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微波電路講義
微波電路講義Microwave Circuits Notes
瞿大雄教授國立臺灣大學電機工程學系
微波電路講義
1. Class: EE5009(921U0560) Tue. 9:10-12:10, EE-II 144
2. Textbook: Microwave Engineering, 4th ed., D.M. Pozar, John Wiley &
Sons, 2011
3. Scopes: basic principles and design formula of passive and active
microwave linear circuits using transmission line theory in terms of V(z)
and I(z) (V+(z), V-(z), I+(z), I-(z)) representation and microwave network
analysis in terms of S-parameters
4. Contents:
Ch.2 Transmission line theory, 2.1, 2.3-2.7
Ch.3 Transmission lines and waveguides, 3.1, 3.5, 3.7, 3.8, 3.11
Ch.4 Microwave network analysis, 4.1-4.6
Ch.5 Impedance matching and tuning, 5.1-5.9
Ch.6 Microwave resonators, 6.1, 6.2, 6.5, 6.6
Ch.7 Power dividers and directional couplers, 7.1-7.3, 7.5-7.9
1
General Information
微波電路講義
Ch.8 Microwave filters, 8.3-8.8
Ch.9 Theory and design of ferrimagnetic components, 9.6
Ch.10 Noise and nonlinear distortion, 10.1-10.4
Ch.11 Active RF and microwave devices, 11.1-11.4
Ch.12 Microwave amplifier design, 12.1-12.5
Ch.13 Oscillators and mixers, 13.2, 13.5
Ch.14 Introduction to microwave systems, 14.1-14.6
5. Estimated time table: 48hrs
2
Date Notes Date Notes
1 9/13 ~2-9 10 11/15 university carnival
2 9/20 ~2-34 11 11/22 ~8-53
3 9/27 ~4-11, Quiz#1(ch.2) 12 11/29 ~10-16, Quiz#4(ch.7)
4 10/4 ~4-27, Quiz#2(ch.3) 13 12/6 APMC
5 10/11 ~5-5 14 12/13 ~11-11. Quiz#5(ch.8)
6 10/18 ~5-36, Quiz#3(ch.4) 15 12/20 ~11-38
7 10/25 ~6-20 16 12/27 ~12-36, Quiz#6(ch.10,11)
8 11/1 ~7-25 17 1/3 ~14-14
9 11/8 Midterm Exam (~ch.6) 18 1/10 Final Exam
微波電路講義
6. Grades: review quiz 15%, midterm exam. (Ch.2-Ch.6) 40%, final
exam. (Ch.7-Ch.14) 45%
7. Office hour: Tue. 14:00~16:00 @room 541, or [email protected]
8. Related readings:
3
(1) Foundations of interconnect and microstrip design, 3rd ed., T.C. Edwards, M.B. Steer,
John Wiely & Sons, 2000.
(2) Microwave engineering using microstrip circuits, Fooks and Zakarevicius, Prentice
Hall, 1990.
(3) RF/microwave circuit design for wireless applications, U.L. Rohde and D.P. Newkirk,
John Wiley & Sons, 2000.
(4) Microwave and rf design, 2nd ed., M. Steer, Scitech, 2013.
(5) Radio frequency and microwave electronics, M. M. Radmanesh, Prentice Hall, 2001.
(6) Microwave and RF engineering, R. Sorrentino and G. Bianchi, John Wiley & Sons,
2009.
*(7) Foundations for microwave engineering, R.E. Collin, McGraw-Hill, 2nd ed., 1992.
9. Notes (including solved problems and ADS examples) are available at
http://cc.ee.ntu.edu.tw/~thc/ or comm.ntu.edu.tw/faculty.htm
微波電路講義
Chapter 1 Introduction1. Definition
Microwave: designating or of that part of the electromagnetic
spectrum between the far infrared and some lower frequency limit:
commonly regarded as extending from 300,000 to 300 megahertz.
