# finding output resistance of CB amplifier with ro

I have this CB amplifier circuit below

This is the small signal equivalent circuit arranged to find the output resistance R_out with the r_o included

Here is my analysis below to find the output resistance R_out

$$v_x\:=\:r_o\left(i_x-g_mv_{\pi }\right)+R_e\left(i_x+\frac{v_{\pi }}{r_{\pi }}\right)$$ $$v_x\:=\:i_x\left(r_o+R_e\right)+\left(\frac{R_e}{r_{\pi }}-g_mr_o\right)v_{\pi }$$ $$v_{\pi }+R_e\left(i_x+\frac{v_{\pi }}{r_{\pi }}\right)\:=\:0$$ $$therefore\:\:v_{\pi }\:=\:-\frac{r_{\pi }}{r_{\pi }+1}R_ei_x$$ after some rearragements $$\frac{v_x}{i_x}\:=\:R_{out}\:=\:r_o+R_e+\left(g_mr_o-\frac{R_e}{r_{\pi }}\right)\left(\frac{r_{\pi }R_e}{r_{\pi }+1}\right)$$ I tried to manipulate my results but it does not come out to the same formula in the book which is below: $$\frac{v_x}{i_x}\:=\:R_{out}\:=\:r_o+R_e\backslash \backslash r_{\pi }+\left(R_e\backslash \:\backslash \:r_{\pi \:}\right)g_mr_o$$

can somebody please tell what am doing wrong?? thanks in advance for all the help.

For those who are not familiar with a small-signal linearized T-model take look here: http://www.ittc.ku.edu/~jstiles/412/handouts/5.6%20Small%20Signal%20Operation%20and%20Models/The%20Hybrid%20Pi%20and%20T%20Models%20lecture.pdf

http://www.ittc.ku.edu/~jstiles/412/handouts/5.6%20Small%20Signal%20Operation%20and%20Models/section%205_6%20%20Small%20Signal%20Operation%20and%20Models%20lecture.pdf

$$v_\pi+R_e \left( I_x+ \frac{v_{\pi}}{r_\pi} \right) = 0$$

Solve for $v_\pi$

$$v_\pi+R_eI_x+ \frac{R_ev_{\pi}}{r_\pi} = 0$$

$$v_\pi r_\pi + R_e I_x r_\pi + R_e v_{\pi} = 0$$

$$v_\pi r_\pi + R_e v_{\pi} = - I_x R_e r_\pi$$

$$v_\pi (r_\pi + R_e) = - I_x R_e r_\pi$$

$$v_\pi = - I_x \cdot \frac{ R_e \cdot r_\pi}{r_\pi + R_e} = -I_x\cdot R_x$$

Also, you can look here:

BJT common-base output resistance derivation

Calculation of output impedance of CE emitter bias configuration( unbypassed) with r_0

• Woaw, comic sans. Nice. Commented Jun 2, 2018 at 9:41
• @G36, thank you so much for your help. I knew I was heading in the right direction. Turned out it's a simple algebra mistake. Thank you very much for spotting the error and appreciate your help. Commented Jun 2, 2018 at 13:46

Your dependent current source is in the wrong place in your AC equivalent diagram, it should be in parallel with r_o, going from C to E junction.

• This is a screenshot from one of the most famous electronics textbooks, Sedra Smith. What are you talking about?? plus your reasoning is exactly what is on the diagram. Commented Jun 2, 2018 at 1:36
• Accidents happen. If you look at the hybrid pi model even here on Wikipedia, you can see that the dependent current source in the equivalent circuit of a BJT goes between the C and E junctions, not C and B. Commented Jun 2, 2018 at 1:58
• But this is the T-model, and this is the convention used throughout the book so far. Commented Jun 2, 2018 at 2:02
• @Raykh I don't have sedra/smith, nor have I read it. By "T-model" do you mean a linearized version of the "transport model" as shown here at three equivalent Ebers-Moll models of the BJT? Just curious.
– jonk
Commented Jun 2, 2018 at 2:43
• @jonk these look like more of a device physics models. The T-model is another version for the hybrid pi model Commented Jun 2, 2018 at 3:10