Electrical circuit

An electrical circuit connects components that carry electric current and sustain electric potential difference differences. Its dynamics may couple electrical resistance, inductance and capacitance.

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 $R>0$.