Chapter 5 Transmission Lines
5-1 Characteristics of Transmission Lines
Transmission line: It has two conductors carrying current to support an EM wave, which is TEM or quasi-TEM mode. For the TEM mode, HaZE nTEM
ˆ ,
EaZ
H n
ˆ
1
TEM
, and TEMZ .
The current and the EM wave have different characteristics. An EM wave
propagates into different dielectric media, the partial reflection and the partial
transmission will occur. And it obeys the following rules.
Snell’s law: 2
1
2
1
1
2
1
2
2
1
2
1
sin
sin
r
r
p
p
i
t
v
v
n
n
and θi=θr
The reflection coefficient: Γ=0
0
i
r
E
E and the transmission coefficient: τ=
0
0
i
t
E
E
)sin(
sincos2
coscos
cos2
cos/cos/
cos/2
)sin(
)sin(
coscos
coscos
cos/cos/
cos/cos/
21
1
12
2
21
21
12
12
it
ti
ti
i
it
t
it
it
ti
ti
it
it
nn
n
nn
nn
for perpendicular polarization (TE)
)cos()sin(
sincos2
cos/cos/
cos/2
coscos
cos2
)tan(
)tan(
cos/cos/
cos/cos/
coscos
coscos
21
1
12
2||
21
21
12
12||
titi
ti
ti
t
it
i
it
it
ti
ti
it
it
nn
n
nn
nn
for parallel polarization (TM)
In case of normal incidence,
12
2//
12
12//
2
, where η1=1
1
and η2=2
2
.
Equivalent-circuit model of transmission line section:
)/( mR , , , )/( mHL )/( mSG )/( mFC
Transmission line equations: In higher-frequency range, the transmission line model
is utilized to analyze EM power flow.
t
tzvCtzGv
z
tzitzzit
tziLtzRi
z
tzvtzzv
),(),(
),(),(
),(),(
),(),(
t
vCGv
z
it
iLRi
z
v
Set v(z,t)=Re[V(z)ejωt], i(z,t)=Re[I(z)ejωt]
)()(
)()(
zVCjGdz
dI
zILjRdz
dV
)()())(()(
)()())(()(
22
2
22
2
zIzICjGLjRdz
zId
zVzVCjGLjRdz
zVd
where γ=α+jβ= ))(( CjGLjR zzzz eIeIzIeVeVzV 0000 )(,)(
Characteristic impedance: Z0=CjG
LjR
CjG
LjR
I
V
I
V
0
0
0
0
Note:
1. International Standard Impedance of a Transmission Line is Z0=50Ω.
2. In transmission-line equivalent-circuit model, G≠1/R.
Eg. The following characteristics have been measured on a lossy transmission
line at 100 MHz: Z0=50Ω, α=0.01dB/m=1.15×10-3Np/m, β=0.8π(rad/m). Determine
R, L, G, and C for the line.
(Sol.) CjG
LjR8
8
102
10250
, 1.15×10-3+j0.8π= )102(50))(( 8CjGCjGLjR
)/(8050102
8.08
mpFC
, )/(103.210
50
15.1 53 mSG ,
)/(0575.02500 mGR , )/(2.02500 mFCL
Eg. A d-c generator of voltage and internal resistance is connected to a lossy
transmission line characterized by a resistance per unit length R and a
conductance per unit length G. (a) Write the governing voltage and current
transmission-line equations. (b) Find the general solutions for V(z) and I(z).
(Sol.) (a) RGCjGLjR ))((0
)()(
),()(
2
2
2
2
zRGIdz
zIdzRGV
dz
zVd
(b) zRGzRGzRGzRG eIeIzIeVeVzV 0000 )(,)(
Lossless line (R=G=0):
0,,1
,,0 00000 XC
LRjXR
C
LZ
LCvLCLCjj p
Low-loss line (R<<ωL, G<<ωC):
)](2
11[
1,),(
2
1)](
2
11(
0 C
G
L
R
jC
LZ
LCvLC
C
LG
L
CR
C
G
L
R
jLCjj p
Distortionless line (R/L=G/C):
C
LZ
LCvLC
L
CRLjR
L
Cj p 0,
1,,)(
Large-loss line (ωL<<R, ωC<< G):
21
21
)1()1())((G
Cj
R
LjRGjCjGLjR
)](2
1[G
C
R
LjRG
)(2
,G
RC
R
GLRG
, )(2
1
G
RC
R
GLv p
21
21
0 )1()1(
G
Cj
R
Lj
G
R
CjG
LjRZ
)](2
1[G
C
R
Lj
G
R
Eg. A generator with an open-circuit voltage vg(t)=10sin(8000πt) and internal
impedance Zg=40+j30(Ω) is connected to a 50Ω distortionless line. The line has a
resistance of 0.5Ω/m, and its lossy dielectric medium has a loss tangent of 0.18%.
