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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 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.

The PDF gives $V=H(t)$, the Heaviside step function; the TeX incorrectly changes this to $V=I(t)$. A unit voltage step has derivative the Dirac delta function, so the current is the causal Green function for $L{d}^2/d{t}^2+Rd/dt+1/C$.

The current must be continuous at zero: a jump would create a delta derivative through $L{I}^{\prime \prime }$, absent on the right-hand side. Integrating across zero therefore gives $L[I^{\prime }]+R[I]=1$, so $I(0^{+})=0$ and $I^{\prime }(0^{+})=1/L$. PutThese distinct negative characteristic roots give the overdamped RLC responseTogether with $I(t)=0$ for $t<0$, this solves the step response. The eventual current is zero: the capacitor charges and blocks steady current in the series RLC circuit.

For a resonator, the quality factor is $2\pi$ times the stored energy divided by the energy dissipated per cycle, using a specified steady-state stored-energy convention. A series RLC circuit at resonance has $Q=L{\omega }_0/R$, because its maximum magnetic energy is $L{I}_M^2/2$ and one cycle of Joule heating dissipates $\pi R{I}_M^2/{\omega }_0$.

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 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 $R>0$.