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I have this CB amplifier circuit below

enter image description here

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

enter image description here

Here is my analysis below to find the output resistance R_out

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

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

JordenSH
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2 Answers2

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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

You made a mistake here

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

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

G36
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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.

Avid Pro Tool
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  • 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. – JordenSH Jun 02 '18 at 01:36
  • Accidents happen. If you look at the hybrid pi model [even here on Wikipedia](https://en.wikipedia.org/wiki/Hybrid-pi_model), 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. – Avid Pro Tool Jun 02 '18 at 01:58
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    But this is the T-model, and this is the convention used throughout the book so far. – JordenSH Jun 02 '18 at 02:02
  • @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](https://electronics.stackexchange.com/questions/252197/why-is-vbc-absent-from-bjt-equations/252199#252199)? Just curious. – jonk Jun 02 '18 at 02:43
  • @jonk these look like more of a device physics models. The T-model is another version for the hybrid pi model – JordenSH Jun 02 '18 at 03:10
  • either way looks like there was an algebra mistake when getting v_pi in the work you've shown. There should be an R_e in the denominator instead of that 1. – Avid Pro Tool Jun 02 '18 at 03:18
  • @Raykh No, device models are deeper and would involve things like current densities, variable doping profiles, etc., whether 1D, 2D, or 3D. The link I provided here are for large signal behavior models (meaning, non-linear) that work over a wide range. The hybrid-\$\pi\$ I mentioned there is the ***non-linear*** large signal model (*before* being linearized.) Many folks, not even aware of the fuller model, may write "hybrid-\$\pi\$" but actually would mean to say "small signal linearized hybrid-\$\pi\$" if they knew better. – jonk Jun 02 '18 at 04:10
  • I think @Raykh is referring to the T model described [here](https://wiki.analog.com/university/courses/electronics/text/chapter-8) in section 8.8. I'm not very familiar with it either and it threw me off since I'm used to small signal linearized hybrid pi. But I think it's just an algebra mistake here in the original question. – Avid Pro Tool Jun 02 '18 at 04:20