At resonance in parallel RLC, LC branch presents infinite impedance — circuit draws minimum current from source.
• Q factor for parallel RLC: Q = R/ω₀L = R·√(C/L). Higher R means higher Q in parallel (opposite of series).
• Initial conditions: Vc(0+) = Vc(0−) and iL(0+) = iL(0−) both apply for finding A and B constants.
• Exam trap: Never use series damping formula α = R/(2L) for a parallel RLC circuit. This is the most common GATE mistake.