3
Small Signal Amplifier
Design and Measurement
3.1 Introduction
So far device models and the parameter sets have been presented. It is now
important to develop the major building blocks of modern RF circuits and this
chapter will cover amplifier design. The amplifier is usually required to provide
low noise gain with low distortion at both small and large signal levels. It should
also be stable, i.e. not generate unwanted spurious signals, and the performance
should remain constant with time.
A further requirement is that the amplifier should provide good reverse
isolation to prevent, for example, LO breakthrough from re-radiating via the aerial.
The input and output match are also important when, for example, filters are used
as these require accurate terminations to offer the correct performance. If the
amplifier is being connected directly to the aerial it may be minimum noise that is
required and therefore the match may not be so critical. It is usually the case that
minimum noise and optimum match do not occur at the same point and a circuit
technique for achieving low noise and optimum match simultaneously will be
described.
For an amplifier we therefore require:
1.

Maximum/specified gain through correct matching and feedback.
2.

Low noise.
3.

Low distortion.
4.


y
V
=+
(3.1)
I
y
V
y
V
2
22
2
21
1
=+
(3.2)
From equation (3.1):
Small Signal Amplifier Design and Measurement 99
1
212
11
1
1
V
Vy
y
V
I
Y
in

y
I
y
V
y
y
V
I
Y
in
−
+==
(3.5)
Dividing top and bottom by
V
2
:
yy
yy
y
V
I
Y
L
in
22
2112
11
1
1

Y
in
can therefore be seen to be dependent on the load admittance
Y
L
. Similarly
Y
out
is dependent on the source admittance
Y
S
. The effect is reduced if
y
12
(the reverse
transfer admittance) is low. If
y
12
is zero, Y
in
becomes equal to
y
11
and
Y
out
becomes
equal to
y
22

s
and
g
L
the real part of
Y
in
is greater than
zero and the real part of
Y
out
is greater than zero. The imaginary part can of course
be positive or negative. In other words the real input and output impedance is
always positive for all source and loads which are not negative resistances. Note
that when an amplifier is designed the stability should be checked at all
frequencies as the impedance of the matching network changes with frequency.
An example of a simple stability calculation showing the value of resistor
required for stability is shown in the equivalent section on
S
parameters later in this
chapter.
John Linvill [13] from Stanford developed the Linvill stability parameter:
C
yy
gg yy
=
−
12 21
11 22 12 21
2Re()

stringent as it only guarantees stability for the specified loads. Care needs to be
taken when using the stability factors in software packages as a large
K
is
sometimes used to define the inverse of the Linvill or Stern criteria.
3.2.1.1 Summary for Stability
To maintain stability the Re(
Y
in
)
≥
0
and the Re(
Y
out
)
≥
0
for all the loads
presented to the amplifier over the whole frequency range.
The device is unconditionally stable when the above applies for all Re(
Y
L
)
≥
0
and all Re(
Y
S
)

yY
I
YY
I
V
L
S
S
inS
S
+
−+
=
+
=
(3.11)
()
21122211
22
1
))((
yyyYyY
yYI
V
LS
LS
−++
+
=
(3.12)

As
V
2
is also equal to –
I
2
/
Y
L
then
I
2
= -
V
2
Y
L
and:
22
1212
2
y
VyYV
V
L
−−
=
(3.14)
102 Fundamentals of RF Circuit Design
V

1
1
=−






+












(3.16)
L
Y
y
V
y
V
+

2
|
2

G
L
where
G
L
is the real part of
Y
L
:
()()
yyy
Y
y
y
y
GI
P
L
s
LS
L
211222
11
2
21
2

G
LS
LS
AVS
L
T
2112
2211
2
21
2
4
−
++
==
(3.21)
Small Signal Amplifier Design and Measurement 103
For maximum gain we require a match at the input and the output; therefore
Y
S
=
Y
in
*
and
Y
L
=
Y
out

11
2
I
+−=
(3.23)
LS
GG
g
g
=
22
11
(3.24)
()
g
yy
b
B
L
11
2112
m
22
2
I
+−=
(3.25)
LLLSSS
jBGYjBGY
+=+=

P
G
LS
LS
AVS
L
T
2112
2211
2
21
2
4
−
++
==
(3.27)
104 Fundamentals of RF Circuit Design
2211
21
2
4
gg
y
G
T
=
(3.28)
This is the maximum unilateral gain often defined as GUM or MUG and is another
figure of merit of use in amplifier design. This enables fairly easy calculation of

