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The Rapid Design of RF Ampliﬁers

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Alexander N. Stewart

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6/01 c:/ClRC/DateDue.p65-p.15

The Rapid Design of RF Ampliﬁers
By

Alexander N. Stewart

A THESIS

Submitted to
Michigan State University
in partial fulﬁllment of the requirements

for the degree of
MASTER OF SCIENCE
Department of Electrical and Computer Engineering

2004

ABSTRACT

The Rapid Design of RF Ampliﬁers
By

Alexander N. Stewart

Today’s electronics market has seen an increase for high frequency, wireless tech-
nology. Associated with this trend is the need for suitable RF ampliﬁers to handle the
reception and transmission of these signals. Unfortunately, the cost of these ampli-
ﬁers from traditional vendors is quite high. It is therefore advantageous to investigate
methods to reduce the acquisition cost, such as a custom design.

The design of RF ampliﬁers is a time consuming and repetitive process if the
calculations were to be done by hand. There are commercially available packages
which can aid in the design process; however, this is an expensive alternative for
creating these types of ampliﬁers.

The purpose of this thesis will be to develop such a design methodology and tool.
First, all of the necessary background in electromagnetic theory is needed to develop
the models used in RF ampliﬁer design. Then there is a discussion on how to design
these types of ampliﬁers while pointing out the shortcomings of doing these designs
by hand. Next, a Matlab based GUI is introduced which will reduce the time required
in the design process. Other software packages are also discussed used in the design
process. Finally, results are shown to verify the design process. It is the intent of this

work to provide an educational tool for undergraduate students.

Copyright © by
Alexander N. Stewart

2004

To my parents, Robert and Jane

iv

ACKNOWLEDGMENTS

First, I would like to thank my family for their support and giving me a hand
when I needed it. Without it, college would have been near impossible.

I would also like to thank my graduate advisor, Dr. Leo Kempel, for taking me on
as his student and allowing me the opportunity to attend graduate school at Michigan
State University.

Next, I would like to thank Catherine Beauduy and her family for their support.
They have been there to offer their advice and support and have been like a second
family to me.

Finally, I would like to thank the National Science Foundation (NSF) for their

support by funding my way through graduate school with grants ECS-0134236 and
DUE—0231312.

TABLE OF CONTENTS

LIST OF TABLES

LIST OF FIGURES

1 Introduction

2 Theory
2.1 Transmission Lines ............................
2.2 Microstrip .................................
2.3 Two Port Networks ............................
2.3.1 S-Parameters ...........................
2.3.2 Power Waves ...........................
2.4 Matching Networks ............................
2.5 Signal Flow Graphs ............................
2.6 Noise ....................................
2.6.1 Noise Sources ...........................
2.6.2 Noise in Systems .........................

3 Ampliﬁer Design

3.1

3.2

3.3

3.4

3.5

3.6

3.7

Stability ..................................
Standing Wave Ratio ...........................
Bilateral Conjugate Match ........................
3.3.1 Design Procedure .........................
Operating Power Gain ..........................
3.4.1 Design Procedure .........................
Available Power Gain ...........................
3.5.1 Design Procedure .........................
SWR Circles ................................
3.6.1 Design Procedure .........................
Noise Figure Circles ............................

vi

viii

ix

1

3

3

9
12
12
15
21
27
36
36
37

40
40
43
46
48
48
50
51
52
53
55
57

3.7.1 Design Procedure .........................

4 Software Tools with Updated Design Procedures

4.1 Using the Design Program ........................
4.2 Design using the GUI ...........................
4.3 Circuit Design ...............................
4.3.1 Microstrip Implementation ....................
4.3.2 Lumped Element Design .....................

5 Designs and Analysis

5.1 S-Parameter Measurement ........................

5.2 Ampliﬁer Designs

5.2.1 AT—41486 Design .........................
5.2.2 MPSH10 Design ..........................
5.2.3 MPSH17 Design ..........................

5.2.4 Summary
6 Conclusion

BIBLIOGRAPHY

ooooooooooooooooooooooooooooo

vii

58

60
60
64
67
67
71

73
73
81
81
87
94
96

98

101

LIST OF TABLES

5.1 Transistor S Parameters used in Designs ................ 81
5.2 Design Values ............................... 81
5.3 Measured Values Summary ........................ 97

viii

LIST OF FIGURES

2.1 Transmission Line Model ......................... 3
2.2 Finite 'Itansmission Line ......................... 7
2.3 Microstrip Geometry ........................... 10
2.4 Two Port Network ............................ 13
2.5 BJT Two Port Model ........................... 15
2.6 Two Port Model with Load ....................... 16
2.7 Reactance of an Open Transmission Line ................ 22
2.8 Reactance of a Shorted Transmission Line ............... 23
2.9 Smith Chart ................................ 25
2.10 Two port signal ﬂow graph model .................... 27
2.11 Port 1 equivalent circuit and signal ﬂow graph ............. 29
2.12 Port 2 equivalent circuit and signal ﬂow graph ............. 30
2.13 Two port signal ﬂow graph ........................ 31
2.14 Model for PIN computation ........................ 32
2.15 Model for P Avg computation ....................... 33
2.16 System noise ﬁgure model ........................ 37
3.1 Reﬂection coefﬁcient diagram of a two port model ............ 40
3.2 Transistor ampliﬁer with input and output matching networks. . . . . 41
4.1 Screenshot of the Microwave Ampliﬁer Design Program ........ 61

5.1 DC Bias Circuit Used in S—Parameter Measurement for the MPSH10 74
5.2 DC Bias Circuit Used in S-Parameter Measurement for the MPSH17 74

5.3 8'11 of MPSH10 .............................. 76
5.4 512 of MPSH10 .............................. 77
5.5 821 of MPSH10 .............................. 77
5.6 822 of MPSH10 .............................. 78
5.7 511 of MPSH17 .............................. 78
5.8 5'12 of MPSH17 .............................. 79
5.9 521 of MPSH17 .............................. 79

ix

5.10 8'22 of MPSH17 .............................. 80

5.11 Schematic of AT-41486 Ampliﬁer Bias Circuit ............. 82
5.12 Layout of AT-41486 Ampliﬁer on Microstrip .............. 83
5.13 511 and SWR,“ of Compensated AT-41486 Ampliﬁer .......... 84
5.14 521 of Compensated AT-41486 Ampliﬁer ................ 85
5.15 822 and SWR“, of Compensated AT—41486 Ampliﬁer ......... 87
5.16 Biconjugate Match Network for MPSH10 at 100MHz ......... 88
5.17 Sn of Biconjugate Matched MPSH10 Ampliﬁer ............ 89
5.18 521 of Biconjugate Matched MPSH10 Ampliﬁer ............ 90
5.19 822 of Biconjugate Matched MPSH10 Ampliﬁer ............ 90
5.20 Mismatched Network for MPSH10 at 100MHz ............. 91
5.21 511 of Mismatched MPSH10 Ampliﬁer ................. 92
5.22 321 of Mismatched MPSH10 Ampliﬁer ................. 92
5.23 822 of Mismatched MPSH10 Ampliﬁer ................. 93
5.24 Biconjugate Matched Network for MPSH17 at 100MHz ........ 94
5.25 5'11 of Biconjugate Matched MPSH17 Ampliﬁer ............ 95
5.26 521 of Biconjugate Matched MPSH17 Amplifier ............ 95
5.27 322 of Biconjugate Matched MPSH17 Ampliﬁer ............ 96

CHAPTER 1

Introduction

Ampliﬁers are common parts of any electronic circuit. In wireless systems, they
are typically the ﬁrst element after the receive antenna. When used in this sort of
application, it is important that the input impedance of the ampliﬁer be very close
to the impedance of the antenna and add little noise to the system. This increases
the dynamic range of the receiver, thereby improving its performance [6]. Finding a
commercial ampliﬁer which meets these criteria may be difﬁcult, leaving the designers
to develop their own ampliﬁer.

The theory of designing RF transistor ampliﬁers requires a broad range of topics
to be discussed. Chapter two is dedicated to giving the reader the basic knowledge
of these topics including transmission line theory, two port networks, and signal ﬂow
graphs, to name a few.

Chapter three pools all of this basic knowledge together and creates the framework
on how to design an RF ampliﬁer without the aid of any design tools. This leads to
the discussion of how difﬁcult the design process can be without the aid of a design
tool.

Chapter four presents the reader with the design tool developed by the author
of this thesis and explains how to use this program. This design tool computes

design parameters necessary to develop the ampliﬁer, and proceeds to explain how to

implement this design. Also, other software packages are discussed which aid in the
design process.
Chapter ﬁve presents a few designs that were made by the author and discusses

the results of these designs as compared to the design goals.

CHAPTER 2

Theory

2. 1 Transmission Lines

For a better understanding of the means in which microstrip geometries will be used,
it is necessary to include a brief explanation about transmission lines The model
that will be used to develop the theory is shown in Figure 2.1 where the wave is TEM

with propagation in the 2 direction [2].

 

 

 

|(z,t) |(z+A2.t)
—> R A2 L AZ —>
__/VV\/___/YYY\
+ +
V(z,t) V(z+A,t)
0 A2 I: G Az
- l -
l A‘ A.

 

Figure 2.1. Transmission Line Model

It will be assumed that there will be a series resistance R per unit length as well
as an inductance per unit length L. Between the two conductors there will be some
capacitance per unit length C as well as an conductance per unit length G. Applying

Kirchhoff ’s voltage law around the loop yields

 

 

 

 

 

 

 

 

v(z + Az, t) -— v(z, t) _ . Bi(z, t)
A2 —— Rz(z,t) L at (2.1)
Taking the limit of (2.1) and allowing Az —-> 0 shows
80(z,t) _ . 6i(z,t)
02 — Rz(z,t) L at (2.2)
Performing the Kirchhoff current law at the v(z + A2, t) node gives
i(z + A2, t) — z'(z, t) _ 811(2 + A2, t)
A2 -— C at Gv(z + Az, t) (2.3)
Taking the limit of (2.3) as A2 -—> 0 leads to
0i(z,t) _ 0v(z,t)
62 — C 8t Gv(z, t) (2.4)

It is convenient to assume that the system involved will be time-harmonic, there-

fore phasor notion can be used to simplify the equations by deﬁning two new terms:

v(z, t) = Re[V(z)ej“’t]

i(z, t) = Re[I(z)ej“’t]

and so the time convention 63"“ will be assumed and suppressed. The motivation for
this is to turn a partial differential equation into an ordinary differential equation.

The simultaneous solving of equations (2.2) and (2.4) give the following two second

order ordinary differential equations

 

 

 

‘12:?) = We) (25)
([21:22) — 721(2) (2 6)

where
7 = a + jﬂ = (/02 + ij)(G + M) (27)

Equation (2.7) has the terms a and [3 which represent the attenuation constant and
the phase constant respectively.

The solutions to (2.5) and (2.6) are

V(z) = V+e‘TZ+V‘eTz (2.8)

1(2) = Fee—"+1?” (2.9)

where the superscripts represent the direction of travel for each wave. The positive
superscript is for wave travel in the +2 direction and negative superscript is for wave
travel in the —2 direction when referring to Figure 2.1.

Rewriting equation (2.2) into the time-harmonic form yields

dV(z)
dz

 

= (—R — ij)I(z) (2.10)

Taking equations (2.8) and (2.9) then substituting into (2.10) gives

7(V+e"’z — V‘s") = (R + ij)(I+e‘7z + I'elz)
7(V+e‘7’ — V7272)

= 1+ -'72 I— 72
(R+ij) e + e

 

(G -l- ij)

(R +ij) (Viki-72 — V‘s”) = I+e‘7‘ + I—eTZ (2.11)

Equation (2.11) is similar to Ohm’s Law if we deﬁne

(R + ij)
(G + ij’)

where Z0 is the characteristic impedance of the transmission line. This will allow the

writing of equation (2.11) in the form

(V+e"7’ — V‘s”)
Zo

 

= (I‘Le'TZ + I—e‘ﬂ) (2.12)

which now resembles Ohm’s Law as desired. Going one step further and grouping

the forward traveling wave quantities and reverse traveling wave quantities together

 

 

yields
V+(Z) _ +
Z0 — I (z) (2.13)
V'(Z) _ _
— Zo — I (z) (2.14)

Equations (2.8) and (2.9) show that there are four unknowns and only two equa-
tions, but if the relationships of (2.13) and (2.14) are used, this reduces the number

of unknowns down to two with the resulting equations

 

 

V(z) = V+e‘7z + V‘s” (2.15)
V+ V—
: _7z — 72
I(z) 20 e Zo e (2.16)

These equations then allow a solution to be found for any arbitrary position along
the transmission line assuming that at least two boundary conditions are provided.

Using Figure 2.2, by letting z = l and then solving for V+ and V‘ gives

W = %(VL+ILZ,)ei‘ (2.17)

6

length=l >

Iin —> IL

 

Vs Q V(0) (20.1)

_———————.-4

Pl
If

1 z’=z-l z=0

Ill (5(—

Figure 22. Finite Transmission Line

1
v- = §(VL — [Lane-7‘ (2.18)
Knowing that I LZ L = VL and substituting into (2.17) and (2.18) gives the following

1
W = 51,,(Z,,+Z.,)e‘11 (2.19)

V_ = -;-IL(ZL—Zo)€_vl (2.20)

Since 2 = l is at the same location as 2’ = 0 along the transmission line, they have
the same electrical properties. Taking the ratio of equations (2.20) and (2.19) gives
a term known as the reﬂection coefﬁcient and is shown mathematically as

V— ZL - Z0

”177:2”;

 

(2.21)

This quantity represents the electric ﬁeld intensity that is reﬂected due to an
impedance mismatch at z = Z. Note that if Z, = ZL, there is no reﬂection; a
condition which is referred to as a perfect match. This quantity will also be used to
represent current reﬂection as well.