The line is 50m long and is terminated in a matched load. Find the instantaneous
expressions for the voltage and current at an arbitrary location on the line.
(Sol.) 0.18%= GCC
G 21021.2
, Vg=10j
∵ Distortionless, ∴ mHLG
C
R
L/1011.1 2 ,
0Z
R
L
CR =0.01Np/m,
mradZ
L
L
CLLC /58.5
0
, γ=α+jβ=0.01+j5.58
53
5
0
00 j
ZZ
VZV
g
g
, , 00 V zjz ejeVzV )58.501.0(
0 )53
5()(
)6.7158.58000cos(3
105])(Re[),( 01.08000 zteezVtzV ztj
)6.7158.58000cos(102
1),(),( 01.0
0
teZ
tzVtzI z
Relationship between transmission-line parameters:
21
21
)1()1())((
jj
Cj
GLCjCjGLjR G/C=σ/ε
and LC=με
Two-wire line: saJI 2 , )2
(2
1 2
a
RIP s
c
cs f
aa
RR
1
)2
(2
Coaxial-cable line: sosi bJaJI 22 , )2
(2
1 2
a
RIP s
i , )2
(2
1 2
b
RIP s
o
)11
(2
1)
11(
2 ba
f
ba
RR
c
cs
Eg. It is desired to construct uniform transmission lines using polyethylene
(εr=2.25) as the dielectric medium. Assume negligible losses. (a) Find the distance
of separation for a 300Ω two-wire line, where the radius of the conducting wires
is 0.6mm; and (b) find the inner radius of the outer conductor for a 75Ω coaxial
line, where the radius of the center conductor is 0.6mm.
(Sol.) Two-wire line: )2/(cosh 1 aD
C
, )2
(cosh 1
a
DL
, a=0.6mm, ε=2.25ε0
mmDa
D
C
LZ 5.25
1036
125.2
104)2
(cosh300
9
71
0
Coaxial line: )/ln(
2
abC
, )ln(
2 a
bL
a=0.6mm,
9
7
0
1036
125.2
104
2
)ln(75
a
b
C
LZ b=3.91mm
Parallel–plate transmission line:
xz
yz
HxeE
xH
EyeEyE
ˆˆ
ˆˆ
0
0
0
, ,jj
At y=0 and y=d, Ex=Ey=0, Hy=0
At y=0, , ya ˆˆ n
zjzslsl
zjyslsl
eE
zHzJJHy
eEEDy
0
0
ˆˆˆ
ˆ
At y=d, , ya ˆˆ n
zjxsusu
zjysusu
eE
zHzJJHy
eEEDy
0
0
ˆˆˆ
ˆ
HjE
, xy Hj
dz
dE
d
x
d
y dyHjdyEdz
d00
)/()(])()[()()(
mHw
dLzLIjwzJ
djdzJj
dz
zdVsusu
EjH
, yx Ej
dz
dH
w
y
w
x dxEjdxHdz
d00
)/()(])()[()()(
mFd
wCzCVjdzE
d
wjwzEj
dz
zdIyy
w
d
w
d
C
L
zI
zVZ
LCvLC p
)(
)(
11,
0
)()(
)()(
22
2
22
2
zLCIdz
zId
zLCVdz
ZVd
CVjdz
dI
LIjdz
dV
Lossy parallel–plate transmission line: d
wCG
Surface impedance: c
cssc
x
z
s
ts
fjjXR
H
E
J
EZ
)1(
RIw
RIRJZJP s
ssussu2222
2
1)(
2
1
2
1)Re(
2
1
)/(2
)(2 mf
ww
RR
c
cs
Eg. Consider a transmission line made of two parallel brass strips σc=1.6×107S/m
of width 20mm and separated by a lossy dielectric slab μ=μ0, εr=3, σ=10-3S/m of
thickness 2.5mm. The operating frequency is 500MHz. (a) Calculate the R, L, G,
and C per unit length. (b) Find γ and Z0.
(Sol.) (a) )/(11.12 0 m
f
wR
c
, )/(108 3 mSd
wG
)/(1057.1 70 mH
w
dL , )/(1012.2 10 mF
d
wC
(b) γ= ))(( CjGLjR =18.13∠-0.41°, ω=2π×500×106, Z0=CjG
LjR
=27.21∠0.3°
Eg. Consider lossless stripline design for a given characteristic impedance. (a)
How should the dielectric thickness d be changed for a given plate width w if the
dielectric constant εr is doubled? (b) How should w be changed for a given d if εr
is doubled? (c) How should w be changed for a given εr if d is doubled?
