I'm working to characterize the power NPN-based power stage in the DC Lab Power Supply I'm designing.
I got a really helpful answer from gsills to a prior question I asked on that topic; now I'm just trying to follow along step-by-step to be able to derive the results for myself, so I can apply the same reasoning to other possible pass devices. Also, it's a good opportunity for me to solidify my understanding of BJT small signal analysis.
The place I'm having trouble is in deriving the transfer function for the common-emitter (CE) small signal model to determine the location of the pole due to the transition frequency (\$f_T\$) of the pass device. From gsills' answer:
What about the poles? First let's look at the pole caused by \$\beta\$ rolloff to \$f_T\$. In the model, eliminate all the capacitors and write the transfer function. It's kind of big, but there is just one pole, which after solving for the root gives the pole frequency for \$\beta\$ rolloff.
$$f_{p-\beta } = \frac{f_T \left(r_b+(\beta +1) \left(r_e+R_E\right)+R_4\right)}{\beta \left(r_b+r_e+R_4+R_E\right)}$$
Here's a redrawn small-signal model with all the capacitors removed (also I changed \$R_4\$ to the perhaps more conventional \$R_s\$:
I guess where I'm struggling is there are no capacitors to bring in a \$1/sC\$ element, so I don't see how to end up with something where frequency is a factor. I'm kind if imagining there's some sort of \$\beta(s)\$ expression that gets substituted, but I can't really discern an approach that seems like it will get me there. I do note that \$f_{p\beta}\$ is proportional to \$f_T/\beta\$, the best case beta rolloff pole, and also that the rest of the numerator exactly matches the denominator of the DC gain of the model. But that's all I've got.
How can I get from here to an \$f_{p\beta}\$ expression like the one gsills provided?