Now to ﬁnd the voltage or current at any arbitrary point (z’ = z — l) on the

transmission line and using (2.8) and (2.9) with (2.19) and (2.20) gives

 

V(z’) = I?!“ (Z1, + Zo)e"’ + (ZL — Zak-7"] (2.22)
1(2’) = 21; [(ZL + Zo)eT"' + (ZL — Zo)e_"z'] (2.23)

To ﬁnd the impedance of a wave that is terminated by some ZL translated by a
distance 2’, divide (2.22) by (2.23). Using the identities 67" + 6—72, = 2cosh(yz’) and

6”" — 6’7" = 2sinh('yz’) gives

ZL cosh('yz’) + Z0 sinh(yz’)

Z ’ = 0
(Z ) Z0 cosh(7z’) + Z L sinh(7z’)

 

which can further simplify down to

ZL + 20 tanh(7z’)

 

 

2(2 ) = Z0 Z0 + ZL tanh(7z’) (224)
Since the transmission line is of length l, a better deﬁnition of (2.24) is
Z,’ _ Z ZL + Zo tanh('yl) (2'25)

_ 0Z0 + ZL tanh('yl)

where Z, is the input impedance looking into the transmission line terminated by some
arbitrary impedance, Z L. Equation (2.25) is known as the impedance transformation.

Before moving on to the discussion of microstrip transmission lines, it will be
assumed that the transmission lines are lossless. When referring to Figure 2.1, it
is possible to assume that the attenuation constant a = 0 in the special case that
G = R = 0. This is not an impossible assumption since the lengths of the conductors
on microstrip are relatively short causing R z 0, and that the conductivity of the
dielectric is quite low making G % 0. Using this assumption allows ’7 to be rewritten

as

 

7 =15 = \/(jWL)(ij) = jun/L—C

This allows us to deﬁne ﬂ = (in/LC. Also note that equation (2.25) is now changed.

This makes tanh(ﬁl) = j tan(ﬁl) giving the ﬁnal working equation of (2.25)

._ Z ZL + jzotanwz)
1 — 0Z0 +jZL tan(18l)

 

(2.26)

Equation (2.26) is the basis by which most of the networks for the transistor ampliﬁers

will be designed.

2.2 Microstrip

Some of the transistor ampliﬁers for this thesis were built with microstrip transmis—
sion lines. The main reason for this is that the lumped element concept of discrete
components is no longer valid since the phase change through the element is not
negligible at higher frequencies. These discrete components also have an associated
quality factor (Q) to them where they may no longer act optimally. For example, a
surface mount resistor will have a capacitance associated with it due the the metallic
pad on each end of the resistor. These disadvantages of discrete components can
be overcome since equation (2.26) allows one to design a component which has the
desired impedance at a certain point with the use of transmission lines.

To use the equation (2.26), the ﬁrst assumption made was that the mode of prop-
agation was TEM. This however is not valid in the strictest sense for the microstrip.
The reason this is the case is because microstrip has half of the conductor exposed to
freespace and the other side is in contact with the dielectric. Figure (2.3) shows the

geometry where W is the width of the transmission line, h is the distance between

the ground plane and the transmission line, and 6,. is the relative permittivity of the

dielectric. Since there is a dielectric with a permittivity 6,. between the two layers

 

 

 

 

 

8r

 

 

 

Figure 2.3. Microstrip Geometry

of copper, then the majority of the ﬁelds emitted from the top conductor will tend
to concentrate inside the material. However, Pozar [6] states that the exact ﬁeld
distribution is a hybrid TE—TM wave, but since the dielectric layer is electrically thin
(h < /\), then the ﬁeld distribution inside the dielectric is almost identical to the
ﬁelds outside creating a quasi-TEM mode.

The propagation velocity in free-space is deﬁned as

 

where c is the speed of light, no is the permeability of free-space, so is the permittivity
of free-space. If the propagation is in a material other than free-space with some

(6,, pr), then the velocity is scaled by the following relationship

C

V/J'r€r

Since the microstrip being used is non-magnetic, then 11,. = 1. This changes equation

 

up:

(2.27)

10

(2.27) to

 

(2.28)

In freespace c = /\ f , where f is the frequency of the wave, but as was shown in
equation (2.28) the velocity of the wave is scaled by the dielectric constant for that
material,and thus the wavelength /\ is dependant on the material properties since the

frequency does not change and is shown by

v, = = ,\ f (2.29)

 

To use equation (2.26) for microstrip, it is desirable to ﬁnd an approximate value for
the effective dielectric constant which exists for the microstrip. An approximation for

the effective dielectric constant is given [6] below

_6,.+1 6,—1 1

e. _ — +
2 2 ./1+12h/W

where 1 < 68 < 6,. if 6,- > 1 assuming that the microstrip structure is immersed in

 

(2.30)

free-space. Equation (2.30) allows the computation of the wavelength of the wave that
travels on the microstrip allowing us to use (2.26) for designing microstrip lines with
the desired impedance. The only thing missing that is needed for full implementation
of (2.26) is the characteristic impedance of the transmission line.

Approximate closed form equations for the characteristic impedance are given by

 

 

60 8h W W
= -— — —— <
Z0 @111 (W +4h) for h _1
(2.31)
1207r W
o = f — >
Z ./ee[W/h + 1.393 + 0.667ln(W/h + 1.444)] or h " 1

11

where W is the width of the trace and h is the height of the substrate. There
will be times where it will be necessary to design a transmission line for a desired

impedance. If that is the case then use the following set of equations

 

 

88A W
W _ m for 7 S 2
7,— "— 2 e, — 1 0.61 W
_ _ __ _ _ . _ __ >
rt B 1 ln(2B 1) + 26,. {ln(B 1) + 39 6r for h __ 2
(2.32)

where

 

 

 

Z0 6r+1 er—l 0.11
A = — .
6O 2 J“.s.+1(023+ a.)

377w
2Z.../E:

 

In general, the larger the dielectric constant means that the traces can be thinner
versus boards with smaller dielectric values. The transmission line thickness becomes
important when deciding which transistor to use since the transistor pin spacing may
be narrower than the transmission line width being used and this makes mounting

the device quite difficult.

2.3 Two Port Networks

2.3.1 S—Parameters

The transistors that are used for the ampliﬁers being designed will be treated as two
port networks. This has the advantage of allowing the designer to not worry too
much about the actual physical phenomena such as junction capacitances and lead
inductance that occurs inside the transistor. Instead, all of the physical phenomena

are accounted for by the measurement of its two port parameters. The two port pa-

l2

rameters that are of primary concern are the scattering parameters, or S-parameters,

which allows one to measure the wave-like behavior of the device.

 

 

 

8.1 ___) 32
Port 1 2 P9” Port 2
z... devrce Z02
bl <—— __. b2

 

 

 

 

 

Figure 2.4. Two Port Network

Figure (2.4) is a block diagram of a two port device. On port 1, a1 represents the
incoming wave to the device and ()1 represents the wave leaving the device. Similarly
on port 2, a2 represents the incident wave into the device and b2 represents the wave
leaving the device. In general, there will be some transmission through the network
so both b1 and b2 will have some of its energy come from (12 and a1, respectively.
There will also be some reﬂection at either port 1 and port 2, which also comprise a
portion of b1 and b2, due to the waves impinging on these respective ports.

To express the above mathematically

b1 311 512 01
= (2.33)
b2 521 S22 02
where
()1 Reﬂection coefficient at port 1 with
$11 = — = (2.34)

0’1 a2=0 a matched load on port 2

l3

Transmission coefﬁcient output to

 

 

b
512 — CT: = port 1 due to an input on port 2 (2-35)
a1=0
with a matched load at portl
Transmission coefficient output to
b
521 - 2131 = port 2 due to an input on port 1 (2-36)
02—0
with a matched load at port2
()2 Reﬂection coefficient at port 2 with
322 — a— = (2.37)
2

 

a1=0 a matched load on port 1

The entries of the scattering matrix are typically complex numbers and are frequency
dependant. This means that a phase plane must be established for a reference point
at each port. If the phase plane is elsewhere along the transmission line, then the
given S—parameter must be multiplied by em, where l is the distance moved from the
given reference plane.

Using S-parameters to model the transistors as wave-like devices ﬁrst requires
one to choose the conﬁguration for which the transistor will be operated. The most
fundamental conﬁgurations for a BJT ampliﬁer are the common-emitter, common-
collector, and common-base. Manufacturer data sheets typically give the values of
the S—parameters when oriented as a common-emitter ampliﬁer. Other properties
which are given are the collector-emitter voltage and collector current. These are the
bias conditions and they have a direct effect on the S-parameters as well as the noise
ﬁgure of the transistor.

Below is the model that will be used for measuring the transistor’s 2—port prop—
erties conﬁgured as a common-emitter. Notice the two port model does not have the
external bias circuitry included. This does not invalidate the model since by using ei-

ther an RF choke or other means, it is possible to make externally connected circuitry

14

appear as if it were disconnected at higher frequencies.

 

 

0,——
a1——" *— a2
Port 1 Port 2
b1 ——’ b2
0 O

 

Figure 2.5. BJT Two Port Model

2.3.2 Power Waves

Since we are interested in the wave-like nature of the transistor, it is also reasonable
to discuss the amount of power that is either being transmitted or reﬂected by the
device. Ideally, the characterization will be related to the already known S—parameters
of the device. Some notation must be deﬁned ﬁrst before proceeding. It should be
noted that the following derivations come from [3]. Referring back to equations (2.8)
and (2.9), V(z) and I (z) will be normalized by J70 and the assumption of a lossless

transmission line (i.e. '7 = j B) will be used creates the following deﬁnitions

 

 

a(z) = VJékqu
b(2) = V_(Z)ejﬁz

 

The terms 22(2) and 2(2) are the voltage and current respectively, whereas a(2) and
b(2) are the forward and reverse traveling waves as functions of position. These
deﬁnitions are consistent with equations (2.13) and (2.14). Using these equations we

can rewrite (2.8) and (2.9) as the following

 

 

 

 

 

 

 

 

 

11(2) = (2(2) + b(2) (2.38)
2(2) = (2(2) — b(2) (2.39)
which again will allow equations (2.17) and (2.18) to be rewritten as
a<z> = g 2(2) +1<z>1 = Z—lﬁ W) + 2.12)] (2.40)
1 . l
b(2) = 51222) —z<z>1 = 2./z‘ [V(z) — 2.112)} (2.41)
21 mg “‘31.. ﬂ
+ + 2 port +
20 o 20
E1 V(0)1 b (231 dewce b(2). W». 22
Port 1' Port 1 Port 2 Poft 2'
ll: 0 "51:1 22:12 22: 0

Figure 2.6. Two Port Model with Load

Figure 2.6 shows a source connected to a two port network and a load. The source
provides the network with power. It is desirable to ﬁnd out how much power is being

delivered into the system as well as how much power is received at whatever load is

16

attached to the network, which in this case is Z2. The average power of an incident

.wave is found in the following way

P(Z) = R8[V+(Z)(I+(Z))’l

= $119. [V+(z)(L;(:’l:]

1|V+(2)l2
2 Z,

1 2
= §la(3)| (2-4?)

The reﬂected wave’s power can be derived in a similar manner as, the incident wave’s

power; speciﬁcally

P12) = $1112)? (243)

From Figure 2.6, a Kirchhoff Voltage Law expression at 21 = 0 gives
V(0)1 = E1 — ZolI(0)1 (2.44)

if Z01 = Z1. Substituting this expression into (2.40) gives

E1

 

 

 

0 = 2.45
a1( ) 2 Z01 ( )
or
2 _ IE1|2
|a1(0)| — 42... (2.46)
Substituting (2.46) into (2.42) yields
1 E 2
131(0) = §|a1(0)|2 = |8Z10|1 = PAVS (2-47)

The result of (2.47) represents the power available from the source (P AV 5) when the

17

source impedance equals the transmission line impedance (i.e. Z01 = Z1, or a matched
source). The term P Avg is important and will appear later when discussing network
power gains in Section (2.5). It is also worth noting that the power input from the
source reaches 21 = 11 without any loss since there is no power loss in the transmission
line.

Suppose that the impedance is not matched at 21 = 0 (i.e. Z01 aé Z1). In this case
there is a reﬂection at 21 = 0 as given by equation (2.21). This results in P1(0) not
equaling P Avg. It is thus desirable to ﬁnd what fraction of power does reach 21 = I].
Start again with ﬁnding the input power to the system using equations (2.40) and

(2.44) this power is given by

1

5'01 (0M2:

 

8Z0. (ll/1(0 )|2+2.111(0)V1‘(0)+ZoiV1(0)II0( )+Zo21|11(0 )l2 ) (2-48)

Similarly using equations (2.41) and (2.44)

1 2
'2'lb1(0)|=

 

8201(le(0 )l —Z..I.(0)V.*(0)—2.11/1(0)I;(0)+zg,|1,(o )1) (2.49)

Taking the difference between the above equations gives

2|01(0)I2_%|b1(0)l2 = '1'(11(OV1) (0)+If(0)V1(0 ))

= §Re[11(0)V1’(0)l (2-50)

which represents the power delivered to the two port network at 21 = 0 which is the
same as the power delivered to 21 = l1. The left had side of (2.50) will be denoted by
P1(0) = P1(l1) = PIN. The term PIN is another term that will appear often later in

the work, and is deﬁned as the power input to the network. Therefore (2.50) can be

18

rewritten with a substitution from (2.47) as

1 2

§|b1(0)| = PAVS '- P1(0) (2.51)
which is equivalent to

1
5119101)? = PAVS - P101) (2.52)

Using the deﬁnition of the S-parameters at 21 = 11 shows

|b1(ll)l2 : PAVS - P101)

5 2:
I “l |(11(l1)|2 PAVS

 

 

which can be rearranged to

P101) = PAvs(1 — I511I2)

The above equation relates Sn in power terms to show how much power is reﬂected
from the port due to an input at port one.
Now a derivation for how much power is transmitted through the two port network

will be found by relating it to 5'21. First using the deﬁnition of 521, it is found that

$21 = 5202) _ «7512-02)

a1(l1) 02(12)=0 _ VZOIIr(ll) I;(12)=0
V 2021202)

x/Z—ollﬂll)

 

 

(2.53)

 

12+(12)=o

For the rest of the derivation, assume that there is no transmission line between the
the voltage source and the 2 port network of Figure 2.6 and that the location they
meet is at 21 = ll as well as letting Z2 = Z02 at 22 = 12. This allows the rewriting of

(2.40) as

0,101) = 1
201 2201

 

 

11+(ll) = [V1(l1)+ Z0111(l1)] (2.54)

19

Another KVL is written at 21 2 ll as

“(11) 2 E1 — Zolllal) (2.55)

Substituting (2.55) into (2.54) gives

 

 

E1
+ _
1, (ll) — 2201 (2.56)
At port 2 of Figure 2.6, the current is found to be
V l
4202) = 2; 2) (2.57)
02

Substituting (2.56) and (2.57) into (2.53) yields

_2\/Zolv2(12)
821-— «Z; E1 (2.58)

 

Finding the power equivalent term of (2.58) gives

%|‘/2(l2)|2/ZOZ

s 2:
' 21' IE1I2/8Z01

 

(2.59)

Recognizing that the denominator is what was deﬁned earlier as P Avg and that the
numerator represents the power that is delivered to the load, a new term is deﬁned
called the transducer power gain and is denoted as CT.

