Bipolar cascode current mirror full small signal analysis

[megaknaller]

Hello everyone!

This is my first post and hope I can get some help/feedback about one question that has been annoying me since some time ago.

The problem is quite simple. I´m performing a full small signal analysis of the typical bipolar cascode as shown in the figure in order to calculate the output resistance of the current mirror



The result is well known to be R_out = \[\beta\]₀*r_o / 2, but when i´m performing the analysis i simply get stuck in the equations and cannot find a functional relation between my test output voltage and my test output current. My original small signal circuit, and after some typical assumptions, looks like this



And here is where i got stuck I can´t simply find how to relate the output voltage and current with respect to i_b. Could you maybe provide some help how to solve it or where can I found a full explanation of this analysis. All the books and documents I´ve found state that "after doing the full analysis" the result is the one i mentioned before, but no detail can be found

Any help would be really appreciated!

Thanks a lot,

Regards,

Megaknaller

 

[godfreyl]

cannot find a functional relation between my test output voltage and my test output current.

There isn't a simple relationship. You need to worry about things like Early effect.

For a given collector current, Vbe and hFE both change with changing collector voltage.

BTW, I think that mirror works better if you connect the bases of Q1 and Q3 to Q1's collector instead of Q3's collector

 

[megaknaller]

Dear godfreyl,

From my understanding, the Early effect is considered when we include a finite ro at the output of each transistor. Assuming that all the transistors work in the active region, Vbe could be considered mainly as the Vbe(on) and hfe to be defined by the linear relationship ic = hfe*ib. From the equivalent circuit that I derived, which are then the assumptions that have to be made in order to arrive to the expected result Rout = beta*r0/2? Maybe I´m lacking them, but in this case, i´m mainly interested in the theoretical solution of this circuit.

Many thanks for your help.

 

[godfreyl]

Vbe could be considered mainly as the Vbe(on)

Mainly, but not entirely, otherwise life would be simple.

and hfe to be defined by the linear relationship ic = hfe*ib.

It's not a linear relationship. Also, hFE varies with Vce.

the assumptions that have to be made

If you make "simplifying" assumptions, you get "simplified" answers e.g. output impedance = infinity.

Seriously, look at some datasheets for e.g. BC547 transistors, and pay attention to the characteristic curves.