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IV. Microwave Amplifier Design
1.
-
Microwave Amplifier Specifications:
Frequency range
Power Gain
Gain flatness
Input voltage standing wave ratio VSWR
Output voltage standing wave ratio VSWR
Noise Figure NF
Output Power at 1dB Gain Compression
2.
-
Amplifier Topologies (see next page)
Reflective Match Amplifier
Feedback Amplifier
Balanced Amplifier
Distributed Amplifier
3.
-
Reflective Match Technique
For narrowband applications ( <10% bandwidth)
Good input/output VSWR over the narrowband
Best noise performance
Stability problem at out-of-band frequencies and needs to be
controlled in the design of matching networks
49
50
4. Microwave Small Signal Transistors
Two commonly used categories of microwave transistors:
i. Silicon Bipolar transistor (BJT)
ii. Galium Arsenide Field Effect Transistors (GaAs FET)
- Metal Semiconductor Field Effect Transistor (MESFET)
- High Electron Mobility Transistor (HEMT)
Comparison between BJT and GaAs FET
i. For application below 4GHz, BJTs have several advantages:
- It has a well established technology and hence lower cost
& proven reliability.
- Low noise BJTs generally operate with lower bias currents
and lower figures.
- BJTs are easier to match over a wide bandwidth and are
normally unconditionally stable below 4GHz.
- The optimum source impedance for low-noise operation of
BJTs leads to a smaller trade-off between noise
performance and gain.
ii. Advantages of GaAs FET over BJTs
- Can operate at even higher frequencies, higher than 10
GHz.
- Higher gain & lower noise figure (HEMTs have the best
noise performance).
Transistor Packing
Transistor packaging parasitic are significant at microwave
frequency, in general
i. Packaged transistors for narrowband design.
ii. Unpackaged transistors (die form) for wideband and ultrawideband designs.
51
5. S-Parameters
Like the other two-port network parameters such as y, z, h, or
ABCD parameters, at high frequencies, we prefer to use the s
parameters to characterize high frequency two-port networks,
such as a high-frequency amplifier circuit.
As shown in the figure below, the s-parameters are defined in
terms of the incident and reflected power waves rather than the
voltage and current, which are sometimes difficult to be defined
at high frequencies.
1
2
Two-port
network
a1
b1
1'
[ss ss ]
11
12
21
22
a2
b2
2'
The s parameters are defined as:
b1  s11a 1  s12a 2
b 2  s 21a 1  s 22a 2
where
a1 = square root of the power incident at port 1
a2 = square root of the power incident at port 2
b1 = square root of the power reflected from port 1
b2 = square root of the power reflected from port 2
The reflection coefficient at a certain point in a circuit is defined
as

Z  Z o square root of the power reflected

Z  Z o square root of the power incident
where Z is the impedance at the point and Zo is the system
impedance. In the above formula, Z can be the load impedance,
the source impedance, the input impedance, or the output
impedance. The corresponding reflection coefficient will then
be called the load, the source, the input, and the output reflection
coefficients, respectively.
52
6. Amplifier Power Gain Equations
i.
An amplifier circuit can be modeled as a two-port network
as shown below. s is the source reflection coefficient and
ℓ is the load reflection coefficient.
One of the important parameters of the amplifier is the
transducer power gain, which is defined as:
Gt 
P
Pavs
where Pℓ the power delivered to the load and Pavs is the power
available from the source. The S-parameters, s and ℓ all
contribute to the transducer power gain. The transducer power
gain can be expressed as:
2
Gt 