If Z1 = Z2 = Z01 = Z02, then it follows from (2.58) that

 

 

V2
821 = (EL)
2
which leads to
G — (s (2 — V2 2
T _ 21 — E1/2

 

20

Similar derivations of equations (2.59) and (2.47) starting at port 2 can be used to

ﬁnd equations for 512 and 522.

2.4 Matching Networks

In most cases, it is desirable to deliver power to a load in the most efﬁcient manner.
The concepts presented thus far have shown that power can be reﬂected at the inter-
face between a transmission line and source impedance, or at the interface between
the transmission line and the two port network. It possible to match the transmis-
sion line to the either the source impedance or the two port network, but typically
the transmission line will be matched to the source impedance since that is usually
a system wide impedance. This means that there will be a mismatch between the
transmission line and the 2 port network interface. The mismatch can be overcome
by the design of matching networks which will transform the impedance of the 2 port
network to appear as a matched load which maximizes power transfer. This is accom-
plished by creating transmission line elements which act as circuit elements by means
of equation (2.26) when assuming a lossless transmission line. The primary method
that equation (2.26) will be implemented is with microstrip matching networks.

For microstrip geometries, it is practical to use shunt elements which are either
open or shorted to the ground plane. This then deﬁnes what the load impedance is

in (2.26). Exploring the open case (Z L = 00) shows

_ . 1+ J'(Zo/ZL) tanwl)
Z," — 2113110020 (Zo/ZL) +jtan(ﬁl)
jZo
—tan(ﬁl)
= —jZo cot(/31) (2-60)

 

 

Figure 2.7 shows a plot of the above equation. Note that that values repeat over

21

Reactance of an Open Line

 

‘J'OOKBD

 

 

 

 

 

0 0.2 0.4 0.6 0.8 1
0 SIS 1

Figure 2.7. Reactance of an Open Transmission Line

a period of n/\/ 2 where n is any integer and that the values are both positive and
negative over the interval.

Examining the case of the short circuit(ZL = 0) shows

 

. ZL + jZo tanwl)
Zin : l o .
211,120 Z Z, + 3ZL tan(ﬁl)

jZO tan(,Bl) (2.61)

with a graph of the resulting equation in Figure 2.8. Just as in the case of the open
line, the reactance is periodic every n/\ / 2 but is shifted to the left by /\ / 4. This implies
that when designing a matching network that either the shorted line or open line can
give the same desired reactance but one tradeoff then is the transmission line may be
longer. The other major tradeoff lies in the fact that it is easier to realize a short as
compared to a true open.

Another transmission line that is of note is of length A/4. This changes the

22

Reactance of a Shorted Line

 

jtan(B|)

 

 

 

 

 

 

0 0.2 0.4 0.6 0.8 1
0 S I S 1

Figure 2.8. Reactance of a Shorted Transmission Line

argument of tangent to the following

Al»

(at:

was

where tan(7r/2) is undefined. Through some manipulation, equation (2.26) reduces

down to

 

= _0_ (2.62)

Equation (2.62) has a special name called a quarter-wave transformer, and is usually
used only if Z0 and Z L are real valued. If Z L and Z, are real valued, then Z," will also

be real valued, thereby transforming from one resistance to another. All the designer

23

must do is ﬁnd the appropriate Z, to transform between the two desired impedances.
Since microstrip is being used, then the designer uses equations (2.32) to ﬁnd the
appropriate width to height ratio.

The last case of note is a transmission line of length A / 8. The value of tan(7r / 4)
or cot(7r/4) simply equal 1, which reduce equations (2.61) to Z," = jZo and equa-
tion (2.60) to Zn = —jZo. Again the designer just needs to design for the desired
impedance of the transmission line to create the purely reactive element. The length
does not need to be restricted to l = A/ 8. Values of l = (2n — 1))\/ 8 where n is any
integer are also acceptable. The only thing that will change will be the sign of Z,"
depending on which quadrant ,8l ends up in.

Matching networks will be designed using various forms of the above equations
as either series elements or parallel elements. For series elements, it makes sense to
use the given impedance equations since impedances add in series. In practice, there
will typically be some element that will be in parallel since these element themselves
are purely reactive when used as opens or shorts on microstrip, thereby changing
the reactive component of of a desired impedance. The problem is that the total

impedance of elements in parallel have the form

Z1 Z2
Z] + Zz

 

Ztotal =3 (2.63)

which when dealing with complex numbers is quite cumbersome mathematically. To
make the computation of elements in parallel easier, ﬁrst change the impedance of

the element to an admittance which is done by the relation
y =

i
Z

Using the relationship of impedance to admittance and equation (2.63), it can be

24

shown that admittances add when in parallel.

2

 

Z.,.2_
+M+
21 1_1
._ __
131W
2 Z

 

— Y1+Y2

Yum:

Finally, most all of the matching network design will be done on a Smith chart.

A Smith chart shown in Figure 2.9 is a tool which allows for graphical solutions of

    

           

 

Figure 2.9. Smith Chart

equation (2.26). The circles which are completely contained in the chart represent

25

the real part of the impedance, whereas the circles which are partially contained are
the reactances with those circles in the upper half of the plane being positive valued
(e.g. inductance) and those in the lower half being negative (e.g. capacitance). The
outer circle represents a reﬂection coefﬁcient I‘ = 1 while I‘ = 0 when at the origin.
Hence, a short would be located on the centerline at the extreme right while a open
lies on the centerline of the extreme left. Note that from equation (2.21), I‘ can be
complex valued and therefore also has an angle associated with it when in polar form.
The numbers that are drawn on the outside of the circle represent these angles.

To use the Smith chart, the impedance of a load or source must be normalized by
the characteristic impedance of the transmission line. The impedance is then found
on the Smith chart and graphed. A line is drawn from the center of the circle through
the impedance. The linear distance from the center of the circle and the angle that the
ray passes through represent the reﬂection coefﬁcient of that load when normalized
to the given impedance. If it is desired to see what the input impedance would be if a
wave traveled of length l, then another circle is drawn centered about the origin in a
clockwise manner until the length I has been traveled. The length l is normalized by
the actual wavelength in the media of propagation. The value where the circle stops
being drawn is the normalized input impedance as seen by the source. Multiplying
this number by the characteristic impedance gives the actual resistance value. Also,
the reﬂection coefﬁcient magnitude did not change; however, the polar angel did. This
method may sound complicated at ﬁrst, but when compared to the complex math
which is necessary for equation (2.26), matching network design is usually quicker

and easier when using a Smith Chart.

26

2.5 Signal Flow Graphs

The S—parameters described in section 2.3.1 can be implemented by means of a signal
ﬂow chart. By using this method, it is possible to deﬁne additional terms which are
associated with RF transistor ampliﬁers. This is by no means the only way to derive
the following information (see [3] for two derivation methods), but it allows for insight
without being overly concerned with the details.

The signal ﬂow model for a two port device is shown in Figure 2.10 The nodes

 

 

81 A 331 22 %
Sn [522
b1 512 8.2

Figure 2.10. Two port signal ﬂow graph model

represent signals and the branches represent systems [5]. Some general rules that will

apply to signal ﬂow graphs are listed below [3]
1. Each variable is designated as a node.
2. The S-parameters and reﬂection coefﬁcients are represented by branches.

3. A branch enters dependent variable nodes and emanates from independent vari—
able nodes. The independent variable nodes are the incident waves and the

reﬂected waves are dependent variable nodes.

4. A node is equal to the sum of the branches entering it.

27

5. Travel from node to node is only permitted in the direction as indicated by the

arrow, and when that does occur, the signal is multiplied by the value adjacent

to that arrow.

For example,

b2 = 01521 + 02522

which is identiﬁed as part of the deﬁnition of S parameters.

Moving between nodes in Figure 2.10 is quite simple. There will be other systems
which are more complicated and require more complex methods for analysis. One way
to do this is through Mason’s Rule [5] which provides a formulaic way of analyzing
these systems. Mason’s Rule gives the transfer function of a system when starting

from an independent node and traveling to another node and is shown as

 

T A
__ Ending Node _ ; k k (2 64)
” Starting Node _ A '
where
k = the number of forward paths
T), = the kth forward-path gain
A = 1 — 2 loop gains + E non-touching-loop gains taken two at a time

—2 non-touching-loop gains taken two at a time + . . .

A)c = A — 2 loop gain terms in A that touch the kth forward path

This equation can become quite complex for large systems, but since there are rela-
tively few forward paths in the two port model being used, the equation (2.64) does
not become unmanageable.

A representation of Figure 2.6 as a signal ﬂow graphs must be made to allow for

28

use of Mason’s rule. To modify port one, an equivalent circuit must ﬁrst be developed

which is shown in Figure 2.11. Performing a KVL on the circuit shows

 

Figure 2.11. Port 1 equivalent circuit and signal ﬂow graph

E, = I Z, + Vg
Recalling that a V = V+ + V‘ and I = 1+ — I ‘ for waves gives
9 9

Z. _ _
E.=Z(v+-v>+(vg++vg)

Solving for V; yields
V9+ V; Z, — Z0 Es\/Zo

 

 

 

= 2.65
«‘2. mz,+z,+z,+z, ( l
which can be rewritten as
(11 = bll‘s + b8 (2.66)
where
V+
a1 = g
x/ZT
V—
()1 = g

 

29

 

 

b _ EMZ
’ _ Z.+Z.
z—z
Pa = 8 O
Zs+Zo

Equation (2.66) can be represented in signal ﬂow graph terms and is shown on the

right side of Figure 2.11.

The output signal ﬂow graph must also be derived. The model for the output

is shown in Figure 2.12. Similar to the derivation for the input signal ﬂow graph, a

is l:
S+
It]

Figure 2.12. Port 2 equivalent circuit and signal ﬂow graph

KVL is performed around the circuit giving the following equation

VL = —ILZL

when written in terms of traveling waves gives

Solving for VL+ and normalizing by V70

v; _ V; ZL—Zo (2 67)
\/—Z:-\/Z—OZL+ZO .

 

 

30

This can be rewritten in terms of wave quantities as

a2 = 1321“,, (2.68)

where

V3
«70
V;
x/Z

ZL " Z0
ZL + 20

 

D
to
I

 

8"
II

A composite model can now be deﬁned from the two port network in Figure 2.10 and

the networks developed in Figures 2.11 and 2.12 and is shown in Figure 2.13.

 

 

b1 512 a2

Figure 2.13. Two port signal ﬂow graph

An important quantity for ampliﬁers is the input reﬂection coeﬂicient, F m, due
to an arbitrary termination at the output as shown in Figure 2.14 The mathematical
deﬁnition of the input reﬂection coefﬁcient is F m = b1/ (11. This is in the form that
allows for use of Mason’s rule since a1 is now an independent variable. The ﬁrst step

for Mason’s rule is to ﬁnd all of the forward paths from a; to b1. There are two, one

31

 

FIN SI 1 S22 If.

 

< v <
b] 5]: a2

Figure 2.14. Model for FIN computation

of which is T1 = Sgll’ L512 and the other is T2 = Sn. Loop gains must also be found.
There is only one loop gain, which is SngL. Next, A must be found which is simply
1 — SggFL. Finally, A,c must be found for each Tk, so A1 = 1 and A2 = 1 — ngFL.

Substituting into equation (2.64) gives

511(1— 522R) + SIZS2IPL

 

I‘ __
IN 1 - 5'2er 1 - S22P L
_ 512S2IFL
PIN —- $11 + 1 - :52er (2.69)

A similar derivation can be obtained at the output to ﬁnd the output reﬂection
coefﬁcient which is deﬁned as Four = bz/az where the input has an arbitrary source
impedance, b, = 0, and a2 is an independent node. By applying Mason’s rule again,

the following equation results

312521Ps

F = S ——
OUT 22 + 1 _ SIIPS

(2.70)

The power delivered to the load is just the power of the incident wave minus the

power reﬂected from the load, or

1 I 1
PL = §Ib2|2 — §|a2|2 = Elbzlzu — [FL|2) (2-71)

32

If the circuits from Figures 2.11 and 2.12 were not connected to a two port network,

then the signal ﬂow graph in Figure 2.15 would be created.

Figure 2.15. Model for P Avg computation

Recall that the deﬁnition for P Avg requires that a matched load be attached to
the network. The P Avg is equal to the power leaving the source minus the power that

is returned to the source giving,
PAVS = Elall - §[bl[ (2.72)

If the load attached is a matched load (i.e. I‘L = I‘g), then a1 = b, + I‘Sl’gal and

bl = I‘gal, which gives

be
= ————-- 2.
a1 1 _ IFS|2 ( 73)
bsl‘g
= —— 2. 4
Substituting equations (2.73) and (2.74) into (2.72) yields
1
Elbslz
(2.75)

AVS = 1_ [PSI2

33

Recalling that the transducer power gain (GT) is the ratio of the power delivered to
the load (P L) to the power available from the source (P AV 3), the following equation

is now created
: PL = [[32]2
PAVS Ibs|2

 

 

GT (1‘ [FL]2)(1_ IFSI2) (2-76)

The ratio of bg/bs can be solved with Mason’s rule and after some simpliﬁcation is

 

 

 

given by
b2 S21
__ = 2.77
b3 (1 — SllPS)(1— 5'2er) — SizszirLrs ( )
Substituting equation (2.77) into equation (2.76) gives
a. : |S21|’(1-|I‘L|’)(1 — II‘slz)
[(1— SIIPle - S22FL) — 51:25'21FL1‘SI2
which simpliﬁes to
1"II‘SI2 2 1- [1‘le
G = —— S 2.78
T (1—rmrsll 2" |1—322PL|2 ( )

The operating power gain (GP) is deﬁned as the ratio of the power delivered to
the load, P L, to the power input into the network, Pm. Describing P, N in terms of

signal ﬂow graphs is shown here.
1 1
PIN = §IGII2 " §lbll2

PIN has a similar deﬁnition as P Avg, but the distinction lies in the fact there is no

restriction placed on the load impedance, therefore the deﬁnition is modiﬁed to

1 1 1
Pm = §lail2 - 5W2 = 5(1— |P1N|2)

where I’m is a function of the S parameters as well as the load impedance.