(Webster’s dictionary)
f: 300MHz - 300GHz l : 100cm - 0.1cm
electromagnetic spectrum (p.2, Fig.1.1)
2. Why use microwaves?
(1) Antenna gain is proportional to the electric size of the antenna.
f ↑, gain ↑(p.665, eq.(14.12))
miniature microwave system possible
(2) f ↑ available bandwidth ↑(p.671, Sec.14.2)
e.g., TV BW=6MHz
10% BW of VHF @60MHz for 1channel
1% BW of U-band @60GHz for 100 channels
1-1
max 2
4 AD
l
微波電路講義
(3) Line of sight propagation and not affected by cloud, fog,…
frequency reuse in satellite and terrestrial communications
(frequency division duplexing, FDD)
(4) Radar cross section (RCS) is proportional to the target electrical size.
frequency ↑, RCS ↑(p.696, Table 14.3)
radar application, see p.690
(5) Molecular, atomic and nuclear resonances occur at microwave
frequencies (p.703, Fig.14.29)
astronomy, medical diagnostics and treatment, remote sensing
and industrial heating applications
3. Biological effects and safety
non-ionized radiation thermal effect
IEEE standard C95.1-2005 (p.707, Fig. 14.32)
Excessive radiation may be dangerous to brain, eye, genital, stomach
organs,….. cataract, sterility, cancer,…..
http://www.fda.gov/Radiation-
EmittingProducts/RadiationEmittingProductsandProcedures/HomeBusinessandEntertainment/CellPhones/default.htm
1-2
(14.39)s
t
P
S
微波電路講義
Chapter 2 Transmission Line Theory
2.1 The lumped-element circuit model for a transmission line
transmission line or telegrapher equation,
traveling wave solution
2.3 The terminated lossless transmission line
Zin, G, VSWR
time-average power flow
2.4 The Smith chart
Zin-plot conformal mapped on G-plot
2.5 The quarter-wave transformer
frequency response, TDR
2.6 Generator and load mismatches
impedance match, conjugate match
2.7 Lossy transmission lines
low loss line, distortionless line
perturbation method 2-1
微波電路講義
R, L: conductor resistance, inductance/unit length (series elements)
G, C: dielectric conductance, capacitance/unit length (parallel elements)
KVL, KCL (2.1)
time-domain form of transmission line equation, or telegrapher equation
( , ) ( , )( , )
( , ) ( , )( , )
v z t i z tRi z t L
z t
i z t v z tGv z t C
z t
2-2
i(z, t)
+
-
2.1 The lumped-element circuit model for a transmission line
• Transmission line equation
TEM lines: coaxial line, parallel line and stripline
+
-v(z, t)
z
v(z, t)
i(z, t)
z
R z L z
G z C z
i(z+ z, t)
+
-v(z+ z, t)z
微波電路講義
time-harmonic form ejwt wave equation
• Traveling wave solution
)()()(
)()()(
zVjwCGdz
zdI
zIjwLRdz
zdV
2 22 2
2 2
( ) ( )( ) 0, ( ) 0
( )( )
propagation constant
d V z d I zV z I z
dz dz
R jwL G jwC j
( ) ( ) ( )
1( ) ( ) ( ) ( )
: characteristic impedance
z z
o o
z z z z
o o o o
o
o oo
o o
V z V z V z V e V e
I z I z I z I e I e V e V eZ
R jwL V V Z
G jwC I I
V¯(z)V(z)
z
I(z)
I+(z)
I¯(z)incident
wave
reflected
wave
2-3
( )
( )= ( ) ( ) ( ) ( )= ( )
1( )= ( ) ( )
z z z z z z
o o o o o o
z z z z
o o o o o
o
derivation
dV z G jwCV e V e R jwL I z I z V e V e V e V e
dz R jwL R jwL
R jwL R jwLV e V e I e I e I z I z Z
Z G jwC
V+(z)
( ) Re[ ( ) ], ( ) Re[ ( ) ]jwt jwt v z,t V z e i z,t I z e
微波電路講義2-4
time-domain solution
( ) cos( ) cos( )
( ) cos( ) cos( )
αz αz
o o o o
αz αz
o o o o
v z,t V e wt βz V V e wt βz V
i z,t I e wt βz I I e wt βz I
Discussion:
1. attenuation constant
(1 ) (0 )
(0 ) (0 )ln (Neper) 20log (dB),1Np 20log dB 8.68dB
(1 ) (1 )
V m V m e
V m V me
V m V m
)(zV
z
ze
0m 1m
2( ) (0) zP z P e
微波電路講義
3. characteristic impedance (wave impedance)
input impedance
4.
