About the math of paralleled power transistors and current hogging

[jneutron]

Is device gain vs temperature of concern? For a single die, a hotspot can hog current by higher gain.

John

 

[bwaslo]

You have derived a formula for the temperature rise caused by a temperature rise. All you have to do is put this result in the form of the normal feedback equation (for positive feedback), then you can see what the effect will actually be for a given initial fluctuation or imbalance. I think you will find that if you choose a resistor twice the calculated value, then the temperature deviation will be twice the stimulus. Three times the calculated value gives 150% of the stimulus etc.

Hi DF,

Check the (hard to read) math from my latest writeup, you'll see that it is in the form of a feedback equation:

I(N) = [(Vbg - Vbe(N) -2.1E-3 * (I(N) * Vce *Rth(N)] / R

Collect I(N) terms, becomes:

I(N) = (Vbg - Vbe(N))/(R - 2.1E-3 * Vce * Rth(N))​

(note the "R minus stuff" term in the denominator)

But I realized last night that I have an error in the way I calculated Vgd to set the overall current -- Vgd is being set without considering any effect from either of the junctions heating. That makes a pessimistic prediction for the imbalance for non-matched thermal resistances (total current will be too high). Shouldn't affect things for differing Vbe, though. When I get a chance today or tomorrow, I'll fix that. Or at least make it less wrong by setting things so that BJT#2 has half the target current. Not quite the same scenario, but shouldn't be too far off when the "percent hogging" isn't large.

I could set it up in a "solve" loop or a succession of calculations to get the total current to equal the target, I guess, but I don't think it's necessary for engineering work.

 

[bwaslo]

I tested twelve Fairchild D44H11 power transistors from lot code "C20AB" using the test fixture illustrated below. It took fifteen minutes.

The sample standard deviation of "VBE" was 2.69 millivolts. I will let other readers consider how to turn that into an estimate of the population standard deviation. Want raw data? Spend fifteen minutes measuring.

That fifteen minutes assumes I have a large enough batch of power transistors to make meaningful statistical measurements (I don't)!

I did find an older thread where someone measured a bunch of Motorola/ONsemi power NPNs and was suprised to find all of their Vbe within 1mV! He mentioned that to get such consistent results he had to probe the transistor leads separately from the biasing circuit, Kelvin-style (his fixture was also arranged to allow collector current, though, so contact resistances would play a bigger part than with your open-collector method). Someone on that thread mentioned that PNPs tend to vary more than NPNs. All anecdotal data, though, more so since I didn't mark that thread to find it again (reading too late last night).

I don't think you can take the std deviation from 12 samples from the same batch (and I assume, wafer) and use just that to get a population estimate --- can you?

 

[bwaslo]

Is device gain vs temperature of concern? For a single die, a hotspot can hog current by higher gain.

John

John, as you know, the gain (transconductance) of a BJT is pretty much always Ic/(kT/q), and, T being in Kelvin units, doesn't change a lot for the kinds of temp differences I was thinking of.

As long as we're ignoring base current, and feeding all the devices from a common base bias point, the effective Vbe drop, R value, and that bias point should completely determine the emitter current current. Though if the beta is low so base current gets significant, then that will certainly play into it.

 

[Mark Johnson]

I don't think you can take the std deviation from 12 samples from the same batch (and I assume, wafer) and use just that to get a population estimate --- can you?

Phrase it as a Monte Carlo question and then answer the question using Monte Carlo techniques.

Alice samples twelve datapoints from a gaussian distribution. What is the probability that the sample standard deviation of Alice's twelve points, is within ±10% of the "true" (population) standard deviation? What is the probability that the sample standard deviation of Alice's data, is within ±20% of the true standard deviation? In general, what is the probability that the sample standard deviation of Alice's data, is within ±X% of the true standard deviation? Make a plot of probability versus X.​

 

[jneutron]

Sounds good. As I think of it, on a die the emitter resistance is actually the spreading resistance of the aluminum on the top of the chip, so the resistance is very very low equivalent. Side to side temp differences on a chip essentially have no degenerating resistance to help out.

Separates in discrete form shouldn't be such a concern for that.

John

 

[bwaslo]

Phrase it as a Monte Carlo question and then answer the question using Monte Carlo techniques.

Alice samples twelve datapoints from a gaussian distribution. What is the probability that the sample standard deviation of Alice's twelve points, is within ±10% of the "true" (population) standard deviation? What is the probability that the sample standard deviation of Alice's data, is within ±20% of the true standard deviation? In general, what is the probability that the sample standard deviation of Alice's data, is within ±X% of the true standard deviation? Make a plot of probability versus X.​

But that would apply (assuming I figured out how to do that, Prob&Stats is many years back for me!) only for samples from that wafer, no?

 

[bwaslo]

Spreadsheet, 2nd try

OK, try again...