34

Now G p can be written as

 

G _ PL _ [b2[2(1-|FL[2)
P _ — - 2 2
PIN [all (1‘ [PIN] )

Dividing the numerator and denominator by b, gives

 

b2 2 2
P ,— (1— 1m >

GP = F—L— = as (2.79)
“V 7,1 <1 — Irmm

 

 

then ﬁnding the ratios of bg/bs and a1 /bs by Mason’s rule gives the ﬁnal form of G p

which is
_ 1

G
P |1 — 5221“,]?

[521]2

 

(2.80)

One last power term which needs to be deﬁned is the power available from the

network which is the power delivered to a conjugately matched load and is written as

1 l
PAVN = PLlrL=rgUT = [5W2]2 - §|a2]2] =
r‘Lzl‘bu'r

[$22120 — mm] = glam — Iran?) (281)

FL=FOUT
This allows one more gain term to be deﬁned, which is called the available gain and

is shown as

P b 2
G. = A” = '—2'—,<1—Ir0m12><1—Irsl2) (2.82)
PAVS [b3]

 

Using Mason’s rule again on the ratio bg/bs, substituting into equation (2.82) gives

1

1— lf‘sl2
G _
A 1— [POUT12

— [1- .32ng]2

 

 

[S2ll2 (2.83)

35

2.6 Noise

Noise is a phenomenon which is both naturally generated as well as man made. In
either case it is undesirable when developing systems which require a high sensitivity
to low powered signals. Noise is typically classiﬁed into two broad categories [11].
One category refers to the types of noise generated by devices inside of a receiver
such as resistors, electron tubes, and semiconductors. The other category refers to
those noise sources which are external to the receiver and examples of these would
be lighting discharge and cosmic radiation. Obviously the designer has no control of

the latter of these types of noises, but can control those of the former.

2.6.1 Noise Sources

One source for noise is thermal noise which is deﬁned as the noise generated by the
random motion of electrons in a conductive medium. Nyquist was one of the ﬁrst
people to have studied thermal noise. His theorem on noise states that the root-
mean-square (RMS) noise voltage appearing across a resistor of value R ohms at

temperature T in kelvin across a bandwidth B in hertz is given as

aims = 4kTRB (2.84)

where k is Boltzmann’s constant.

Another type of noise is called shot noise. Shot noise is caused by the quantized
and random nature of current ﬂow [4]. When a DC current is ﬂowing in a device,
there is some average current ﬂowing in it, Io, but does not indicate any variation
or frequency of this variation that is indicated from the average current. Schottky’s

theorem relates this average current to the noise generated as shown below

36

where e is the charge of an electron in coulombs and B is the bandwidth in hertz of
the circuit.

Other types of sources of noise are generation-recombination, temperature ﬂuc-
tuation, and ﬂicker noise. Generation-recombination is caused by electron hole pairs
being recombined in a semiconductor and is typically classiﬁed as a shot noise. Tem-
perature ﬂuctuation noise is caused by the temperature interaction of the device with
its environment. Flicker noise is a type of noise that is characterized by its Spectral

density that increases with decreasing frequency [11].

2.6.2 Noise in Systems

Figure 2.16 shows two ampliﬁers which are cascaded and have available gains of GA,

and G A2. The resistor is acting as a noise source and has a signal input power PM

U2

P i : 11,1‘m8 = k
N 4R ToB

 

where To is a temperature of 290°K (for room temperature). Each ampliﬁer has
its own contribution to the noise of the system and is denoted as Pnl and Pu2 for

ampliﬁers one and two respectively. The noise output from the system is simply

labeled as PNO.

 

 

 

 

Ampliﬁer 1 Ampliﬁer 2
. + o
G“ Rd: (3:1. R1 G52 1:1: ZL
Pnl P112

 

 

 

 

 

 

 

Figure 2.16. System noise ﬁgure model

37

One characteristic of an ampliﬁer is its noise ﬁgure which deﬁned as the ratio of
the signal-to-noise ratio (SNR) power at the output of the ampliﬁer to the signal-
to—noise ratio power at the input of the ampliﬁer [10], or expressed mathematically

as
__ PSi/PNi

F—
PSO/PNO

am)

where P5, is the input signal power and P30 is the output signal power. Equation

(2.86) can be rewritten by using the fact that GA = Pgo/PS, as

PNo

 

 

 

 

F = 2.87
PNiGA ( )
Referring to Figure 2.16, it is easily seen that the output noise power is
PNO = GA2(GA1PN1 + Pn1)+ Png (2.88)
Dividing both sides by G A20 AIPN, gives
PNO P711 P112
F=-——=1+-—+—— 2.89
GAZGAIPNi GAIPNi GA2GA1PNi ( )
or alternatively
F — 1
F=E+ 2 am)
Gm
where
Pnl
F = 1
1 + PNiGAl
P112
F = 1
2 + PNiGAZ

F1 and F2 are called the individual noise ﬁgures of the ﬁrst and second ampliﬁers

respectively.

38

Closer inspection of equation (2.90) shows that the overall system noise ﬁgure is
reduced by the gain of the ﬁrst stage. The problem is that the higher the gain, the
higher the noise ﬁgure for that stage typically. Thus, there is a tradeoff in the ﬁrst
stage ampliﬁer; does one increase the gain of the ﬁrst stage to reduce the noise contri-
bution from the second stage or does one design for reducing the noise added by the
ﬁrst stage ampliﬁer by lowering the gain at the cost of having a greater contribution

from the second stage? The answer to this lies in a term called the noise measure

(M) and is found to be

 

1 (2.91)

1__
GA

If equation (2.91) is computed for each stage, the ampliﬁer with the lower M should
be the ﬁrst stage ampliﬁer. This will ensure that the system noise ﬁgure be at its

lowest.

39

CHAPTER 3

Ampliﬁer Design

3.1 Stability

A stable ampliﬁer by deﬁnition is one which does not oscillate. A two port network
is unconditionally stable if the reﬂection coefﬁcients at both ports are less than one.
When referring to Figure 3.1, [PIN] and [Four] are these port reﬂection coefﬁcients.

A value of greater than one represents that the port being measured has a negative

 

 

 

 

 

Figure 3.1. Reﬂection coefficient diagram of a two port model.

40

resistance, which is shown by solving equation (2.21) for Z L.

1 + I‘
Z = Z,—
L 1 — I‘
If Z L is real valued and F > 1, then the denominator is negative which in turn implies
Re[ZL] is negative as well.

Figure 3.2 shows the transistor ampliﬁer with both input and output matching

networks attached. Recall that FIN (equation 2.69) is a function of the S-parameters

 

 

 

 

 

 

 

 

 

 

 

 

Io... 2.
28 Z...
Z’ Input Output
E1 Matching ’ 9 Matching Z2
New“ 1 r l r Network
| l f l l
Pa 1; Fm 1801‘ 1f. 1“b

Figure 3.2. Transistor ampliﬁer with input and output matching networks.

of the transistor as well as FL and that Four (equation 2.70) is a function of the
S-parameters of the transistor as well as F3. The designer will have some nominal
frequency when designing the ampliﬁers, so the S-parameters for the transistor will
not vary at that frequency. The only values that can change for a given frequency
are PS and FL. The arbitrary selection of these could cause an instability, thus it is
desirable to know which values of F3 and I}, will prevent this from happening. Since
most of the design process will be done on the Smith chart, it is desirable to be able

to show what regions where these passive terminations could cause an instability.

41

The derivation of the stability circles is given in [3], but the general procedure is
to let [FIN] = [Pom] = 1 and ﬁnd for which values those correspond to for F1, and F3

in equations (2.69) and (2.70) respectively. The FL stability circle is given by

 

 

 

 

_ 512521
"L ‘ lav—mm ‘3'”
CL ‘ lav—mm (3")

where TL is the radius of the circle, CL is the location of the center of the circle is on
the Smith chart, and

A8 = 511522 - 51252]

Similarly for F5

 

 

 

 

_ 5.23..
’5 ‘ ISnIZ-lAslz (3'3)
CS ‘ lam—M (3'4)

where 73 is the radius of the circle, CS is the location of the center of the circle, and
A, has the same deﬁnition as above.

The drawing of these circles on the Smith chart will likely create an overlap with
the Smith chart. The region of overlap is the area in which the ampliﬁer is unstable
as long as Sm, < 1, where Sm, is either Sn or $22. The region of stability is anywhere
else on the Smith chart. If the stability circles do not overlap on any region of the
standard Smith chart, then the transistor is then classiﬁed unconditionally stable at
that frequency since no selection of a passive FL or F3 network will cause an instability.

The necessary and sufﬁcient means of showing unconditional stability is given in

42

Appendix B of [3] but here is summarized as
K > 1

and

IA.I < 1

where
= 1- |511|2 — [3:22]2 + |A3|2

K
2[512321]2

 

(3.5)

If neither of these two conditions are met, then the transistor is potentially unstable

and the stability circles must be plotted to ﬁnd which regions are considered stable.

3.2 Standing Wave Ratio

One important aspect of any ampliﬁer design is the standing wave ratio (SWR) on
that transmission line. The deﬁnition of SWR is the ratio of the maximum voltage
(or current) to the minimum voltage (or current) on a transmission line. For VSWR.

this is shown as
leaxl _ 1+ IFI

SWR = -—
[Vminl 1 - IF]

(3.6)

Recall that F in equation (2.21) is a function of the characteristic impedance as
well as the load impedance. If F = 0 then it is easily seen that equation (3.6) is equal
to one. If Z L aé 20 then the reﬂection coefﬁcient is non-zero. Any non-zero value for
I‘ causes (3.6) to become a value greater than one. Horn this, it is intuitive that an
SWR value near one is desirable as it implies that the impedances are matched.

In transistor ampliﬁers, a SWR can not only be due to an impedance mismatch,
but is also due to the actual matching networks which are connected to the input and

outputs of the transistor, as will be shown below.

43

Before explaining how the input and output networks affect SWR, a small dis-
cussion must be made about gain and power relationships. This will lead to the
derivation of how SWR is affected by these networks. A summary of the gain terms

of importance deﬁned in section 2.5 is given by

P power delivered to the load
PMS power available from the source
_ P _ power delivered to the load
GP _ P7,; — power input to the network

_ P power available from the network
GA _ PAVS power available from the source

 

GT:

 

 

If the ratio of CT and GP is taken, the P L terms will cancel and the following expres-

sion is found
a: = Pm
GP PAVS

 

If this ratio is equated to some constant called the mismatch factor, then the following

equations are deﬁned

PIN = PAVSMs (3-7)

GT -_- cpM, (3.8)

where M,3 is called the source mismatch factor and can be written as

1 — IPSI2)(1— [FIN|2)
I1 _. FSFINI2

 

M, = ( (3.9)

Similarly, an output mismatch term can be found by taking the ratio of CT and

GA with the ﬁnal resulting equations

PL = PAVNML (3-10)

CT = GAML (3.11)

44

where M L is called the load mismatch factor and can be written as

(1 - IFL|2)(1- IFOUTI2)
M = 3.12
L [1" 1"ou'rI‘le ( )

 

Referring to Figure 3.2, the term Fa is a reﬂection coefﬁcient as seen by the source
for the entire transistor network. It follows from here that the input SWR will be
deﬁned as

 

(3.13)

The input power into the system from the source is P IN = P Avs(1 — |F3|2). Using

the above equation and equation (3.7) gives the following result
M, = 1 — |1‘,|2

01'

(m = (/1— M, (3.14)

Substituting the deﬁnition of M3 into (3.14) gives

 

_ 2 _ 2 _ *
111.1: )1} lFsl)(1 Irm|)_lrm 1‘. (3,5)

]1_PSFIN[2 _ l—FINFS

 

Examination of equation (3.15) shows that SWR,n is a function of PIN and F3, where
I‘m itself is a function of the S-parameters of the transistor as well as the output
matching network, FL. Equation (3.15) also shows that if I‘m = g, then [Pa] = 0
which will cause SWR,n to equal one.

The output SWR is deﬁned as

1+ [Pb]

SWROUt : 1_ [Pb]

 

(3.16)

45

where H, is the reﬂection coefﬁcient at the output of the transistor network as seen
by the load. Using the concept of the load mismatch factor then [Pb] can be written

as

1‘1. = 1— ML (3.17)

Substituting (3.12) into (3.17) gives

 

POUT — PE
1 — POUTPL

 

 

111.1: \/, _ <1 — 111.190 — [Pourl2l ___ (3,8)

I1 — POUTPLIZ

 

 

This shows that [R] is a function of Four and I}, where Pour is a function of the
transistor S-parameters as well as the input matching network PS. If I‘L = Pbur:

then [R] = 0 thereby causing SWR“, to equal one.

3.3 Bilateral Conjugate Match

By deﬁnition, a bilateral conjugate match occurs if
I‘S = IN

and

I‘L = FOUT

Recalling equations (2.69) and (2.70) and using the conditions for a bilateral match

give

SIZSZIFL
r* = s ___— .1
S “+1—-522rL (3 9)
S12S21Fs
11* = ___ .2
1. 522+1_ 5111.3 (3 0)

46

The simultaneous solutions of (3.19) and (3.20) give the following results

131 :l: B? — 4(0112

 

 

PM, = 201 (3.21)
PM, = 32* 28:;— 4IC’P (3.22)
31 = 1+l511|2-|522|2-|As|2 (3-23)
32 = 1+|522|2—|511|2-|As|2 (3-24)
01 = 5.1—6.8;, (3.25)
02 = 822—833;, (3.26)

where I‘M, and F11“, represent the bilateral conjugate matched values. The next
question to ask is what sign is used in equations (3.21) and (3.22). The proofs for
which sign to use are in Appendix E of [3]. The results say that if [Bl/201] > 1 and
81 > 0 for equation (3.21), then the solution with the minus sign is appropriate to
produce |FM3| < 1. If [Bl/201] > 1 and BI < 0, then the plus sign is used to produce
ll‘Msl < 1. Similar requirements are used for equation (3.22) on which sign to choose.

