In an oscillating series RLC circuit calculate UU the fracti

In an oscillating series RLC circuit, calculate U/U, the fraction of the energy lost per cycle of oscillation. Assume that L = 1.5 mH, C = 3.4 F and R = 4.1

Solution

In an RLC circuit:

At time t = 0 sec the capacitor is fully charged with a charge equal to Q0 and the energy stored in the capacitor is Given by:

U(t = 0) = Q0^2/(2*C)

After one cycle (t = 2*pi/omega) the maximum charge on the capacitor has decreased. This implies that also the energy stored on the capacitor has decreased. After this time the energy stored in capacitor is given by:

U(t = 2*pi/w) = Q0^2*[exp(-2*pi*R/wL)]/(2*C)

dU/U = [U(t = 2*pi/w) - U(t = 0)]/U(t = 0)

dU/U = [Q0^2*[exp(-2*pi*R/wL)]/(2*C) - Q0^2/(2*C)]/Q0^2/(2*C)

dU/U = [exp(-2*pi*R/wL)] - 1

after expansion of exponential and cancelling higher order terms

dU/U = -2*pi*R/wL

dU/U = 2*3.14*4.1/(1*1.5*10^-3) = 17165.33