Assume that:

• Vbe(typ) is typical Vbe for linear operation if at the heatsink temp, somewhere around 0.7V
• Itotal is the approximate total current for the bank of bipolar transistors
• Vbg is the voltage between the bases and the lower net on the drawing
• dVbe is the voltage variation that BJT#1 has in Vbe, relative to Vbe(typ). (<=0)
• Vbe2 (for BJT#2) is assumed to be Vbe(typ); Vbe1 is assumed to be Vbe1+dVbe.
• ThetaJC is the thermal resistance of each BJT package (assumed equal)
• ThetaCH is the typical thermal resistance from package to heatsink
• thMult is a multiplier on ThetaCH, for BJT#1 (>=1)
• Rth1 is the sum of thermal resistances for BJT#1, equals ThetaJC+ThetaCH*thMult
• Rth2 is the sum of thermal resistances for BJT#2, equals ThetaJC+ThetaCH
• Vce is the approximate Collector-Emitter voltage of the BJTS (>1), assumed close enough to equal for all
• I1 is the emitter current of BJT#1
• I2 is the emitter current of BJT#2
• base currents are ignored
• Vbe varies by -2.1mV/K of the junction temp;

The value of interest is I1/I2, or the percentage that I1 exceeds I2. IOW, the amount of current hogging!

The current in either BJT is the voltage across its resistor R, divided by R. The power dissipation in either BJT is its current times Vce. The temperature of each junction rises above the heatsink temp by an amount equal to the power dissipation times its Rth, which causes a corresponding decrease in its effective Vbe. So

I(N) = [(Vbg - Vbe(N) -2.1E-3 * (I(N) * Vce * Rth(N)] / R
Collect I(N) terms as in feedback equation:
I(N) = (Vbg - Vbe(N))/(R - 2.1E-3 * Vce * Rth(N))
then
I1/I2 = (Vbg-Vbe1)*(R - 2.1E-3 * Vce * Rth2)/((Vbg-Vbe2)*(R - 2.1E-3 * Vce * Rth1))​

If you buy all these approximations, then a few things of note:
1) If the Rth values are equal, then the result doesn't depend on that value of Rth, but can depend on the operating current.
2) If the Vbe values are equal, then the result doesn't depend on the operating current.

See the attached v2 spreadsheet if you want to play with the relationships. Attached is a graph of a situation I'm looking at with Vce~= 24V, Itotal = 2A, ThetaJC=0.83K/W, ThetaCH=0.25K/W. The y axis is "% hogging", the x axis is the value of R. The blue curve is for an assumed decrease of 25mV for Vbe of BJT#1. The green curve is for a decrease of 4mV for Vbe of BJT#1. In the blue and green curves, I hand-adjusted the target Itotal value to keep the calculated Itotal near 2A for all R values. The red curve is for an assumed increase of 50% for the thermal resistance of the Case to Heatsink for BJT#1 (might be about right for too much grease or a lump?). No target Itotal tweak needed for the red curve.

I picked the "4mV" value as something that might be practical matching for some 2SA1943N PNP transistors. Does that seem reasonable?

 

[Mark Johnson]

But that would apply ... only for samples from that wafer, no?

The question may suggest that all attempts at measurement could be worthless and futile. Because we cannot possibly measure every individual in the population and we certainly cannot measure individuals that are born/made a day, week, month, year after we quit measuring. So there's always a nagging uncertainty about the measured data: is it, or is it not, representative of a small sample of the population taken at some point in the future?

A number of designers find solace when using conservatively derived upper-bounds and lower-bounds; however others consider this a misleading and dangerous practice, which wastes resources, time, and effort slaying imaginary dragons. "Polishing the wrong coprolite" (synonym for * U R D)

 

[bwaslo]

Mark, I certainly wasn't disparaging the value of measurement in general. But data from only a strongly related set of samples may only be representative of members of that set, and not very reliable if you are trying to come up with statistics that might be used by people who only have access to different sets than you had and maybe not even be able to get all their parts from a common set!

Though probably helpful if you can stipulate that all parts to be used in a project are to come from a common wafer. Though, even then, if the question were "what's the variance you can expect in Vbe of transistors that come sourced from any common wafer?", the sample size would be only = 1 (wafer). Variance of variances, iow.

 

[Mark Johnson]

Ten parts with the same date code only means they popped out of final test during the same work week. There's no guarantee, and no reason to expect, that they came from the same wafer or even from the same 24-wafer batch. Probably they did come from the same factory; so if your Portland fab tends to err on the side of "epi too thick" while your Austin fab tends to err on the side of "epi too thin", you can be confident that die with the same date code came from either Portland or from Austin but not both. You got either "epi too thin" or "epi too thick" but not both.

To alleviate the fear that your sample of parts are too much alike, I suggest buying from 3 different distros in the US, plus 2 different distros in Europe, plus 1 distro in Asia, plus one or two semi-legitimate (but not factory authorized) hobbyist suppliers, like B&D or Jameco or Anchor or Futurlec. Increase the probability that they all didn't come from the same Fab during the same work week.

 

[bwaslo]

Makes sense, if I were doing all the research myself. Which is why I was asking whether there already were some such info out there (not to minimize your data, just hoping there might be some exensive manufacturer or industry data).

I need to find that Portland fab place, maybe its' in the back room of one of the zillion coffee shop here.

 

[Mark Johnson]

Gresham and Aloha.

 

[PRR]

> I need to find that Portland fab place

Was a fine fab in Portland MAINE for half a century. Fairchild, but one of the other big boys shared it for a while. In recent years it only did prototypes, all real fabbing done at the usual offshore fabs. Last year the Fairchild operation was pending sale to I think ON Semi, but there was another party in the dickering.