A bilateral conjugate match can occur only if the the transistor is unconditionally
stable for that given frequency. This is the case since only in an unconditionally
stable transistor will all of the Smith chart be available for the selection of I‘M, and
I‘ML. Under a bilateral conjugate match condition, then the transducer power gain,

equation (2.78), then can be simpliﬁed to

 

G max = S 2 3.27
T 1— IrM.|2' 21' 11— 5221‘»!le ( )
which after substituting equations (3.21) and (3.22) reduces to
_ [$21] ,/ 2
GTmax "" W(K — K _ 1) (328)
12

47

As a side note, under bilateral conjugate match conditions CT = G p = G A, which
also implies GTmax = Gpmax = GAmax.

There will always be cases where the transistor is not unconditionally stable, but
it is desirable to know what the maximum gain could be and still be stable. Under
this scenario then a different type of measure is used to determine what the maximum

Stable gain is. The maximum gain is Slmply found by
G 3.29

3.3.1 Design Procedure

The design procedure for unconditionally stable transistor is quite straight forward.
1. Compute PMS and I‘ML from equations (3.21) and (3.22) respectively.
2. Design an appropriate matching network to the values found for I‘M, and I‘ML.

The properties associated with a bilateral conjugate matched ampliﬁer are SWR“,
= 1, SWRout =1, and that the maximum gain possible is being used. If these are
the design criteria then this is the ideal, but it will be shown later that an ampliﬁer
in this conﬁguration will not always have the best noise performance. Also, a bilat-
eral conjugate match will not always be possible, so other design methods must be

developed to achieve the desired operating performance.

3.4 Operating Power Gain

When the bilateral conjugate match is not necessary or if the transistor is potentially
unstable for the desired operating frequency, then a design utilizing operating power

offers a viable alternative. Gonzalez says that since the operating power gain is inde-

48

pendent of the source impedance, then this is the recommended method for practical
ampliﬁer design [3].

Starting with equation (2.80) and substituting equation (2.69) for FIN gives

(1-

which is rewritten in compact form

|521|2(1 — WW)

2
M > '1 _ S22FL|2

 

GP:

1 - 5221—1

 

 

 

GP = |S21l29P
where
1- |PL|2
9,, I1 — 522FLI2 - '511 — AsFle
1 — II‘LI2

 

= 3.30
1- |~S‘11|2+|1‘L|2(|522|2 - IA.|2> — meme.) ( ’

Notice that 9,, is a function of FL and the transistor S-parameters. Since the
S-parameters are ﬁxed for a given frequency, the only true variable is I‘L; thus, it
necessary to ﬁnd which values of PL will give the desired value for gp. A mapping of
these values of FL will be graphed on a Smith chart to aid in the design. A derivation
of these equations is in Appendix G of [3] and are merely presented here. The radius
of the circle drawn will be denoted as no and the location of the circle center is given
as Cp. Mathematically

ll‘L — Cpl = rp (3.31)
where

C = 911C;
p 1+ £ho(|522|2 — lAsl2)

 

(3.32)

49

and

 

\/1— 2K|521512Igp +|S21512|29122
— 1 + 9p(lS22l2 — IASlZ)

 

T19

(3.33)

3.4.1 Design Procedure

There are two possible cases that must be considered when designing by this method

and those are if the transistor is unconditionally stable or not.

Unconditionally Stable

1. Choose a G' p < 0pm,, and ﬁnd gp.

2. Find the center and the radius for the given operating power gain by equations

(3.32) and (3.33) respectively and plot the circle onto the PL Smith chart.
3. Select the desired I‘L somewhere along the plotted circle.
4. Compute I‘m from equation (2.69).

5. Maximum power transfer will occur if FfN = PS so this is the value that should

be chosen for PS.

6. Since both PS and I}, have been found, design the matching network to these

values.

Conditionally Stable

1. Choose a GP < GMSG and ﬁnd gp.

2. Find the center and the radius for the given operating power gain by equations

(3.32) and (3.33) respectively and plot the circle onto the PL Smith chart.

3. Select the desired I‘L somewhere along the plotted circle that is inside the Smith

chart and away from the instability circle.

50

4. Compute I‘m from equation (2.69).

5. Since the transistor is only conditionally stable, the choice of PfN = F3 may not
be available due to the fact that the region of instability may be covering that
value of PS. Instead choose a value of F3 far enough away from the stability
circle but still somewhat near I‘fN. This will help keep SWR,“ as small as

possible.

6. Once a value for PS has been chosen, then design the input and output matching

networks for the values found for PS and FL.

Suppose that FL is chosen such that I}, aé F M L for the design speciﬁcation of some
G p < Gpmax. This will create an SWR“, > 1. This shows that there is a trade-off
in one parameter for another, where in this case the gain is desired to be reduced
but has the overall effect of changing SWRout. These decisions will be the theme
throughout any transistor ampliﬁer design.

Also, the choice of an operating power gain does not imply that is the total gain
of the ampliﬁer. Instead it will be the maximum possible gain the ampliﬁer could
have. This is also true for the available power gain scenario to be discussed in the
next section. The true gain of the ampliﬁer is the transducer power gain, GT, and

corresponds to the measured 5'21 on a network analyzer.

3.5 Available Power Gain

Available power gain design procedures are similar to that of the operating power
gain. The difference lies in the fact that available power gain can be found to be

independent of the load impedance attached.

51

Starting with equation (2.83) and substituting equation (2.70) for Four gives

5 21— P 2
GA 2 l 21' ( l 3| ) : lS21|2ga (334)

2
(1_ M )l1_5,1r8|2

1—511Ps
_ GA _ l—Il‘sl2
9“ 182.12 1— |522|2 + IPsl2(lSul2 - lAslz) - 2Re<rscl>

 

 

 

where

 

The term 90 is a function of only 1‘3 and the transistor S-parameters. Similar to the
operating power design, the only true variable for the available power gain is I‘s since
the S-parameters are ﬁxed for a given frequency. Circles will be drawn onto a Smith
chart to ﬁnd all values of PS which will solve for the desired go. The radius of the

circle and center of the circle will be deﬁned as 7“,, and Ca respectively and are found

 

 

 

to be
1-2K a S S 2 2
Ta 2 \/ |521512|§ +| 2: ml 9.. (3.35)
l1+ga(lSll| — lAsl )I
C, gacl (3.36)

_ |1+ 9.2(ISuI2 - IA3|2)|

3.5.1 Design Procedure

There are two possible cases again that one must consider and those are if the transis-
tor is unconditionally stable or not. Both procedures are similar, however the choices

for PS and FL are limited in the conditionally stable case.

Unconditionally Stable

1. Choose a desired GA < CAM; then ﬁnd 90.

2. Compute Ta and Ca from equations (3.35) and (3.36) respectively and plot the

circle onto the PS Smith chart.

52

3. Select a value for PS along the constant gain circle.
4. Compute Four from equation (2.70).

5. Choose Pour = I], since this maximizes the power transfer as well as makes

SWR-out = 1.

6. Design the matching network from values found for I}, and PS.

Conditionally Stable

1. Choose a desired G A < GMSG then ﬁnd go.

2. Compute Ta and Ca from equations (3.35) and (3.36) respectively and plot the

circle onto the F3 Smith chart.

3. Select a value for PS along the constant gain circle which is in the stable region

and is not too close to the unstable region circle.
4. Compute Four from equation (2.70).

5. Choose a I}, in the stable region near Fbu'r but not too close to the stability

circle. This will help to minimize SWRout.

6. Design the matching network from the values found for I}, and PS.

3.6 SWR Circles

As was seen in the potentially unstable cases for the designs of both the operating and
available power gains, a conjugate match could not be obtained in most cases. Instead
the designer had to chose between ampliﬁer gain versus SWR. Suppose that there was
a speciﬁcation for a ﬁxed SWR at either the input or output of the ampliﬁer, then it

would be nice to know for which values of FL or F3 would give that SWR to meet the

53

design requirements. Since all designs are done on the Smith chart, it is desired to
ﬁnd an equation which will draw constant SWR circles onto their respective planes;
SWR,n circles will be plotted on F5 and SWR“, circles will be plotted on I},

The equations used to draw the constant SWR,n circles on the PS plane are

PIN(1 - lFalz)

 

 

CV,- = 1 _ IFaP1N|2 (3.37)
_ I1“..|(1 —|I‘1N|2)
TV.- — 1 _ IPaFIle (3.38)
where CV,- is the circle center location, rv, is the radius of the circle, and
SWR,n — 1
IP.I — ___—SWR... H (3.39)

The circles that are drawn represent the all of the values for a chosen I‘s which will
give a constant SWR,n at the desired level.

Similarly for the output, the equations are

PBUTU - IFbl2)

 

C = 3.40

v., 1 - IFbI‘ourl2 ( )
I‘ - I‘ 2

TVO l b|(1 l OUTI ) (3'41)

 

1 — leFOUTIZ

where CV0 is the location of the constant SWRout circle center, Tvo is the radius of

this circle, and
SWRout "' 1
SWRout + 1

 

lrbl = (3.42)

The circles that are drawn represent those values for a chosen I}, which will give a

constant SWRout at a desired value.

54

3.6.1 Design Procedure

Designing for a desired SWR value can be quite complicated. There many different
values that one can chose for either F5 or I‘L, all which will have an effect throughout
the design. The procedure is more or less a guideline since there is no set way to do

this.

Suppose that SWR,n is the restricted parameter in the design and SWR“, is less

important. The general procedure then would be as follows:

1. Choose a value for 0;: such that G p < G pm“ if the transistor is unconditionally

stable or G p < G MSG if the transistor is conditionally stable.

2. Find the center and the radius for the given available power gain by equations

(3.32) and (3.33).
3. Choose a value for I‘L along the G’ p circle that is in the stable region.
4. Compute FIN from equation (2.69).

5. The ideal choice for F3 is the case where F5 = I‘fN. That choice may not be
available since the transistor may be conditionally stable. If SWR,“ has a design

limit, use that value in equation (3.39) to solve for II‘al

6. Use equations (3.37) and (3.38) to compute the constant SWR,” circle. Choose

a value for F3 that is in the stable region not too close the stability circle.
7. Compute Pour from equation (2.70) using the value selected for F3.

8. Compute SWRout by ﬁrst ﬁnding IFbI from equation (3.18) then use equation
(3.16).

As can be seen, the procedure is quite involved. Even worse, initial choice for

FL and the ﬁnal answer of Four may cause an unacceptable result for the output

55

SWR. In this case then one would have to start almost completely over to ﬁnd values
for I}, and Four which would provide an acceptable result for SWRout. However,
there will still be a limit to how well SWR“, can be. One way to achieve a better
SWR“, is to loosen the requirements on the input SWR. This will only work so well,
but it illustrates again this concept of give and take on operating parameters for the
ampliﬁer.

A similar procedure is developed in the case where the output SWR needs to be

a certain value.

1. Choose a value for G A such that G A < G Am” if the transistor is unconditionally

stable or G A < G MSG if the transistor is conditionally stable.

2. Find the center and the radius for the given available power gain by equations

(3.36) and (3.35).
3. Choose a value for F3 along the G A circle.
4. Compute Four from equation (2.70).

5. The ideal choice for FL is the case where I‘L = PbU'r- That choice may not
be available since the transistor may be conditionally stable. If SWRout has a

design limit, use that value in equation (3.42) to solve for II‘bI

6. Use equations (3.40) and (3.41) to compute the constant SWRout circle. Choose

a value for I‘L that is in the stable region not too close the stability circle.
7. Compute I‘m from equation (2.69) using the value selected for FL.

8. Compute SWR,n by ﬁrst ﬁnding II‘al from equation (3.15) then use equation
(3.16).

56

3.7 Noise Figure Circles

One ﬁnal design consideration given for transistor ampliﬁers is the noise ﬁgure. In
some applications, such as telecommunications, it is desirable to have a low noise
receiver so that a minimal amount of noise is added to the signal for processing of
the information. In this regard, a designer may deem the noise performance more
important than those discussed above which again adds another layer of complexity
to the design of the ampliﬁer.

The noise ﬁgure of a two port ampliﬁer is found to be

Tn
F = Fmin + g— lye - yoptl (3'43)

8

The derivation of this equation is found in Appendix L of [3]. The goal is to ma-
nipulate equation (3.43) in a way such that it can be plotted onto the Smith chart
such that a graphical approach can be maintained in the design. The ﬁrst step is to

express ya and ymin as reflection coefﬁcients deﬁned as

 

_ 1—rS
ys _ l-l-Ps

_ 1—1‘opt
y0pt — 1+Popt

Substituting these equations into (3.43) gives

47'an3 — Poptlz

F = Fmin +
(1" lPSl2)l1+ I‘optlz

 

(3.44)

The terms Fmin, 7‘", and F01), are all provided on the data sheets for a given RF

transistor. Note that sometimes Fm,“ is called Nfo by some manufacturers. Allowing

57

F = E, where F,- is the desired noise ﬁgure and using

Fl ‘- Fmin
N,- = Tu + I‘.,,,,|2 (3.45)
in equation (3.44) gives
IFS - Poptlz
—— = N,- 3.46
1 _ IFSIZ ( )

where N,- is referred to as the noise ﬁgure parameter. Equation (3.46) can be manip-

 

 

 

 

 

 

 

ulated to
rs — 11:?“ 2 = N? +3913:th (3.47)
which is easily identiﬁed as a circle centered at
CF.- = 13"]; (3.48)
with radius
m = 1+1N.- W + N,(1 — (romp) (3.49)

Also from the left hand side of equation (3.47), the noise ﬁgure circles are plotted
on the F3 plane. To use equations (3.48) and (3.49), equation (3.45) is ﬁrst used to
ﬁnd a value for Ni. Equation (3.45) requires the desired design noise ﬁgure F,- must

be chosen such that it is greater than Fmin.

3.7 .1 Design Procedure

If noise is the most important aspect of a design and must be at a minimum, then the
choice for F3 is quite easy. It is given on the data sheet. The procedure for available
gain circles is then followed. The draw back to this choice for F3 is that the input
SWR may be beyond the desired speciﬁcation. If the restrictions on the noise ﬁgure

are relaxed, then one can tune the other performance characteristics.

58

There is no true procedure that can be broken down into discrete steps for this
design. Instead, adapting the idea that the noise ﬁgure circles are more of a guide

will be beneﬁcial. The only part that can be done procedurally is as follows
1. Choose a value for a target noise ﬁgure (E).
2. Compute N,- from equation (3.45).