(Sol.)
w
d
C
LZ 0
(a) dd 22 , (b) 2
2w
w
(c) wwdd 22
Attenuation constant of transmission line: α =)(2
)(
zP
zPL , where PL(z) is the
time-average power loss in an infinitesimal distance.
]))((Re[)Re( CjGLjRj
Suppose no reflection, , zjeVzV )(0)( zje
Z
VzI )(
0
0)(
zeRZ
VzIzVzP 2
02
0
20*
2)]()(Re[
2
1)( ze 2
)(2
)()(2)(
)(
zP
zPzPzP
z
zP LL
Microstrip lines: are usually used in the mm wave range.
ff
p
c
,
C
L
Cp
1,
ff
p
f
Assuming the quasi-TEM mode:
Case 1: t/h<0.005, t is negligible.
Given h, W, and εr, obtain Z0 as follows:
For W/h1:
h
W
W
h
ff
25.08ln60
,
where
22/1
104.01212
1
2
1
h
W
W
hrrff
For W/h1: 444.1ln667.0393.1
120
hW
hW
ff
,
where 2/1
1212
1
2
1
W
hrrff
Given Z0, h, and εr, obtain W as follows:
For W/h2: 2
82
A
A
e
heW , where
rr
rr
11.0
23.01
1
2
1
600
For W/h>2:
rr
r BBBh
W
61.039.01ln
2
112ln1
2, where
r
02
377
Case 2: t/h>0.005. In this case, we obtain Weff firstly.
For W/h 21 :
t
h
h
t
h
W
h
Weff 2ln1
For W/h 21 :
t
W
h
t
h
W
h
Weff
4ln1
And then we substitute Weff into W in the expressions in Case 1.
Assuming not the quasi-TEM mode:
ffeff fW
f
h377
, where 2
1
0
p
effeff
ff
WWWfW ,
hp 8f (h in cm)
and 00
0ff
effW
377h
, G=0.6+0.009Z0, 2
1
p
ffrrff
ffG
f
(f in GHz)
The frequency below which dispersion may be neglected is given by
1
3.00
rhzGf
, where h must be expressed in cm.
Attenuation constant: α=αd +αc
For a dielectric with low losses:
tan
1
13.27
r
ff
ff
rd (
cm
dB)
For a dielectric with high losses:
2/1
1
134.4
rff
ffd (
cm
dB)
For W/h → ∞: sc RW
68.8
, where
fRs
For W/h 21 :
W
t
t
W
W
h
W
h
h
PR
effeff
sc
4ln1
2
68.8
For 21 <W/h2: PQ
h
Rsc
2
68.8, where
2
41
h
WP eff
and
h
t
t
h
W
h
W
hQ
effeff
2ln1
For W/h2:
94.02
94.02
2ln268.8
2
hW
hW
h
W
h
We
h
W
h
QR
eff
effeffeffeffs
c
Eg. A high-frequency test circuit with microstrip lines.
5-2 Wave Characteristics of Finite Transmission Line
Eg. Show that the input impedance is Zi=
tanh
tanh)(
0
00
'0
L
L
zz ZZ
ZZZZ
.
(Proof) ,
)2.....()(
)1...()(
00
00
zz
zz
eIeIzI
eVeVzV
0
0
0
00
I
V
I
VZ
Let z=l, V(l)=VL, I(l)=IL
eZIVV
eZIVV
eZ
Ve
Z
VI
eVeVV
LL
LL
L
L
)(2
1
)(2
1
00
00
0
0
0
0
00
])()[(2
)(
])()[(2
)(
)(0
)(0
0
)(0
)(0
zL
zL
L
zL
zL
L
eZZeZZZ
IzI
eZZeZZI
zV
)'cosh'sinh()'(
)'sinh'cosh()'(
])()[(2
)'(
])()[(2
)(
00
0
'0
'0
0
'0
'0
'
zZzZZ
IzI
zZzZIzV
eZZeZZZ
IzI
eZZeZZI
zV
LL
LL
zL
zL
L
zL
zL
L
'tanh
'tanh)'(
0
00 zZZ
zZZZzZ
L
L
, Zi=
tanh
tanh)(
0
00
'0
L
L
zz ZZ
ZZZZ
Lossless case (α=0, γ=jβ, Z0=R0, tanh(γl)=jtanβl): Zi=ljZR
ljRZR
L
L
tan
tan
0
00
Note: In the high-frequency circuit, the input current Ii=Lg
g
ig
g
ZZ
V
ZZ
V
: the
value in the low-frequency case. And the high-frequency Ii is dependent on the length
l, the characteristic impedance Z0, the propagation constant γ of the transmission line,
and the load impedance ZL. But the low-frequency Ii is only dependent on Z0 and ZL.
