UNIVERSITY OF CALIFORNIA AT BERKELEY
College of Engineering
Department of Electrical Engineering and Computer Sciences
Discussion Notes: Degeneration
Tuesday, October 16, 2007
EE 105
Prof. Liu
1
Fall 2007
Derivation of the Output Resistance of a BJT with Emitter
Degeneration
Let’s derive the output resistance of a BJT with emitter degeneration, as shown in Figure 1. First, let’s
draw the small-signal model as in Figure 2. Note that we’ve grounded the input and applied a test voltage
at the output.
+
vout
+
−
vin
RE
−
Figure 1: BJT with emitter degeneration
+
rπ
gm vπ
vπ
−
ro
vt +
−
it
RE
Figure 2: Small-signal model of a BJT with emitter degeneration
π
The first equation we should note is that it = − REvkr
, or vπ = −it (RE k rπ ). Now, let’s write KCL at
π
1
the collector and use the relation between it and vπ to simplify:
it = gm vπ +
vt + vπ
ro
= −gm it (RE k rπ ) +
(1)
it (RE k rπ )
vt
−
ro
ro
RE k rπ
vt
it 1 + gm (RE k rπ ) +
=
ro
ro
RE k rπ
vt
= ro 1 + gm (RE k rπ ) +
Rout =
it
ro
= ro + gm ro (RE k rπ ) + RE k ro
(2)
(3)
(4)
(5)
We often assume that gm ro (RE k rπ ) dominates the output resistance and leave off the other terms as an
approximation.
2