Video Transcript
An NPN transistor is connected to a power supply with a voltage 𝑉 CC. A power supply with a voltage 𝑉 BE is connected across the transistor’s emitter and
base terminals, as shown in the diagram. There is a current 𝐼 C between 𝑉 CC and the collector terminal, a current 𝐼 E
between 𝑉 BE and the emitter terminal, and a current 𝐼 B between 𝑉 BE and the
base terminal. An external resistance 𝑅 C is placed between 𝑉 CC and the collector terminal. And an external resistance 𝑅 B is placed between 𝑉 BE and the base terminal. The potential difference across the collector and emitter terminals is 𝑉 CE. The value of 𝑅 B is varied. Which colored line on the graph shown most correctly represents the variation in 𝐼 C
with 𝑅 B?
This question is essentially asking us to find the relationship between the current
into the collector, 𝐼 C, and the resistance of the resistor along the path to the
base, 𝑅 B. If we can find this relationship, then we should be able to determine which line on
the graph is the correct one. Let’s begin by recalling the equation for current gain, which can help us relate 𝐼 C
and 𝐼 B. The current gain of a transistor, 𝛽 sub 𝑒, is equal to the collector current
divided by the base current.
This equation looks fine, but we’re being asked to compare 𝐼 C, a current, with 𝑅
B, a resistance. If we use Ohm’s law for the components for the base of the transistor, then we will
have that the potential difference of the power supply 𝑉 BE is equal to the product
of 𝐼 B and 𝑅 B. Since 𝑉 BE is a constant power supply, the potential difference won’t change. This means that an increase in 𝑅 B will have to result in a decrease of 𝐼 B so that
the total potential difference remains the same.
If we then look at the equation for the current gain, we’ll see that a decrease in 𝐼
B must also result in a decrease in 𝐼 C if the current gain remains constant as
well. Therefore, an increase in 𝑅 B, and subsequently a decrease in 𝐼 B, will result in a
decrease in 𝐼 C.
Looking back at the graph, we see that all five lines are showing some form of
decrease, so now we just have to determine which one is correct. We may be tempted to choose the red line, since the relationship seems obvious. If we cut 𝐼 B in half, then we should expect 𝐼 C to similarly get cut in half in
order to maintain the same current gain. This would mean that there should be a more-or-less linear relationship, which is
what the red line shows.
However, this is not actually the case because the current gain is not always
constant. Specifically, there are two different scenarios in which current gain is more or less
constant: when a transistor is acting as a closed switch and when it is acting as an
open switch. Otherwise, current gain changes depending on the values of 𝐼 B and 𝐼 C.
To see why this is, let’s recall that a transistor is made up of three
semiconductors, labeled differently as the collector, the base, and the emitter. In order for there to be a current across the transistor, all three parts of it must
allow conduction. However, the base will not allow any charge through it by itself. Current must be driven into the base in order for the base–collector boundary to
become forward biased, which will allow a current to be conducted through the
transistor. This means that the transistor cannot have a current and thus no collector current 𝐼
C when there is no current through the base independently, 𝐼 B.
The current gain at these very small values of 𝐼 B is quite high, meaning even small
increases in 𝐼 B will result in large increases in 𝐼 C. We see this represented by the blue, black, and pink lines in the graph here. They have sharp curves that indicate a high current gain at the region to the
right. Remember that a high base resistance 𝑅 B indicates a low base current 𝐼 B. So since only these three lines indicate a proper change in current gain, the red and
green lines must not be correct.
Continuing to look at these three lines, we see that as 𝐼 B increases, which is to
say as 𝑅 B decreases, the collector current 𝐼 C levels off. This is because the current gain becomes lower as 𝐼 B increases.
But which colored line is correct? Let’s look at each individually. The blue line shows a very steep current gain at high 𝑅 B levels that levels off as
𝑅 B decreases. The collector current value gradually increases beyond the initial sharp increase but
never hits a particular limit. This line then does not accurately portray what happens to the collector current
because it actually does hit a current limit.
In transistors, there is a limit to the output current, called the saturation
current, for high values of input current. For this transistor setup, the output current is the collector current. The input current is the base current, meaning that we should see a current limit in
this graph here. However, for the blue line, it is always gradually increasing as the base current
increases, which is not the actual behavior of transistors.
So the blue line can’t be it, nor can the pink line be correct, as we see that the
line is very straight, with a distinct break where the current gain apparently
changes. These straight lines aren’t representative of the behavior we would expect of a
transistor, which involves sharp yet granular changes in current gain. So the pink line isn’t correct either.
This means the only line that can be correct is the black line, which we see
correctly shows a large current gain at high values of 𝑅 B and reaches a limit in
the current gained, the saturation current, as 𝑅 B decreases. The saturation current and sharp changes in current gain for transistors is desirable
for their use in electronics. A small change in 𝐼 B can cause a much greater change in 𝐼 C, allowing a circuit
with a transistor to effectively switch 𝐼 C on or off, either having no current or
reaching a specific saturation current. This means that the colored line on the graph that most accurately represents the
variation in 𝐼 C with 𝑅 B is the black line.