3. Compute the circle center and radius from equations (3.48) and (3.49) respec-

tively.
4. Plot the circle on the F3 plane.

The next steps (such as gain and SWR) are up to the designer as to which aspects
are most important. Use any of the above procedures as a guide on what to do next.
The only addendum to the above procedure is that the choices for PS should be near

the desired noise ﬁgure circle if all other choices for F3 are otherwise equally as good.

59

CHAPTER 4

Software Tools with Updated

Design Procedures

Last chapter dealt with design procedures which were to be done by hand on Smith
charts. As was touched on before, this technique can be, and is, quite tedious. Even
after going through one of the design procedures mentioned, the results may be un-
satisfactory for the design requirements which leads to going through the procedure
again however many times until an acceptable design is found. This sort of repetitive
nature of the above design process lends itself perfectly to the use of a computer

application to aid in the design.

4.1 Using the Design Program

Matlab was chosen for the computing environment due to it’s presence on university
campuses. In addition to having the main program ﬁles, there are some thirty other
ﬁles necessary for various function calls and scripts. Ensure that the folder location
for all of the ﬁles has been added to the path, or change the active directory to this
folder location. Once the path has been set, type “AmpDesignerm” in the command

window. The GUI then opens as well as two other ﬁgures which have more detailed

60

 

 

ROM“
00m

 

 

(‘1‘ .
it "so.” "

 

«15
22mm

m1
! «mm
69 «me

cm

was

. 418?

um
4m

Gum-S
mm
“L
mt.
Mb
Mad
"the

I

6‘ QQWO‘

it!

*. again
"In“
1.-

 

 

 

Gmsme-OL
Gan-Lhoms

 

 

sumac
C “.1

Figure 4.1. Screenshot of the Microwave Ampliﬁer Design Program

 

mm

mm?
J

lol—
N‘0 (a)

 

 

 

ﬁ..____|

L 1,7

womb'rémm

 

 

 

L 1— . f
. a»; 1; . ., ()5.
a °l ‘3 32L E§ lgEE
I 93 E} E}? E 33‘ £3 E
a .‘ :‘JcJ
. A

 

wwmmna

-_‘2__J .

 

 

 

 

 

 

 

Smith charts for PS and I}.

Figure 4.1 is a screenshot of the program that is used for the design of microwave
transistor ampliﬁers. The upper left hand corner is where the user inputs the S
parameters for the desired frequency. Underneath the S-parameters group box is the
place to enter the the noise parameter data. Once the information has been entered,
it is possible to save the data so that it does not need to be retyped. Simply click on
“File —» Save Transistor Parameters.” Under the same menu is the “Load 'IYansistor
Parameters” used to ﬁll in the saved S parameters and noise parameters into the
appropriate boxes.

As a minimum, the S parameter data is required for any ampliﬁer design. The
noise parameters are optional and will not affect the operation of the program if they
are blank; the user will simply not be able to plot constant noise ﬁgure circlas.

Once the information has been entered into appropriate boxes, then the user
should click on the “Compute Stability Values” box. This will do two things; it plots
the stability circles onto the F3 and FL planes, and it causes two boxes to appear.
One of these boxes is called “Results” and the other is called “Circle Computations”.
Inside the “Results” box, one of two things will happen: the values for I‘M, and I‘ML
will appear with the associated transducer gain (GT) for the unconditionally stable
transistor, otherwise the maximum stable gain (MSG) is given for the ampliﬁer for
the conditionally stable transistor. In the “Circle Computations box”, a text box
appears on the left with a pull-down menu on the right. To plot a 14 dB operating
gain circle on the FL plane, simply enter 14 into the text box and select the proper
option on the pull down menu to the right. Pressing the “Plot” button will then plot
the circle on the on appropriate plane. Two options that are in the window which
are not discussed are the “Ga on Gamma L” and “Gp on Gamma S” options. As
the name implies, these are for plotting the gain circles on the plane other than what

they are normally plotted on. Consult [3] on designs using these circles.

62

There is a color code to the circles that are plotted on F3 and FL. Maroon circles
are plotted for G A on F3 and G p on I‘L. Green circles are plotted for G A on FL
and G p I‘s. Yellow circles represent constant SWR circles. Finally, noise circles are
graphed in red.

Underneath the F3 plane, there are two buttons: “Gamma S to Gamma L” and
“Gamma L to Gamma S”. These are used to activate the continual updating feature
of the cursor’s location on either the F3 or I‘L plane. Use of this feature will be
discussed more in detail later.

At the bottom right is where the user will input a desired input or output SWR
value and selects which type to plot via the radio buttons. It is important to note,
that a gain is required for the plotting of these functions. The way that this feature
operates is that it reads the value from the gain box at the bottom left of the GUI
inside the “Circle Computations” box prior to computing the SWR circle. When
these constant SWR circles are plotted, the user will see ten red stars appear on both
F3 and FL. These points represent the ideal solutions for a given constant SWR. For
example, if a constant SWR circle is plotted on PI, for a speciﬁc G p, then on the FL
plane, these red stars appear at various points along the constant gain circle. The
ten points on along the constant gain circle have a one-to—one correspondence to to
those values that appear on the F3 plane and represent the minimal values for SWR,n
for a constant SWR at the output.

There are two buttons under each Smith chart. The “Clear Gamma X” button is
for clearing that Smith chart if it becomes to cluttered in both the ﬁgure and main
GUI. The button called “Enlarge Gamma X” is there in case the ﬁgure that represents
that ﬁgure is closed, then this button provides a means of recreating that ﬁgure in

the background.

63

4.2 Design using the GUI

As was discussed in Chapter 3, there are two ways to design an ampliﬁer: start with
either 0,; or with G p and then incorporate the SWR limitation if needed. Below is

a procedure starting with a desired G p.

1. Enter the values for the transistor S parameters in the appropriate box. If
available, enter the noise ﬁgure data as well. It is recommended that these

values be saved.
2. Press the “Compute Stability Values”.

3. If the transistor is unconditionally stable and that is what is desired, then
simply read the values for I‘M, and I‘ML; your design is done. If a gain other
than the MAG is desired or for noise considerations, then continue on with the

procedure.

4. Choose a value for G p in dB less than the gain shown in the “Results” box and

enter it into the “Circle Computations” text box.

5. On the pull-down menu select “Gp on Gamma L” and click on “Plot”. This
graphs the constant operating power gain circle onto the FL plane. Repeat this

step as many times as necessary for different gain values.

(a) If noise ﬁgure data has also been entered, then enter noise ﬁgure values

greater than those that have been entered into the box labeled “Nfo (dB)”.

6. Since the procedure being considered is using 0;: as a starting point and is

plotted on FL, then click on the button labeled “Gamma L to Gamma S”.

7. The user should notice that the value for “Gamma L” is now changing in the

middle of the GUI with the location of the cursor. Choose some point along

64

10.

the desired constant C p circle by right clicking on the PL Smith chart. Right
clicking on I}, will ﬁx the value for FL, and now shifts control to the F3 plane.
Also, the user should see that the values for F3, Four, I‘ IN, SWRin, SWRout,

and noise are now all changing as the cursor changes its location.

. To plot a constant SWR circle, ensure that value for the gain circle that was

clicked on in FL is in the gain value box of the lower left hand corner. Ensure
the radio button for “Input” is selected in the “VSWR Circles” box and select
a value. This step can be repeated any number of times by selecting a different

I‘L. Simply right click on a new value for PI, and click on “Plot” again.

Choose a value for F3 based on the circles that have been plotted by pressing
the space bar. If noise is a consideration, choose the value which has a red star
near the desired constant noise ﬁgure circle. Pressing the space bar will ﬁx all
of the values that are updated on the GUI as well as tell the designer what the

values are for GT, GA, and CT.

The two Smith charts in the background have been updating all actions that
have been occurring on the main GUI. At this point, the designer may wish to

print off these Smith charts to complete the design of the ampliﬁer.

That is all for the procedure, but there are few things on which to comment.

First, notice that the value for G p is the same as was used in the design. This would

simply be just a self check for the user to ensure that points chosen during the design

procedure are indeed the value which was desired.

Also note that if a point for I}, is chosen on one of the red stars that appear during

the plotting of a constant SWR circle, the new constant SWR,n circle touches only

one of these points. If the user were to move to this choice for I}, to an adjacent red

star in the clockwise direction and plotted a new constant SWR,n circle, then the new

65

SWR,In circle would also be adjacent to the previous choice in a clockwise direction
as well.

Finally, when constant SWR circles are graphed and have centers that are in the
origin, then that represents a match at either the input or the output (i.e. R; = I‘fN
or Pour = Ff). Speciﬁcally this can be shown to be true if the following equation

constant input SWR equation is examined

= IPA1 — IPINI2ll
1 — lPaFIN|2

 

 

lFS _ IN(1 _ lPalz) (41)

1 - |I‘,,,I‘1N|2

 

where I} is deﬁned as it is in equation 3.15 and is rewritten here for convenience.

PIN - Pg

1“,: _—
ll—PINPS

 

(4.2)

Since a calculation for constant SWR,n circles is being considered, a value for H, has
likely been chosen and PIN has been calculated and is ﬁxed. When the user is moving
the cursor, the variable that is changing in equation 4.1 is I‘s. As can be seen in
equation 4.2, Pa = 0 if I‘g = P IN. When I}, = 0, the term on the right hand side of
equation, or the radius, 4.1 goes to zero. Also, the second term on the left hand side
of equation 4.1 equals I‘fN since I‘a = 0 and the denominator equals one. A similar
proof can be given for the constant output SWR circle equation.

The design procedure for constant GA circles is essentially the same as above.
The only difference would be to click on the “Gamma S to Gamma L” button. The
advantage of starting with available gain circles is that it allows the designer to choose

Fs immediately, thereby setting the noise ﬁgure at the desired value.

66

4.3 Circuit Design

Once the designer has the two design values for F3 and FL, then it is up to the designer
as how to implement the design; will the design be done on microstrip or will the

design be done using lumped elements. Each design technique will be discussed here.

4.3.1 Microstrip Implementation

If the designer is implementing a microstrip based design, then the Smith charts
which have been plotting in the background can be used. These Smith charts have
degree measurements around the perimeter as well as a linear scale at the bottom to
accommodate the use of a compass.

When using the Smith chart to match to either I‘S of FL, take note on the deﬁ-
nition of these terms (Figure 3.2); they are the reﬂection coefﬁcients looking into the
matching network and terminate at the system characteristic impedance. This means
that the ﬁrst rotation will be in the “Wavelengths towards load” direction from this
impedance/ admittance. Depending on what type of network being used, the goal is
to move all of the components added toward the center of the Smith chart.

The values necessary for the matching network should be extracted such as the
lengths in fraction of wavelengths as well as what type of transmission lines used.
Next, ﬁnd the effective wavelength of the for the design frequency on the board to be
used. This allows the designer to turn the normalized lengths into actual lengths. If
using impedances other than those at the system impedance, such as for quarter wave

transformers, use equation 2.32 to calculate the necessary transmission line width.

Ansoft Designer

This next step is optional, but is recommended as a sort of veriﬁcation of the design.

For this thesis, a program called Ansoft Designer has been used for this step.

67

When the program opens, by default a new project opens. Click on the icon which
looks like a transistor at the top. This opens a dialog box which asks what type of
microstrip is being used. If the type is not there that is being used, then choose any
type that is listed. It can be overridden later.

A blank schematic sheet now appears as well as more icons at the top of the
page. The next thing to do is add a two port network to the schematic. Click on
‘Draw—>N-Port...’. A dialog box appears. On the ﬁrst tabbed sheet, there are three
radio buttons under ‘Data Source’. Select ‘Enter data in spreadsheet’. Select the
tab labeled ‘Network Data’. Select the proper units for the frequency. On the ﬁrst
row, ﬁrst column of the spreadsheet enter the frequency that the operation will be
performed. Tab to the next square and enter Sn in that block formatted as the
magnitude-space-angle. Do this for the other S parameters noting that 5'12 is in the
ﬁrst row, last column and 5'21 is in the second row, middle column. Select the tab
‘Noise Data’ if there is noise data to enter. Press ‘OK’ and place the network on the
schematic.

On the left hand side of the program, there are a set of three tabs labeled, project,
components, and search. Click on ‘Components’. A list of different types of elements
then become available including microstrip elements and lumped elements. Select
the folder labeled microstrip. There are three options here that will be of use to the
designer: shorted stubs, Open stubs, and transmission lines. Add whatever elements
are necessary to the schematic. Select the type out of these three menus that are
denoted as near ground and electrical length.

Double click on any one of the transmission lines just added. This will bring up
another dialog box. Click on the box on the line labeled ‘SUB’. This is where the
microstrip speciﬁc information can be added. This will pop up another dialog box.
Click on ‘New’. Enter all of the pertinent information for the board being used. Click

‘Ok’ twice. Finally enter the information necessary for all of the microstrip lines such

68

as electrical length and width.

Next, RF sources need to be added. Select ‘Draw—->Interface Port’ and add two
of these. Double click on each of these ports and select the radio button which turns
the port into a microwave port.

After connecting all of the components, it is time to simulate the circuit. Under
the menu ‘Circuit’, select ‘Add Analysis Setup...’. Click on ‘Next’. Click on the
‘Add’ button. Select the radio button labeled ‘Single Value’ and enter the frequency
at which to simulate. Press the button labeled ‘Add>>’. Click ‘Ok’ then ‘Finish’.
To simulate the circuit, go to ‘Circuit—>Analyze’. A box at the lower middle will tell
if the analysis has been successful.

To view a report, go to ‘Circuit—+Create Report...’ and simply add what variables
are of concern. There are options such as GT, 321, and SWR. Simply add what is
desired to look at. If these values are close to what is expected, then the design is as

it should be and is ready for the next step.

Sonnet

To model the matching networks on microstrip, the program Sonnet was used to tune
the network to the desired impedance values. This was done by drawing the geometry
to the approximate values given by using the Smith charts. When using Sonnet, it is
important make sure that the proper dielectric is being used and that the cell sizes
are reasonable for the given frequency. Also, make sure that there is sufficient space
all around the board to minimize interactions with the edges as well as for leaving
room for changing the transmission line lengths as will be discussed later.