Eg. A 2m lossless air-spaced transmission line having a characteristic impedance
50Ω is terminated with an impedance 40+j30(Ω) at an operating frequency of
200MHz. Find the input impedance.
(Sol.) 3
4
pv, , 500R 3040 jZ L , m2
87.93.26)2
3
4tan()3040(50
)23
4tan(50)3040(
50 jjj
jjZi
Eg. A transmission line of characteristic impedance 50Ω is to be matched to a
load ZL=40+j10(Ω) through a length l’ of another transmission line of
characteristic impedance R0’. Find the required l’ and R0’ for matching.
(Sol.)
105.0'),(7.381500''tan)1040(
'tan104050 0'
0
'0'
0
R
jjR
jRjR
Eg. Prove that a maximum power is transferred from a voltage source with an
internal impedance Zg to a load impedance ZL over a lossless transmission line
when Zi=Zg*, where Zi is the impedance looking into the loaded line. What is the
maximum power transfer efficiency?
(Proof) gi
i ZZ
VI
, V
ZZ
ZV
gi
ii
])()[(2*]Re[
2
1)(
22
2
gigi
iiiout XXRR
VRIVPower
When and , ∴ gi RR gi XX MaxPower out )( , *gi ZZ
In this case, g
out R
VPower
4)(
2
, g
is R
VVIP
2*]Re[
2
12
, s
out
P
Powere
)(
2
1
Transmission lines as circuit elements:
Consider a general case: Zi=
tanh
tanh
0
00
L
L
ZZ
ZZZ
1. Open-circuit termination (ZL→∞): Zi=Zio=Z0coth(γl)
2. Short-circuit termination (ZL =0): Zi=Zis=Z0tanh(γl)
∴ Z0= isi ZZ 0 , γ=0
1tanh1
i
is
Z
Z
3. Quarter-wave section in a lossless case (l=λ/4, βl=π/2): L
i Z
RZ
20
4. Half-wave section in a lossless case (l=λ/2, βl=π): Li ZZ
Eg. The open-circuit and short-circuit impedances measured at the input
terminals of an air-spaced transmission line 4m long are 250∠-50°Ω and
360∠20°Ω, respectively. (a) Determine Z0, α, and β of the line. (b) Determine R,
L, G, and C.
(Sol.) (a) 6.778.289360250 20500 jeeZ jj ,
jj
235.0139.050250
20360tanh
4
1 1
(b) 3.575.580 jZLjR , )/(812.03.573.57
mHc
L
45
0
1076.8105.24 jZ
CjG , )/(4.12
1076.8 4
mpFc
C
g. Measurements on a 0.6m lossless coaxial cable at 100kHz show a capacitance E
of 54pF when the cable is open-circuited and an inductance of 0.30μH when it is
short-circuited. Determine Z0 and the dielectric constant of its insulating
medium.
(Sol.) (a) )/(1096.0
1054 1112
mFC
, )/(1056.0
103.0 76
mHL
C
LRZ 00 =74.5Ω, 05.400 rrr LC Lossless
General expressions for V(z) and I(z) on the transmission lines:
Let Γ= j
L
L eZZ
ZZ
0
0 , z’=l-z
]1[)(2
)'(
]1[)(2
)'(
'2'0
0
'2'0
zzL
L
zzL
L
eeZZZ
IzI
eeZZI
zV
]1[)(2
)'(
]1[)(2
)'(
)'2('0
0
)'2('0
zjzL
L
zjzL
L
eeZZZ
IzI
eeZZI
zV
For a lossless line, V(z)= ]1[ )(2
0
0 zjzj
g
g eeZZ
VZ
Eg. A 100MHz generator with Vg=10∠0° (V) and internal resistance 50Ω is
connected to a lossless 50Ω air line that is 3.6m long and terminated in a
25+j25(Ω) load. Find (a) V(z) at a location z from the generator, (b) Vi at the
input terminals and VL at the load, (c) the voltage standing-wave radio on the
line, and (d) the average power delivered to the load.
