Electronic – Calculating the gain and phase shift of a phase shifting circuit

circuit analysisgainoscillatorphase shift

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

schematic

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.

Best Answer

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}\$)