# 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? Mar 8, 2015 at 19:06
• No actually, I want to derive a common formula. Not to calculate the value. Mar 8, 2015 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. Mar 8, 2015 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. Mar 8, 2015 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. Mar 9, 2015 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 $Z=\frac{1}{jwC}$)

• Why don't use the properties of "iterative quadripole" ? Jul 12, 2021 at 18:53
• res-nlp.univ-lemans.fr/NLP_C_M14_G02/co/Contenu_21.html Jul 13, 2021 at 7:02
• Without a schematic it's not possible to know how i1, i2 or i3 are defined, so your answer is of no use to an outsider.
– dirk
Jun 20, 2022 at 9:46