Chart.
A tapped
C
matching circuit is shown in Figure 3.2a. The aim is to design the
component values to produce the required input impedance, e.g. 50
Ω
for the input
impedance of the device which can be any impedance above 50
Ω
. To analyse the
tapped C circuit it is easier to look at the circuit from the high impedance point as
shown in Figure 3.3.
C
C
2
1
Y
in
R
Figure 3.3
Tapped C circuit for analysis
The imaginary part is then cancelled using the inductor. Often a tunable capacitor
is placed in parallel with the inductor to aid tuning. Y
in
is therefore required:
Y
in
=
Real + Imaginary parts
= G + jB

sCRCC
s
Y
in
(3.31)
The real part of
Y
in
is therefore:
()
21
22
2
2
2
1
CCR
RC
Y
in
++
=
ω
ω
(3.32)
The shunt resistive part of Y
in
is therefore R
in
:

2
2
1
1








+=
C
C
RR
in
(3.34)
The imaginary part of Y
in
is:
()
()
()
2
21
22
221
2
21

CC
CC
C
T
+
=
(3.36)
This is equivalent to the two capacitors being added in series.
Small Signal Amplifier Design and Measurement 107
In conclusion the two important equations are:
2
2
1
1








+=
C
C
RR
in
(3.34)
and
21







−=
1
21
R
R
CC
in
(3.38)
As:
22
22
1
1
CC
R
R
CC
R
R
C
in
in
T
+








−








−
=
R
R
C
R
R
C
in
in
T
(3.40)
Therefore:


T
in
in
C
R
R
R
R
C








−








=
1
2
(3.42)


=
1
(3.43)
Small Signal Amplifier Design and Measurement 109
3.3.1 Tapped C Design Example
Let us match a 50
Ω
source to a 5K resistor in parallel with 2pF at 100MHz. A
block diagram is shown in Figure 3.4. A 3dB bandwidth of 5 MHz is required.
This is typical of the older dual gate MOSFET. This is an integrated four terminal
device which consists of a Cascode of two MOSFETS. A special feature of
Cascodes is the low feedback
C
when gate 2 is decoupled.
C
feedback for most
dual gate MOSFETs is around 20 to 25fF. An extra feature is that varying the DC
bias on gate 2 varies the gain experienced by signals on gate 1 by up to 50dB. This
can be used for AGC and mixing.
Figure 3.4
Tapped C design example
To obtain the 3dB bandwidth the loaded
Q
,
Q
L
is required:
5
100

(3.45)
Therefore to obtain
C
T
f
LC
=
1
2
π
(3.46)
110 Fundamentals of RF Circuit Design
so:
()
2
2
1
f
LC
π
=
(3.47)
As
L
= 200nH at 100MHz
C
res
= 12.67pF (3.48)
C
T

=
(3.51)
Thus:
C
1
= 9C
2
(3.52)
and:
T
C
CC
CC
CC
CC
=
+
=
+
12
12
22
22
9
9
(3.53)
T
CC
=
9

2
(
C
1
+
C
2
)
2
should be much greater than one for the
approximations to hold. Also ensure that C
2
<<
ω
2
C
1
C
2
R
2
(C
1
+C
2
) for the
approximations to hold.
3.4 Selectivity and Insertion Loss of the Matching Network
It is important to consider the effect of component losses on the performance of the
circuit. This is because the highest selectivity can only be achieved by making the





−++
==
C
LjRZ
Z
VS
LOSS
out
ω
ω
1
2
2
0
0
21
(3.58)
At resonance:
LOSS
RZ
Z
S
+
=
0
0

+
=
ω
(3.62)
Q
L
ZR
L
LOSS
ω
=+
0
2
(3.63)
Therefore at resonance:
L
Q
Q
Q
LS
L
L
ω
ω







Q
jQ
df
f
df
L
L
o
21
0
1
1
12
=−






±






(3.66)
Small Signal Amplifier Design and Measurement 113
This can be used to calculate the frequency response further from the centre

=
R
S
:
()
2
21
2
SV
P
P
G
out
AVS
L
T
===
(3.70)
As
21
SV
out
=
for
V
= 2 source voltage:
2
0
1


L
Q
loss
ω
=
0
(3.60)
It is interesting to investigate the effect of insertion loss on this input matching
network. For a bandwidth of 5 MHz,
Q
L
= 20. If we assume that
Q
0
= 200,
G
T
=
(0.9)
2
= -0.91dB loss. The variation in insertion loss versus
Q
L
/
Q
0
is shown in
Figure 3.6 for four different values of
Q
L

