lectures microwave engineering
TRANSCRIPT
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Microwave Circuits andSystems
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Objectives
Develop understanding of fundamentals of RF/microwave circuits
Analyze and design microwave circuits
Learn transmission line, scattering parameters, the Smith chart,
passive and active devices Simulate microwave circuits using popular EDA tools:
Advanced Design System (ADS), CST (Computer SimulationTechnology)
Use Vector Network Analyzer for S parameter measurement
Study and Analyze Microwave communication systems
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Aim
After this part the participants should be able to:
• Apply electromagnetic theory to calculations regarding waveguides andtransmission lines• Describe, analyze and design simple microwave circuits and devices eg
matching circuits, couplers, antennas and amplifiers.• Describe and coarsely design common systems such as radar andmicrowave transmission links.• Describe common devices such as microwave vacuum tubes,high-speedtransistors.
• Handle microwave equipment and be able to make measurements(VNA).
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Textbook
Pozar
David M Pozar, Microwave Engineering- 2nd Ed.,John Wiley , 1998
ReferencesReference materials such as component datasheet, FCCregulations, wireless standards, papers, useful links, etc.:
in class
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Analysis
L j R ),( t z I
dz
dI z I
),( t zV C jG dz
dV zV
zdt
dI L z RI z
dz
dV
dt
dI L RI
dz
dV
zdt
dV C zGV z
dz
dI
dt
dV C GV
dz
dI
From Kirchoff Voltage Law Kirchoff current law
zdt
dV C zGV z
dz
dI I I
z
dt
dI L z RI z
dz
dV V V
(a) (b)
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Analysis
Let’s V=Voe jt , I = Ioe jt
Therefore
V jdt
dV
I jdt
dI
then
I L j R
dz
dV V C jG
dz
dI
1
2
Differentiate with respect to z
dz
dI L j R
dz
V d
2
2
V C jG L j Rdz
V d
2
2
V dz
V d 2
2
2
dz
dV C jG
dz
I d
2
2
I C jG L j Rdz
I d
2
2
I dz
I d 2
2
2
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Analysis
The solution of V and I can be written in the form of
where
C jG
L j R Z o
Let say at z=0 , V=VL , I=IL and Z=ZL
Therefore
and L
L
L Z
I
V
5 6
3 4
C jG L j R j and
z z Be AeV
B AV L 0 Z
B A I L
0 Z
Be Ae I
z z
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Analysis
Solve simultaneous equations ( 5 ) and (6)
Inserting in equations ( 3) and (4) we have
22)(
z z
o L
z z
L
ee Z I
eeV zV
22)(
z z
o
L z z
L
ee
Z
V ee I z I
2
o L LZ I V
B
2
o L L Z I V A
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Analysis
2)cosh(
z z ee z
2)sinh(
z z ee z
We have
)sinh()cosh()( z Z I zV zV o L L
)sinh()cosh()( z Z
V z I z I
o
L L
)sinh()cosh(
)sinh()cosh(
)(
)()(
z Z
V z I
z Z I zV
z I
zV z Z
o
L L
o L L
and
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Analysis
)sinh()cosh(
)sinh()cosh()(
z Z z Z
z Z z Z Z z Z
Lo
o Lo
)tanh(
)tanh()(
z Z Z
z Z Z Z z Z
Lo
o Lo
Or further reduce
or
For lossless transmission line , = j since 0
)tan(
)tan()(
z jZ Z
z jZ Z Z z Z
Lo
o Lo
)cos()cosh( z z j
)sin()sinh( z j z j
C
L Z o
C jG
L j R Z o
Lossy line
Lossless line
C jG L j R
LC
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II-Loaded Transmission Line
Example: Antenna is a load (Feeder to antenna)
Load impedance: the input impedance of the antenna
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Standing wave
)()( jBd d j eeV d V
)2
cos()sin(2),( t d V t d v
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Voltage:
Current
Impedance
)sin(2)( d jV d V
)cos(2
)(0
d Z
V d I
)tan()( 0 d jZ d Z in
Special terminal conditions
Input impedance of short circuit transmission line
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What is about open circuit transmission line?
