Whites, EE 320
Lecture 14
Page 1 of 9
Lecture 14: BJT Small-Signal
Equivalent Circuit Models.
Our next objective is to develop small-signal circuit models for
the BJT. We’ll focus on the npn variant in this lecture.
Recall that we did this for the diode back in Lecture 4:
ID
⇒
rd =
nVT
ID
In order to develop these BJT small-signal models, there are two
small-signal resistances that we must first determine. These are:
1. rπ: the small-signal, active mode input resistance between
the base and emitter, as “seen looking into the base.”
2. re: the small-signal, active mode output resistance between
the base and emitter, as “seen looking into the emitter.”
These resistances are NOT the same. Why? Because the
transistor is not a reciprocal device. Like a diode, the behavior
of the BJT in the circuit changes if we interchange the terminals.
© 2009 Keith W. Whites
Whites, EE 320
Lecture 14
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Determine rπ
Assuming the transistor in this circuit
(Fig. 5.48a)
is operating in the active mode, then
⎛
⎞
i
1⎜
IC ⎟
iB = C =
I
vbe ⎟
+
N
⎜ NC
β (5.84) β ⎜ DC V
T
⎟
N
AC ⎠
⎝
I
g
so that
ib = C vbe = m vbe
βVT
β
(1)
(5.90),(5.91),(2)
The AC small-signal equivalent circuit from Fig. 5.48a is
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Lecture 14
Page 3 of 9
(Fig. 1)
Notice the “AC ground” in the circuit. This is an extremely
important concept. Since the voltage at this terminal is held
constant at VCC, there is no time variation of the voltage.
Consequently, we can set this terminal to be an “AC ground” in
the small-signal circuit.
For AC grounds, we “kill” the DC sources at that terminal: short
circuit voltage sources and open circuit current sources.
So, from the small-signal equivalent circuit above:
(Fig. 2)
we see that
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Lecture 14
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vbe
ib
(5.92),(3)
rπ =
Hence, using (2) in (3)
rπ =
β
[Ω]
(5.93),(4)
gm
This rπ is the BJT active mode small-signal input resistance of
the BJT between the base and the emitter as seen looking into
the base terminal. (Similar to a Thévenin resistance, this
statement means we are fictitiously separating the source from
the base of the BJT and observing the input resistance, as
indicated by the dashed line in Fig. 2.)
Determine re
We’ll determine re following a similar procedure as for rπ, but
beginning with
i
I
i
(5)
iE = C = C + c
α N
α α
N
DC
AC
The AC component of iE in (5) is
i
IC
ie = c =
vbe
N
α (5.85) αVT
(5.96),(6)
or with I E = I C α ,
ie =
IE
vbe
VT
(5.96),(7)
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Lecture 14
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As indicated in Fig. 1 above, re is the BJT small-signal
resistance between the emitter and base seen looking into the
emitter:
Mathematically, this is stated as
ve
(8)
−ie
Assuming an ideal signal voltage source, then ve = −vbe and
v
re ≡ be
(5.97),(9)
ie
Using (7) in this equation we find
V
(5.98),(10)
re = T
IE
But from (5.87)
I
αI
V
α
gm = C = E ⇒ T =
I E gm
VT
VT
re ≡
Therefore, using this last result in (10) gives
Whites, EE 320
Lecture 14
α
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1
[Ω]
(5.99),(11)
gm gm
This is the BJT active mode small-signal resistance between the
base and emitter seen looking into the emitter.
re =
It can be shown that
≈
rπ = ( β + 1) re [Ω]
(5.100),(12)
It is quite apparent from this equation that rπ ≠ re . This result is
not unexpected because the active mode BJT is a non-reciprocal
device, as mentioned on page 1 of these notes.
BJT Small-Signal Equivalent Circuits
We are now in a position to construct the equivalent active
mode, small-signal circuit models for the BJT. There are two
families of such circuits:
1. Hybrid-π model
2. T model.
Both are equally valid models, but choosing one over the other
sometimes leads to simpler analysis of certain circuits.
Hybrid-π Model
• Version A.
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Lecture 14
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(Fig. 5.51a)
Let’s verify that this circuit incorporates all of the necessary
small-signal characteristics of the BJT:
9 ib = vbe rπ as required by (3).
9 ic = g m vbe as required by (5.86), which we saw in the last
lecture.
9 ib + ic = ie as required by KCL.
We can also show from these relationships that ie = vbe re .
• Version B. We can construct a second equivalent circuit by
using
g m vbe =
g m rπ ib = β ib
N g m ( ib rπ ) = N
(3)
(4)
Hence, using this result the second hybrid-π model is
(Fig. 5.51b)
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Lecture 14
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T Model
The hybrid-π model is definitely the most popular small-signal
model for the BJT. The alternative is the T model, which is
useful in certain situations.
The T model also has two versions:
• Version A.
(Fig. 5.52a)
• Version B.
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Lecture 14
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(Fig. 5.52b)
The small-signal models for pnp BJTs are identically the same
as those shown here for the npn transistors. It is important to
note that there is no change in any polarities (voltage or current)
for the pnp models relative to the npn models. Again, these
small-signal models are identically the same.