# Unable to model Transresistance amplifier with feedback correctly

I spent the whole day trying to figure why transresistance amplifier modelled with z-parameters does not match with nodal analysis results but I sill can't figure out. I desperate need help on this...

I write down step by step what I did for a very simple transresistance amplifier here and hope someone can point out what I did wrong:

The following is a very simple transistor I use as a transresistance amplifier example here:

and the following is the corresponding π model assume output resistance $$r_o = ∞$$ just to make it simple:

I converted the circuit to a z-parameters model as follows:

with the following z-parameters value from the circuit:

With nodal analysis, I got

$$Z_{11} = \frac {v_1} {i_1}\Bigm|_{i_2=0}=r_π$$

$$Z_{21} = \frac {v_2} {i_1}\Bigm|_{i_2=0}=-g_mr_πR_C$$

$$Z_{12} = \frac {v_1} {i_2}\Bigm|_{i_1=0}=0$$

$$Z_{22} = \frac {v_2} {i_2}\Bigm|_{i_1=0}=R_C$$

so my z-parameters model ends up like this:

When I added feedback to the π model as follows:

and did nodal analysis, I got:

$$Nodal\ analysis:$$

$$1)\ i=\frac {v_i} {r_π}+\frac {v_i-v_o} {R_f}$$

$$2)\ g_mv_i\ +\frac {v_o-v_i} {R_f}\ +\frac {v_o} {R_C}=0$$

$$\Rightarrow \frac {v_o} {i}=\frac {-R_Cr_π(g_mR_f-1)} {R_C+R_f+r_π+g_mr_πR_C}$$

However, when added the same feedback to the equivalent z-parameters model as follows:

and did nodal analysis:

$$Nodal\ analysis:$$

$$1)\ \frac {-g_mr_πR_Ci-v_o} {R_C}=\frac {v_o-v_i} {R_f}$$

$$2)\ i=\frac {v_i-v_o} {R_f}\ +\frac {v_i} {r_π}$$

$$\Rightarrow \frac {v_o} {i}=\frac {-R_Cr_π(R_fg_m + g_mr_π - 1)} {R_C+R_f+r_π}$$

The results do not match. I really have no idea what I have missed. Any helps or pointers will be greatly appreciated!!

• @ Matt - in the z-parameter model you have TWO sources: A voltage source and a current source (I2). Can you explain?
– LvW
Commented Mar 9 at 9:45
• The standard z-parameters model is in this form and that's how you obtain all z parameters from the circuit.
– Matt
Commented Mar 9 at 22:20
• However, in your calculation you set i2=0, correct?
– LvW
Commented Mar 10 at 9:13