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L L
Lin
Z Z
l jZ Z l jZ Z Z
2
0
0
0
)4tan()4tan()4 / (
Quarter-wave transmission line
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)1( ininin V V )1(0
inin
in Z V I
}Re{2
1 *in
inin
I V P
)1(2
1 2
0
2
in
in
in Z
V P
)1(1
1
8
1 2
2
2
0
2
in
inS
SG
in Z
V P
Power considerations
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Two special cases:
Load and sourceMatched line
Mismatch at source,But match at load
How to measure power?
00 S
00
0
2
8
1
Z
V P
G
in
2
0
2
18
1S
G
in Z
V P
mW
mW PdBmP
1
][log10][
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Return and insertion losses
Return loss:
Insertion loss:
][log20log10)log(102
dBP
P RL inin
t
r
][1log10)log(10)log(102
dB
P
PP
P
P IL in
i
r t
i
t
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III- The Smith chart
What for ?When ?
How to understand ?How to determine microwave transmission parameters ????
Note: Please print the complete Smith chart .
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Normalized impedance
ir
ir inin
j
j
d
d jxr z Z d Z
1
1
)(1
)(1 / )( 0
ir
d j j
jeed L
2
0)(
Real part of normalizedimpedance
Imaginary part of normalizedimpedance
22
22
)1(
1
ir
ir r
22)1(
2
ir
i x
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Inversion of complex reflection coefficient(constant normalized resistance)
222
)1
1()1( r r
r ir
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Inversion of complex reflection coefficient(constant normalized resistance)
222
)
1
()
1
()1( x xir
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Generic Smith Chart computation
• Normalize load impedance
• Find reflection coefficient
• Rotate reflection coefficient
• Record normalized input impedance
• De-normalize input impedance
L L z Z
0 L z
)()( d Z d z inin
)(d zin
)(0
d
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Admittance Transformation(Smith Chart)
• Impedance represetation in Smith Chart
• Admittance representation in Smith Chart
)(1
)(1
d
d
jxr zin
)(1
)(1
)(1
)(11
0 d e
d e
d
d
zY
Y y
j
j
in
inin
180 degreePhase shift
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Transformation2
1
2
111 j y j z inin
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Alternative: re- interpretation
Instead of rotating the reflection coefficient about 180 degree, we keep the
location fixed and rotate the entire Smith Chart by 180 degree.
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Re-interpretation leads to ZY-Smith Chart
The Smith Chart inits original form iskept for impedance
display,but a second SmithChart is rotated by180 degree forAdmittance display.
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IV- Some microwave transmission lines
Two wirecable Coaxial
cable
Microstripeline
Rectangularwaveguide
Circularwaveguide
Stripline
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Parallel wire cable
d a for ad or ad L / ln2 / cosh 1
d a for
ad or
ad C
/ ln2 / cosh 1
ad Z o 2 / cosh1 1
Where a = radius of conductord = separation between conductors
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Coaxial cable
abC
/ ln2
ab L / ln2
ab Z o / ln
2
1
Where a = radius of inner conductorb = radius of outer conductorc = 3 x 108 m/s
r
cc
ck f 2
ba
k c
2
a
b
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Micro strip
whe r
t
t=thickness of conductor
Substrate
Conducted strip
Ground
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Characteristic impedance of Micro-strip line
hwwh Z hwFor eeeff
o / 25.0 / 8ln601 /
444.1 / ln667.0393.1 /
3771 /
hwhw Z hwFor
eeeff o
25.0 / 104.0 / 121
2
1
2
1hwwh ee
r r eff
5.0
/ 1212
1
2
1
e
r r
eff wh
1
2ln
t
ht ww e
e
t hhe 2Where
w=width of striph=height andt=thickness
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Micro-strip width
r r
r r o Z A
11.023.011
21
60
2 / 1
r o Z
B
260
2)2exp(
)exp(8 /
A
AhW
r r
r Blb B BhW
61.039.01
2
112ln1
2 /
For A>1.52
For A<1.52
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Simple Calculation
