Thus if we go back to the
circuit model for the common emitter transistor, and
re-draw it as a small signal model it would look
something like Figure 1. Here we have replaced the
diode with a linear element (a resistor, called
r
π
r
π
)
and we have changed the notation for the currents from
I
B
I
B
and
I
C
I
C
to
i
b
i
b
and
i
c
i
c
respectively, to remind us that we are now talking about small signal
ac quantities, not large signal ones. The bias currents
I
B
I
B
and
I
C
I
C
are still flowing through the device (and we will leave it to ELEC 342
to discuss how these are generated and set up) but they do not appear
in the small signal model. This model is only used to figure out how
the transistor behaves for the ac signal going through it, not have
it responds to large DC values.
Now
r
π
r
π
the equivalent small signal resistance of the base-emitter diode
is given simply by the inverse of the conductance of the
equivalent diode. Remember, we found
r
π
=1qkT
I
B
=1qkT
I
C
β=β40
I
C
r
π
1
q
k
T
I
B
1
q
k
T
I
C
β
β
40
I
C
(1)
where we have used the fact that
I
C
=β
I
B
I
C
β
I
B
and
qkT=40V-1
q
k
T
40
V
-1
.
As we said earlier, typical values for
β
βin a standard bipolar transistor will be around
100. Thus, for a typical collector bias current of
I
C
=1mA
I
C
1
mA
,
r
π
r
π
will be about 2.5 kΩ.
There is one more item we should consider in putting together our
model for the bipolar transistor. We did not get things completely
right when we drew the common emitter characteristic curves for the
transistor. There is a somewhat subtle effect going on when
V
CE
V
CE
is increased. Remember, we said that the current coming out of the
collector is not effected by how big the drop was in the reverse
biased base-collector junction. The collector current just depends
on how many electrons are injected into the base by the emitter,
and how many of them make it across the base to the base-collector
junction. As the base-collector reverse bias is increased (by
increasing
V
CE
V
CE
the depletion width of the base-collector junction increases as
well. This has the effect of making the base region somewhat
shorter. This means that a few more electrons are able to make it
across the base region without recombining and as a result
α α and hence
β β increase somewhat. This then
means that
I
C
I
C
goes up slightly with increasing
V
CE
V
CE
.
The effect is called base width modulation.
Let us now include that effect in the common emitter
characteristic curves. As you can see in Figure 3,
there is now a slope to the
I
C
V
CE
I
C
V
CE
curve, with
I
C
I
C
increasing somewhat as
V
CE
V
CE
increases. The effect has been somewhat exaggerated in Figure 2, and I will now make the slope even bigger so
that we may define a new quantity, called the Early
Voltage.
Back in the very beginning of the transistor era, an engineer
at Bell Labs, Jim Early, predicted that there would be a slope to the
I
C
I
C
curves, and that they would all project back to the same
intersection point on the horizontal axis. Having made that
prediction, Jim went down into the lab, made the measurement, and
confirmed his prediction, thus showing that the theory of
transistor behavior was being properly understood. The point of
intersection of the
V
CE
V
CE
axis is known as the Early Voltage. Since
the symbol
V
E
V
E
,
for the emitter voltage was already taken, they had to label the
Early Voltage
V
A
V
A
instead. (Even though the intersection point in on the negative
half of the
V
CE
V
CE
axis,
V
A
V
A
is universally quoted as a positive number.)
How can we model the sloping I-V curve? We can do almost the
same thing as we did with the solar cell. The horizontal part of
the curve is still a current source, and the sloped part is
simply a resistor in parallel with it. Here is a graphical
explanation in Figure 4.
Usually, the slope is much less than we have shown here, and so
for any given value of
I
C
I
C
,
we can just take the slope of the line as
I
C
V
A
I
C
V
A
,
and hence the resistance, which is usually called
r
o
r
o
is just
V
A
I
c
V
A
I
c
.
Thus, we add
r
o
r
o
to the small signal model for the bipolar transistor. This is
shown in Figure 5. In a good quality modern
transistor, the Early Voltage,
V
A
V
A
will be on the order of 150-250 Volts. So if we let
V
A
=200
V
A
200
,
and we imagine that we have our transistor biased at 1 mA, then
r
o
=200V1mA=200kΩ
r
o
200
V
1
mA
200
kΩ
(2)
which is usually much larger than most of the other resistors you
will encounter in a typical circuit. In most instances,
r
o
r
o
can be ignored with no problem. If you get into high impedance
circuits however, as you might find in a instrumentation
amplifier, then
v
be
v
be
has to be taken into account.
Sometimes it is advantageous to use a mutual transconductance
model instead of a current gain model for the transistor. If we
call the input small signal voltage
v
be
v
be
,
then obviously
i
b
=
v
be
r
π
=
v
be
β40
I
C
i
b
v
be
r
π
v
be
β
40
I
C
(3)
But
i
c
=β
i
b
=β
v
be
β40
I
C
=40
I
C
v
be
≡
g
m
v
be
i
c
β
i
b
β
v
be
β
40
I
C
40
I
C
v
be
g
m
v
be
(4)
Where
g
m
g
m
is called the mutual transconductance of the transistor.
Notice that ββ
has completely cancelled out in the expression for
g
m
g
m
and that
g
m
g
m
depends only upon the bias current,
I
C
I
C
,
flowing through the collector and not on any of the physical
properties of the transistor itself!
Finally, there is one last physical consideration we should make
concerning the operation of the bipolar transistor. The
base-collector junction is reverse biased. We know that if we
apply too much reverse bias to a pn junction, it can breakdown
through avalanche multiplication. Breakdown in a transistor is
somewhat "softer" than for a simple diode, because once a small
amount of avalanche multiplication starts, extra holes are
generated within the base-collector junction. These holes fall up,
into the base, where they act as additional base current, which,
in turn, causes
I
C
I
C
to increase. This is shown in Figure 7.
A set of characteristic curves for a transistor going into
breakdown is also shown in Figure 8.
Well, we have learned quite a bit about bipolar transistors in a
very short space. Go back over this chapter and see if you can
pick out the two or three most important ideas of equations which
would make up a set of "facts" that you could stick away in you
head someplace. Do this so you will always have them to refer to
when the subject of bipolars comes up (In say, a job interview or
something!).
"Accessible versions of this collection are available at Bookshare. DAISY and BRF provided."