(Sol.) , )(010 VVg )(50 gZ
)
, , ,
,
)(108 Hzf )(500 Z
(4525 Z L 36.3525 j
)(6.3 m , )/(3
2
103
1028
8
mradc
, )/(4.2 mrad
648.0447.050)2525(
50)2525(
0
0
j
j
ZZ
ZZ
L
L , 0g
(a) ]447.0[5]1[)( )152.03/2(3/2)(2
0
0
zjzjzjzj
g
g eeeeZZ
VZzV
(b) )(43.806.7)447.01(5)0( 152.0 VeVV ji
(c) )(5.4547.4]447.0[5)6.3( 248.04.0 VeeVV jjL
(d) 62.2447.01
447.01
1
1
S , )(200.025)
36.35
47.4(
2
1
2
1 2
2
WRZ
VP L
L
Lav
Eg. A sinusoidal voltage generator Vg=110sin(ωt) and internal impedance
Zg=50Ω is connected to a quarter-wave lossless line having a characteristic
impedance Z0=50Ω that is terminated in a purely reactive load ZL=j50Ω. (a)
Obtain the voltage and current phasor expressions V(z’) and I(z’). (b) Write the
instantaneous voltage and current expressions V(z’,t) and I(z’,t).
(Sol.) (a) 4
,0,5050
5050,110
0
0
0
0
ZZ
ZZj
j
j
ZZ
ZZjV
g
gg
L
Lg
(c) )(55)1()'( '''2'
0
0 zjzjzjzj
g
g jeejeeZZ
VZzV
,
)(1.1)1()'( '''2'
0
zjzjzjzj
g
g jeejeeZZ
VzI
),0'(),0'( tzItzVP )2cos(5.60 t , 01
0
T
av PdtT
P
(b) )]'cos()'[sin(55])'(Im[),'( ztztezVtzV tj )]'cos()'[sin(1.1])'(Im[),'( ztztezItzI tj
Eg. A sinusoidal voltage generator with Vg=0.1∠0° (V) and internal impedance
Zg=Z0 is connected to a lossless transmission line having a characteristic
impedance Z0=50Ω. The line is l meters long and is terminated in a load
resistance ZL=25Ω. Find (a) Vi, Ii, VL錯誤! 尚未定義書籤。, and I ; (b) the
standing-wave radio on the line; and (c) the average power delivered to the load.
L
(Sol.) (a) 3
1
0
0
ZZ
ZZ
L
L , 00
0
ZZ
ZZ
g
gg , )
3
11(
2
1.0)'( 2 j
i ezVV
)3
11(
100
1.0)'( 2 j
i ezII
jjL eezVV
30
1)
3
11(
2
1.0)0'(
eezII jL 750
1)
3
11(
100
1.0)0'(
(b) 21
1
S , (c) WIVP LLav
51022.2*]Re[2
1
Eg. Consider a lossless transmission line of characteristic impedance R0. A
time-harmonic voltage source of an amplitude Vg and an internal impedance
Rg=R0 is connected to the input terminals of the line, which is terminated with a
load impedance ZL=RL+jXL. Let Pinc be the average incident power associated
with the wave traveling in the +z direction. (a) Find the expression for Pinc in
terms of Vg and R0. (b) Find the expression for the average power PL delivered
to the load in terms of Vg and the reflection coefficient Γ. (c) Express the ratio
PL/Pinc in terms of the standing-wave ratio S.
(Sol.) )(1
)(,)( 000
00zjzjzjzj eVeV
RzIeVeVzV ,
2)0( 0
ginc
VVzV ,
00 2
)0(R
VIzI g
inc
(a) 0
2
0
2
0
00 82)*](Re[
2
1
R
V
R
VIVP g
inc
(b) 2
1)])(Re[(
2
1)](*)(Re[
2
1 2
0
2
00
*0
*000
0
VVR
eVeVeVeVR
zIzVP zjzjzjzjL
18
12
2
0
22
0
2
0
R
V
R
Vg
(c) 2
22
)1(
4)
1
1(11
S
S
S
S
P
P
inc
L
Case 1 For a pure resistive load: ZL=RL
'sin)/('cos)'(
'sin)/('cos)'(
'sin'cos)'(
'sin'cos)'(
220
2
220
2
0
0
zRRzIzI
zRRzVzV
zR
VjzIzI
zRjIzVzV
LL
LL
LL
LL
1
1S ,
1
1
S
S1. Γ=0S=1 when ZL= Z0 (matched load)
2. Γ=-1 S=∞ when Z L=0 (short-circuit), 3. Γ=1S=-∞ when ZL=∞ (open-circuit)
minmax & IV occurs at nz 2'2 max
maxmin & IV occurs at )12('2 min nz
If ....3,2,1,0,2
',00 max0 nn
zRRL
If 2
',0 min0
nzRRL
If 2
'max
nzRL
Eg. The standing-wave radio S on a transmission line is an easily measurable
quality. Show how the value of a terminating resistance on a lossless line of
known characteristic impedance R0 can be determined by measuring S.