Place two ports onto where the transmission line will be attached and where the
transistor will be attached. Double click on the port that will be attached to the
transistor. Make the port an autoground since the port is fed from the ground plane

to the trace. This allows the user to modify the port impedance. To ﬁgure out

69

the values to be entered for the impedance, simply enter Re[I‘S] times the character-
istic impedance in the ‘Resistance’ block and enter -Im[I‘s] times the characteristic
impedance into the reactance block. The simulations should show how well the initial
design compares to the actual layout.

Sonnet also comes with an optimization feature. To use this feature, there must be
elements on the design called ‘parameters’ for Sonnet to vary. To add a parameter, go
to ‘Tools—>Add Parameter’. From there the user has the choice of either anchored or
symmetric. The anchored option sets one point which will not move from that point
on the board and uses this as a reference when changing the other dimensions. The
symmetric case changes the lengths equally in opposite directions about it’s midpoint.

Once the parameter components have been placed, it is required to setup the
program to optimize. Go to ‘Tools-—>Setup...’. Under the box labeled “Analysis
Control”, select the option called Optimization. This will cause two other boxes
underneath to appear. The upper box is called ‘Parameters’. This is where the user
will set the constraints on the parameters that have been added to the drawing. Select
the ‘Edit’ feature. Another dialog box appears and allows the user to select which
length will be varied as well as what values to limit the possible lengths. Once the
constraints have been entered, click on ‘OK’. The second box which appears is the
place where the user enters the design goals. Next, the user should click on ‘Add...’.
Here the user can enter a frequency range over which to optimize the design as well
as what the design goal should be. It is suggested that only one frequency be entered
and that Sn be set to less than -30dB.

If a designed stub happens to be a shorted stub, then it is possible to create a
short by adding a via to the board in Sonnet. When using the optimization routine,
ensure that all corners of the via have been selected while adding a parameter. This
will move the via and maintain it’s position at the edge of the stub.

Once networks have been developed, then merge all of the components onto one

70

board to be made. There are practical aspects to be considered before completing
the design. First, the vias that were used to simulate shorts for the shorted stubs
need to be removed. If a DC bias is applied to the shorted stub, then there will be a
short circuit to ground. The way to get around this problem is to make the shorted
stubs open but then attach a bypass capacitor from the stub edge to a grounded pad
near the edge of the stub. This will preserve the DC bias, but will act as a short at

the higher operating frequency of the ampliﬁer.

4.3.2 Lumped Element Design

Microstrip designs start being practical when the design frequency is above lGHz, be-
cause the free-space wavelength is about 30cm. This is still large, but the wavelength
can be signiﬁcantly reduced by choosing microstrip board with higher dielectric val-
ues, but still may be large. Thus, for frequencies less than about 1GHz, it makes
sense to use lumped elements.

After obtaining the design values for F3 and FL, then it is better, if not a necessity,
to use an admittance-impedance Smith chart (or sometimes called ZY Smith chart.)
A ZY Smith chart has a normal impedance Smith chart on as well as a mirror image
superimposed on top of the impedance Smith chart.

To explain the reason for using this type 08 Smith chart, it is better to examine
what can be done on the standard Smith chart. The circles which have origins along
the central axis represent constant Re[Z]. To add a series reactive element then it is
only required that the designer move along the perimeter on one of these circles. To
add a shunt reactive element on the standard Smith chart is not as easy. There are no
circles on which to traverse. By adding the admittance equivalent of the Smith chart,
then the designs have been reduced to traveling along circles for shunt elements.

The key to the design with lumped element is to start at the design value for F3

or FL and move towards the center to the Smith chart. Clockwise motion along a

71

constant Re[Z] circle represents adding a series inductor, whereas counterclockwise
rotation represents adding a series capacitor. Likewise on a constant Re[Y] circle,
clockwise rotation represents adding a shunt capacitor and counterclockwise rotation
represents adding a shunt inductor. If starting at the wrong point, then the wrong
element will be inserted.

Once the values for the inductors and capacitors are found, then it is possible
to test this design in Ansoft as well. Instead of using microstrip elements there is a

folder labeled “lumped” which then allows the user to add these types of elements.

72

CHAPTER 5

Designs and Analysis

A design tool or procedure is only as good as the the ﬁnished product that is produced
as well as its ease of use. To verify the design tool works, a few designs were made
with different transistors and different parameters.

The foundation for any of the designs that are to be built in this manner are
the S parameters. If these values are incorrect, then the rest of the design will
be wrong; thus, it is important to have accurate S parameter measurement. The S
parameters for the Agilent transistor AT-41486 were extracted from the datasheet [9].
This might seem contrary as to the previous statement about accurate S parameter
measurement, but also notice that only one design was made using this information.
On the other hand, the S-parameters for the MPSH10 and MPSH17 transistors were
actually measured for the designs made, and as will be seen, made the measured

values very close to the design goals.

5.1 S-Parameter Measurement

Accurate measurement of the transistor S-parameters will improve the accuracy of
the design of the ampliﬁer. Figures 5.1 and 5.2 are the bias circuits used in measuring

these transistor’s S-parameters. This circuit is designed to give a 5V collector-emitter

73

voltage with a collector current of 10mA. The ports labeled “NA” on either end of
the circuit are where the network analyzer was attached. Capacitor C1 is used as a
bypass capacitor to effectively short the emitter to ground, and capacitors C2 and
C3 are used as DC blocks. Both circuits had identical current gains of approximately

100, so the DC bias circuits end up being the same.

 

 

 

v1
R3 gm,
R1 <: <: _'I_
3,215» #470 -.-
03
NA cg {F—ﬁ—C
-‘ 680 r NA
6801”" MPSH10 p
R2 l R4 4 C,
5.616“ 470:» #680101,

 

 

 

 

J.—
-0

Figure 5.1. DC Bias Circuit Used in S—Parameter Measurement for the MPSH10

 

 

v1
R3 élsv
R1 3 ‘
3.2};1: 1'470 7';

NA 02 ———Caf—<:1
D—lH—K Q1 680 F NA
6301’” MPSH17 p

R2 ‘: R4 ‘: CI
5.616: 470» $680131,

 

 

 

 

Figure 5.2. DC Bias Circuit Used in S-Parameter Measurement for the MPSH17

74

Neither of these designs have RF chokes to isolate the transistor from the bias
circuitry. This means that the bias circuitry was included in the measurement of the
networks S-parameters. This approach seemed to work well for when the matching
networks were placed on the network analyzer side of the blocking capacitors and
treating the entire ampliﬁer with bias circuitry as a two port network. If the desire is
to ﬁnd the actual transistor S parameters, then it would be necessary to include the
RF chokes.

The circuit was built by using the “dead bug” or ground plane technique as dis-
cussed in [7]. This technique eliminates most stray capacitance which may appear
if the circuit were to be built in a prototyping board or perforated board. A short
coaxial cable was then cut and stripped down to expose the center conductor such
that a connection could be made to the circuit. The exposed length of the center con-
ductor was minimized so that the cable could maintain it’s characteristic impedance.
The length of the transmission line was short to ensure that the attenuation in the
transmission line was negligible.

To ensure that a proper phase measurement was obtained, the center conductor
of the coaxial cable was shorted to the outer braided shielding while connected to the
network analyzer. Since the shorted transmission line is known to have a reﬂection
of 14180°, the user can then change the electrical delay of the network analyzer to
this value at the desired frequency. When the cable is open circuited, it should have
a reﬂection coefﬁcient near 140° and should have angle measures near these points.
It is important that both cables be near identical. This means that the lengths of the
cable and the amount that has been stripped should be about the same. This should
yield similar values for the electrical delay of each cable. If the shorted and opened
transmission lines have electrical delay values, then the cables must be adjusted until
they agree. Once a value for electrical delay has been found for Sn and 5'22, then

ensure that the delays for $21 and 812 are adjusted to these same values. Figures 5.3

75

through 5.6 are the measured S-parameters of the circuit shown in 5.1, and Figures

5.7 through 5.10 are the MPSH17 S-parameters measured using the circuit in Figure

5.2.

Magnitude (dB)

Phase

S11 magnitude and phase of MPSH10

 

  

 

 

 

 

 

 

 

 

1 0 I T I I
0
-10
_20 1 1 1
0 100 200 300 400 500
Frequency (MHz)
0 1 Y TI T
-20 - .
-40 r- a
-60 ~ 4
-80 . .-
_100 1 i L 1
100 200 300 400 500
Frequency (MHz)

Figure 5.3. Sn of MPSH10

76

Magnimde (dB)

Phase

Phase

S12 magnitude and phase of MPSH10

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

o I T I I
_100 1 1 1 1
0 100 200 300 400 500
Frequency (MHz)
150 . . . .
_100 1 1 1 1
0 100 200 300 400 500
Frequency (MHz)
Figure 5.4. Sn of MPSH10
$21 magnitude and phase of MPSH10
20 . r r . . . .
a
3
O
“D
52‘
C
U)
(B
2
o l l l l 4 1 1
50 100 150 200 250 300 350 400
Frequency (MHz)
200 I I I I
100 - -
O F .
400R .
_200 1 1 1 4
0 100 200 300 400 500
Frequency (MHz)

Figure 5.5. 321 of MPSH10

77

$22 magnitude and phase of MPSH10

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

5 I I I I
a
'3 o - _
0
'O
.5
2
_10 1 1 1 1
0 100 200 300 400 500
Frequency (MHz)
0 I I I I
-50 - .
3
2
a.
-100 ~ 4
_1 50 1 1 1 1
0 100 200 300 400 500
Frequency (MHz)
Figure 5.6. $22 of MPSH10
S11 magnitude and phase of MPSH17
10 I I I I W
8
E o -
O
13
.3
E
g -10 ~ -
2
_20 1 1 1 1 1
50 100 150 200 250 300
Frequency (MHz)
0 I I 1 I I
_50 .. .4
33
2
a.
-100 ..
_150 1 1 1 1 1
O 50 100 150 200 250 300
Frequency (MHz)

Figure 5.7. 5'11 of MPSH17

78

S12 magnitude and phase of MPSH17

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6
E
O
‘O
.3
C
U)
m
2
_80 1 1 1 1 1
0 50 100 150 200 250 300
Frequency (MHz)
200 I I I ﬁ I
100 ‘
3 N\,_\\
2 o .- -.
e.
-100 - E
_200 1 1 1 1 1
0 50 100 150 200 250 300
Frequency (MHz)
Figure 5.8. $12 of MPSH17
S21 magnitude and phase of MPSH17
20 . . . . .
8
3
0
13
.3
C
O)
N
2
0 l L l l L
50 100 150 200 250 300
Frequency (MHz)
200 I I I I I
100 ~ ~
0.
“100 N .
_200 1 1 1 1 1
0 50 100 150 200 250 300
Frequency (MHz)

Figure 5.9. 821 of MPSH17

79

Phase

Magnitude (dB)
.5

-100

-150
0

$22 magnitude and phase of MPSH17

 

 

 

 

 

 

 

 

50 100 150 200 250 300
Frequency (MHz)

50 100 150 200 250 300
Frequency (MHz)

Figure 5.10. 522 of MPSH17

80

 

5.2 Ampliﬁer Designs

Table 5.1 has extracted S-parameter values for the MPSH10 and MPSH17 transistors,

whereas the values for the AT-41486 were from the datasheet [9]. Table 5.2 represents

the target design values given the transistor S parameters and the values for F3 and

FL. The “AmpDesigner” program was used to compute all of the design values.

 

 

 

 

 

 

 

 

 

 

 

I Transistor ] AT-41486 (2GHz) MPSH10 (100MHz) MPSH17 (100MHz)
Sn 0.624 152 0.4734 -77.6 0.5154 -71.0
512 0.0584 43 0.0154 38.0 0.0274 42.1
Sm 3.614 56 5.554 91.23 6.3354 86.26
522 0.424 -39 0.6504 -32.42 0.6004 -44.2
Table 5.1. Transistor S Parameters used in Designs
[ Transistor AT-41486 MPSH10 MPSH10 MPSH17 I
Center Frequency 2GHz 100MHz 100MHz 100MHz
GT (dB) 14.43 18.2 14.8 19.2
I‘S 0.4704 -148.2 0.4444 88.3 0.4904 55.6 0.5204-71.0
FL 0.5574 41.1 0.6384 36.4 0.5104 -37.5 0.6004556
SWRm 3.07 1 1.5 1
SWRout 1.59 1 5.59 1

 

 

 

 

 

 

 

 

Table 5.2. Design Values

5.2.1 AT-41486 Design

Figure 5.11 shows the DC bias circuitry with the RF amplifier attached. This is

known as an active biasing circuit since there is another transistor used in the bias

circuit. The data sheet states that the bias conditions on the AT-41486 for the S

81

parameters listed in Table 5.1 required that the collector current be 10mA and a
collector-emitter voltage of 8V. This circuit will allow the user to adjust both of these

values by adjusting the two potentiometers in the circuit.

 

 

 

R1 R4
3.3k 330k
:=__Vl R2 8
T: 15V 3.3k 0-500
.1 C1
0‘ 1.10mi
2N3906 --
- VQZ
120 C2
".5 *IUUn

 

 

Figure 5.11. Schematic of AT-41486 Ampliﬁer Bias Circuit

This design was done at 2GHz so a microstrip implementation was utilized for the
ampliﬁer. Figure 5.12 shows the microstrip layout. The microstrip board is Taconic
TLC-32 with a thickness 20 mils. On the left edge is the input to the ampliﬁer.
The gap that is seen immediately to the right is for a DC blocking chip capacitor.
The length of microstrip before the open stub is simply a 509 transmission line. In
the center of the board is a gap for the transistor to be placed. The two pieces of
microstrip above and below this gap are pads for the transistor to be soldered to and
are grounded. On the far right edge is where the output port was attached with a
gap for a DC blocking capacitor to be placed again. The stub on the right was to be

shorted for this design. To implement a short at the higher frequencies but maintain

82

 

  
 
 

   
    
   

//./, .

,I/ .I 11

/
A

'- 7/;

«111'

 

/ [Viv/y v’ 717 j
14(1,’/1 4'

 

 

 

 

 

 

 

 

 

 

 

Figure 5.12. Layout of AT-41486 Ampliﬁer on Microstrip

the DC bias level, two grounded microstrip pads were placed on the ends of the stubs
where capacitors could be placed to act as a short. Finally, the devices which look
like a bow tie are actually broadbanded stubs [1, 8]. These are used in conjunction
with high impedance quarter-wave transformers. These elements together create a
microstrip version of an inductor, which at high frequencies will block RF energy
from getting onto DC supply lines.

