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29th December 2016, 02:59 AM  #1 
Old guy with soldering iron
diyAudio Member

About the math of paralleled power transistors and current hogging
I'm trying to get a good estimate of the needed resistance of ballast resistors when paralleling a bank of output BJTs. I'm not sure if I'm looking at this the right way (trying to do a maybe toosimplified estimate?), would appreciate any direction others here may have. Hoping to avoid trying to dredge up semiconductor physics. Anyone want to slog through this? Or can the correct calculation for this be found somewhere I haven't looked?
Here's the situation: Two (or more) BJTs, all have their collectors tied together, their bases tied together, and each emitters has its own ballast resistor R to a common emitter point. The collector voltage is constant above the common emitter voltage and the whole set is controlled (via the base to the common emitter voltage) to regulate the total current. One of the BJTs gets a little hotter (deltaT degrees) than the others, and the ballast value needs to be large enough to prevent the hot transistor from going into thermal runaway. The extra temperature results in the Vbe of the hot transistor to tend to drop (approximately, according to EbersMoll model) 2mV for every degree K that its heat exceeds that of the others. Which if unregulated, would increase the drop across resistor R and cause an increase in that transistor's current of about 2m*deltaT/R. But since the total current of the whole bank is being regulated, it doesn't go quite that high  if there are two in the bank, one would increase by approximately 1m*deltaT/R, the other would decrease by that amount. (If the bank has more than two transistors in it, then the current increase of the hot transistor would be something like (11/N)*2m*deltaT/R. I think.). So assuming that is approximately right, and only two devices in the bank (for example) then the additional power dissipation of that transistor would be about Vce*1m*deltaT/R. The junction will have a thermal resistance between it and the heatsink of Theta. The total power of the bank doesn't change so the heatsink temperature should stay about the same (neglecting thermal resistance between the devices where they mount  is that a big mistake to assume?). So the Theta of interest is the Theta(jc) [K/W] of the package plus the thermal resistance of the package to the heatsink. The extra power should raise the junction temperature again by Theta*Vce*1m*deltaT/R. In order to not run away infinitely (edit: actually, not infinitely, only till it hogs all the current), this has to stay lower than the deltaT that started the whole change. So, require that: Theta*Vce*1m*deltaT/R < deltaT Adjust the 1m value above as needed for more than 2 devices used in the bank. And of course R would have to be significantly bigger than the resulting value, since the calculation is for when it just barely doesn't take off.R > (Theta*Vce*1m/R) Is that anywhere near correct? I can see one (mitigating) factor that is not included, which is the internal resistance that might already exist in each emitter leg. And if the overall heatsink temp doesn't rise about the same for each BJT, that would also skew the calculation (probably would be an addition to the Theta value). A concern is that using this calculation, I get R values quite a bit smaller than I'd expect and usually see (is that to allow for the possibility of a weaker thermal interface on one of the devices from the others  to avoid failure from one misapplied bit of thermal goo?). Or is this approach just too simplistic, or did I miss something? What's the right way? Set me straight.
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[W9MJE] Horn spreadsheet SynergyCalc/; SmallSyns SmallSyns; Crossover design Xsim; Depot diffusor supereasy diffusors Last edited by bwaslo; 29th December 2016 at 05:25 PM. 
29th December 2016, 09:21 AM  #2  
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29th December 2016, 05:18 PM  #3 
Old guy with soldering iron
diyAudio Member

Double sounds like a decent margin for error. I agree that the R calculated only results in preventing total thermal runaway. Or, at least up to where the one hot device draws ALL the current... I guess that's an asymptotic limit to it, and the junction temperature would reach some sort of equilibrium regardless.
But in this scenario (which is an actual real situtation), I'm not concerned about "ringing" (at least not ringing of the current) as the total current doesn't change and the temperature/individualcurrent difference will only ever be toward increasing. But too much "positive feedback" into the thermal runaway mechanism will make the "hot" device creep up to to be still hotter (than the other devices) than if the "positive feedback" were less. But I can't see a way to quantify how much. I don't use actual calculus a lot anymore, but can't see a way to quantify the limit from what calculus I do remember. But surely there must be a way.
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[W9MJE] Horn spreadsheet SynergyCalc/; SmallSyns SmallSyns; Crossover design Xsim; Depot diffusor supereasy diffusors Last edited by bwaslo; 29th December 2016 at 05:29 PM. 
29th December 2016, 05:36 PM  #4 
diyAudio Member