S21 1  
2
1   
2
s
1 S11 s  1 S22    S21s S12  2
53
Other forms of Gt are:
Gt 
Gt 
where
1  s
1  in s
2
1  s
2
1  S11 s
2
in  S11 
out  S22 
S21
2
S21
1  
2
2
1  S22 
1  
2
2
2
1  out 
2
S12S21
1  S22
S12S21s
1  S11s
ii. If S12  0 , it is called the unilateral case.
Transducer power gain
Gt 
1  s
2
1  S11 s
2
S21
1  
2
2
1  S22 
2
Maximum unilateral transducer power gain can be obtained
when S = S11* and ℓ = S22*.
Because
in  S11 
out  S 22 
S12S 21
1  S 22
S12S 21s
1  S11s
54
 S11
S12 0
 S 22
S12 0
therefore
G tu,max 
S21
2
1 S 1 S 
2
2
11
22
iii. If the two-port network is terminated in a Zo system, where
S = 0 and ℓ =0, then
Transducer power gain
G t  S21
2
iv. If the system is unconditionally stable, maximum transducer
power gain happens when in = S* and out = ℓ*. The
expression of the transducer power gain will be derived
later.
55
Example:
A GaAs MESFET has the following S-parameters measured
with Vds =4V and Ids= 30mA at 8GHz with a 50 Ω reference.
S11 = 0.55 158o
S12 = 0.01 -5o
S21 = 1.95 9o
S22 = 0.46 -148o
s = 0.20 0o
ℓ = 0.33 0o
Calculate the transducer power gain.
Solution:
The transducer power gain equation is
2
Gt 

S21 1  
2
1   
2
s
1 S11 s  1 S22    S21s S12  2
2
1.95

1  0.33 2  1  0.20 2 






1  0.55 x 0.2158  1  0.46   148 x 0.33   0.01  5 x 1.959 x 0.20x 0.33



= 2.08
= 3.18dB
56
2
7. Amplifier Stability
Consider an amplifier circuit shown below:
The amplifier is unconditionally stable if
in  1 and out  1
Using the above condition, stability circles which define
graphically stable values of source and load reflection
coefficients can be derived.
1. Output Stability Circle:
2. Input Stability Circle:
 values for in 1
s values for out 1
r 
c
S12S21
2
S22  
S

 radius
2

* *
22
 S11
2
S22  
2
rs 
 center
cs
57
S12S21
2
S11  
S

 radius
2

* *
11  S 22
2
2
S11  
 center
The input and output reflection coefficients are given by:
in  S11 
S12S21
1  S22
ℓ (output) stability circle
indicates the values of all
ℓ that provide |in | 1.
out  S22 
S12S21s
1  S11s
S (input) stability circle
indicates the values of all
S that provide |out | 1.
1
(on ℓ-plane)
(on s-plane)
The stability criteria of an amplifier can be more conveniently
expressed in terms of the stability factor K and the delta factor 
which have the following expressions:
1  S11  S22  
K
2 S12 S21
2
2
 = S11S22 - S12S21
58
2
The stability criteria in general are stated as:
(i)
Unconditionally Stable:
K  1 and   1
||CS| - rS | > 1
for |S22| < 1
||Cl| - rl | > 1
for |S11| < 1
(ii)
Potentially Unstable:
K > 1 and || > 1
K < 1 and || < 1
or
(iii) For the unilateral Case:
K>1
 = S11 S22 – S12 S21 =S11 S22
||= |S11| |S22| <1
It is always stable!
59
Example:
A certain GaAs MESFET has the following S parameters
measured at 9GHz with a 50 Ω reference.
S11  0.64  170
S12  0.0515
S21  2.1030
S22  0.57  95
Compute:
a. the delta factor,Δ
b. the stability factor, K
c. plot the input and output stability circles.
Solution:
a. The delta factor,
 = S11S22 - S12S21
 0.64  170 o x 0.57  95 o  0.0515 o x 2.1030 o
 0.30110 o
  0.30  1
2
  0.09
b. The stability factor K,
2
K
2
1  S11  S22  
2 S12 S21
2
K
2
2
1  0.64  0.57  0.30
2
2 0.05 x 2.10
1.71  1
60
c. Input and output stability circles,
Center of input stability circle :
cs 
S
11
 S 22
2
S11  

