expo mix up

[Magnus Danielson]

From: Tom May <tom at go2net.com>
Subject: Re: expo mix up
Date: 07 Oct 1999 16:56:48 -0700

Hi Tom!

You really got all the opertunities to put me out on the thin ice here, it's
been over than a year that I did this last time. I also ended up reviewing my
old notes.

> Magnus, could you go into a little more detail here?  It seems that
> the ref transistor draws come current that loads whatever is driving
> its base.  Since the collector current of the ref transistor is fixed,
> I'd expect the base current to be fixed except for beta-dependent
> temperature variations, whereas the expo transistor base current will
> vary with the CV, but it's not clear how rBE fits into this or where
> the matching of rBE comes in.
>
> And what exactly do you mean by rBE?  The lowercase "r" suggests a
> dynamic (small-signal) resistance, which according to the Ebers-Moll
> equations (and the moog ladder) should be a function of the collector
> current and therefore fixed for the ref transistor and variable for
> the expo transistor.

Let's start with approximate an transistor with the formula

            I
      kT     C
V   = -- ln -- + I r
 BE    q    I     C BE
             S

where

V   is the Voltage between Base and Emitter
 BE

r   is the bulk resistance between Base and Emitter
 BE

I   is the Collector current
 C

I   is the extrapolated collector current for zero V
 S                                                  BE

T   is the absolute temperature of the PN-junction measured in degrees Kelvin

                                            -23
k   is the Boltzmann's Constant (1.380658*10    J/K)

                                      -19
q   is the unity charge (1.60217733*10    C)

Do you know understand what I mean by rBE?

I did revist my notes, just to find out that I cheated and approximated rBE of
both transistors to be near zero and then simply dropped the terms.
Shame on me!

However, stuffing to of those transistors together you would get a relation
like this:
             I          I
     k        C1         C2
V  = - (T ln --- - T ln ---) + I  r    - I  r
 3   q   1   I      2   I       C1 BE1    C2 BE2
              S1         S2

Where V3 is the voltage of the ref-transistor Q1 and Q2 is the expo transistor.

So you could eliminate the effect of the bulk resistance by feedback the
current Ic2 as a suitable level voltage to sum up V3.

While my original comment was incorrect, the addition of resistance is still
an issue. I will do some more homework some day.

Cheers,
Magnus