How do I prove that the Pi and T models of a BJT are equivalent?

Question

I want to prove the equivalence of the Pi and T-models of an npn BJT.

The only way I can think of doing this is by showing that both circuits have the same Z parameters. To do so, you would need to use the images bellow.

I have no idea how to extract the Z parameters from the circuits above. Here's my best attempt for the circuits in figure 2:

$$V_{1}=r_{\pi}I_{1}\Rightarrow Z_{11}=r_{\pi} $$

$${Z_{21}=?}$$

$$I_{2}=g_{m}V_{1}\Rightarrow{Z_{12}=\dfrac{1}{g_{m}}}$$

$${Z_{22}=?}$$

Here's my best attempt for the circuits in figure 3: $${I_{1}+g_{m}V_{1}-\dfrac{V_{1}}{r_{e}}=0}$$

$${\Rightarrow Z_{11}=\dfrac{V_{1}}{I_{1}}=\left(\dfrac{1}{r_{e}}-g_{m}\right)^{-1}}$$

$$Z_{21}=?$$

$${I_{2}=g_{m}V_{1}\Rightarrow\dfrac{V_{1}}{I_{2}}=\dfrac{1}{g_{m}}}$$

$${Z_{22}=?}$$

Answer 1

The equivalence between the Pi model and the T model can be proven with some relatively straightforward manipulations of the small-signal circuit diagram. Let's start by drawing the basic Pi model:

where rpi is meant to be the input resistance at the base, normally written as \$r_{\pi}\$, except that CircuitLab doesn't have Greek letters and subscripts, so I couldn't do the \$\pi\$ subscript. The small-signal voltage between base and emitter terminals is labelled vbe.

Next we make what amounts to a simple cosmetic change, and run the current source from the collector to the base, and then base to emitter. We haven't changed anything by doing this, because the current through rpi has not changed:

We can now observe that the voltage-controlled current source that runs between the base and emitter terminals is equivalent to a resistor, because it is controlled by the same voltage that is across its terminals. Therefore it is equivalent to a resistor of value \$1/g_m\$:

We can now combine the parallel resistances into a single resistor \$r_e = \frac{r_{\pi}}{1+g_m r_{\pi}}\$ (remember \$r_{\pi}\$ = rpi in diagram) and reorient the components to see this as the T model:

Answer 2

Since these are linear circuits, they must be modelable with Thevenin models. Or Norton. Or any method combining these.

I suggest you short the input, then drive the output with voltage sources and also a current source, in 2 phases. Show the equivalence.

Then open the input, and drive the output with voltage souces and then a current source, in 2 phases. Show the equivalence.

OR you may prefer to short the output, and drive the input with the 2 phases. Then open the output, and drive with the 2 phases.

Answer 3

To find z parameters:

Drive port 1 with a one amp source. Leave port two open.

     Z11= voltage at port one. 
     Z21= voltage at port two. 

Drive port two with a one amp source. Leave port one open.
      Z12= voltage at port one.
      Z22= voltage at port two.