* *
2
*
 0.64  170 o  0.30111.45 x 0.5795 o 
  1.50176.42 o

2
2
0.64  0.30
Radius of input stability circle :
rs 
S12S 21
2
S11  

2
0.05 x 2.10
2
0.64  0.30
 0.33
2
Center of output stability circle :
c 


*
S 22  S11*
2
2
S 22  
*
 0.57  95 o  0.30111.45 x 0.64170 o 
  1.70103.32 o
 
0.57 2  0.30 2
Radius of output stability circle : r 
S12S 21
2
S 22  
61
2

0.05 x 2.10
2
0.57  0.30
2
 0.48
8. Constant Gain Circles
i. Unilateral Case, S12 =0
Gt 
1  s
2
1  S11 s
2
S21
1  
2
2
1  S22 
2
For any arbitrary value of s and ℓ, the transducer power gain
may vary between its maximum and minimum values. Gain
equation can be used to plot input (Gs) and output (Gℓ) gain
circles for various s and ℓ values. Gt can be re-written as:
G t  Gs Gd G
where
Gs 
1  s
2
1  S11 s
2
, G d  S21 , G  
2
1  
2
1  S22 
2
The maximum value of Gs is obtained when s = S11*.
G s max 
1
1  S11
Define
2
gs 
Gs
G s max
Mapping constant gs values into the Γs plane form constant input
gain circles where the radius and the center of the circles are
given by:
Rs 

1  g s 1  S11
1  S11
2
2
 1 gs 

*
g s S11
Cs 
1  S11  1  g s 
62
ii. Bilateral case , S12  0
It is tedious to plot constant gain circles for bilateral case where
in is a function of ℓ, and out is a function of s. In practice,
constant operating power gain circles (for any arbitrary chosen
ℓ, conjugately match in to s) or constant available power gain
circles (for any arbitrary chosen s, conjugately match out to ℓ)
are plotted, as shown below.
a. Constant Operating Gain Circles
Transducer power gain:
Gt 
1  in s
 S   
When s  in*   11
 1  S22
G p  S21
2
1  s
2
S21
1  
2
2
1  S22 
2
*

 , it is called the operation gain.

1  
2
 S 

1  11

1  S22

2
2
 S21 g p
2
 1 S 
22 


2
Mapping gp into the ℓ-plane gives constant gain circles with
centers given by:
C p  U p  j Vp
where
U p 
 
g p Re C*

2
1  g p S 22  
2
,
Vp 
*
C  S22   S11
63
 
g p Im C*

2
1  g p S 22  
2

and radii given by:
1  2K S


0.5
2
2
12S 21 g p  S12S 21 g p
rp

2
1  g p S 22  
2

b. Constant Available Gain Circles
Transducer power gain:
Gt 
*
  out
When
Ga 
1  s
2
1  S11 s
2
1  out 
2
*
 S   s
  22
 1  S11s

 , it is called the available gain.

1  s
 S 
s
1  22

1  S11s

S21
2
1  
2
2
2

 1 S 
11 s


2
S 21  g a S 21
2
2
Mapping ga into the Γs-plane gives constant gain circles with
centers given by:
Cas  U as  j Vas
where
U as 
 
g a Re C*s

2
1  g a S11  
2

, Vas 
 
g a Im C*s

2
1  g a S11  
2

, Cs  S11   S*22
and radii given by:
ras
1  2K S

2
2
12S 21 g a  S12S 21 g a

2
1  g a S11  
64
2


0.5
9. Maximum Available Gain
Transducer power gain:
2

S21 1  
Gt 
2
1   
2
s
1 S11 s  1 S22    S21s S12  2
For an unconditionally stable device, the maximum transducer
power gain can be obtained and is known as the maximum
available power gain (MAG)
K 
S21
G MAG 
S12
K2 1

