= Undamped resonant forcing has no bounded sinusoidal steady state
<Undamped resonant forcing has no bounded sinusoidal steady state> because the forcing continually adds <energy> to a mode that has no dissipation. For a series <LC circuit> driven from rest by $V_0\cos\omega t$, let $q$ be its <electric charge> and $\omega_0=1/\sqrt{LC}$. When $\omega\ne\omega_0$, imposing $q(0)=\dot q(0)=0$ on the <linear ordinary differential equation> gives
$$
q(t)=\frac{V_0}{L(\omega_0^2-\omega^2)}(\cos\omega t-\cos\omega_0t).
$$
Both frequencies persist: the undamped <homogeneous solution> never disappears. Taking the limit $\omega\to\omega_0$ at fixed time produces the growing resonant response described in <sinusoidal steady state>. Its factor proportional to time cannot be removed by adding a bounded <homogeneous solution>.
For an original numerical example, choose $L=10\,\mathrm{mH}$, $C=1\,\mathrm{\mu F}$ and $V_0=1\,\mathrm V$. Then $\omega_0=10^4\,\mathrm{rad\,s^{-1}}$. At exactly ten resonant periods, $t=20\pi/\omega_0\approx6.283\,\mathrm{ms}$, the resonant solution has $q=0$ but
$$
i=\dot q=\frac{V_0t}{2L}\approx0.3142\,\mathrm A,\qquad
\frac12Li^2\approx0.4935\,\mathrm{mJ}.
$$
At twenty periods the <electric current> doubles and the stored magnetic <energy> quadruples. These are ideal-model values; a real source and winding have finite <electrical resistance> and ratings.
With positive series resistance $R$, the resonant <phasor> calculation instead gives current <amplitude> $V_0/R$. The DOE handbook explicitly identifies resistance as the remaining series <electrical impedance> at <resonance>.<DOE Fundamentals Handbook Electrical Science>[Volume 3, Module 8, Resonance, printed p. 19, equation 8-14 and Series Resonance]{ref} For weak <damping>, decay takes a time of order $2L/R$, which grows as resistance decreases. These <noncommuting damping and observation-time limits at resonance> explain why an infinite steady-state amplitude is a failed limiting calculation, not the actual finite-time response. The late-time transient assumption underlying sinusoidal analysis must be checked.<6.200 Notes: Sinusoidal Steady State by Karl K. Berggren>[Printed p. 3, transient assumption]{ref}
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