Noncommuting damping and observation-time limits at resonance

Noncommuting damping and observation-time limits at resonance means that the late-time approximation of a damped circuit is not uniform as its dissipation tends to zero. A precise comparison needs an observable: the unnormalized resonant sinusoid itself has no ordinary limit as time tends to infinity. Use the series RLC circuit with zero initial conditions, source $V_0\cos {\omega }_{0}t$, and ${\omega }_0=1/\sqrt{LC}$. For $R>0$, its steady-state electric charge and electric current areThe remaining electrical resistance sets the resonant amplitude, as in the DOE series-resonance model.^[ref]

Define a dimensionless state error relative to that amplitude:For each fixed $R>0$, the homogeneous solution decays and $D_R(t)\to 0$. Conversely, on a fixed finite observation interval the actual circuit solution converges to the finite undamped solution as $R\downarrow 0$. The two terms containing $R$ then vanish, giving $D_R(t)\to 1$. Thus the genuine iterated limits areThis states the failure without treating an oscillating quantity as though it had a time limit.

For weak damping, put $\gamma =R/(2L)$ and ${\omega }_d=\sqrt{{\omega }_0^2-{\gamma }^2}$. The exact zero-state chargeboth verifies the decaying transient and recovers the undamped resonant response in undamped resonant forcing has no bounded sinusoidal steady state by a fixed-time expansion.

For an original timing comparison, $L=10\mathrm{mH}$ gives envelope decay times $2L/R=0.1\mathrm{s}$ at $R=0.2\Omega$ and $1\mathrm{s}$ at $R=0.02\Omega$. At a fixed $50\mathrm{ms}$ the respective exponential multipliers are about $0.607$ and $0.951$. A smaller electrical resistance can make the sinusoidal late-time approximation worse over a fixed acquisition window. MIT's steady-state notes explicitly require the startup transient to have disappeared.^[ref]