Follower Circuits: Output Resistance
• Follower ROut = (RiE || RL) ~ 1/gm
• But, does driving resistance matter?
– Consider extreme case: Base driven by current source.
RL
19
€
Follower Circuits: Output Resistance
Current is injected
into emitter of BJT.
αi R i
ve = o + th
g m β o +1
€
%
(
v
x
x *
i = −i − β i =
− β '−
'
x
o R +r
o R +r *
& th π )
th π
R +r
r
R
th
π
π
∴ RiE ≅
=
+ th
β +1
β +1 β +1
o
o
o
v
R
R
1
o
th
+
≅
+ th
RiE =
g
β +1 g
β +1
m
o
m
o
α
€
Current αoi coming out of collector must
be supported by veb = αoi/gm, given by the
first term. ib = -i/βo+1 creates a voltage
drop in Rth given by second term
1
=
In case of FET,
RiS
g
m
Thus equivalent resistance looking into
emitter or source of a transistor is
€ 1/ gm.
approximately
20
Emitter (Source) Degeneration
• Emitter (source) resistor often used in
CE (CS) amps.
• What is its effect?
• Examine transconductor by itself.
• Assume VC is small-signal ground.
RC
vy
vx
RE
23
Emitter (Source) Degeneration
• We will view NPN+RE as
vx
transconductor
• From CD analysis: gain from vx to ve?
– gmRE/(1+gmRE)
• For Gm = ic/vx, find vbe/vx
• vbe/vx = 1 - gmRE/(1+gmRE)
vx
= 1/(1+gmRE)
rπ
• ic = gmvbe, so Gm = gm/(1+gmRE)
• For gmRE >> 1, Gm → 1/RE
• Why reduce Gm? Sensitivity, linearity
24
ic= gm(vx-ve)
RE
ro
gmvbe
RE
Emitter (Source) Degeneration
• RIn?
• vbe = vx/(1+gmRE)
• ib = vbe/rπ
= vx/[rπ(1+gmRE)]
• RIn = vx /ib = rπ(1+gmRE)
vx
ic= gm(vx-ve)
RE
vx
• ROut?
rπ
ro
gmvbe
RE
25
Emitter (Source) Degeneration
• ROut? (Ignore ib)
• Find ic
ic =
gm ve + (vc
= vc /ro
ve )/ro
ve (gm + 1/ro )
• Get rid of ve? KCL @ ve
vx
rπ
⇥
1
1
vc
ve
+
+ gm =
RE
ro
ro
vc
ve =
ro (1/RE + 1/ro + gm )
ro
gmvbe
RE
26
Emitter (Source) Degeneration
• ROut? (Ignore ib)
• Combine
⇥
1
gm + 1/ro
i
c = vc
ro
1 + gm ro + ro /RE
1/ro + gm + 1/RE gm 1/ro
=
1 + gm ro + ro /RE
ic
1
=
v
c
R E + g m ro R E + r o
ROut ⇥ ro (1 + gm RE ) If ro >> RE
• Error significant if RE >> rπ
27
vx
rπ
ro
gmvbe
RE
Terminal Impedances - Collector
RiB
RB
rπ
RiC
ro
gmvbe
RE
ic = gm vbe + (vc
RC
RiE
• Current across RE
decreases iE, increasing
RiC.
• Some “escapes” through
base, increases vBE &
limiting RiC
ve )/ro
ve = ic RE ||(RB + r )
RE (RB + r )
R E + RB + r
RE
v b = ic
RB
R E + RB + r
R E R B R E R B RE r
vbe = ic
RE + RB + r
⇥
RE (RB + r )
1
RE r
ic
+ vc /ro
ic = g m ic
RE + R B + r
ro
RE + R B + r
⇥
gm RE r + 1/ro (RE (RB + r ))
vc /ro = ic 1 +
RE + RB + r
⇥
vc
+ (RB + r )/ro
RiC =
= ro 1 + R E
ic
R E + RB + r
⇥
vc
+ RB /ro + /(gm ro )
RiC =
= ro 1 + R E
ic
RE + R B + r
⇥
vc
+ RB /ro
RiC =
= ro 1 + R E
ic
RE + RB + r
= ic
28
Terminal Impedances - Collector
RiB
RB
rπ
RiC
ro
gmvbe
RE
RC
RiC = ro
RiE
RiC
ro
If
• Expression is cumbersome
and unhelpful. Make some
approximations.
• Important when trying to
maximize gain
RiC
If
RiC
29
+ RB /ro
1 + RE
RE + R B + r
⇥
RE
1+
RE + RB + r
⇥
⇥ ⇤ (MOS); recall r = /gm
⇥
ro 1 + R E
= ro (1 + gm RE )
/gm
finite and RE >> r
ro (1 + )