What Is BJT Beta? Understanding the Current Gain of a Bipolar Junction Transistor
If you’ve ever been confused about the different terms related to the gain of a bipolar junction transistor, read on.
Let’s say that we’re working with a simple circuit consisting of an npn bipolar junction transistor (BJT) and a couple resistors, connected like so:
If you apply a voltage VIN that is high enough to forward-bias the base-to-emitter junction, current will flow from the input terminal, through RB, through the BE junction, to ground. We’ll call this IB. Current will also flow from the 5 V supply, through RC, through the collector-to-emitter portion of the transistor, to ground. Call it IC. Assume that IC is small enough to leave a relatively high voltage at the collector terminal—a voltage high enough, that is, to keep the base-to-collector junction reverse-biased.
Base Current and Collector Current
You probably won’t be surprised to learn that IC will be significantly larger than IB. I suspect that you also have an idea about the mathematical relationship between these two currents. The word is already emerging from the recesses of your mind, ascending to the conscious realm, conjuring images of circuits and symbols, equations and graphs, letters both English and Greek. Perhaps you’ve already whispered it. Yes, beta. That well-traveled path which leads us from IB to IC, ah yes, we know it well … or do we?
What Is BJT Beta?
Some might say that it’s the current gain of a particular bipolar junction transistor, and they might offer β = 100 as a typical value. That’s a good start, but we need to be more precise: beta is the factor of proportionality between the base current and the collector current of a bipolar junction transistor that is operating in the forward active mode.
That’s better, but we’re not done yet, because the situation is actually more complicated than that. The question that serves as a subheading for this section is a bit misleading, because if we want to be technically rigorous, we really should speak of BJT betas, in the plural.
First of all, what we usually think of as β should perhaps be written as βF, where the subscripted “F” indicates that this is the IC-to-IB ratio in forward active mode. It is possible to use a BJT in reverse active mode, and in this case the IC-to-IB ratio is denoted by βR. One of my textbooks even suggests a beta for saturation mode: βforced, where “forced” refers to the fact that the IC-to-IB ratio has been imposed by external circuit conditions rather than established by the transistor. βforced is always smaller than βF.
Small Signals and Large Signals
Before you started reading this article, you perhaps thought that a transistor had one beta. Now you are acquainted with three. And we’re not done yet.
In the context of transistor circuits, the terms “small signal” and “large signal” don’t simply specify amplitude. They refer to distinct analytical domains, and if you’re not familiar with this concept, you might want to read my article on BJT small-signal models.
When we’re establishing biasing conditions or using a transistor as a switch, we’re working in the realm of large-signal DC quantities, and the transistor has an IC-to-IB ratio that goes by the name of βDC. Fortunately, βDC is the same as βF, which is the same as the nondescript “beta” that arises frequently in BJT discussions. Thus, when we talk about “beta,” we are (perhaps unintentionally) referring to the factor of proportionality between the large-signal base current and the large-signal collector current of a bipolar junction transistor operating in the forward active mode.
I know what you’re thinking. “If there’s a beta for large-signal operation, there must be a beta for small-signal operation.” Correct! Beta number 5, denoted by βAC, is the IC-to-IB ratio for small-signal AC quantities. The value of βAC and βDC for a given transistor are similar, but not identical.
When Beta Isn’t β
If you’ve ever searched a BJT datasheet for beta and come up empty-handed, you’re not alone. Manufacturers have contributed an additional layer of confusion by frequently using the symbols hFE and hfe instead of β. The “h” portion comes from “hybrid” and refers to the h-parameter approach to characterizing a two-port network; “f” stands for forward gain and “e” stands for common emitter.
The only challenge here—unless you skipped directly to this section and didn’t read about small-signal and large-signal beta!—is learning the terminology, because hFE is the same as βDC and hfe is the same as βAC. The different subscripts in hfe and hFE reflect the convention of using lowercase letters for small-signal quantities and uppercase letters for large-signal quantities.
Here’s an example from the datasheet for the P2N2222A from ON Semiconductor:
Table taken from this datasheet.
Perhaps you now know more about beta than you ever wanted to. Well, at least that’s better than knowing less than you need to.
You might have noticed something unsettling in that datasheet excerpt. Beta isn’t a fixed value that changes only when we move from one BJT part number to another. Not at all. It actually varies rather wildly according to operating conditions. Ah, beta—so much has changed since we first met. The next article will explore the variability of the parameter that we thought we knew.
OK, I have understood that “beta is the factor of proportionality between the base current and the collector current of a bipolar junction transistor”. But I would like to know why at all there is such a proportionality between them… why the relation I = beta x Ib is linear…