I'm looking for the derivation of the formula for the stability factor (\$S''\$ or \$S(\beta)\$) of BJT bias circuits, particularly the ones with emitter resistance. The book I'm using says that the derivation is too complex and so they didn't include it. I tried to derive it myself but I can't get the same result. Does anyone know where I can find the derivation? I tried googling, but the sources I found say the same thing as my book.
\$\begingroup\$
\$\endgroup\$
2
-
\$\begingroup\$ For my opinion, googling gives a lot of (good and detailed) references (example: iitg.ac.in/apvajpeyi/ph218/Lec-7.pdf). \$\endgroup\$– LvWCommented Jun 19, 2014 at 9:08
-
\$\begingroup\$ s′′ has a unit, as, by definition, \frac{dI_{C}}{d\beta} has Amperes as unit \$\endgroup\$– user179765Commented Mar 1, 2018 at 7:08
Add a comment
|
1 Answer
\$\begingroup\$
\$\endgroup\$
4
It seems I made some error in the manipulation of variables which is why I didn't get the same formula as in the book. I thought I needed to find partial derivatives, but it turns out that algebra is all that is needed. I'll just place the first few equations of the derivation if someone else wants to know since it's kind of long.
For an emitter-bias configuration: $$V_{CC} - \frac{I_C}{\beta} R_B - V_{BE} -\frac{\beta + 1}{\beta} I_C R_E = 0$$ $$I_C = \frac{\beta (V_{CC} - V_{BE})}{R_B+(\beta + 1)R_E} \tag1$$
Since \$s''= \frac{\Delta I_C}{\Delta \beta}\$, compute \$\Delta I_C = I_{C_2} - I_{C_1}\$: $$\Delta I_C = I_{C_2} - I_{C_1} = \frac{\beta_2 (V_{CC} - V_{BE})}{R_B+(\beta_2 + 1)R_E} -\frac{\beta_1 (V_{CC} - V_{BE})}{R_B+(\beta_1 + 1)R_E}$$
Simplifying (a lot) and then using eq. 1 and then dividing \$\Delta I_C\$ by \$\Delta \beta\$ you will get the formula: $$s'' = \frac{I_{C_1}(R_B + R_E)}{\beta_1(R_B + (\beta_2+1)R_E)}$$
-
\$\begingroup\$ :I didn´t recalculate your result. However, I am afraid - it is not correct. I think, the stability figure must NOT have a unit (instead it must be dimensionless). In your case, the figure "s" is given as a current ("Amperes"). \$\endgroup\$– LvWCommented Jun 19, 2014 at 13:07
-
\$\begingroup\$ More than that, you have replaced the derivative by the ratio of two finite differences. This can be done, however, if you know proper values for Ic1, Ic2 and the beta values only, do you? \$\endgroup\$– LvWCommented Jun 19, 2014 at 14:41
-
\$\begingroup\$ Sorry - forget my last comment. Only now I have realized that you are interested in s'' or s(beta) only. \$\endgroup\$– LvWCommented Jun 19, 2014 at 15:36
-
\$\begingroup\$ I have to apologize, because I didn´t realize that the stability S with respect to beta was wanted. Therefore, both of my comments do not apply anymore. \$\endgroup\$– LvWCommented Jun 19, 2014 at 17:09