where K is the stability factor.
in = s*
= sm*
MAG is obtained when
and out = ℓ*
= ℓm*
where
sm 
B1  B12  4 C1
2
m 
2 C1
B1  1  S11  S22  
2
2
2
2
2
2
B2  1  S11  S22  
B2  B22  4 C 2
2 C2
C1  S11  S22*
C 2  S22  S11*
For positive B1 value, take - from .
For negative B1 value, take + from .
For positive B2 value, take - from .
For negative B2 value, take + from .
65
2
Example:
A certain GaAs MESFET has the following S-parameters
measured at 8GHz with a 50 Ω reference.
S11  0.26  55
S12  0.0880
S21  2.1465
S22  0.82  30
Find its maximum transducer power gain for the unconditionally
stable case.
Solution:
i. Compute the delta factor  and the stability factor K.
  0.26  55x 0.82  30  0.0880x 2.1465
 0.3488  62.9
  0.3488  1
2
  0.1217
2
K
2
1  0.3488  0.26  0.82
2 0.08x 2.14
2
 1.1148  1
ii. Calculate the maximum available gain.

2.14
1.1148  1.1148 2  1
0.08
16.645
G MAG 

12.2dB
iii. Γsm and Γℓm can also be determined.
sm  0.487147.6
m  0.89332.6
66
10. Steps for Narrowband Design
Four categories of narrowband reflective match amplifier
design:
i. Maximum gain design
ii. Optimum noise figure design
iii. Design for maximum gain with a specific noise performance
iv. Design for minimum input reflection coefficient and
maximum gain with a specific noise performance
Reflective match amplifier design steps:
Step1:
Step 2:
Step 3:
Step 4:
Step 5:
Step 6:
List specifications of the amplifier to be designed.
Choose a device and amplifier topology that will
meet the specifications.
Measure the S-parameters of the device against the
specifications at the respective bias point.
Check the stability conditions.
Compute s and ℓ.
Design the input and output matching networks.
Unconditionally Stable
1. Compute Γsm and ℓm.
2. Calculate MAG.
Conditionally Stable
1. Plot stability circles and
operating/available circles
on a Smith Chart.
2. Determine ℓ and Γs that
will result ℓ and Γs in the
stable region and the
combination of ℓ and Γs
will result in the optimum
power gain.
67
V. Microwave Amplifier Implementation
Single stub microstrip line matching - for narrowband
design.
 There are two types of microstrip line matching stubs:
open stub
short stub
 For microstrip line stub-matching, we consider only parallel
stubs.
 The load impedance is normalized and converted into
admittance for the purpose of parallel stub-matching. The
Smith Chart is thus used as an admittance chart.
 On the admittance Smith Chart, the open-end admittance is
located at the left-most point of the greatest circle and the
short-end admittance is located at the right-most point of the
greatest circle.
 The length of a stub is measured in the clockwise direction.
 The conversion formulas between reflection coefficients and
NORMALIZED impedance are:

