What are the loop gain and the transfer function of this transconductance amplifier feedback circuit?

Question

Related to my previous questions, I would like to know what the approach for determining the loop gain and transfer function of the circuit presented below is:

The circuit is highly simplified. The VCCS would in reality be a differential amplifier loaded by a current mirror and would have big input and output impedances. VG1 is meant to represent the unregulated input voltage of the regulator with a DC level and a significant AC ripple. V1 would also be a Zener voltage reference.

This representation is closer to the reality of the circuit:

For DC conditions I calculated the closed loop gain to be:

$$V_{o}=\frac{g_{m}(\beta + 1)(R_{1}+R_{2})}{1+g_{m}(\beta + 1) R_{2}}\cdot V_{ref}$$

Simplifying for

$$g_{m}(\beta + 1) R_{2}\gg 1$$

gives the usual

$$V_{out}=\left(1+\frac{R_{1}}{R_{2}}\right)\cdot V_{ref}$$

For the low frequency closed loop gain / open loop gain the result is the same. I obtained it by breaking the loop, passivizing all DC sources and replacing the transistor with the small signal equivalent model.

I am almost certain I made some mistakes in my derivations.

What is the correct approach?

Answer 1

Here is my approach for a (rough) loop gain calculation:

• The loop is opened at the base node of T2

• The voltage gain at the collector node of T2 is

  G=v_c2/v_b2=-(1/2)gm_2 * r_load * 2=-gm_2 * r_load. (the factor 2 results from the current mirror)

• The dynamic load resistance is r_load=hie_7(1+gm_7 * R_e) with R_e=R1+(R2||r_in2)

• Because of the voltage follower (T7) we can set v_c2=v_e7 and the voltage v_f across the feedback resistance r_f=(R2||r_in2) can be found using the voltage division between r_f and R1.

• Therefore, the loop gain is G_loop=G * [R2||r_in2]/[R1+(R2||r_in2)]