Transistor hFE equation question

[TFM]

Homework Statement

Showing how to get:

$$Z_{out} = \frac{Z_{in}}{h_F_E + 1}$$

from

$$\Delta I_E = \Delta V_B/R$$

Homework Equations

$$\Delta I_E = \Delta V_B/R$$

$$Z_{out} = \frac{Z_{in}}{h_F_E + 1}$$

$$\Delta V_E = \Delta V_B$$

The Attempt at a Solution

I am trying to prove the above, but the book makes quite a large jump (again, "The Art of Electronics")

It goes from:

$$\Delta I_E = \Delta V_B/R$$ - (1)

straight to

$$\Delta I_B = \frac{1}{h_{FE} + 1}\Delta I_E = \frac{\Delta V_B}{R(h_{FE} + 1)}$$ -(2)

Can anyone help show how they've gone from (1) to (2)?

Thanks,

TFM

 

[MATLABdude]

Homework Statement

Showing how to get:

$$Z_{out} = \frac{Z_{in}}{h_F_E + 1}$$

from

$$\Delta I_E = \Delta V_B/R$$

Homework Equations

$$\Delta I_E = \Delta V_B/R$$

$$Z_{out} = \frac{Z_{in}}{h_F_E + 1}$$

$$\Delta V_E = \Delta V_B$$

The Attempt at a Solution

I am trying to prove the above, but the book makes quite a large jump (again, "The Art of Electronics")

It goes from:

$$\Delta I_E = \Delta V_B/R$$ - (1)

straight to

$$\Delta I_B = \frac{1}{h_{FE} + 1}\Delta I_E = \frac{\Delta V_B}{R(h_{FE} + 1)}$$ -(2)

Can anyone help show how they've gone from (1) to (2)?

Thanks,

TFM

They don't. They substitute (1) into the equation relating $$\Delta I_E$$ and $$\Delta I_B$$ (the first half of (2)). I believe AoE uses $$h_{FE}$$ in place of $$\beta$$

 

[TFM]

Okay so:

$$\Delta I_E = \Delta V_B/R$$

substitute into:

$$\Delta I_B = \frac{1}{h_{FE} + 1}\Delta I_E$$

gives:

$$\Delta I_B = \frac{1}{h_{FE} + 1}\frac{\Delta V_B}{R}$$

$$\Delta I_B = \frac{\Delta V_B}{(h_{FE} + 1)R}$$

R is the load, so I am assuming that this is the R output. Multiply it out:

$$R\Delta I_B = \frac{\Delta V_B}{(h_{FE} + 1)}$$

divide by I_B

$$R = \frac{\Delta V_B}{(h_{FE} + 1)\Delta I_B}$$

V = IR
R = V/I

thus:

$$R_{output} = \frac{R_{input}}{(h_{FE} + 1)}$$


$$Z_{out} = \frac{Z_{in}}{h_F_E + 1}$$

is also the same as:

$$R_{out} = \frac{R_{in}}{h_F_E + 1}$$

Z is just a complex version of R

Does this look correct?

TFM

 

[MATLABdude]

I think that's okay. Though you might want to make some justifications as to what the input and output (and input and output impedances) are of the BJT (unless this was part of the setup for the question). And you should probably start with complex impedances (Z=V/I) instead of changing from R to Z mid-way through.

Just my 2c.

 

[TFM]

Okay, Thanks for all your assistance

Thanks,

TFM