The background is, I'm trying to understand how well a typical transistor obeys the Ebers-Moll equation at constant (junction) temperature.

See this snapshot taken from the Fairchild datasheet of the 2N3904 transistor. 2N3904 Vbe against Ic

My question is what is meant by 25°C? Is it

  1. ambient (room) temperature?
  2. The temperature of the epoxy casing?
  3. The actual temperature of the silicon (junction) itself?

Also, how is constant temperature maintained? Will a manufacturer publish information on how it took its measurements for datasheets?

In general, there is a temperature gradient from ambient to casing to junction. So the junction is hotter than the casing which is hotter than ambient. The temperature in the Ebers-Moll model refers to junction temperature.

My own speculations:

Of course if the answer is 3., we expect all of the lines to be straight since Ebers-Moll gives

$$I_C = I_S \exp(V_{BE}/V_T)$$

where \$I_S\$ and \$V_T\$ are constant at constant temperature and \$V_{CE}\$. Note the logarithmic scale for \$I_C\$. So there is definite deviation at 125°C near \$I_C= 100 \text{mA}\$.

Is this deviation really due to increased junction temperature due to heat dissipated by \$I_C\$, or is there genuine deviation from the model?

I am thinking that it can't be 1. because the junction would get hotter with increasing \$I_C\$ and the line would deviate strongly from straight. For the 2N3904 with \$V_{CE}=5V\$, I calculated that junction temperature increases with \$I_C\$ as 1°C/mA at constant ambient temperature. (\$I_S\$ and so \$I_C\$ increase by about 9% per °C.)

Perhaps they can achieve 3. by taking all the measurements automatically in a fraction of a second, before giving it a chance to heat up.

Any ideas?


2 Answers 2


It's the junction temperature \$T_J\$, and yes, they would typically take such a measurement in microseconds before the junction temperature changes.

The upward slope of at high base and emitter currents is due to resistive (non-ideal) behavior that is temperature-sensitive, or due to the reduction in beta at high current (so the base current is higher for a given collector current than if the beta was constant). Note that the beta reduction is temperature-sensitive.

enter image description here

  • \$\begingroup\$ Thanks for the answer. Is it some resistance in series with the base to emitter path which causes this? \$\endgroup\$
    – Michael
    Commented Aug 1, 2014 at 20:39
  • \$\begingroup\$ Not sure which it is or a combination, see my edit. Typically base resistance should be under 100 ohms. Beta reduction is quite pronounced too. \$\endgroup\$ Commented Aug 1, 2014 at 20:55

A cold soak to ensure even temperature distribution and then testing at temperature. Remember that the heat equation does not allow for instantaneous changes so the values are tracked as the die heats up. With enough data you can back calculate that out for verification, but more importantly there is very little self heating at the low current values.


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