RLC circuit

An RLC circuit combines electrical resistance, inductance and capacitance. Stored energy can oscillate between magnetic and electric forms, while Joule heating dissipates it.

In a series RLC circuit, the same electric current $I=q^{\prime }$ flows through a resistor, inductor and capacitor. Summing their electric potential difference drops gives $L{q}^{\prime \prime }+R{q}^{\prime }+q/C=V$, and differentiating givesThe homogeneous damped harmonic oscillator has angular frequency ${\omega }_0=1/\sqrt{LC}$ and decay rate $\gamma =R/(2L)$.

For a series RLC circuit driven by $V(t)=V_M\sin ({\omega }_{0}t)$ with $R>0$ and ${\omega }_0^2=1/(LC)$, substitution shows that $I_0=V_M\sin ({\omega }_{0}t)/R$ solves the current equation. The inductive and capacitive terms cancel at this resonance. Every homogeneous contribution decays, so every solution approaches this periodic response. Its Q factor is $L{\omega }_0/R$.

When $R^2>4L/C$, the homogeneous series RLC circuit has two distinct negative characteristic roots, $[-R\pm \sqrt{R^2-4L/C}]/(2L)$. Its current is a linear combination of the corresponding decaying exponential functions.

When $R^2=4L/C$, the homogeneous series RLC circuit has one repeated characteristic root, $-R/(2L)$. Its current is $e^{-Rt/(2L)}(A+Bt)$, with constants determined by the initial data.

When $R^2<4L/C$, the homogeneous series RLC circuit response oscillates with angular frequency $\sqrt{1/(LC)-R^2/(4L^2)}$ and envelope $e^{-Rt/(2L)}$. Its characteristic roots form a complex-conjugate pair with negative real part when $R>0$.