[SOLVED] Simplified BJT model question (sedra smith)

[peterpops]

Hello!

Another question regarding a problem in Microelectronic circuits 5th ed by Sedra Smith (exercise 1.20)

Please see the attached image. Everything except the green arrow and label is in the original figure


Problem:
Find the input resistance between terminals B and G in the circuit shown in the attached image. The voltage v_x is a test voltage with the input resistance R_in defined as R_in = v_x / i_x

My approach
i_x = i_b

the currents sum in the point above E in the figure is
i_x + βi_x - i_e = 0

EQ 1: i_x = i_e / (1 + β)

i_e = v_e / R_e

v_e = v_x * (R_e / (R_e + r_π))

i_e = v_x / (R_e + r_π)

inserted into EQ 1

i_x = (v_x / (R_e + r_π)) / (1 + β) = v_x / ((R_e + r_π)*(1 + β))

And finally

R_in = v_x/i_x = (R_e + r_π)*(1 + β)

However

Answer by the book is R_in = r_π + (β + 1)R_e

So I have one "βr_π" to much.. what's wrong with my calculations??

 

[LvW]

The equation Ve=... is not correct. This equation assumes that Ve is determined by Vx only - and you forgot the current source.

 

[peterpops]

The equation Ve=... is not correct. This equation assumes that Ve is determined by Vx only - and you forgot the current source.

Ahh okay, but how do I incorporate that into the Ve-equation? I've been looking around at different methods (DC network analysis methods) but I can't get anyone to work. Could you point me in the right direction?

 

[LvW]

The circuit diagram in post#1 is a LINEAR small-signal circuit. Therefore, you can apply the superposition method.
It`s one of the fundamental laws which is used to calculate the voltage-current distribution in a circuit with more than one sources.

 

[peterpops]

The circuit diagram in post#1 is a LINEAR small-signal circuit. Therefore, you can apply the superposition method.
It`s one of the fundamental laws which is used to calculate the voltage-current distribution in a circuit with more than one sources.

Thanks! I tried a some of the other methods without success but the superposition worked! Thanks!