# Is the beta of a BJT really the same as h_FE?

I am currently re-reading my old textbook Microelectronics by Millman and Grabel. Attached two pages. Here is a somewhat theoretical question on large signal parameters for the BJT.

Starting with Ebers-Moll equations (3-6) and (3-7) and assuming $$\V_{CB} = 0\$$ they easily derive the common-base forward short circuit current gain $$\h_{FB}\$$ which is defined as $$h_{FB} \equiv \left. -\frac{I_C}{I_E} \right|_{V_{CB} = 0}$$ In $$\h_{FB}\$$ the F stands for FORWARD (normal use of transistor) and B for common-BASE. Short-circuit means that $$\V_{CB} = 0\$$, i.e. the collector is connected to common (base, as in this case) .

It is easy to see that $$\h_{FB} = \alpha_F\$$ as in equation (3-8). All good so far.

Then there is the common-emitter forward short-circuit current gain, $$\h_{FE}\$$ which is defined as $$h_{FE} \equiv \left. -\frac{I_C}{I_B} \right|_{V_{CE} = 0}$$ i.e. the gain of collector current versus base current when the collector is grounded to the common emitter.

In (3-13) they define $$\\beta_F = \frac{\alpha_F}{1-\alpha_F}\$$ and then simply claim that $$\h_{FE} = \beta_F\$$. In other words, they claim that $$h_{FE} = \frac{\alpha_F}{1-\alpha_F}$$

But when I do the math to derive $$\h_{FE}\$$ from Ebers-Moll, by using emitter as common and assuming $$\V_{CE} = 0\$$, I get $$h_{FE} = -\frac{\alpha_F - \frac{\alpha_F}{\alpha_R}}{1 - \alpha_F + (1 -\alpha_R) \frac{\alpha_F}{\alpha_R}} \neq \frac{\alpha_F}{1-\alpha_F}.$$ What is right here? Can you derive $$\h_{FE} = \beta_F\$$?

Images from Microelectronics 2nd ed. by Jacob Millman & Arvin Grabel, McGraw-Hill 1987, pages 89 and 90.

• You get $h_{FE} \neq \frac{\alpha_F}{1-\alpha_F}$. OK, not that, so what do you get? Please edit it into the question. Mar 28 at 1:11
• Going back as far as I've done (decades before my Millman from '79), $\beta_{_\text{F}}=h_{_\text{FE}}$. Note that the 'h' is from 'hybrid' and it refers to the early (1950's) hybrid-$\pi$ model, which was a later re-analysis developed from the earlier transport and injection versions published before it. (They are mathematically equivalent, though notation is different, but the hybrid-$\pi$ is used today as computer programs can linearize it with less effort and its modeling of the low-current variation of $\beta$ is also easier to handle.) Mar 28 at 1:24
• Here's a question: How do you design an experimental setup and then rigorously process the experimental results in order to find $h_{_\text{FE}}$ at $V_{_\text{CE}}=0\:\text{V}$? I'm curious what you imagine here. Mar 28 at 1:27
• @periblepsis: This is DC analysis, before the hybrid-pi (small signal) model is even introduced in the text. The h at this point refers to the regular hybrid two-port parameters en.wikipedia.org/wiki/…. Maybe Millman is confusing the large-signal beta and small-signal beta? I don't know how I would set it up experimentally and I don't believe it is relevant since this is just model-based theory. Mar 28 at 7:53
• @periblepsis now that I think about it more, I believe the problem here is that Millman erroneously applies two-port concepts (h-parameters) to the DC model, which is non-linear. Two-ports are for linear networks only, such as the hybrid-pi. Mar 28 at 8:16

They are assuming the diode currents in Figure 3-11 are negligible when the diodes are reverse biased. $$\h_{FE}\$$ is defined with the collector-base junction reverse biased, therefore, $$\I_{CD}\$$ is negligibly small, which in turn means the $$\\alpha_R\$$ dependent current source is turned off.
Then $$\I_C\$$ = $$\\alpha_F * I_{ED} = -\alpha_F * I_E\$$ and the rest follows directly from that.