ParallelRLCia

ParallelRLCia refers to a parallel resistor-inductor-capacitor circuit model used in electrical engineering to study impedance, transient behavior, and filtering characteristics of parallel RLC networks. In this configuration, a resistor R, an inductor L, and a capacitor C are connected in parallel across a common node, and the network is driven by a source that supplies either a current i_in(t) or a voltage v_in(t).

The governing relations are based on Kirchhoff’s current law at the node. The currents through the elements are i_R = v/R, i_C = C dv/dt, and i_L with di_L/dt = v/L. The input current sum satisfies i_in = i_R + i_C + i_L. For zero input (homogeneous case), the node voltage satisfies the second-order differential equation C d^2v/dt^2 + (1/R) dv/dt + (1/L) v = 0, which can be written in standard form as d^2v/dt^2 + (1/(RC)) dv/dt + (1/(LC)) v = 0. The natural angular frequency is ω0 = 1/√(LC), and the damping factor is α = 1/(2RC). The complex impedance of the network is Z(ω) = 1 / (1/R + jωC + 1/(jωL)); at resonance, where ω = ω0, the reactive terms cancel and the impedance equals R.

Key properties include the parallel quality factor Qp = R√(C/L), which characterizes the sharpness of the resonance. The circuit behaves as a tunable impedance element and can form RF filters, impedance matching networks, and resonant tanks in oscillators. In practice, ParallelRLCia is implemented in circuit simulators as a configurable block with parameters R, L, and C, enabling both time-domain and frequency-domain analyses. See also parallel RLC circuits, impedance, resonance.