Question

In the given circuit, what is the closest approximate frequency at which the ratio vout/vin is 1/root 2

Answer 1

Answer:

The closest approximate frequency is 950 MHz to 1 GHz at which the ratio Vout/Vin is 1/root 2.

Explanation:

The problem statement seems to suggest that Vin / Vout = 1/root 2, however, in this circuit the resistance is smaller than the impedance. That means V in > V out.

Now,  

V in / V out = root 2 . Voltage equation: V = q/C + (dq/dt) R can be put in the form:  

(V in / I in) cos(wt – a) = (1/(wC)) cos( wt –pi/2) + R cos( wt ) where w = 2 pi f  

This looks, the equation for adding the x components of two vectors to get the x component  of the resultant vector, (all of them rotates CCW about the origin). These vectors can be sketched in the complex number plane or  in the xy plane. Freezing the rotation at t = 0 permits one to calculate  the impedance:  

V in/ I in = ( R^2 + (1/wC)^2 )^1/2 where I in = V out /R = V in / (R root 2)  

So:  

2 R^2 = R^2 + (1/ wC )^2  

R = 1/ wC  

w = 2 pi f = 1/ RC = 6 x 10^9  

f = 9.55 x 10^8 hz