z 1
1 
and z 
1 
z 1
where z can be zs, zℓ, zin, zout, and Γ can Γs, Γℓ, Γin, Γout.
 In general, it is easier to locate a normalized
impedance/admittance into the Smith Chart than a reflection
coefficient.
The matching procedure will be illustrated by the following
examples.
68
Example 1:
A GaAs MESFET amplifier has the following source and load
reflection coefficients measured at 9GHz with a 50ohm
system:s = 0.606 155o ℓ = 0.707 120o.
Design the input and output matching networks by using opencircuited and short-circuited shunt stubs.
Solution:
For input matching network:
1. Convert the reflection coefficient s into the normalized
source impedance zs and locate it as point A on the Smith
Chart. Locate also the normalized source admittance ys
which is diagonally opposite point A and is denoted as
point B on the Smith Chart.
zs = 0.26+j 0.21
ys = 2.33-j 1.88
2. Draw an arc with O-B as the radius and read the value at
point C, which is the meeting point of the arc with the unit
resistance circle.
yc = 1+j 1.70
3. From point O at the origin to point C, a shunt opencircuited stub with a length 0.166λ is needed.
lshunt = 0.166λ
69
4. From point C to point B a series line of 50 Ω with a length of
0.105λ is needed for transforming the admittance at point C to
that at point B.
lseries = 0.105λ
For output matching network:
5. Convert the reflection coefficient ℓ into the normalized load
impedance zℓ and locate it as point A on the Smith Chart.
Locate also the normalized load admittance yℓ which is
diagonally opposite point A and is denoted as point B on the
Smith Chart.
zℓ = 0.22+j 0.56
yℓ = 0.65-j 1.55
6. Draw an arc with O-B as the radius and read the value at point
C, which is the meeting point of the arc with the unit
resistance circle.
yc = 1 -j 2.0
7. From point O at the origin to point C a shunt short-circuited
stub with a length 0.074λ is needed.
lshunt = 0.074λ
8. From point C to point B a series line of 50 Ω with a length of
0.02λ is needed for transforming the admittance at point C to
that at point B.
lseries = 0.02λ
The completed matching equivalent networks are shown:
70
Example 1a
Example 1b
71
Example 2:
A GaAs MESFET amplifier has the following parameters
measured at 6GHz with a 50 Ω reference.
Source reflection coefficient for maximum power gain:
sm = 0.761 177o
Load reflection coefficient for maximum power gain:
ℓm = 0.719 104o
Maximum available power gain
GMAG = 12dB
Design the input and output matching networks by using 50 Ω
microstrip lines as shown below:
72
Solution:
1. Convert sm and ℓm into zsm and zℓm and locate them on
the Smith Chart.
zsm = 0.14+j 0.003
zℓm = 0.26+j0.75
2. Locate the normalized admittances from the Smith Chart.
ysm = 7.50-j1.60
yℓm = 0.40-j1.20
3. Use the distances from the origin of the Smith Chart to ysm
and yℓm as radii and draw arcs in the counter-clockwise
direction (toward the load) and stop at points A and B on
the unity resistance circle. Read
yA = 1+j2.40 at point A
yB = 1-j2.10
at point B
4. Realize two series lines. If alumina substrate is used, with
εr of 10 and w/h of 0.95 (Zo = 50 Ω). The wavelength is:
λ=0.39 λo=0.39x5=1.95cm at 6GHz.
The length of the input and output series 50 Ω lines are:
lA =0.055λ=0.055x1.95=0.11cm
lB =0.046λ=0.046x1.95=0.09cm
5. Realize two shunt elements. Two shunt open stubs are
needed to contribute j2.40 and –j2.10 for neutralizing the
susceptance.
The length of the input shunt open stub for j2.40 is:
ls =0.187λ=0.187x1.95=0.36cm
The length of the output shunt open stub for -j2.10 is:
lo =0.321λ=0.321x1.95=0.63cm
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The completed matching equivalent networks are shown:
Example 2
74
Example 3:
A GaAs MESFET has the following S-parameters measured
with Vds= 4V, Ids =30mA, at 1 GHz with a 50 Ω system.
S11 = 0.614 –160o
S21 = 4.8  80o
(i)
(ii)
(iii)
S12 = 0
S22 = 0.682 –97o
Design a narrowband amplifier for maximum power
gain. Use a transmission line section and an opencircuited shunt stub for the input matching network, a
transmission line section and a short-circuited shunt
stub for the output matching network, respectively. The
characteristic impedance for all these lines are taken to
be 50 Ω. (l1 = 0.159 , l2 = 0.099 , l1’= 0.077 , l2’ =
0.051 )
Defined the maximum unilateral transducer power gain
Gtmax of a transistor, and hence find the G tmax of above
amplifier designed. (18.4dB)
If the amplifier is built on an Al2O3 substrate with r =
10 and h=0.625mm. Find the lengths and widths of all
the lines in the matching circuit.
(g= 11.4cm,
w=0.625mm)
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Solution:
Maximum power gain is obtained when s = S11* and ℓ = S22*.
(i) For the input matching network:
1. Convert the source reflection coefficient s =0.614160°
into a normalized impedance and enter as point A on the
Smith Chart. Read the normalized admittance at point B:
ys =2.8-j 1.8
2. Draw an arc with OB as the radius and read the value at
point C.
yc =1 +j 1.55
3. From point O at the origin to point C a shunt opencircuited stub with a length l 1 =0.159λ is needed.
4. From point C to point B, a series line of 50 Ω with the
length of l2=0.099λ is needed.
(ii) For the output matching network:
5. Convert the load reflection coefficient ℓ = 0.68297°
into a normalized impedance and enter as point D on the
Smith Chart. Read the normalized admittance at point E:
yℓ =0.45-j1.09
6. Draw an arc with OE as the radius and read the value at
point F.
yF =1 - j1.9
7. From point O at the origin to point F a shunt shortcircuited stub with a length l1’ = 0.077λ is needed.
8. From point E to point F a series line of 50ohm with the
length of l2’ = 0.051λ is needed.
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(iii) The unilateral transducer power gain of the device is
defined as the forward power gain having its S12  0 or
Gt 
1  s
2
1  S11 s
2
1  
2
S21
2
1  S22 
2
and the maximum unilateral transducer power gain is obtained
when s = S11* and l = S22*,
G t max 
S21
2
1 S 1 S 
2
2
11
22
1
1
(4.8) 2
2
1  (0.614)
1  (0.682) 2
 69.125 18.4dB

