Low Frequency Response of BJT amplifier(effect of bypass capacitor)

Question

Doubt related to impedance seen by the bypass capacitor:

or

What is the impedance seen by the bypass capacitor \$C_E\$

To calculate the impedance as seen by \$C_E\$, we attach a Thevenin volatge source as shown.

Applying Kirchoff's current law:

$$ \frac{V_T}{\beta r_e+R_S||R1||R_2}-\beta I_B+\frac{V_T}{R_E}=I_T$$ $$\frac{V_T}{\beta r_e+R_S||R1||R_2}+\beta \frac{V_T}{\beta r_e+R_S||R1||R_2} +\frac{V_T}{R_E}=I_T$$ $$V_T[\frac{(1+\beta)}{\beta r_e+R_S||R_1||R_2}+\frac{1}{R_E}]=I_T$$ $$V_T[\frac{1}{\beta r_e+R_S||R1||R_2}+\frac{1}{r_e+\frac{R_S||R_1||R_2}{\beta}}+\frac{1}{R_E}]=I_T$$

From here I get the resistance as $$\frac{1}{R_e}=\frac{1}{\beta r_e+R_S||R1||R_2}+\frac{1}{r_e+\frac{R_S||R_1||R_2}{\beta}}+\frac{1}{R_E}$$

However in book, the resistance has been given as:

$$R_e=R_E||(\frac{R_s||R1||R2}{\beta}+r_e)$$

It seems the first term in \$\frac{1}{R_e}\$ vanishes! Where might have I gone wrong?

(I have referred to the following text book: Electronic Devices and Circuit Theory, by Boylestad and Nashelsky.)

Answer 1

Soumee, I cannot identify any error in your calculation.

However, if you replace in the third line of your calculation the term (1+β) by β, your result will be identical to the expression as given in the book. As you know the current gain β is relatively large (mostly > 100), not a constant but dependent on Ic and - more important - equipped with large tolerances.

Therefore, we often simplify (1+β) to β (without expecting not acceptable errors) - and this seems to be the only reason for the discrepancy you have observed.

Answer 2

Here put It=0 because according to KCL algebraic sum of currents in the node is zero. \$ 1+b = b \$ (it doent make any change)