( ) ( )( ) ( )
( ) ( )
o oo
o o
V V z V V zZ
I I z I I z
( ) ( ) ( )( ) ( )
( ) ( ) ( )
z z
o oin z z
o o
V z V z V z V e V eZ z
I z I z I z I e I e
2. phase constant wavelength
phase velocity group velocity
2
p
p
w
v
wv
l
1
2
( )g
dv
dw
l
2-5
2*
* * 2
2*
* * 2
( ) ( ) ( )
( ) ( ) ( ) ( )
oz z z z zoo o o
o o
oz z z z zoo o o
o o
VVP z V z I z V e I e V e e e
Z Z
VVP z V z I z V e I e V e e e
Z Z
5. Transmission line equation can be derived from the Maxwell’s
equations (Sec. 2.2 on coaxial line example).
微波電路講義
2.3 The terminated lossless transmission line
standing wave solution
( )
( )
j z j z
o o
j z j z j z j zo oo o
o o
V z V e V e
V VI z I e I e e e
Z Z
-l 0 z
ZL
I(-l)
G(-l) GL
V(-l)+
-
Zo
( )inZ l
source matched to line
(Zg=Zo)
Gs=0
2-6
Zo,β
( ) ( ), or ( )inc in transP l P l P l
(0)reflP
Vg
( )reflP l
微波電路講義
2 2
2 2
0
( )
(1 ) 1( )( )
( ) (1 ) 1
1(0)(0)
(0) 1
( )
j l j l j l j l j l
o o o L Lin o o oj l j l j l j l j l
o o o L L
lL oL
in L o L
L L o
j l j l
Lin o j
derivation
V e V e V e e eV lZ l Z Z Z
I l V e V e V e e e
Z ZVZ Z Z
I Z Z
e eZ l Z
e
G G
G G
G G
G
G
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
cos sin tan
sin cos tan
j l j l j l j l j l j l
L o L o L oo ol j l j l j l j l j l j l j l
L L o L o L o
L o L oo o
L o o L
Z Z e Z Z e Z e e Z e eZ Z
e Z Z e Z Z e Z e e Z e e
Z l jZ l Z jZ lZ Z
jZ l Z l Z jZ l
G
2( )( )
( )
(0)
j lj l j l j lo
L Lj l
o
oL
o
V eV ll e e e
V l V e
V
V
G G G
G G
• Reflection coefficient (applications in measurement, radar, and
remote sensing)
• Input impedance (application in circuit design)
2-7
tan( )( )
( ) tan
L oin o
o L
Z jZ lV lZ l Z
I l Z jZ l
微波電路講義
• Time-average input or transmitted power flow
L
L
1 ΓVVSWR
V 1 Γ
max
min
2-8
2
2*1 1 1( ) ( , ) ( ) Re ( ) ( ) (1 )
2 2
constant time-average incident power time-average reflected power
(lossless line)
T o
in L inc refl transo
o
VP z v z t i z,t dt V z I z P P P
T Z
G
• Voltage standing wave ratio
2 2
2
max
min
( )
( ) (1 ) (1 )
( ) 1
122 ,2
(2 1) 2(2 / ) 21
o L
L
j V jj l j l j l j l j l j l
o o o L o L
j j l
o L
o L
L
o L
derivation