(Sol.) If , 0RRL 0 , maxV occurs at 0'z and minV occurs at 2' z .
LVV max , L
L R
RVV 0
min , LII min , 0
max R
RII L
L , 0min
max
min
max
R
RS
I
I
V
VL or
. 0SRRL
If , 0RRL , minV occurs at 0'z , and maxV occurs at 2' z .
LVV min , L
L R
RVV 0
max , LII max , 0
min R
RII L
L .LR
RS
I
I
V
V0
min
max
min
max or
S
RRL
0
Case 2 For a lossless transmission line, and arbitrary load:
ZL=mm
mm
jRR
jRRR
tan
tan
0
00
, zm’+lm=λ/2
Find ZL=?
1. 1
1
S
S, 2. At θΓ=2βzm’-π, V(z’) is a minimum.
3. ZL=RL+jXL =
1
1
1
100 R
e
eR j
j
Eg. Consider a lossless transmission line. (a) Determine the line’s characteristic
resistance so that it will have a minimum possible standing-wave ratio for a load
impedance 40+j30(Ω). (b) Find this minimum standing-wave radio and the
corresponding voltage reflection coefficient. (c) Find the location of the voltage
minimum nearest to the load.
(Sol.)
3
1500,
1
1,]
30)40(
30)40([
3040
30400
0
2122
0
220
0
0
0
0
RdR
dSS
R
R
jR
jR
RZ
RZ
L
L
, 23
190
3
1
3090
3010
0
0
j
j
RZ
RZ
L
L , 2
2 S
8)
2(
2
1'min
z , 8
3
82
m
Eg. SWR on a lossless 50Ω terminated line terminated in an unknown load
impedance is 3. The distance between successive minimum is 20cm. And the first
minimum is located at 5cm from the load. Determine Γ, ZL, lm, and Rm.
(Sol.)
52
,4.02.02
m
'2
05.0',5.013
13mmm zz
=0.15m
5.0'2 mz , 2
5.0 5.0 jee jj
mm
mmL jR
jRj
j
j
ZR
tan50
tan50504030
)2
(1
)2
(150,500
)(7.163
50 mR
Eg. A lossy transmission line with characteristic impedance Z0 is terminated in
an arbitrary load impedance ZL. (a) Express the standing-wave radio S on the
line in terms of Z0 and ZL. (b) Find the impedance looking toward the load at the
location of a voltage maximum. (c) Find the impedance looking toward the load
at a location of a voltage minimum.
(Sol.) (a) '2
00
'200'2
0
0
1
1,
zLL
zLLz
L
L
eZZZZ
eZZZZSe
ZZ
ZZ
(b) )'(1
1)'(,2'2
max
0
'2
'2
0max'2'2
maxmax
max
maxmax
zS
Z
e
eZzZeeenz
z
zzzj
(c) '2'2min
minmin)12('2 zzj eeenz
)'(1
1)'(
min
0'2
'2
0minmin
min
zS
Z
e
eZzZ
z
z
5-3 Introduction to Smith Chart
1
1
1/
1/
0
0
0
0
L
L
L
Lj
L
L
z
z
RZ
RZe
RZ
RZ ir j
j
j
Le
ez
1
1
1
1jxr
, 22)1(
2
ir
ix
22
22
)1(
1
ir
irr
222 )1
1()
1(
rr
rir
: r-circle, 222 )
1
xe ()
1()1(
xir : x-circl
Several salient properties of the r-circles:
1. The centers of all r-circles lie on the Γr-axis.
d centered at the origin, is the largest.
as r increases from 0 toward ∞,
se for x>0 (inductive reactance)
ose for x<0 (capacitive reactance) lie below the
cle becomes progressively smaller as |x| increases from 0 toward ∞,
pen-circuit.
r radii vary uniformly from 0 to
ough the point representing zL equals θΓ.
dio S.
2. The r=0 circle, having a unity radius an
3. The r-circles become progressively smaller
ending at the (Γr=1, Γi=0) point for open-circuit.
4. All r-circles pass through the (Γr=1, Γi=0) point.
Salient properties of the x-circles:
1. The centers of all x-circles lie on the Γr=1 line, tho
lie above the Γr–axis, and th
Γr–axis.
2. The x=0 circle becomes the Γr–axis.
3. The x-cir
ending at the (Γr=1, Γi=0) point for o
4. All x-circles pass through the (Γr=1, Γi=0) point.
Summary
1. All |Γ|–circles are centered at the origin, and thei
1.
