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bandgap and Vbe

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kell

Sun Apr 18 2010, 10:54PM

Registered Member #142 Joined: Sat Feb 11 2006, 01:19PM
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Vbe kind of mystified me for a long time. I finally dove into a good textbook and picked apart the equations.

This is actually the math for a bjt (diode-connected), and it has a term related to base doping level, so it may not apply to a diode that consists of a single pn junction.

But that's just as well; it makes more sense in most applications to use a diode-connected transistor anyway.

The expression for Vbe is a linear function of the log of absolute temperature.

Vbe = Vg + (kT/q)( ln(E*Ic) - (4-n) ln T)

where Vg = band gap voltage for silicon

k/q = 86.17 microvolts per degree Kelvin (Boltzmann's constant divided by electron charge)

E is a bunch of device parameters bundled together that doesn't depend on temperature, according to the text. I got an expression for E in terms of the fundamental device parameters but it's nasty.

Ic = collector current

-n is the exponent in the power law that relates average electron mobility in the base to absolute temperature. A possible value for n, according to the text, is 0.8

Taking the derivative,
dVbe/dT = (k/q) (ln(E*Ic) - (4-n) (1+ ln T))

Ok, so the temperature coefficient is still a linear function of ln T. Now the question that always interested me was, what happens to the temperature coefficient as the temperature varies across a range, say -55C to 125C?
According to the equation, tempco changes 166uV. If the Vbe is 2mV per degree Kelvin, then diode tempco changes about 460 ppm per degree centigrade. It changes about 8 percent from -55C to 125C.

So for example, say you are using a diode-determined voltage or current to cancel a thermal-voltage determined source in a bandgap circuit.

The thermal voltage is perfectly linear over temperature, in fact it is directly proportional to absolute temperature.

The fact that diode voltage is not quite perfectly linear points up a systemic error in the bandgap circuit architecture, and explains why it may not be possible to get a perfectly flat voltage or current reference with respect to temperature from a bandgap circuit, even in theory. But it's still pretty darn good. And there are ways of compensating even the small curvature in a bandgap reference, apparently.
Another interesting fact the equations reveal: at very low temperatures the temperature coefficient begins to shift dramatically.

radiotech

Mon Apr 19 2010, 01:24AM

Registered Member #2463 Joined: Wed Nov 11 2009, 03:49AM
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Wow! we just used 26mV/mA to determine Re to add to RE for the effective emitter resistance. The constant seemed to work with all bjt's.

kell

Mon Apr 19 2010, 02:27PM

Registered Member #142 Joined: Sat Feb 11 2006, 01:19PM
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On further reflection, the argument of a log function should be dimensionless, so there's something suspect about those equations.
I think I'll try to rework it. I could use Boltzmann's constant and the bandgap voltage to cancel degrees Kelvin.
Vg*q/k equals about 13 thousand degrees Kelvin. I can use that as the new base unit for temperature. Call it Ta, for apocalypse temperature. Then temperature expressed in terms of that base unit would be dimensionless, and ln (T/Ta) would make sense.

Steve Conner

Mon Apr 19 2010, 03:14PM

Registered Member #30 Joined: Fri Feb 03 2006, 10:52AM
Location: Glasgow, Scotland
Posts: 6706

Interesting! I enjoy messing around with solid-state audio amps, and the thermal stability of them depends quite a lot on minutiae like these.

For instance, ONSemi and Sanken make output transistors with built in diodes for thermal sensing and compensation. But nobody, not even the guys who write the ONSemi application notes, seems to know whether the diode actually has the same tempco as the transistor Vbe it's supposed to be compensating! Let alone how the tempco varies with current, and what current you're supposed to run it at.

I always just assumed that Vbe had a tempco of -2.1mV per degree C in theory. In practice it doesn't, but the deviations from theory I always treated as manufacturing defects that you can't model. Non-linearity of the tempco with temperature is a second-order effect: it's the tempco of the tempco if you like.

Bob Pease presents a much more detailed method that takes account of how the tempco varies with current and temperature.

Interestingly, he points out that a bandgap reference works by extrapolating Vbe back to a temperature of absolute zero.

There are ICs used to provide a digital readout from thermal sensing diodes, and the datasheets and app notes for these ICs go into the theory in quite some detail.

kell

Tue Apr 20 2010, 02:17AM

Registered Member #142 Joined: Sat Feb 11 2006, 01:19PM
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Well, that's mighty interesting reading in the link you gave.
We have a new Robert A. Pease constant, or Crap for short, on the order of 10^11 depending on device.
Set the classic ideal diode equation for Ic equal to the equation for Ic that Pease gives at the bottom of the article.
Comes out Crap equals the saturation current times exp (Vg/Vt), but Crap has very small temperature dependency and Is has a very heavy temperature dependency. Pease's equation is so much more useful. But even it is not a perfect model -- he states in the article that the Vbe versus temperature graphs are not actually perfectly linear like the model he's using. When I have some free time I'll try to see if the slight nonlinearity he describes is congruent with the results I got working from the other direction, starting with the saturation current and deriving that goofy equation that uses log T. Perhaps there will be some distant convergence point at very high T (that apocalypse temperature) analogous to the voltage convergence at absolute zero.

[edit] Hey radiotech, I'm not sure what you mean by 26 mV per mA. That's just 26 ohms.
Do you mean thermal voltage divided by emitter current: 26mV/Ie -- is that the value of the emitter resistance? I'll have to look it up.

Steve Conner

Tue Apr 20 2010, 09:42AM

Registered Member #30 Joined: Fri Feb 03 2006, 10:52AM
Location: Glasgow, Scotland
Posts: 6706

kell wrote ...

Do you mean thermal voltage divided by emitter current: 26mV/Ie -- is that the value of the emitter resistance? I'll have to look it up.

Yep, to a first approximation, the internal emitter resistance of a transistor is 26/Ie ohms, with Ie in mA.

So, in a common-emitter amp, the small-signal input impedance at the base is beta times that.

And the small-signal transconductance, gm, is just the inverse of that: Ie/26.

Funny how the UK and Canada use 26, but some American texts use 25. I guess it's colder over here...

kell

Tue Apr 20 2010, 04:25PM

Registered Member #142 Joined: Sat Feb 11 2006, 01:19PM
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Pease's equation at the very end of that article includes a term for emitter resistance. I guess I'd like to know whether that's emitter resistance in the small-signal definition, or does it carry some other meaning?

radiotech

Tue Apr 20 2010, 04:51PM

Registered Member #2463 Joined: Wed Nov 11 2009, 03:49AM
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Bias stabilization calculations were done in the past while various techniques were being perfected to move away from germanium and over to silicon. You can bypass the external emitter resistor but not the intrinsic.
So when you think about alpha always being <1, at one point there was a 'hook transistor' with a>1.

This was the thinking about 50 years ago- Reference Data for Radio Engineers 4th Ed. International Telephone and Terlegraph Corp. New York (1962)- note how 'c' was wandering then.

radiotech

Tue Apr 20 2010, 05:19PM

Registered Member #2463 Joined: Wed Nov 11 2009, 03:49AM
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sorry abour dounle post image vanished

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