2
377
h
w Z
r
o
2377
/ or Z
hw
Approximation only
Using ADS (Advanced Design System): LinCal
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Micro-strip components
Capacitance
Inductance
Short/Open stub
Open stub
Transformer
Resonator
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Capacitance
Zo Zo Zoc
1c Z C
oc
12sin
2
c Z C
oc
smc
r
/ 103 8
1
For8
For
8
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Inductance
Zo Zo ZoL
1c
Z L oL
1
sinc
Z L oL
smc
r
/ 103
8
1
For8
For
8
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Short Stub
Zo
Z
Zo
Zo ZL
o L Z X / tan 1 eff
o
360
tano Lsc jZ X Z
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Open stub
Zo
Z
Zo
Zo ZL
oc Z X / cot 1 eff
o
360
cotococ jZ X Z
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Short-circuited /2 lossy line
n /2
Zin Zo
o Z R
o
o Z L
2
LC
o2
1
22
R
LQ o
2where
= series RLC resonant cct
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Open-circuited /2 lossy line
n /2
Zin Zo
22
RC Q o
= parallel RLC resonant cct
o Z R
oo Z C
2
C L
o2
1
2where
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Short-circuited /4 lossy line
/4
Zin Zo
o Z R
= parallel RLC resonant cct
oo Z C
4
C L
o2
1
24
RC Q o
2where
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Smith Chart: Homework+ Exercises
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Chapter III- MicrowaveNetwork Analysis
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Network representations
• For a linear system should be a set of (possibly
frequency dependent) parameters that relate “inputs” to“outputs” • n-port representations
- Impedance matrix (Z parameters) • “input”:currents • “output”:voltage
- Admittance matrix(Y parameters) • input:currents • output:voltage
- h-parameters matrix - Scattering matrix (S parameters) • input: incident traveling (voltage) wave • output: “reflected” traveling wave
- ABCD matrix
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What “parameters” should you use?
• it depends on the type of network…
•Example: view voltages as “inputs”, currents as “outputs” - units for Y parameters: (ohm)-1
If it’s a linear network then we have
N
j
NN Nj N N
jN jj j j
j
N j
N
j
V
V
V
V
Y Y Y Y
Y Y Y Y
Y Y Y
Y Y Y Y
I
I
I
I
2
1
21
21
22221
111211
2
1
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0
0
0
1
21
21
22221
111211
2
1
NN Nj N N
jN jj j j
j
N j
N
j
Y Y Y Y
Y Y Y Y
Y Y Y
Y Y Y Y
I
I
I
I
Y parameter representation
• how would you measure these?
- SHORT circuit ALL ports but port 1, apply “unit” voltage toport 1, measure all currents- Repeat for other ports
+ source : second index+response : first index
11212111 ,...,,...,, N N j j Y I Y I Y I Y I
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Y parameter applications
• “shunt” connected two-port networks
total
total
total
total
V
V
Y Y
Y Y
Y Y
Y Y
I
I
2
1
2221
1211
2221
1211
2
1
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N
j
NN Nj N N
jN jj j j
j
N j
N
j
I
I
I I
Z Z Z Z
Z Z Z Z
Z Z Z Z Z Z Z
V
V
V V
2
1
21
21
22221
111211
2
1
Z parameters
• it depends on the type of network…
•Example: view current as “input”, voltages as outputs -Units for Z parameters: ohmsIf it’s a linear network then we have
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0
0
0
1
21
21
22221
111211
2
1
NN Nj N N
jN jj j j
j
N j
N
j
Z Z Z Z
Z Z Z Z
Z Z Z
Z Z Z Z
V
V
V
V
11212111 ,...,,...,, N N j j Z V Z V Z V Z V
Z parameter representation• how would you measure these?-OPEN circuit ALL ports but port 1, apply “unit” current port 1, measure
all voltages-Repeat for other ports• source : second index• response : first index
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total
total
total
total
I
I
Z Z
Z Z
Z Z
Z Z
V
V
2
1
2221
1211
2221
1211
2
1
Z parameter applications
• “series” connected two-port networks
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Hybrid (H) parameters
• input : i1 , v2
• onput : v1 , i2 • units:- h1 : ohms- hi , hf : dimensionless- ho : mhos
• measurement:
- short “output” (port 2) + hi : input impedance at port 1+ hf : current gain from 1 to 2
- open at input (port 1)+ hr : reverse voltage transfer ratio+ ho : output admittance
2
1
02
1
V
I
hh
hh
I
V
f
r i
0
1
02
1 I
hh
hh
I
V
f
r i
202
1 0
V hh
hh
I
V
f
r i
1211,. I h I I hV f i
2211, V h I V hV
or
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•“Chain” (ABCD) parameters • input: V2 , - I2• output: V
1, - I
1
• units:• - A, D : dimensionless• - B :ohms• - C : mhos
• measurement :- short port 2
+ B : “transfer” impedance + D : reverse current gain
-open port 2
+ A : reverse voltage transfer ratio+ C : ”transfer” admittance
2
2
1
1
I
V
DC
B A
I
V
21
1 0
I DC
B A
I
V
0
2
1
1 V
DC
B A
I
V
2121 , DI I BI V
2121 , CV I AI V
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total
total
total
total
I
V
DC
B A
DC
B A
I
V
2
2
1
1
Application of ABCD parameters
• Cascaded two ports
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Scattering (S) parameters• view travelling waves into the network as “inputs”
• view travelling waves out of the network as “outputs”
• units for S parameters : dimensionless
If it’s a linear network then we have
N
j
NN Nj N N
jN jj j j
j
N j
N
j
a
a
a
a
SSSS
SSSS
SSS
SSSS
b
b
b
b
1
1
21
21
22221
111211
2
1
j j
j Z V a 0 /
j j
j Z V b 0 /
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0
0
0
1
21
21
22221
111211
2
1
NN Nj N N
jN jj j j
j
N j
N
j
SSSS
SSSS
SSS
SSSS
b
b
b
b
S parameter representation
• how would you measure these?
-MATCH ALL ports but port1, apply “unit ” input wave to port 1, measureall reflected waves- repeat for other ports
+source : second index
+response : first index
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2
2
2221
1211
1
1
a
b
T T
T T
b
a
2
2
1
1
abT T
ba
Chain S (transfer T) parameters
•Two port networks only•“mixed” representaion -Port two wavves are “inputs” -Port two waves are “outputs” • units for T parametters : dimensionless
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Scattering Parameters (S-Parameters)
In this course, we focus on S MatrixFor simple, we consider 2-portnetwork.
The behavior of the network can becompletely characterized by itsscattering parameters (S-parameters),or its scattering matrix, [S].
S matrix is used to representmicrowave devices, such as
amplifiers and circulators, and areeasily related to concepts of gain, lossand reflection.
11 12
21 22
S S
S S S
Scattering matrix
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The scattering parameters: ratios ofvoltage waves entering and leaving theports (If the same characteristic
impedance, Zo, at all ports in the networkare the same).
1 11 1 12 2.V S V S V
2 21 1 22 2.V S V S V
11 121 1
21 222 2
,S SV V
S SV V
In matrix form this is written
.V S V
2
1
11
1 0V
V S
V
1
1
12
2 0V
V S
V
1
2
22
2 0V
V S
V
2
2
21
1 0V
V S
V
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Properties:
The two-port network is reciprocal if thetransmission characteristics are thesame in both directions (i.e. S21 = S12).
IA network is reciprocal if it is equal to
its transpose. Stated mathematically,for a reciprocal network
,t
S S
11 12 11 21
21 22 12 22
.
t
S S S SS S S S
12 21S SCondition for Reciprocity:
1) Reciprocity
By inspection :If the network is symmetrical
Reciprocal
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A lossless network does not contain any
resistive elements and there is noattenuation of the signal.
In terms of scattering parameters, anetwork is lossless if
2) Lossless Networks
*
,
t
S S U
1 0
[ ] .0 1U
where [U ] is the unitary matrix
For a 2-port network:
2 2 * *
11 21 11 12 21 22*
2 2* *
12 11 22 21 12 22
1 0
0 1
t S S S S S S
S S S S S S S S
2 2
11 211S S If the network is reciprocal and lossless
* *
11 12 21 220S S S S
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S12 = S21 =0 (Reciprocal)
10
0
0
0
0
011
Z
Z
Z Z
Z Z S
in
inin
0
222 18011 inS
10
01S
Lossless condition check
12
21
2
11 Ss
Example
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