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
In order to develop theseBJT_Small-signal models, there are two small-signal resistances that we must first determine. These are:
1. r p: 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.
Determine r p
Assuming the transistor in this circuit
is operating in the active mode, then
The AC small-signal equivalent circuit from
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:
we see that
Hence, using (2) in (3)
This r p 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.)
We’ll determine re following a similar procedure as for r p, but beginning with
The AC component of iE in (5) is
or with IE =I c|c ,
As indicated in Fig. 1 above, re is theBJT_Small-signal
resistance between the emitter and base seen looking into the emitter:
Mathematically, this is stated as
Assuming an ideal signal voltage source, then e be ve=-vbe and
Using (7) in this equation we find
But from (5.87)
Therefore, using this last result in (10) gives
This is the BJT active mode small-signal resistance between the base and emitter seen looking into the emitter.
It can be shown that
It is quite apparent from this equation that e r r p . . 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- p model
2. T model.
Both are equally valid models, but choosing one over the other sometimes leads to simpler analysis of certain circuits.
Hybrid- p Model
Let’s verify that this circuit incorporates all of the necessarysmall-signal characteristics of the BJT:
ib= vbe |rp as required by (3).
ic= gm vbe as required by (5.86), which we saw in the last lecture. 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
Hence, using this result the second hybrid- p model is
The hybrid- p 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:
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.