2. The angle, measured from the positive real axis, of the line drawn from the origin
thr
3. The value of the r-circle passing through the intersection of the |Γ|–circle and the
positive-real axis equals the standing-wave ra
Application of Smith Chart in lossless transmission line:
]1
1[
)'(
)'()'(
'2
'2
0 zj
zj
i e
ez
zI
zVzZ
,
j
j
zj
zji
i e
e
e
e
Z
zZzz
1
1
1
1)()'(
'2
'2
0
when
'2 z
keep |Γ| constant and subtract (rotate in the clockwise direction) an angle
'4
'2z
z from θΓ. This will locate the point for |Γ|ejφ, which determine Zi.
Increasing z’ wavelength toward generator in the clockwise direction
A change of half a wavelength in the line length 2
'
z A change of
2)'(2 z in φ.
Eg. Use the Smith chart to find the input impedance of a section of a 50Ω lossless
transmission line that is 0.1 wavelength long and is terminated in a short-circuit. (Sol.) Given , ) , 0Lz (500 R 1.0'z
1. Enter the S tion ofmith chart at the intersec r=0 and x=0 (point on the
e chart
scP
extreme left of chart; see Fig.)
2. Move along the perimeter of th )1( by 0.1 “wavelengths toward
generator’’ in a clockwise direction to P1.
At P1, read r=0 and 725.0x , or 72.0jzi , )(3.36)725.0(50 jjZi . 5
g. A lossless transmission line of length 0.434λ and characteristic impedance
j180
j1.8 (point P2 in Fig.)
E
100Ω is terminated in an impedance 260+j180(Ω). Find (a) the voltage reflection
coefficient, (b) the standing-wave radio, (c) the input impedance, and (d) the
location of a voltage maximum on the line.
(Sol.) (a) Given l=0.434λ, R0=100Ω, ZL=260+
1. Enter the Smith chart at zL=ZL/R0=2.6+
60.02 OP2. With the center at the origin, draw a circle of radius .
( scOP =1) traight line and extend it to P2’ on the periphery. Read 3. Draw the s 2OP
0.22 on “wavelengths toward generator” scale. = 21 ,
2160.0je .
circle intersects with the positive-real axis at r=S=4. ocOP(b) The 60.0
(c) To find the input impedance:
1. Move P2’ at 0.220 by a total of 0.434 “wavelengths toward generator,” first to
0.500 and then further to 0.154 to P3’.
2. Join O and P3’ by a straight line which intersects the 60.0 circle at P3.
3. Read r=0.69 and x=1.2 at P3. )(12069)2.169.0(1000 jjzRZ ii .
(d) In going from P2 to P3, the 60.0 circle intersects the positive-real axis
ocOP at PM, where the voltage is a maximum. Thus a voltage maximum
appears at (0.250-0.220) or 0.030 from the load.
Application of Smith Chart in lossy transmission line
jz
jz
zjz
zjz
i ee
ee
ee
eez
'2
'2
'2'2
'2'2
1
1
1
1
∴ We can not simply move close the |Γ|-circle; auxiliary calculation is necessary for
the e-2αz’ factor.
Eg. The input impedance of a short-circuited lossy transmission line of length 2m
and characteristic impedance 75Ω (approximately real) is 45+j225(Ω). (a) Find α
and β of the line. (b) Determine the input impedance if the short-circuit is
replaced by a load impedance ZL= 67.5-j45(Ω). (Sol.) (a) Enter 0.360.075/)22545(1 jjzi in the chart as P1 in Fig.
Draw a straight line from the origin O through P1 to P1’.
Measure 211 89.0'/ eOPOP , )/(029.0)124.1ln(
4
1)
89.0
1ln(
2
1mNp
Record that the arc is 0.20 “wavelengths toward generator”. '1PPsc 20.0/ ,
8.0/42 . )/(2.04
8.0
2
8.0
mrad
.
(b) To find the input impedance for:
1. Enter 6.09.075/)455.67(/ 0 jjZZz LL on the Smith chart as P2.
2. Draw a straight line from O through P2 to P2’ where the “wavelengths toward
generator” reading is 0.364.
3. Draw a –circle centered at O with radius 2OP .
4. Move P2’ along the perimeter by 0.2 “wavelengths toward generator” to P3’ at
0.364+0.20=0.564 or 0.064.
5. Joint P3’ and O by a straight line, intersecting the –circle at P3.
6. Mark on line a point P3OP i such that 89.0/ 23 eOPOPi .
7. At Pi, read . 27.064.0 jzi )(3.200.48)27.064.0(75 jjZi
5-4 Transmission-line Impedance Matching
Impedance matching by λ/4-transformer: R0’= LRR0
Eg. A signal generator is to feed equal power through a lossless air transmission
line of characteristic impedance 50Ω to two separate resistive loads, 64Ω and
25Ω. Quarter-wave transformers are used to match the loads to the 50Ω line. (a)
Determine the required characteristic impedances of the quarter-wave lines. (b)
Find the standing-wave radios on the matching line sections.
