Derive an expression for the ratio of the output voltage over the 
input voltage, Vout/Vin, in the circuit shown in Figure 1, in terms of 
the resistances, R1 and R2, the capacitances, C1 and C2, and the 
frequency, omega, of the input signal.   Use complex representation of the 
impedances in the circuit and, thus, express the ratio, Vout/Vin, as a 
complex number. 

Fig. 1:

     +---R1---+ 
+----+        +----+------+ 
|    +---C1---+    |      | 
|                  |      | 
|                +-+-+    | 
Vin              |   |    | 
|                R2  C2   Vout 
|                |   |    | 
|                +-+-+    | 
|                  |      | 
|                  |      | 
+------------------+------+ 
| 
| 
Ground 
 
Thanks!

Hi,

Thank you very much for your question. I appreciate the innovative
pains you took at reproducing that circuit. I have tried to give you a
detailed analysis below.

Consider the R1C1 arm:

The equivalent impedance value can be calculated this way.

( R1/jwC1 )/( R1 + (1/jwC1) )          --- (1)

where R1 is the resistance. 1/jwC1 is the impedance of the capacitence
arm. Note 1: w = Omega ( the frequency)
Note 2: j = imaginary part operator in a complex number

(1) can be reduced to 

R1 / ( jwC1R1 + 1 )                     --- (2)

The same analysis done above holds for the R2C2 arm:

The equivalent impedance value can be calculated this way.

( R2/jwC2 )/( R2 + (1/jwC2) )          --- (3)

(3) can be reduced to 

R2 / ( jwC2R2 + 1 )                     --- (4)

Now in an impedance division network, the Vout = k2/(k1+k2) times the
Vin.  Where k1 and k2 are the two arms discussed above.

Hence,

(Vout / Vin) = k2/(k1+k2)

= (R2 / (jwR2C2 + 1) ) / [(R2 / (jwR2C2 + 1) ) + (R1 / (jwR1C1 + 1) )]


= R2 / { R2 +  R1 [(jwR2C2 + 1)/(jwR1C1 + 1)]  }


= R2(jwR1C1 + 1) / { R2(jwR1C1 + 1) +  R1(jwR2C2 + 1) }


= (R2 + jwR1R2C1) / { (R1 + R2) + jwR1R2 (C1 + C2) }       -- (5)


This can be conjugated and expressed in the form of a complex number.
If numerical vaules are given it would be easier to simplify further.

I hope this helps!

Thank you once again for your question.

Warmest Regards,
Shiv Reddy

Thanks! Spot on.