We all know that in a BJT, a rise in temperature causes a drop in Vbe and an increase in collector current, when working in the active region. We can verify that in most transistor layouts using Kirchoff's laws. But what about the actual collector current equation: \(I_c=I_s\cdot({e^{\frac{V_{\tiny{BE}}}{nV_{\tiny{T}}}}) \) ? Having in mind that \( V_{\tiny{BE}} \) drops when temperature increases, shouldn't Ic drop too? I know some other variable in the equation must be affected by the temperature, but I don't know which one and how it affects the whole situation. I examined V\(_{\tiny{T}}\) but it doesn't seem to affect Ic enough to compensate for the change of \( V_{\tiny{BE}} \).

Any thoughts? Thanks in advance.

 

If \(\small{T}\) is the variable, you can either:
1. Hold \(\small{V_{BE}}\) constant and examine what happens to \(Ic\) as \(\small{T}\) changes:

\(I_c=I_s\cdot(e^{\frac{qV_{BE}}{{\eta}kT})\)

2. Hold \(I_c\) constant and examine what happens to \(\small{V_{BE}}\) as \(\small{T}\) changes:

\(V_{BE}=\frac{{\eta}kT}{q}\cdot(ln\frac{I_c}{I_s})\)

All this assumes \(I_s\) is constant, but unfortunately, it is also a function of temperature.

 

It seems I did a little mistake, and for the worse:
As the exponent of e increases, Ic increases and vice versa (not proportionaly, but exponentialy). Now, both reduction of Vbe and increase of T (and consequently Vt) decrease the exponent. As a result, both reduce Ic. Things are not looking good for our model, as none of the parameters mirror reality.
This is really confusing...

 

It seems I did a little mistake, and for the worse:
As the exponent of e increases, Ic increases and vice versa (not proportionaly, but exponentialy). Now, both reduction of Vbe and increase of T (and consequently Vt) decrease the exponent. As a result, both reduce Ic. Things are not looking good for our model, as none of the parameters mirror reality.
This is really confusing...

I think this excerpt from Art of Electronics (2nd edition, p.81) explains the source of our confusion, and makes my previous post invalid (or irrelevant).
Also see http://www.cmi.ac.in/~ravitej/lab/10-boltz.pdf

 

Is rises very quickly with temperature and dominates the shrinking exponential.

 

I think this excerpt from Art of Electronics (2nd edition, p.81) explains the source of our confusion, and makes my previous post invalid (or irrelevant).
Also see http://www.cmi.ac.in/~ravitej/lab/10-boltz.pdf

Well, I think that is the best answer we are going to get. Thank you for your effort and patience, leafing that book to find the right page. That link was also helpful. I feel a little dissapointed though, since such a common phenomenon has no clear explanation in any reknowned electronics book (I have also searched in Sedra/Smith).
Thanks again, and have a nice day!