Q
L
/
Q
0
as
the insertion loss of the matching circuit will directly add to the noise figure. Note
that for lower transformation ratios this is often not a problem. A plot of
S
21
against
Q
L
/
Q
0
is shown in Figure 3.7 showing that as the insertion loss tends to infinity
S
21
tends to zero and Q
L
tends to Q
0
.
Figure 3.7
S
21
vs
Q
L

amplifier circuit using a dual gate MOSFET is shown in Figure 3.8. The feedback
capacitance is reduced to around 25fF as long as gate 2 is decoupled. Further the
bias on gate 2 can be varied to obtain a gain variation of up to 50dB. For an N
channel depletion mode FET, 4 to 5 volts bias on gate 2 (
V
G2S
) gives maximum
gain.
Figure 3.8
Dual gate MOSFET amplifier
As an example it is interesting to investigate the stability of the BF981. Taking
the Linvill [13] stability factor:
()
C
yy
gg yy
=
−
12 21
11 22 12 21
2Re
(3.67)
where the device is unconditionally stable when C is positive and less than one.
We apply this to the device at 100MHz using the
y
parameters from the data
sheets:
()
33
21

9
9
99
9
1030
10260
10261005.4
10289.1913
−
−
−−
−
×
×
=
×−−×
×+×
=
jj
C
(3.72)
The device is therefore not unconditionally stable as C is greater than one. This is
because the feedback capacitance (20fF) although low, still presents an impedance
of similar value to the input and output impedances.
To ensure stability it is necessary to increase the input and output admittances
effectively by lowering the resistance across the input and output. This is achieved
by designing the matching network to present a much lower resistance across the
input and output. Shunt resistors can also be used but these degrade the noise
performance if used at the input. Therefore we look at Stern [14] stability factor
which includes source and load impedances, where stability occurs for

×−=yy
(3.75)
As the device is stable for k > 1 it is possible to ensure stability by making 2(g
11
+
G
S
) (g
22
+G
L
) > 234 x 10
-9
. One method to ensure stability is to place equal
admittances on the I/P and O/P. To achieve this the total input admittance and
output admittance are each 3.4
×
10
-4
i.e. 2.9k
Ω
. This of course just places the
device on the border of stability and therefore lower values should be used. The
source and load impedances could therefore be transformed up from, say, 50
Ω
to
2k
Ω
. The match will also be poor unless resistors are also placed across the input
and output of the device. The maximum available gain is also reduced but this is

G
A
is the available power gain and
P
ni
is the noise available from the source.
The noise power available from a resistor at temperature
T
is
kTB
, where
k
is
Boltzmann constant,
T
is the temperature and B is the bandwidth. From this the
equivalent noise voltage or noise current for a resistor can be derived. Let us
assume that the input impedance consists of a noiseless resistor driven by a
conventional resistor. The conventional resistor can then be represented either as a
noiseless resistor in parallel with a noise current or as a noiseless resistor in series
with a noise voltage as shown in Figures 3.9 and 3.10 where:

i
kTB
R
n
2
4
=
(3.77)

transformed to the input as a series noise voltage and a shunt noise current as
shown in Figure 3.11.
1
A
2
e
e
n
n
in
noiseless
2 port
Figure 3.11
Representation of noise in a two port
Small Signal Amplifier Design and Measurement 119
It is now worth calculating the optimum source resistance,
R
SO
, for minimum
noise figure. The noise factor for the input circuit is obtained by calculating the
ratio of the total noise at node A to the noise caused only by the source impedance
R
S
.
()
S
SnnS
kTBR
RiekTBR
NF


+
−
=
2
2
2
4
1
n
S
n
S
i
R
e
kTBdR
dNF
(3.81)
Equating this to zero means that:
2
2
n
S
n
i
R
e
=
(3.82)

4
2
1
4
1
2
2
2
+=








+
+=
(3.84)
120 Fundamentals of RF Circuit Design
Therefore:
kTB
ie
F
nn
2
1
min
+=

fact that impedance match and optimum noise match are often at different
positions. In fact this effect is unusually exaggerated in dual gate MOSFETs
operating in the VHF band due to the high input impedance. For optimum
sensitivity it is therefore more important to noise match than to impedance match
even though maximum power gain occurs for best impedance match. If the
amplifier is to be connected directly to an aerial then optimum noise match is
important. In this case that would mean that the aerial impedance should be
transformed to present 2K in parallel with 1.6uH at the input of the device which
for low loss transformers would produce a noise figure for this device of around
0.6dB. Losses in the transformers would be dependent on the ratio of loaded
Q
to
unloaded
Q
. Note that the loss resistors presented across the tuned circuit would
not now be half the transformed impedance (2k) as impedance match does not
occur, but 2k
Ω
in parallel with 22k
Ω
.
There is a further important point when considering matching and that is the
termination impedance presented to the preceding device. For example if there was
a filter between the aerial and the amplifier, the filter would only work correctly
when terminated in the design impedances. This is because a filter is a frequency
dependent potential divider and changing impedances would change the response
and loss.
Small Signal Amplifier Design and Measurement 121
Figure 3.12
Noise circles for the BF981. Reproduced with permission from Philips using