Question

A damped LC circuit loses 3.6% of its electromagnetic energy per cycle to thermal energy.

If L = 65mH and C = 7.80?F , what is the value of R?

Answer 1

6. We assume underdamping, with

                      w' = w_o= 1/(LC)^1/2, and T = 2p/w_o = 2p(LC)^1/2

         The charge on the capacitor is

                       Q= Q_oe^-Rt/2Lcos (w't + f).

         The energy stored in the capacitor and inductor can be expressed in terms of the amplitude of the cosine function:

                       U = Q²/2C = Q_o²e^-Rt/L/2C = U_oe^-RT/L.

          In one period the energy is reduced by 3.6 percent, so we have

                       0.945U_o = U_oe^-Rt/L, which gives ln(1/0.945) = RT/L = 2pR(C/L)^1/2;

                       ln(1/0.945) = 2pR[(1.00 x10^-6F/7.8x10^-3)/(65 x 10^-3H)]^1/2, which gives R = 12.30W

           We can check to see if we have underdamping:

                       R² = (12.30W)² =151.3W²;

                       4L/C = 4(65 x 10^-3H)/(7.8 x 10^-3F) = 33^.333W².

            Thus R << 4L/C.