# Unsaturated BJT

I'm struggling for an answer to part (c), I can do the first two questions.

I know that $i_C = \beta \times i_B$, however, I don't know how to calculate the maximum current gain of the unsaturated transistor.

• Do you understand what it means for a transistor to be "saturated"? May 14, 2016 at 18:27
• yeah current flows freely from the collector to the emitter May 14, 2016 at 18:29
• That definition is too oversimplified to be of use. May 14, 2016 at 18:29
• but the current flowing freely means that it has the same effect of being a closed switch. May 14, 2016 at 18:36
• Knowing that won't help you calculate ß. May 14, 2016 at 18:40

so, from part (a), you know what $i_B$ is and what $i_C$ is.

from the result of (a) and from part (b), you know what $v_{CE}$ and then $R_C$ are.

now using your definition of saturation, which is "current flows freely from collector to emitter", translate that to $v_{CE} = 0$, which might be overstating the case, but not by too much. is $i_B$ affected by this? calculate what $i_C$ is in this case.

from that and what Tom said about saturation: $\beta \ i_B \ge i_C$, you can calculate your minimum $\beta$. (or is it maximum $\beta$ you want??)

Recall that a BJT is saturated when both the base-emitter (BE) and the base-collector (BC) junctions are forward-biased (and not when "current flows freely from the collector to the emitter", which is a meaningless definition). Recall also that the relationship $I_\mathrm{C} = \beta I_\mathrm{B}$ is valid only in the active region, that is, when the BE junction is forward-biased and the BC junction is reverse-biased.

In your circuit the BE junction is always forward-biased, and the base current is approximately constant whatever the value of $\beta$. The BC junction, instead, can be either reverse- or forward-biased according to the value of $I_\mathrm{C}$, which defines the collector-to-base voltage $V_\mathrm{CB}$.

A way to solve point c) is then the following:

1. Assume that the BJT is working in the active region, $V_\mathrm{CB}>0$, and that the relationship $I_\mathrm{C} = \beta I_\mathrm{B}$ holds.
2. Calculate the collector potential $V_\mathrm{C}$ as a function of $\beta$.
3. Find up to which value of $\beta$ the assumption of point 1 is met, that is, find the maximum value of $\beta$ for which $V_\mathrm{CB} = V_\mathrm{C}-V_\mathrm{BE}>0$.

But of course, you can also solve the problem the other way round:

1. Assume that the BJT is saturated.
2. Calculate $I_\mathrm{B}$ and $I_\mathrm{C}$ under the assumption of point 1 and find their ratio. According to what I wrote in the first two paragraphs, what's the meaning of this ratio, then?

In principle, the two ways are equivalent: you assume a condition, saturation or active region, and then you check for which values of $\beta$ the condition holds. In practice, however, one way is much shorter (which one and why?).

• i don't think "current flows freely from the collector to the emitter", is a meaningless definition for transistor saturation, but maybe it's not perfectly accurate. for instance this NPN curve has a constant collector-emitter resistance of nearly 200Ω when the transistor is saturated. 200Ω is not "current flow[ing] freely", but the resistance is rather low for most electronic contexts. it's meaningful and as accurate as "200Ω is a conductor of current". May 15, 2016 at 1:39
• @robertbristow-johnson That definition is not only inaccurate, but wrong because the implication is in the other direction: if the BJT is saturated, i.e., both the BE and BC junctions are forward biased, then $I_\mathrm{C}$ is not constrained by the relationship $I_\mathrm{C}=\beta I_\mathrm{B}$. In addition, such misleading definition doesn't let one understand that, e.g, in a Darlington pair the second transistor never saturates. May 15, 2016 at 8:13
• i am not saying that $i_C$ is or is not constrained by the relationship of $i_C = \beta i_B$ regarding saturation. i am saying that saturation implies that $v_{CE} \approx 0$ which is saying that the collector to emitter path is acting approximately the same as a wire (or a resistor of very low resistance) in the context of the rest of the circuit (and the load line determined by the rest of the circuit). in some sense of semantic, a wire replacing the collector-emitter terminals is similar in meaning to "current flows freely from the collector to the emitter". May 15, 2016 at 16:43
• dunno which transistor you mean byt the "second transistor", but in a Darlington pair, both transistors will saturate if you force enough current into the Darlington base. May 15, 2016 at 16:55
• @robertbristow-johnson Referring to this diagram, Q2 can never saturate, because its base to collector junction is always reverse-biased. May 15, 2016 at 17:04