V l V e V e V e e V e e e e
V l V e e
V Vnl l l
n V V
G
G
G G
G
G lG
l G
微波電路講義2-9
0 0
* *
0
2 * * 2 * *
0
*
( )
1 1( ) ( , ) ( ) Re[ ( ) ]Re[ ( ) ]
1 1 1[ ( ) ( ) ] [ ( ) ( ) ]
2 2
1= [ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )]
4
1[ ( ) ( )
4
T Tjwt jwt
in
Tjwt jwt jwt jwt
Tj wt j wt
derivation
P z v z t i z,t dt V z e I z e dtT T
V z e V z e I z e I z e dtT
V z I z e V z I z e V z I z V z I z dtT
V z I zT
* *
0
*
* * * * *
* ** * * *
2 2 2
2
1( ) ( )] 2 Re[ ( ) ( )]
4
1Re[ ( ) ( )]
2
1 1Re{( )( ) } Re{ }
2 2
1 1 1Re{ } Re{ } Re
2 2 2
(1 )2 2 2
T
o o
o o o
L in
o o o
V z I z dt V z I z TT
V z I z
V V I I V I V I V I V I
V V V VV I V I V I V I P P
Z Z
V V VP
Z Z Z
G
,o L o
c refl L
L oo
V Z ZP
Z ZV
G
Im
lossless line
微波電路講義
• Discussion
1. Zin G
2. Return loss
in
in
Z ( )1 ( )( ) , (z)
1 ( ) Z ( )
oin o
o
z ZzZ z Z
z z Z
G G
G
1 0 1 0
0 20
VSWR 1.22
Le.g., Γ .
RL dB dB dB
1
all incident power is reflected matched load
"no return loss" " return loss"
2-10
L
1
matched load 0 1
VSWR
VSWR
G
3.
4.L L L L
L
passive load 0 1, eg. . . 1, . . 1, matched load 0
active load 1
o c s c G G G G
G
20log ( ) 20log 10logo oL
o o
V PRL Γ dB
V P
Q:VSWR for a lossy line?
“N”
微波電路講義
5. Impedance match
Zin(z)=Zo no reflected wave G(z)=0, VSWR=1, RL= dB
Pin=Pin,max: maximum power delivered to the load
6. l=l/2, Zin(l)=ZL,
l=l/4, Zin(l)=Zo2/ZL impedance inverter or quarter-wave “transformer”
2
2
2
2 2 21:
22
2 2:1
1: , :1
,1/1
11 ,
/
L o
L o
Lin in L
R ZoL L LL o
inn
o L L oL L o
R ZL L o o o L
in Ln
L o L L L o
Rn R n R n R
n
ZR R RR ZR
Z R R Z VSWRR ZVSWR
R Z Z Z RR VSWR R
R Z R R R Z
G
G
1:n or n:1
Rin RL
2-11
Rin RL
Zo
l/4
n VSWR
微波電路講義
7. Mismatch loss (ML)
2-12
z
RL
ZoZo,β
o oI I
LI
LV
o oV V
2 2 2 2
2 2
2
22 2
2 2 2 2
2
@ 0,
,
(1 ) 1
1 1 1,
2 2 2
1 1
2 2
o o L
o LL
o o
o L o L
o o o o refl
inc refl L inc L
o o o inco
oL Lo o trans otrans inc P inc P
L L o L inc Lo
inc
z V V V
V VT
V V
V V T T
V V V V PP P P
Z Z Z PV
VV VZ Z P ZP T P T P T T
R R Z R P RV
P
G
G G
G G
2 2 2 2 2 21
( ) 10log 20log
( ) 10log 20log
, 1, 0 , 2, 0,
0, 1, 0 , 0, 0,
orefl trans inc L inc P inc L P L