(iv) Since all line impedances are 50 Ω and r = 10, w/h=1,
w1= w2= w1’= w2’= 0.625mm
For 50 Ω lines, eff = 6.748, and at 1GHz,
g 
o
 11.4cm
 eff
  1  0.159x11.4cm  18.13mm
 2  0.099x11.4cm  11.29mm
 '1  0.077x11.4cm  8.78mm
 '2  0.051x11.4cm  5.81mm
77
Example 3
(VIIII) ADS design example #7
Evaluate an amplifier circuit which is designed for maximum transducer gain
(unilateral case) at 1 GHz, as on page 70. Now, add on two more sets of S parameters
at 0.8 GHz and 1.2 GHz. The S-parameters in 50 ohms system are:
0.8 GHz S11
S12
S21
S22
1 GHz S11
S12
S21
S22
0.61 -158deg.
0 0deg.
4.9 81deg.
0.671 -95deg.
0.614 -160deg.
0 0deg.
4.8 80deg.
0.682 -97deg.
78
1.2 GHz S11
S12
S21
S22
0.62 -162deg.
0 0deg.
4.8 79deg.
0.685 -100deg
The maximum unilateral transducer power gain at 1 GHz is calculated as on page 72.
The amplifier circuit topology is shown on page 70 as:
Input circuit: (all lines 50 ohms)
98
Source (50 ohms) -> open-circuited shunt stub (0.159 wavelength at 1 GHz) ->
transmission line (0.099 wavelength at 1 GHz) -> MESFET input
Output circuit: (all lines 50 ohms)
Load (50 ohms) -> short-circuited shunt stub (0.077 wavelength at 1 GHz) ->
transmission line (0.051 wavelength at 1 GHz) -> MESFET output
The calculation of the gain (unilateral transducer power gain) of the amplifier at 0.8
GHz and 1.2 GHz is left to students.
To evaluate the unilateral transducer power gain of the amplifier at 0.8 GHz, 1 GHz
and 1.2 GHz by ADS.
Hint: You may use the same approach as ADS design example #5 to simulate and
display. Additional works are on the defining the substrate, editing of 3 sets of Sparameters instead of 1, and connecting the lines and stubs externally to the linear two
port device model before connect it to the ports.
Q10. Calculate the power gain of the amplifier (device plus microstrip line matching
circuits) at 0.8 GHz, 1 GHz and 1.2 GHz. Compare the results with the one shown on
the display by simulation.
79