Electronic – Z(in) for small signal analysis with BJT for unbypassed emitter and r0 in place


I have been studying the small signal analysis of BJT with re model with the emitter terminal not bypassed. I am able to derive those Zi, Zo,Av, Ai long as ro is assumed to be infinite. But I am unable to find Zi with a finite ro. Following is the circuit diagram:

enter image description here

Below is the equivalent re model:
enter image description here

I tried with google, but have not been able to find any. Everywhere the solution is done omitting ro.Kindly help me with the analysis or provide some helpful link. Thanks in advance.


let, beta=B

From my analysis, Vb=Ib[Bre+(B+1)RE]

So, Zb=Vb/Ib= [Bre+(B+1)RE]


But the expression for Zb given in the book by Boylestad and Nashelsky is


which I am not able to understand. Please explain it.

Best Answer

enter image description here
Let the current through r0 be Ir. Applying KCL,

1.  Ie = (B*Ib) + Ib + Ir.
2. Io = Ir + (B*Ib).

Applying KVL,

(Io*Rc) + (Ir*r0) + (Ie*Re) = 0.

From eq 2:

Ir = -[(B*Ib*Rc) + Ie*Re]/r0+Rc.

Ie = (1+B)*Ib - [(B*Ib*Rc) + (Ie*Re)]/(r0+Rc).

Ie(1 + (Re/r0+Rc)) = (1+B)Ib - (B*Ib*Rc)/r0+Rc.

Vb = Ib(B*Re) + Ie*Re.  

Vb= Ib(B*Re) + Ib*Re[(B+1) + (Rc/r0)]/[1 + (Rc+Re)/r0]

Zb = B*Re + Re[(b+1) + (Rc/r0)]/[1+(Rc+Re)/r0]

Please pardon me I don't know which software to use for formatting. I tried to upload an image, but it failed. I have skipped some of the steps which could be easily arranged.