Question

A. Write down the differential equation describing the circuit for an arbitrary time-dependent voltage V (t), in terms of the inductance L, capacitance C and resistance R of the circuit. B. Determine an analytic solution when the voltage is switched off [V(t) = ol. First, express your solution in terms of arbitrary coefficients as appropriate. Then, determine those coefficients for the initial conditions where the current is given by I(に0)-10 and satisfies I'(t = 0) = 0. C. Determine the analytic solution when the voltage is switched on to V(t) = 10 cos ut, first with arbitrary boundary conditions, and then for boundary conditions/(に0) = 1,(に0)-0.

Answer 1

a.

Kirchoff's voltage law,

$L\frac{di}{dt}+iR+\frac{Q}{C}=V$

where Q is the charge that flows onto the capacitor in time t. Differentiate the equation w.r.t. time,

$L\frac{d^2i}{dt^2}+R\frac{di}{dt}+\frac{i}{C}=\frac{dV}{dt}$

b.

Equation becomes

$L\frac{d^2i}{dt^2}+R\frac{di}{dt}+\frac{i}{C}=0$

Roots of auxiliary equation

$a_1,a_2=\frac{-R\pm\sqrt{R^2-\frac{4L}{C}}}{2L}$

Thus, the general solution is

$i=Ae^{a_1t}+Be^{a_2t}$

According to given initial conditions,

$I_0=A+B$

$0=a_1A+a_2B$

$A=\frac{a_2I_0}{a_2-a_1}, B=\frac{a_1I_0}{a_1-a_2}$

C.

Using phasor diagrams,

$i_p=\frac{V_0}{|Z|}\cos(\omega t+\phi)$

where

$|Z|=\sqrt{R^2+(X_C-X_L)^2}$

$X_C=\frac{1}{C\omega}, X_L=L\omega$

and

$\tan\phi=\frac{X_C-X_L}{R}$