# Calculating the gain and phase shift of a phase shifting circuit

I want to calculate the gain and the phase shift of the phase shifting circuit below. My approach is to derive equations to different meshes using Kirchoff's law, and then calculate the ratio between Vin and Vout in the frequency domain.

But obviously, it is too hard to simplify the equations (at least for me). Are there any other (easy) methods to calculate it ?

--Thanks simulate this circuit – Schematic created using CircuitLab

EDIT: I want to derive a formula for the gain and the phase shift. Not to calculate 'a' value.

• You drew it in CircuitLab, why not simulate it there? – jippie Mar 8 '15 at 19:06
• No actually, I want to derive a common formula. Not to calculate the value. – ThisaruG Mar 8 '15 at 19:08
• Fair enough, but you are looking at a 4th order differental equation. Not for the faint of heart. Are you familiar with Laplace s-domain? Maybe I'm overlooking something, see what the others think. – jippie Mar 8 '15 at 19:09
• Oh...... This was supposed to be a simple equation. I mean not a differential equation. It was a quiz we had to do. – ThisaruG Mar 8 '15 at 19:14
• Looking at the accepted answer, I guess what I overlooked is in the fact that all R's and C's are identical. It's been a while for me since I dived into these sort of puzzles. – jippie Mar 9 '15 at 6:35

I think that writing the loop equations would be easier.

The loop equations for first two loops: $$I_1(Z+R) = V_{in}$$ $$- I_1R +I_2(Z+2R) -I_3R = 0$$

Where $Z=\dfrac{1}{Cs}$. From this:

$$I_2(Z+2R) -I_3R = V_{in}\frac{R}{R+Z}\tag1$$ The remaining two loop equations: $$- I_2R +I_3(Z+2R) -I_4R = 0\tag2$$ $$- I_3R +I_4(Z+2R) = 0\tag3$$

Expressing in matrix form:

$$\left[\begin{array}{ccc} &Z+2R &-R &0 \\ &-R &Z+2R &-R \\ &0 &-R & Z+2R \end{array}\right] \left[\begin{array}{c} I_2\\ I_3\\ I_4 \end{array}\right] = \left[\begin{array}{c} \frac{V_{in}R}{R+Z}\\ 0\\ 0 \end{array}\right]$$

Now by Cramer's rule: $$I_4 = \frac{\left|\begin{array}{ccc} Z+2R &-R &\frac{V_{in}R}{R+Z} \\ -R &Z+2R &0 \\ 0 &-R & 0 \end{array}\right|}{\left|\begin{array}{ccc} Z+2R &-R &0 \\ -R &Z+2R &-R \\ 0 &-R & Z+2R \end{array}\right|}$$

$$V_{out} = I_4\times R$$

From this the transfer function can be calculated. Gain and phase shift can be calculated from transfer function. (substitute $Z=\frac{1}{jwC}$)