6.301 Solid State Circuits
Recitation 3: AC Coupling, and Single-Transistor Amplifiers
Prof. Joel L. Dawson
When first presented, AC coupling is kind of a mysterious thing. After seeing it enough times, most
people come to terms with it in an intuitive way, which is certainly good enough. But it’s useful as an
exercise to come to understand it in a way that is more mathematically explicit.
CLASS EXERCISE: Consider the following two-source network:
(1) Write a general expression for V0 using superposition.
(2) Approximate V0 in the cases z1 << z2 and z2 << z1 .
(Workspace)
Now let’s take a look at AC coupling.
VCC
R1
C
Vs
C
VB
R2
VB
Vs
R1 R2
R2
VCC = VA
R1 + R2
6.301 Solid State Circuits
Recitation 3: AC Coupling, and Single-Transistor Amplifiers
Prof. Joel L. Dawson
Using superposition, we can write VB fairly rapidly:
VB =
=
1
V +
( R1  R2 ) Cs + 1 A
1
Cs
R1  R2
VS
+ R1  R2
1
(R1  R2 )Cs
VA +
VS
(R1  R2 )Cs + 1
( R1  R2 ) Cs + 1
So the AC coupling setup low-pass filters the supply and high-pass filters the signal source!
For frequencies ω <<
1
, VB ≈ VA
(R1  R2 )C
For frequencies ω >>
1
, VB ≈ VS
(R1  R2 )C
This is exactly what we wanted!
For you network theory buffs, we can generalize this development to an arbitrary number of sources:
z1
V1
V0
z2
z3
V2
n
V0 = ∑
i =1
V3
⎛
1⎞
⎜∑ z ⎟
⎝ j ≠i j ⎠
−1
⎛
1⎞
zi + ⎜ ∑ ⎟
⎝ j ≠i z j ⎠
n
V =∑
−1 i
i =1
Vi
⎛
1⎞
zi ⎜ ∑ ⎟ + 1
⎝ j ≠i z j ⎠
Notice that if there is one source impedance zi that is small compared to all of the other source
impedances, its corresponding source dominates and V0 ≈ Vi . This is why one tends to think of lowimpedance voltage sources as “strong.”
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6.301 Solid State Circuits
Recitation 3: AC Coupling, and Single-Transistor Amplifiers
Prof. Joel L. Dawson
Now let’s embark on our study of single-transistor stages. Our overall goal will be to collapse the
details of the stage into three easy-to-remember figures of merit:
R0
•
•
•
Input impedance
Gain
Output impedance
→
RI
av vi
vi
We write things this way to simplify our lives when we go to a cascade of multiple stages:
R0
R0
RI
Overall gain:
vi
av vi
vi
RI
av vi
⎛ RI ⎞
⎛ RI ⎞
av ⎜
av = av 2 ⎜
⎟
⎝ R0 + RI ⎠
⎝ R0 + RI ⎟⎠
“Interstage Loading”
Note that sometimes it will be convenient to use the Norton equivalents model. We’ll probably do
this when modeling stages within op-amps:
vi
RI
Gm vi
av = Gm R0
As a warm-up, the familiar Common Emitter Stage:
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R0
6.301 Solid State Circuits
Recitation 3: AC Coupling, and Single-Transistor Amplifiers
Prof. Joel L. Dawson
RL
Rs VA rb
Rs
vs
rπ
Vπ
Input Impedance:
RI = rb + rπ
Output Impedance:
R0 = RL
Voltage Gain:
Vπ =

↓ gmVπ
RL
Vs
⎛ r ⎞
rπ
VA → V0 = −gm RL ⎜ π ⎟ VA
rb + rπ
⎝ rb + rπ ⎠
⎛ r ⎞
av = − ⎜ π ⎟ gm RL
⎝ rb + rπ ⎠
Common Base Stage
rb
RL
VOUT
VOUT
+
↓ gmVπ
rπ
−
VA
Rs
Vs
it ↑
+
v
−t
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VA
RL
6.301 Solid State Circuits
Recitation 3: AC Coupling, and Single-Transistor Amplifiers
Prof. Joel L. Dawson
First, we can calculate the input impedance using a test voltage source:
it =
=
⎛ r ⎞
vt
+ gm ⎜ π ⎟ vt
rb + rπ
⎝ rπ + rb ⎠
1
(1 + gm rπ ) vt
rπ + rb
But gm rπ = β . So
RIN =
vt rπ + rb rπ
1
=
≈ ≈
it
β + 1 β gm
Output impedance is easy: R0 = RL
And for the gain:
Vπ =
⎛ r
⎞
rπ
VA → VOUT = gm ⎜ π VA ⎟ RL
rb + rπ
⎝ rb + rπ ⎠
⎛ r ⎞
⎛r ⎞
VOUT
= gm RL ⎜ π ⎟ ≈ gm RL ⎜ π ⎟ = gm RL
VA
⎝ rb + rπ ⎠
⎝ rπ ⎠
So our model for the common base stage becomes
RL
Rs
Vs
+
+
VA
−
+
gm RL
−
rπ + rb
β +1
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V0
−
6.301 Solid State Circuits
Recitation 3: AC Coupling, and Single-Transistor Amplifiers
Prof. Joel L. Dawson
Now notice two things. First, the input impedance to the common base stage is low.
RIN =
rπ + rb
1
≈
β + 1 gm
Notice that the impact this has on the gain from VS to V0
V0 ≈
1
gm
1
gm
+ RS
VS ⋅ gm RL
V0
1
g R
R
=
gm RL ≈ m L = L
VS 1 + gm RS
gm RS RS
Second, notice how the dependant current source has the effect of “transforming” resistances in the
base so that they appear smaller when seen from the emitter:
RIN =
rπ + rb
β +1
Summary of common base stage: High gain, low input impedance, high output impedance.
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6.301 Solid State Circuits
Recitation 3: AC Coupling, and Single-Transistor Amplifiers
Prof. Joel L. Dawson
Emitter Follower Stage
rb
Rs
Rs
+
Vs
−
RE
VA
+
Vs
↓ β ib
ib ↓ rπ
−
V0
RE
Input Impedance:
RI = rb + rπ + ( β + 1) RE
Resistances in the emitter seem larger when viewed from the base.
Output Impedance:
⎛R +r +r ⎞
R0 = RE  ⎜ S b π ⎟
⎝ β +1 ⎠
Notice how the source resistance effects the
Voltage Gain:
av =
( β + 1) RE
rb + rπ + ( β + 1) RE
R0 if this stage.
≈1
Summary of Emitter Follower Stage: Unity gain, high input impedance, low output impedance
+1
Emitter followers make good voltage buffers:
Common base stages make good current buffers:
I IN
↑ I IN
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