LTSPICE allows you to individually set the temperature of each circuit element. The three transistors can be at three different temperatures and the three emitter resistors can be at three other different temperatures.
Maybe you can set one transistor's temperature to be 5 degrees C hotter than the other two transistors (giving a VBE offest voltage of 10mV) and observe the simulated power dissipation of each device. Monkey around with emitter resistance values until you get results you can live with. Then multiply by a safety guardband (150% ??) because you worry that LTSPICE is a big fat liar. 
29th December 2016, 05:45 PM  #5 
Old guy with soldering iron
diyAudio Member

Mark, thanks for suggesting that approach. I forgot about LTSPICE allowing setting of the individual temperatures. I'll give that a try.
It does still bug me that I can't see a direction to estimate this with algebra or calculus, just on the intellectual question! Edit: on second thought, how would that include the extra heating caused by the current change (cascading effect)?
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[W9MJE] Horn spreadsheet SynergyCalc/; SmallSyns SmallSyns; Crossover design Xsim; Depot diffusor supereasy diffusors 
29th December 2016, 05:48 PM  #6 
Old guy with soldering iron
diyAudio Member

Can I set the temperature of each device in LTSPICE via a formula (i.e., T_heatsink + power_dissipation * Theta)?
Spice works on simultaneous equations (matrices), I should be able to set that up on paper, all the simplified assumptions are for linear behaviours. After a little more morning coffee, that is (slept late!)
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[W9MJE] Horn spreadsheet SynergyCalc/; SmallSyns SmallSyns; Crossover design Xsim; Depot diffusor supereasy diffusors Last edited by bwaslo; 29th December 2016 at 05:55 PM. 
29th December 2016, 05:54 PM  #7 
Got Foam?
diyAudio Member

I think this is solvable with math using a system of differential equations for the heat transfer. You could make some assumptions like a uniform heat sink temperature, assume a mass and thermal conductivity of the transistor to heatsink junction. Each transistor is a heat source given by the equations of dissipation (electrical), then heat carried off by fins and radiated via natural convection needs to be assumed per rating of sink. You could make it complex by adding transistor to transistor conduction and not assuming uniform heatsink. Mathcad or Matlab will almost be essential to solving this. Although it could be modeled as lumped elements as resistor and capacitor networks and all done in LTSpice.
Check out this publication from Infineon, Figs. 8 and 9 in particular to see clues of how to go about this. It shows coupled electrical and thermal models for 1 MOSFET. Now add second and coupling between two via thermal model of heatsink. http://www.infineon.com/dgdl/Thermal...1472fd33c70aa3 Fig. 13 is good too. Last edited by xrk971; 29th December 2016 at 06:08 PM. 
29th December 2016, 06:11 PM  #8 
Old guy with soldering iron
diyAudio Member

Hmm, I guess that could get there (but yielding about zero understanding of what goes on, though!). Are the time aspects (thermal mass, heat transport) relevant or necessary, though? I'm looking only for a steadystate solution.
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[W9MJE] Horn spreadsheet SynergyCalc/; SmallSyns SmallSyns; Crossover design Xsim; Depot diffusor supereasy diffusors 
29th December 2016, 06:21 PM  #9 
Got Foam?
diyAudio Member

It's the timedependent part that causes thermal runaway, which I assume you would like to be able to capture? The heat conduction equation can be solved for SS by setting dT/dt=0.

29th December 2016, 06:22 PM  #10 
Old guy with soldering iron
diyAudio Member

Something that just occurred to me: if I'm to set this up in simultaneous equations or SPICE, there has to be something that starts the temperature difference between the devices. If all are the same, then so are the temperatures. So I guess I have to assign a cause by varying something in one of the BJTs ( I(S)? That would send things right off nonlinear).
Maybe better to give one of them a slightly different Theta to the heatsink?
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