Rc dV0dt V0 Vv where Vi omega sin wt V0 A sin wt B sin

Rc dV_0/dt + V_0 = V_v where V_i = omega sin wt V_0 = A sin wt + B sin wt (steady state) Show that: A = 9/1 + (RC_W)^2; B = -RCwa/1 + (RCw)^2

Solution

Brute Force Method: Start with Kirchhoff\'s loop law:

V(i) = VR(t) + Vo (t)

a sin wt = IR + Q / C

= RdQ(t ) / dt + Q(t) / C

We have to solve an inhomogeneous D.E. The usual way to solve such a D.E. is to assume the solution has

the same form as the input:

Q(t) =a sinwt + b coswt

Plug our trial solution Q(t) back into the differential equation:

                = R . d (a sinwt + b coswt)/ dt + (a sinwt + b coswt) /C

a sin wt   = aRw coswt - bRw sinwt + (a / C)sinwt + (b / C)coswt

= (aRw +b / C)coswt +(a / C - bRw )sinwt

=

a =(a / C - bRw)...................1)

- aRw = b / C……………..2)

using eq 1 and 2 ,

a= =(a / C - ( - aRw .C ) Rw

a =   Ca / (1+ (RCw )2 )

a =   q / (1+ (RCw )2 )

In the same way,

b= - RCw a/ (1+ (RCw )2 )