320-amp-models.tex
Page 1
ECE 320
Amplifier Models
ECE 320 - Linear Active Circuit Design
Phyllis R. Nelson
320-amp-models.tex
Page 2
2-Port Networks
A 2-port network is any circiut with two pairs of wires connecting to the outside world. (Each
“port” is a pair of wires.) The standard notation used for the voltages and currents in a 2-port
network is shown below.
i1
+
v1
-
i2
2-port
network
+
v2
-
Just as there are two completely equivalent models for a “1-port” network (the Thévenin and
Norton equivalent circuits), there are multiple equivalent models for a 2-port network. We will
consider the z , y , g , and h parameter models. (There are also s and abcd parameters.)
ECE 320 - Linear Active Circuit Design
Phyllis R. Nelson
320-amp-models.tex
Page 3
z Parameters
i1
+
v1
-
i2
z11
z21i1 +
+
−
− z12i2
z22
+
v2
-
v1 = z11 i1 + z12 i2
v2 = z21 i1 + z22 i2
ECE 320 - Linear Active Circuit Design
Phyllis R. Nelson
320-amp-models.tex
Page 4
z11 =
z12 =
z21 =
v1 = open-circuit input resistance
i1 i2 =0
v1 = reverse open-circuit transresistance
i2 i1 =0
v2 i1 = forward open-circuit transresistance
i2 =0
z22 =
v2 = open-circuit output resistance
i2 i1 =0
All of the zij are in Ohms.
ECE 320 - Linear Active Circuit Design
Phyllis R. Nelson
320-amp-models.tex
Page 5
y Parameters
i1
+
v1
-
i2
1
y11
y21v1
y12v2
1
y22
+
v2
-
i1 = y11 v1 + y12 v2
i2 = y21 v1 + y22 v2
ECE 320 - Linear Active Circuit Design
Phyllis R. Nelson
320-amp-models.tex
y11 =
y12 =
y21 =
y22 =
Page 6
i1 = short-circuit input conductance
v1 v2 =0
i1 = reverse short-circuit transconductance
v2 v1 =0
i2 = forward short-circuit transconductance
v1 v2 =0
i2 = open-circuit output conductance
v2 v1 =0
All of the yij are in Siemens.
ECE 320 - Linear Active Circuit Design
Phyllis R. Nelson
320-amp-models.tex
Page 7
g Parameters
i2
+
v1
-
1
g11
g21v1 +
−
g12i2
g22
+
v2
-
i1 = g11 v1 + g12 i2
v2 = g21 v1 + g22 i2
ECE 320 - Linear Active Circuit Design
Phyllis R. Nelson
320-amp-models.tex
Page 8
g11 =
g12 =
g21 =
g22 =
i1 = open-circuit input conductance
v1 i2 =0
i1 i2 = reverse short-circuit current gain
v1 =0
v2 = forward open-circuit voltage gain
v1 i2 =0
v2 = short-circuit output resistance
i2 v1 =0
g12 and g21 are dimensionless, while g11 is in Siemens and g22 is in Ohms.
ECE 320 - Linear Active Circuit Design
Phyllis R. Nelson
320-amp-models.tex
Page 9
h Parameters
i1
+
v1
-
i2
h11
h21i1
+
− h12v2
1
h22
+
v2
-
v1 = h11 i1 + h12 v2
i2 = h21 i1 + h22 v2
ECE 320 - Linear Active Circuit Design
Phyllis R. Nelson
320-amp-models.tex
Page 10
h11 =
h12 =
v1 = short-circuit input resistance
i1 v2 =0
v1 v2 = reverse open-circuit voltage gain
i1 =0
h21 =
i2 i1 = forward short-circuit current gain
v2 =0
h22 =
i2 = open-circuit output conductance
v2 i1 =0
h12 and h21 are dimensionless, while h11 is in Ohms and h22 is in Siemens.
ECE 320 - Linear Active Circuit Design
Phyllis R. Nelson
320-amp-models.tex
Page 11
IEEE Alternative Subscript Notation
➠ 11 → i for input
➠ 12 → r for reverse transfer
➠ 21 → f for forward transfer
➠ 22 → o for output
ECE 320 - Linear Active Circuit Design
Phyllis R. Nelson
320-amp-models.tex
Page 12
Amplifiers
An amplifier is a special case of a 2-port network having an input port and an output port.
Amplifiers are considered to be one-directional, producing a scaled copy of the input signal at the
output port.