L
transP
inc
refl
L
inc
L L P
L L P
ZP P P P T P T T
R
PML mismatch loss T
P
PRL return loss
P
R RL dB T T ML dB
R RL dB T T ML
G G G
G
G
G
2
2
2
1 3 50 9 3150 , 50 , , 6 , , , 1.25
2 2 150 4 4
50 1 1 50 1 3, 50 , , 6 , , , 1.25
3 2 2 50 / 3 4 4
50 , 50 , 0, , 1, 1, 0
L o L P
L o L P
L o L P
dB
R Z RL dB T T ML dB
R Z RL dB T T ML dB
R Z RL dB T T ML dB
G
G
G
Vg
微波電路講義2-13
Zo,1 G Zo,2
Te.g. transmission line junction
(p.62, Fig.2.9)
1 2 ,1 ,1 ,2
,1 ,2
,1 ,2 ,1
,1 ,1
2 2 2 2
2 2,1 ,1 ,1 ,1
2
,1 ,1 ,1,1
2
,2 ,1
,2 ,2
@ ,
, (1 ) 1
1 1 1, ,
2 2 2
1
2
o o o
o o
o o o
o o
o o o o refl
inc refl inc
o o o inco
o oo
trans
o o
junction V V V V V
V VT V V V T T
V V
V V V V PP P P
Z Z Z PV
V VZP
Z Z
G G G
G G
2 2
2 2 2 2,2 ,1 ,1 ,1
2
,1 ,2 ,2,1
2 2 2 2 2 2,1
,2
2
,1 ,2
,1 ,2
1,
2
1
50 , 50 , 0, , 1, 1, 0
150 , 150 , ,
2
o o otransinc P inc P
o o inc oo
o
inc refl trans inc inc P inc P
o
o o P
o o
V Z ZPT P T P T T
Z Z P ZV
ZP P P P P T P T T
Z
Z Z RL dB T T ML dB
Z Z
G G G
G
G 23 3
6 , , , 1.252 4
PRL dB T T ML dB
微波電路講義
8. Open-circuited line
2-14
, Zo O.C. GL= 1
l l l l
l
tan
1,
o o
o oin
r
p
l lC
wZ Z c
Z ZZ j
j l l
wwj
wC v c
VVo
2/
jVIZoo
2/
ZXoin
/
l z
0
2 cos
2sin
tan
o
o
o
in in
V V l
j VI l
Z
ZZ jX
j l
微波電路講義
9. Short-circuited line
2-15
, Zo S.C. GL= -1
l l l l
l
tan
,
o ro
in o
r
o
p
Z lZ lL
w c
Z jwL jZ l
wwjZ l
v c
VjVo
2/
VIZoo
2/
ZXoin
/
z-l2 sin
2cos
tan
o
o
o
in o in
V j V l
VI l
Z
Z jZ l jX
0
0
open circuited transmission line
( ) 2 cos 0
, 2 cos , or cos2
2 2( ) sin sin , or sin
2
1 , or
tan
j z j z
o o
o
o
j z j zo o o o
o o o o
inin in
V V e e V z z
Vz l V V l l
V
V j V j V IZI e e z l l
Z Z Z -j V
Z XZ jX
j l Z
tan
similarly, for short circuited transmission line
( ) 2 sin 2 sin , or sin2
2 2( ) cos cos , or cos
2
tan , or
j z j z
o o o
o
j z j zo o o o
o o o o
inin o in
l
VV V e e j V z j V l l
j V
V V V IZI e e z l l
Z Z Z V
XZ jZ l jX
Z
0
tan l
微波電路講義2-16
(derivation of the y-axis scales in Figs.2-6 and 2-8)
微波電路講義2-17
10. Standing wave expression
An ADS example of a terminated transmission line is given in Ex1 in
Ch2_prj.