The results for the ampliﬁer of Figure 5.12 are shown in Figures 5.13 through 5.15.
When comparing the results of these ﬁgures to the target values shown in Table 5.2,

it is easily seen that the values are off by a substantial margin. This is likely due

83

Magnitude (dB)

 

 

 

 

 

 

 

 

 

Figure 5.13

 

Frequency (GHz)

. S“ and SWR,n of Compensated AT-41486 Ampliﬁer

84

 

 

10

Magnitude (dB)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

-5
0.5 3.5
Frequency (GHz)
200 I I I I l I I
100
g o
a.
-100
_200 l I l 1 l L l
0 0.5 1 1.5 2 2.5 3 3.5
Frequency (GHz)

Figure 5.14. 321 of Compensated AT-41486 Ampliﬁer

85

 

 

 

 

 

to the S—parameters used for the design being incorrect.

Under a closer examination, this model of transistor themselves seemed to have
issues that would make them unreliable to achieve a reproducible design. First, the
datasheet for this transistor had a range for the current gain, 6 or h fe, which varied
between 30 and 270 with the average value being about 150. This broad range of
possible current gains would deﬁnitely have an affect on the $21 of the transistor
making the Sn labeled on the datasheet as an approximation at best. An attempt
was made to measure the 6 of each transistor by connecting them to a curve tracer.
Doing this led to an another interesting ﬁnding. In the ideal case, the curve tracer
will give a plot of collector current vs. the collector-emitter voltage for ﬁxed values
for the base current. When connecting the transistor to the curve tracer, the distinct
lines formed for different base currents were not always present. In a few cases, these
lines began to oscillate as the collector-emitter voltage increased. Other transistors
had these constant base current line converge to a point that almost resembled a
diode curve.

The ampliﬁer that is shown in Figure 5.12 had a transistor which did not exhibit
any of these anomalies that were discussed above, but the design is still way off. The
best possible explanation for this is that the S—parameters were not actually measured
for that transistor. Since ,8 had such a large variation between each transistor, there
is no conceivable way that those S parameters were correct. Even if the S-parameters
were correct, the transistor data sheet did not explain where the phase plane for
the measurement of the S—parameters was located. At lower frequencies, one would
expect that the actual location of the phase plane would be less of a factor in the
design, but at higher frequencies, even 1mm could be a substantial enough change in

phase that the matching network could be off margin.

86

 

Magnitude (dB)

 

 

 

 

 

 

 

0 5 1 1 5 2 2.5 3 3 5 4
Frequency (GHz)
5 r
4 _ ..... .
m i
3 3 _ ...... .
w 2
2 ._ .....
1 l l l l l I I

 

 

 

 

 

 

0.5 1 1.5 2 2.5 3 3.5 4
Frequency (GHz)

Figure 5.15. 522 and SWRout of Compensated AT-41486 Ampliﬁer

5.2.2 MPSH10 Design

The S-parameters for this ampliﬁer were measured as described above. The design for
both the MPSH10 and MPSH17 were chosen to be at 100MHz so that the measured
results would in no way be erroneous due to parasitics of the components used in the
design. Also, these designs required the use of hand-wound air core inductors. The
equation used in the fabrication of these inductors is found in [7] and is written here

for convenience

 

n : \/L(18: + 401) (5.1)

 

87

where

n = Number of turns
L = Inductance in microhenrys
d = Coil diameter in inches

l = Coil length in inches

with the assumption that the length of the coil is much greater than the diameter.
The coils that were used in these matching networks were wound to have 0.125”

diameter and were about 0.25” in length.

Biconjugate Match

Figure 5.16 shows the matching network that was used for a biconjugate match to
the circuit shown in ﬁgure 5.1. The capacitor values shown in Figure 5.16 were those
actually used in the measurements and are slightly different than those that were
computed to be necessary for the matching network. The difference between these

capacitor values was at most lpF for this design.

 

Inp‘“ MPSHIO j |

C. e t I l
40pF "cm 17pF outPllt
3411“ 112.3nH

 

 

 

 

 

Figure 5.16. Biconjugate Match Network for MPSH10 at 100MHz

88

S11

 

Magnitude (dB)

 

 

 

50 100 150 200 250 300 350 400
Frequency (MHz)

 

..........................

         

 

 

 

 

50 1 00 1 50 200 250 300 350 400

Frequency (MHz)

Figure 5.17. Sn of Biconjugate Matched MPSH10 Ampliﬁer

The results for this ampliﬁer are shown in Figures 5.17 through 5.19. Comparing
these results to Table 5.2 shows that the predicted does well versus the measured.
The actual measured gain is 16.85 dB compared to a predicted value of 18.2 dB. Also,
SWR,n is negligibly close to one, but SWRout measured is at 1.325 which most likely

accounts for the 1.15 dB difference in the gain.

89

 

 

 

 

 

 

 

 

 

 

 

20 I I I I I I I
8
E
o
'O
.3 . ;
c . .
a, , .
a . 4 .
2 . : ;
o 1 1 1 1 i i 1
50 100 150 200 250 300 350 400
Frequency (MHz)
200 I 7 I I I I I
100 - -
3
2 0 ~
0.
-100 k .
_200 1 1 1 1 1 1 1
O 50 100 150 200 250 300 350 400
Frequency (MHz)

Figure 5.18. 321 of Biconjugate Matched MPSH10 Ampliﬁer

Magnitude (dB)

822

 

I

 

A l l l

 

 

 

150

200 250 350
Frequency (MHz)

 

 

 

I I I I I

 

 

 

50

100

150

200 250
Frequency (MHz)

Figure 5.19. 522 of Biconjugate Matched MPSH10 Ampliﬁer

90

Mismatched Design

Figure 5.20 is the matching network that has been attached to the circuit in Figure
5.1. This design has been made to have parameters other than a biconjugate match to
show that the design parameters can be achieved. As was the case for the biconjugate

match, the capacitor values for the design were rounded to the closest standard value.

107nH

Input
[3 ii MPSH10 l ' ' ' | S

30pF Cll’CLllt Output
84nH 22pF ::

 

 

 

 

 

 

 

Figure 5.20. Mismatched Network for MPSH10 at 100MHz

The results for the other MPSH10 transistor are shown in Figures 5.21 through
5.23. In comparison to the values predicted in Table 5.2, the measured data is quite
close. The target gain for the ampliﬁer was 14.8 dB compared to a measured of 13.4
dB. Also, SWR,-,1 had a target value of 1.5 compared to a measured value of 1.241.
Finally, SWRout was designed to be 5.59 and the measured value is 5.4.

91

Magnitude (dB)

Phase

S11

 

 

 

 

 

 

    

 

 

 

   

 

 

 

 

 

 

 

 

 

 

 

 

50 I I I I I I 7
0 ..
~50 - _
_100 4 1 1 1 1 1 1
O 50 100 150 200 250 300 350 400
Frequency (MHz)
5 ,' .'
5 _. . _
m 4 1.. .........................
3
U) 3 _ ..................
2 .. ..........................................
1 1 1 1 1 1 ' 1
50 100 150 200 250 300 350 400
Frequency (MHz)
Figure 5.21. Sn of Mismatched MPSH10 Ampliﬁer
821
20 f ' r ' 7 1. r
g i
a 10 _. ' . .-
‘E . ,
g’ : . :
z 5 ... . . . . . ........ I .1
50 100 150 200 250 300 350 400
Frequency (MHz)
200 I I I I I I I
\ﬂ—I
100 - -
0 ~ .
-100 _. -
__200 1 1 1 1 1 1 1
0 50 100 150 200 250 300 350 400
Frequency (MHz)

Figure 5.22. 521 of Mismatched MPSH10 Ampliﬁer

92

Magnitude (dB)

822

 

 

 

 

 

 

 

 

 

 

 

 

 

 

5 I I I r I I I
O.
-5... .1
-10... .1
_15 1 1 1 1 1 1 1
0 50 100 150 200 250 300 350 400
Frequency(MHz)
5 ............................. _,
a: ;
34 M. .................. ..
‘0 ;
3... .......................................... -
2 ............... .
1 ' 1 1 1 1 1 ‘1
50 100 150 200 250 300 350 400
Frequency(MHz)

Figure 5.23. 322 of Mismatched MPSH10 Ampliﬁer

93

5.2.3 MPSH17 Design

Figure 5.24 is the matching network used for the circuit shown in Figure 5.2. As was
the case in all the other designs, the capacitor values were rounded to the nearest

standard values but the difference in the capacitor values was less than one.

 

Input I i MPSH17 | 1
C. .t I I
37.7pF 1‘ °‘” 22pF outP“t
61nH 85nH

 

 

 

 

 

Figure 5.24. Biconjugate Matched Network for MPSH17 at 100MHz

Figures 5.25 through 5.27 show the measured results of this ampliﬁer. As can
be seen, the reﬂection coefficient minimums (Sn and 5'22) do not occur at 100MHz
as desired. Attempts to tune the circuit by adjusting the spacing in the inductors’
windings were ineffective. The capacitor values were very close to standard values
so this was likely not the source of the error. Even though these minimums do not
occur at 100MHz, the values measured at 100MHz are still acceptable. The measured
gain was at 18.6 dB when compared to 19.2 dB for the predicted. The SWR,-n was

measured to be 1.46 whereas SWRout was measured to be 1.325.

94

Magnitude (dB)

 

 

 

 

 

50 100 150 200 250 300
Frequency (MHz)

 

 

 

 

 

 

 

 

 

1 1 1 1 1
50 100 150 200 250 300
Frequency (MHz)

Figure 5.25. 811 of Biconjugate Matched MPSH17 Ampliﬁer

Phase

Magnitude (dB)

 

 

 

 

 

50 1 00 1 50 200 250 300

 

 

 

 

 

 

Frequency (MHz)
200 I I I I I
100 - -
0 ..
-100 * 1
_200 1 1 J 1 1
O 50 100 150 200 250 300
Frequency (MHz)

Figure 5.26. 321 of Biconjugate Matched MPSH17 Ampliﬁer

95

$22

 

 

Magnitude (dB)
I
8

 

 

 

 

 

 

 

 

 

 

 

-20
_30 1 1 1 1 1
50 100 150 200 250 300
Frequency (MHz)
6 r
5 .......
1: 4 r
3
(I) 3 .......
2 ... . . . . .
1 1 1 1 1 1
50 100 150 200 250 300
Frequency (MHz)

Figure 5.27. 822 of Biconjugate Matched MPSH17 Ampliﬁer

5.2.4 Summary

Table 5.3 shows a summary of the values attained from the above measurements. for
the most part, the values agree well with the predicted values found in Table 5.2.
For all of the ampliﬁers, other than the AT41486, placing adjustable capacitors will
allow the ability to better tune the circuit to the desired values. Surprisingly, the
inductors made from equation 5.1 were quite accurate and could be slightly adjusted
by changing the spacing between the coils for tuning purposes.

Noting that the ampliﬁer based on the AT—41486 had measured data that was far
away from any of the predicted data shows the importance of accurate S-parameter

measurements. All of the other ampliﬁers had their own S-parameter measurement

96

 

 

 

 

 

 

Transistor AT-41486 MPSH10 MPSH10 MPSH17
Mismatch Biconj ugate Mismatch Biconj ugate
Center Frequency 2GHz 100MHz 100MHz 100MHz
GT (dB) 6.76 16.85 13.4 18.6
SWR,n 1.49 1.07 1.24 1.46
SWRout 3.21 1.325 5.4 1.325

 

Ta

 

data used in the designing process.

97

ble 5.3. Measured Values ‘ummary

 

CHAPTER 6

Conclusion

As can be seen from the ampliﬁers in Chapter ﬁve, the design tool, “AmpDesigner”
will give accurate results for those ampliﬁers which have accurate S—parameter mea-
surements. This program aids in the design process by giving the user all of the
pertinent ampliﬁer information that may be of concern in the design process. Once
the design values for F5 and FL have been obtained from the program, then the user
just needs to concentrate on the physical layout of the ampliﬁer.

However, this program does have some limitations. It will not give the frequency
dependant characteristics, such as bandwidth, of the ampliﬁer with the matching net-
works connected. To implement this type of analysis, a few thing would be needed.
First, the entire spectrum of S-parameters for the transistor without matching net-
works would be required. This is likely not a problem if the S—parameters are measured
by the designer, but if the data is extracted from a data sheet, then this may be a
problem since the the values listed tend to be sparse. Second, the speciﬁc network
that is connected needs to have its frequency dependant characteristics modeled. This
is so the values for F3 and FL can be modeled over the frequency range of interest.
Once this is accomplished, then the ampliﬁer can be characterized over frequency and
showing its true predicted performance.

Despite these drawbacks, “AmpDesigner” will give the user the ability to reliably

98

achieve the desired performance at the frequency of interest.

99

BIBLIOGRAPHY

100

BIBLIOGRAPHY

[1] HA. Atwater. The design of the radial stub: A useful microstrip circuit element.
Microwave Journal, November 1985.

[2] David K. Cheng. Field and Wave Electromagnetics. Addison-Wesley, second
edition, 1989.

[3] Guillermo Gonzalez. Microwave Transistor Ampliﬁers. Prentice Hall, second
edition, 1997.

[4] J .P. Humbert. Microwave measurements. In Bradford L. Smith and Michel-
Henri Carpentier, editors, The Microwave Engineering Handbook, volume 2. Van
Nostrand Reinhold, 1993.

[5] Norman S. Nise. Control Systems Engineering. John Wiley & Sons, Inc., third
edition, 2000.

[6] David Pozar. Microwave Engineering. John Wiley & Sons, Inc., third edition,
2005.

[7] Dana G. Reed, editor. ARRL Handbook. ARRL—The national association for
amateur radio, eighty-ﬁrst edition, 2004.

[8] BA. Syrett. A broad-band element for microstrip bias or tuning circuits. IEEE
Transactions on Microwave Theory and Techniques, MTT-28(8):925-927, Au-
gust 1980.

[9] Agilent Technologies. Up to 6 ghz low-noise silicon bipolar transistor, at—41486.
Technical report, Aglient, 1999.

[10] Agilent Technologies. Fundamentals of rf and microwave noise ﬁgure measure-
ments, application note 57-1. Technical report, Aglient, March 2004.

[11] Rodger Ziemer and William 'IIanter. Principles of Communications. John Wiley
& Sons, Inc., ﬁfth edition, 2002.

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