Standard amplifier models are used in system design in much the same way as the Thèvenin
and Norton models: they provide the simplest possible description of the properties of a more
complex circuit. Thus, the parameters of the standard amplifier models are used in specifications.
ECE 320 - Linear Active Circuit Design
Phyllis R. Nelson
320-amp-models.tex
Page 13
The circuit below shows an amplifier, together with a model for the source that drives it, and the
load. The definitions of a number of commonly-used parameters are also given.
RL
vS
+
−
ii
+
vi
-
+
−
ECE 320 - Linear Active Circuit Design
Av = vvoi
Overall voltage gain: Avs = vvo
S
io
Current gain: Ai = i
i
Transresistance: Rm = vio
i
Input transconductance: Gm = vio
i
po
Input power gain: Ap = p
i
vo io
= vi ii = Av Ai
p
Overall power gain: Aps = po
s
vo io
= vs ii = Avs Ai
Input voltage gain:
io
+
vo
-
RL
Phyllis R. Nelson
320-amp-models.tex
Page 14
Amplifier Models
There are four equivalent amplifier models. Each one can be derived from one of the 2-port
network parameterizations by setting the parameter with the subscript 21 to zero, renaming v1 . v2
and i1 to vi , vo and ii , changing the direction of i2 and naming the new current io . Thus there are
four amplifier models, which are discussed in detail on the next few slides.
ECE 320 - Linear Active Circuit Design
Phyllis R. Nelson
320-amp-models.tex
Page 15
Voltage Amplifier
Ro io
RS
vS
+
vi
+
−
Ri
-
Avo
+
vo
+
Avovi
RL
-
vo = = open circuit voltage gain
vi io =0
The voltage amp comes from the g-parameter 2-port model. The figure below shows how.
io
i2
+ vi
v1
-
1
g11
ECE 320 - Linear Active Circuit Design
Ri
Avovi
g21v1 +
−
g12i2
0
g22
Ro
+ vo
v2
-
Phyllis R. Nelson
320-amp-models.tex
Page 16
Ro io
RS
vS
+
vi
+
−
-
Ri
+
Avovi
+
vo
RL
-
The voltage amplifier, together with its source and load, are used to connect the specific
amplifier model with quantities describing the amplifier’s performance.
RL
Avo = input voltage gain
Ro + RL
vo vi
Ri
vo
=
=
Av = overall voltage gain
Avs =
vS
vi vS
RS + Ri
io
vo /Ro
Ri vo
Ri
Ai = =
=
=
Av = current gain
ii
vi /Ri
Ro vi
Ro
vo
Av =
=
vi
ECE 320 - Linear Active Circuit Design
Phyllis R. Nelson
320-amp-models.tex
Page 17
Current Amplifier
io
RS
vS
+
−
Ro
Ri
ii
Aisii
+
vo
RL
-
io Ais = = short circuit current gain
ii vo =0
io
Ai = =
ii
ECE 320 - Linear Active Circuit Design
Ro
Ro + RL
Ais = input current gain
Phyllis R. Nelson
320-amp-models.tex
Page 18
Transconductance Amplifier
RS
vS
io
+
vi
+
−
Gmsvi
-
Gms
Gm =
Ro
Ri
+
vo
RL
-
io = = short circuit transconductance
vi vo =0
io
=
vi
Ro
Ro + RL
ECE 320 - Linear Active Circuit Design
Gms = input transconductance
Phyllis R. Nelson
320-amp-models.tex
Page 19
Transresistance Amplifier
Ro io
RS
vS
+
−
Rmo
ii
Ri
+
Rmoii
+
vo
RL
-
vo = = open circuit transconductance
ii Io =0
vo
Rm =
=
ii
RL
RL + Ro
ECE 320 - Linear Active Circuit Design
Rmo = input transconductance
Phyllis R. Nelson
320-amp-models.tex
Page 20
Conversion Between Amplifier Models
Let’s find the component values of a voltage amp that is equivalent to a given current amp.
io
RS
vS
+
−
ii
Ro
Ri
Aisii
+
vo
-
Current Amp
Ro io
RS
RL
vS
+
−
+
vi
Ri
-
+
Avovi-
+
vo
RL
-
Voltage Amp
Neither Ro or Ri needs a new value, although Ro changes from parallel- to series-connected.
Avo
vo Ro Ais ii
Ro Ais
Ro
= =
=
=
Ais
vi io =0
vi
vi /ii
Ri
Note that this final expression contains only component values, and that all the voltages and
currents have been eliminated.
ECE 320 - Linear Active Circuit Design
Phyllis R. Nelson