2 2
2
22 2
2
2 2
m
( ) (1 ) (1 )
( ) 1
[1 cos( 2 )] sin ( 2 )
(1 ) 2 [1 cos( 2 )]
(1 ) 4 sin ( )2
( )
o L
L
j V jj z j z j z j z j z j z
o o o L o L
j j z
o L
o L L L L
o L L L
Lo L L
V z V e V e V e e V e e e
V z V e
V z z
V z
V z
VV z
G
G
G G
G
G G G G
G G G
G G G
ax
min
(1 ), 1
2 ( )(1 )2
22 2 ,
(2 1) 2 2(2 / ) 2
o LL
o L
L L
nVz
nV V
nz l l
n
G G
G
lG G
l
eg. an open circuited transmission line
( ) 2 cos ( , ) Re[ ] 2 cos cos( )
2 22( ) sin ( , ) Re[ ] sin sin( ) sin cos( )
2
(
j l j l jwt
o o o o o
o ojwtoo o
o o o
V l V e V e V l v l t Ve V l wt V
V Vj VI l l i l t Ie l wt V l wt V
Z Z Z
1) ( ) : most difficult
( ) ( , ) and ( , ) ( ) and ( ) ( )
( )
(2) ( ) : less difficult
( ) ( , ) into ( , ) and ( , ) ( ) and ( ) ( )
(
in
in
in
in
Z l
V lmeasure v l t i l t V l I l Z l
I l
l
V lseparate v l t v l t v l t measure V l V l l
V
G
G 2
,maxmax
,max ,min
min ,min
1)
(3) VSWR: least difficult
, , ( ) 2 cos
j l
amp
amp amp amp o
amp
el
vVmeasure v v v l V l VSWR
V v
微波電路講義2-18
11. Measurable quantities
微波電路講義
2.4 The Smith chart
Map rectangular plot of onto the polar plot of jxrZZz o
oo
ir
j je 180180 ,1 ),( GGGGGG G
2-19
GL
LZ( )Z l
G(-l)
S.C. O.C.inductive
x=1 r=1
j
-j
G
G
L
G L
x= -1
2 2
2 2
2 2 2 2 2 2
( )
1 1 21 1
1 1 1 (1 )
1 1 1( ) ( ) , ( 1) ( ) ( )
1 1
o r i r i i
o r i r i
r i r i
derivation
Z Z j jzz r jx
Z Z z j
r
r r x x
G G G G G GG
G G G G G
G G G G
capacitive
, γoZ
微波電路講義2-20
微波電路講義
Discussion2( )
max
1. ( ) ( ) ( )
2. r-circle x-circle
3.
j lLl eL L LZ z l z l Z l
VSWR r
G G G G
2-21
max maxmaxmax
min min
2 2
2 2
( )
1 1,
1 1
( ) (1 ) (1 )
( ) 1 , ( ) 1
0
2
o L
L L
L L
L o o L
j V jj l j l j l j l j l j l
o o o L o L
oj jj l j l
o L L
o
L
derivation
V VRVSWR r
V Z I Z
V l V e V e V e e V e e e e
VV l V e I l e
Z
V
l
G
G G
G G
G G
G G
G G
G
max min
min max
(1 ), (1 )
(1 ), (1 )
o
o L L
o
o
o L L
o
VV I
Z
VV V I
Z
G G
G G
4. 2 360ol λ G
Vmax
Imin
Rmax
Vmin
Imax
Rmin
2 l
G
ZL
微波電路講義2-22
6. min, LVSWR l G
L
min L
1 1
1 1
180 2
L
L
o
VSWRVSWR
VWSR
l
G G
G
G
minl
L
min V
min 2 l
LG
min V
max V
V
1
1801 180
180
5. ( ) 180
( )
1 1 1 1 1 ', , '
1 1 1 '1
o
o
o
o
o
o
jj
j
Yz y YZ
Y
derivation
ez z e
z e
G
G G G G G G
G G GG
z
z-1=y
max V
微波電路講義2-23
L
min V
min 2 l
LG
7. Ex. 2.4 VSWR=1.5, find ΓL
L
L min
2 42
1 1 1.5 10.2
1 1 1.5 1
2 180180 2 180 2 1.48 360
4
180 266.4 360 86.4
L
L
oo o o
o o o o
cm cm
VSWRVSWR
VWSR
l
l l
G G
G
G
V
0.2 2.2 4.2
S.C.
