# The Ebers-Moll BJT Model

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Bipolar transistors are one of the basic building blocks of electronics, yet they can be challenging to understand and analyze in circuits. I find the Ebers-Moll model—or at least the “rules of thumb” that derive from it—are pretty much all I need to analyze any large-signal circuit.

Figure 1 shows a basic model for an NPN transistor (a PNP transistor is similar, but with all the polarities flipped). In the active mode, the transistor looks like two diode junctions, with the collector positive with respect to the emitter, the base-emitter diode forward biased and the base-collector diode reverse biased. Note that the collector current does not flow through the reverse-biased base-collector diode. It flows to the emitter via mysterious “transistor action”. We need some kind of mathematical model to understand how this collector current is controlled.

A popular simple model for a bipolar transistor is to think of it as a current amplifier, where the collector current is proportional to the base current. The constant of proportionality, β, is the transistor’s current gain. We write the relationship between base and collector currents:

The current gain model is not really all that useful outside of switching circuits. The gain β could vary over a 5 to 1 range between transistors of the same type in the same batch. It’s also dependent on temperature and collector current. This model is really only useful in determining how much base current is needed to drive a switching transistor into saturation with worst-case β, temperature and voltage—handy when selecting a base resistor when driving a LED or relay, but that’s about all.

The Ebers-Moll model, on the other hand, looks at the transistor as a transconductance device where the collector current is controlled by the base-emitter voltage. In this case the collector current is described by the exponential equation:

Or conversely, the base-emitter voltage can be described a function of collector current.

In these equations VT is the thermal voltage (proportional to absolute temperature) which is about 26mV at room temperature, and IS is the saturation voltage (also strongly temperature-dependent) which is typically 10-15A for a small-signal transistor. Since IC << IS, we can simplify the above equations to:

It’s useful to think about what happens to collector current for a small change in base-emitter voltage, with all else being equal. I will spare you the maths (you can try it yourself if you are so inclined), but this leads to our first two useful rules of thumb.

• Collector currents in two identical transistors with the same VBE (at the same temperature) will be equal.
• Collector current doubles for every ~18mV increase in VBE or increases 10 times for every ~60mV increase in VBE .

Base-emitter voltage decreases with temperature (all else being equal) due to the dominating effect of the temperature dependence of IS. This leads us to the next rule of thumb.

• Base-emitter voltage decreases by ~2.1mV per °C temperature increase

This rule assumes that the collector current remains constant. What happens to the collector current with temperature if the base-emitter voltage is held constant? Again, a little algebra gives us another rule:

• Collector current increases ~9% per °C temperature increase or doubles every ~8 °C temperature increase.

It’s worth noting that there is one other rule that you need to know that is not strictly a product of the Ebers-Moll model, and that is the Early effect. The early effect causes a change to VBE as collector voltage varies due to a change in the width of the base region at the internal PN junction. This is described by the equation:

Were η is typically 10-4 to 10-5 for small-signal transistors.

• Typically, a VCE increase of 10V will reduce VBE by 1mV to 2mV with constant collector current. With a constant VBE collector current will increase ~4% to 8% for a 1V increase in VCE.

With these rules in your toolbox, you are equipped to tackle pretty much any large-signal BJT analysis you are likely to encounter.

References:

“Bipolar Junction Transistor.” In Wikipedia, June 1, 2021. https://en.wikipedia.org/w/index.php?title=Bipolar_junction_transistor&oldid=1026341486.

Horowitz, Paul, and Winfield Hill. The Art of Electronics. Third edition, 11th printing, with Corrections. Cambridge New York, NY: Cambridge University Press, 2017.

“Early Effect.” In Wikipedia, March 21, 2021. https://en.wikipedia.org/w/index.php?title=Early_effect&oldid=1013444812.

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##### Andrew Levido
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Andrew Levido (andrew.levido@gmail.com) earned a bachelor’s degree in Electrical Engineering in Sydney, Australia, in 1986. He worked for several years in R&D for power electronics and telecommunication companies before moving into management roles. Andrew has maintained a hands-on interest in electronics, particularly embedded systems, power electronics, and control theory in his free time. Over the years he has written a number of articles for various electronics publications and occasionally provides consulting services as time allows.

## The Ebers-Moll BJT Model

by Andrew Levido time to read: 3 min