Damped Oscillations in an RLC Circuit — Question 5

Question 5

An energy envelope cannot replace the exact instantaneous loss equation

Consider an underdamped source-free RLC circuit. A proposed exact energy law is $U(t)=U(0)e^{-Rt/L}$ for release from q=Q0, i=0.

1. Test the proposed law at t=0 using the exact energy derivative.

2. Show why this exponential law does hold for energies sampled one damped period apart.

Question 5 - Solution

An energy envelope cannot replace the exact instantaneous loss equation

(a) Method and calculation

$$\dot U=-Ri^2\Rightarrow\dot U(0)=0.$$$$\left.\dot U_{\mathrm{proposed}}\right|_0=-RU(0)/L\ne0\quad\boxed{\text{not an exact continuous-time law}}.$$

(b) Interpretation and further calculation

$$(q,i)(t+T_d)=e^{-\alpha T_d}(q,i)(t),\quad2\alpha=R/L.$$$$\boxed{U(t+T_d)=e^{-RT_d/L}U(t)}.$$

Physical check

The same-phase sampled energy has a simple exponential ratio even though instantaneous energy includes phase dependence.