load
0.72 2.72 4.72
unknown
load
V
maxV
minV
lmin=4.2-2.72=1.48cm=0.37λ
微波電路講義2-24
2.5 The quarter-wave transformer
Discussion
1. Ex.2.5, frequency response (p.73, Fig.2.17)
2. Analysis from multiple-reflection viewpoint (slide 2-28)
2
1matched condition: ( )in o o
L
ZZ f Z
R
1Z
inZ
oZ
4
l
'L
rL
r
4/l
LR
1
1
1
2
1 1 1
' ' '
' '1
LL L L L L
L L
o o o L o L
Rr r r R r Z
Z
R r Z Z Z
Z Z Z r Z R
微波電路講義2-25
2.6 Generator and load mismatches
*
2
2 2
1Re( )
2
1
2 ( ) ( )
in in in
ing
in g in g
P V I
RV
R R X X
Discussion
1. Impedance match
load matched to the line oinLoL ZZZZ G )0(
22
2
)(2
1
ggo
ogin
XRZ
ZVP
ZLGL
Zo,
,in inZ P
Zg
Vg
Gg
Iin
+
Vin
-Gin
( 0)g in inZ Z G
2
2 2
1
2 4( )
g
in g
g g
RP V
R X
source matched to the loaded line
,L LZ P lossless line power delivered to the load
L inP P
微波電路講義2-26
*2*
* *
22 2
2 2 2 2
2
( )
1 1 1 1Re( ) Re( ) Re( )
2 2 2
1 1Re( )
2 2 ( ) ( )
( )
1,
2 (
inin in in in in
in in
in in ing g
in g in ginin g
L o
inin o in g
o g
derivation
time average input power
VP V I V V
Z Z
Z Z RV V
R R X XZZ Z
load matched to the line Z Z
RZ Z P V
Z R
2 2
2 2
2 2 2 2
)
( )
1 1
2 ( ) ( ) 2 4( )
g
g in
g g
in g g
g g g g g g
X
source matched to the loaded line Z Z
R RP V V
R R X X R X
微波電路講義2-27
2. Conjugate match
*
(given a fixed )maximum power transfer from source
( 0, 0)
( 0, 0)
g
in g g in
in in
in in
Z
Z Z
P P
R X
G G
2
,max
*
1 1
2 4
( ) : maximum available power from sourcein g
in g
g
avs inc in Z Z
P VR
P P P
Rg Xg Xin=-Xg
Rin =Rg
Gin 0 Gg 0
Rg
Rin
=Rg
G= 0
Vg
,in inZ P
(12.9)
微波電路講義2-28
3. Reflection coefficient (from multiple-reflection viewpoint)
0 z
ZLGL
Zg
inZ
Gg Gin
1
0
1 1
02 1
0
2 2
03
0
1
1
o g
g
g
g
LL
L
Z Z
Z Z
T
Z Z
Z Z
T
Z Z
Z Z
G
G
G G
G
G G
lj
lj
lj
lj
lj
ljnljlj
ljljljljljlj
e
e
e
e
e
eTTeTeT
TeeeTTeeTTeT
2
31
2
31
2
31
2
1311
2
32
2
231132
2
23
2
11
23
2
23
2
23
2
123
2
23
2
123
2
11in
11
)1()1(
1)(
...Γ
GG
GG
GG
GGGG
GG
GGGGGG
GGGGGGGGGG
G1 Γ2
T2 T1
G3
-l
Zo,
outZ
微波電路講義2-29
2
2
1 3in 2
21 3
2
Γ1
1
( )( ) ( )( )
( )( ) ( )( )
cos sin sin cos
o g j lL o
j lo g L o
j lo g j lL o
o g L o
j l j l
o g L o o g L o
j l j l
o g L o o g L o
o L o g L g o
Z Z Z Ze
Z Z Z Ze
Z Z Z Zee
Z Z Z Z
Z Z Z Z e Z Z Z Z e
Z Z Z Z e Z Z Z Z e
Z Z l Z j l Z Z j l Z Z l
G G
G G
2
1/ cos
cos sin sin cos 1/ cos
( tan ) ( tan )
( tan ) ( tan )
tan
tan t,
tan
tan
o L o g L g o
o L o g o L
