# Calculating ballast resistor values to to control current hogging in paralleled BJTs

I started a thread at https://www.diyaudio.com/forums/solid...t-hogging.html on this subject, and after some time and many helpful suggestions, here is the calculation I came up with.

The goal is to find out how large the ballast resistors must be to prevent current runaway and to control "current hogging" to a given limit. Current hogging is a phenomenon with bipolar transistors when operated in parallel in which the hotter of the set draws more current than the cooler one(s) and so heats up even further. The fix is to put series resistors ("ballasts") in the emitter circuits (see the drawing), but I've not been able to find anything published that says how to calculate the value of the resistors.

The model is for two BJT transistors, with their collectors and bases commoned and their emitters summed through individual ballast resistors. You need to know what the process-variation-caused Vbe imbalance (in volts) is between the transistors, and how much imbalance (current hogging) you are willing to accept. Also, needed is the idle Vce voltage of the devices. You can also estimate the imbalance that would occur if the thermal resistance (from case to heatsink) is different fro the two devices.

Assume that:

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

If you accept these approximations, then note that:

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 (zip compressed) Xcel spreadsheet for particular calculations.

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.

Hope someone finds this useful. (I did)

The goal is to find out how large the ballast resistors must be to prevent current runaway and to control "current hogging" to a given limit. Current hogging is a phenomenon with bipolar transistors when operated in parallel in which the hotter of the set draws more current than the cooler one(s) and so heats up even further. The fix is to put series resistors ("ballasts") in the emitter circuits (see the drawing), but I've not been able to find anything published that says how to calculate the value of the resistors.

The model is for two BJT transistors, with their collectors and bases commoned and their emitters summed through individual ballast resistors. You need to know what the process-variation-caused Vbe imbalance (in volts) is between the transistors, and how much imbalance (current hogging) you are willing to accept. Also, needed is the idle Vce voltage of the devices. You can also estimate the imbalance that would occur if the thermal resistance (from case to heatsink) is different fro the two devices.

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 (summed resistors) on the drawing
- dVbe is the voltage difference of BJT#1's Vbe, relative to Vbe(typ)., assumed negative (<=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 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))

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 accept these approximations, then note that:

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 (zip compressed) Xcel spreadsheet for particular calculations.

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.

Hope someone finds this useful. (I did)

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