o L o g o L
L oo g
in go L L oin o
L o in go g
o L
l
Z Z l Z j l Z Z j l Z Z l l
Z Z jZ l Z Z jZ l
Z Z jZ l Z Z jZ l
Z jZ lZ Z
Z ZZ jZ l Z jZZ Z
Z jZ l Z ZZ Z
Z jZ l
L
an
tan
tanΓ ,
tan
o L
g oL outout o
L out o g
l
Z jZ l
Z jZ lZ ZZ Z
Z Z Z jZ l
4. Impedance match (source is matched to the loaded line)
Zin(z)=Zg(z): no reflected wave, Γin(z)=0, VSWR=1, RL= dB
微波電路講義2-30
2.7 Lossy transmission lines
• low-loss line, R<<wL, G<<wC
• distortionless line RC=LG
1( ), ,
2o o
o
R LGZ w LC Z
Z C
C
LZ
v
lt
LCvLCw
L
CR
o
p
p
constant: ,constant:
1,constant:
• perturbation method
low-loss line (assume Γ(z)=0)
2( ) power loss per unit length: 2 ( )
( ) ( 0)
2 ( ) 2
z
o l
l l
o
PP z P e P P z
z
P z P z
P z P
微波電路講義2-31
Discussion
1. (Ex. 2.6, 2.7 for a coaxial line, p.721 Appendix J)dc
conductor loss current flow inside conductor
in the conductor
2 2
c
o
s oc
o o
ΔH ΔL ΔZ
R dZL
Z Z dl
2. Wheeler incremental inductance rule
a coaxial line
1 1 1 "( ) ( )
2 22 ln
sc d
RC L wR G
bL C a b
a
微波電路講義2-32
Solved Problems
Prob. 2.16
calculate PL
22
2
2
2
2
0 0
0
( / 2)(1) (1 ), , ( )
tan(2) Re( ),
tan
(3) Re( ), (1 ), ( )2
for a lossy line: (
g L oL inc L inc L g o
o L o
g L oL in in in in o
g in o L
gLL L L L L L g o
L
l l
in L
V Z ZP P P Z Z
Z Z Z
V Z jZ lP I R Z Z Z
Z Z Z jZ l
VVP I R Z V V V Z Z
Z
V V e e
G G
G
G(3)
0
tanh) ,
tanh
in L og in o L L
in g o L
Z Z Z lV Z Z V V P
Z Z Z Z l
, 15 , 75 , 75 , 60 40 , 0.7g rms g o LV V Z Z Z j l l
ZLGL
Zo,
Zg
Vg,rms
+
Vin
-
Zin
+
VL
-
Pinc PL
Iin IL
微波電路講義2-33
Prob. 2.18 Calculate Pinc, Prefl, Ptrans
50Ω λ/2
10V
50Ω 75Ω
50Ω
10V 75Ω
2
2 2
50
22
50 50
1 10 1 1010 0.4 , ( ) 50 0.25
2 50 75 2 50 50
1 10 1 10( ) 50 0.16 , ( ) 75 0.24
2 50 75 2 50 75
75 500.25 0.01
75 50
, ( )
g inc
trans in
refl inc
inc refl trans g trans in inc refl g inc i
P W P W
P W P P W
P P W
P P P P P P P P P P P P P
G
50*
For conjugate match: *, ( , impedance match)
0.25 ( 0.24 ) 0.25 , 0.5 , 0in g
n
in g in g
avs in g reflZ Z
Z Z R R
P P W W P W P W P W
I
Pinc
Ptrans (Pin)Prefl
Pg
P50
Pg
P50
Ptrans (Pin)Pinc
Prefl
微波電路講義2-34
Prob. 2.27 Find b/a and Zo for a coaxial line to have minimal αc
1 1( )
2 ln /
sc
R
b a a b
2
2
1 1 1 1 1 1[ ( ) ( ) ( )] 0
2 ln / ln /
1 1( ) ln ,1 ln 3.59
ln 77 , for 12
c s
o r
R
a a b a a b b a a
b b b b ba
a b a a a a a
bZ
a
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