(Sol.) (a) )(1002 021 RRR ii .
)(806410011'01 Li RRR , )(502510022
'02 Li RRR
(b) Matching section No. 1:
11.08064
8064'011
'011
1
RR
RR
L
L, 25.1
11.01
11.01
1
1
1
11
S
Matching section No. 2:
33.05025
5025'022
'022
2
RR
RR
L
L , 99.133.01
33.01
1
1
2
22
S
Application of Smith Chart in obtaining admittance:
LL ZY /1 , LL
LL yYRR
Zz
11
00
, where jbyYRGYYYy LLL 0000 //
Eg. Find the input admittance of an open-circuited line of characteristic
impedance 300Ω and length 0.04λ.
(Sol.) 1. For an open-circuited line we start from the point Poc on the extreme right of
the impedance Smith chart, at 0.25 in Fig.
2. Move along the perimeter of the chart by 0.04 “wavelengths toward generator” to
P3 (at 0.29).
3. Draw a straight line from P3 through O, intersecting at P3’ on the opposite side.
4. Read at P3’: 26.00 jyi , mSjjYi 87.0)26.00(300
1 .
Application of Smith Chart in single-stub matching:
00
1
RYYYY SBi SB yy 1 , where yB=R0YB, ys=R0Ys
∵ 1+jbs= yB, ∴ ys=-jbs and lB is required to cancel the imaginary part.
Using the Smith chart as an admittance chart, we proceed as yL follows for
single-stub matching:
1. Enter the point representing the normalized load admittance.
2. Draw the |Γ|-circle for yL, which will intersect the g=1 circle at two points. At
these points, yB1=1+jbB1 and yB2=1+jbB2. Both are possible solutions.
3. Determine load-section lengths d1 and d2 from the angles between the point
representing yL and the points representing yB1 and yB2.
Determine stub length lB1 and lB2 from the angles between the short-circuit point on
the extreme right of the chart to the points representing –jbB1 and –jbB2, respectively.
Eg. Single-Stub Matching:
Eg. A 50Ω transmission line is connected to a load impedance ZL= 35-j47.5(Ω).
Find the position and length of a short-circuited stub required to match the line. )(500 R , )(5.4735 jZ L , 95.070.0/ 0 jRZz LL (Sol.) Given
–circle centered at O with radius 1OP . 1. Enter on the Smith chart as . Draw a Lz 1P
2. Draw a straight line from through O to on the perimeter, intersecting the 1P 2'P –circle at
which represents . No .109 at “wavelengths toward generator” scale.
3. T io
2P , Ly te 0 2'P on the
wo points of intersect n of the –c with the g=1 circle. ircle
3P : 11 12.11 BB jbjy At . At : 4P 22 12.11 BB jbjy ;
4. Solutions for th sition of the stubs:
For (f to ):
e po
rom3P 2'P 3'P 059.0)109.0168.0(1 d
For 4P (from 2'P to 4'P ): .0(2 d 223.0)109.0332
For (from to , which represents 3P scP 3''P 2.11 jjbB ):
111.0)250.0361.0(1 B
For (from to , w4P scP 4"P hich represents 2.12 jjbB ):
389.0)250.0139.0( 2B
5-5 Introduction to S-parameters
S-parameters: for analyzing the high-frequency circuits.
22
12
21
11
S
S
S
SS
Define xIZxVZ
xa 0
02
1 , xIZxV
Zxb 0
02
1
2212111111 laSlaSlb , 2222112122 laSlaSlb
, where
22
11
22
12
21
11
22
11
la
la
S
S
S
S
lb
lb 0
11
1111 22 lala
lbS ,
0
11
2221 22 lala
lbS ,
0
22
2222S
11 lala
lb, and
0
22
1112 11 lala
lbS .
New S-parameters obtained by shifting reference planes:
10111
jeblb , 10111jeala , 20222
jeblb , 20222jeala
0
0
0
0
2
1
222
12
21
211
2
1
2
21
21
1
a
a
eS
eS
eS
eS
b
bj
j
j
j
, where
=
22
12
21
11
'
'
'
'
S
S
S
S
2
21
21
1
222
12
21
211
j
j
j
j
eS
eS
eS
eS and =
22
12
21
11
S
S
S
S
2
21
21
1
222
12
21
211
'
'
'
'
j
j
j
j
eS
eS
eS
eS
21
11
21
22
12
21
11
1
S
S
S
T
T
T
T
21
221112
21
22
S
SSS
S
S
T-parameters:
lbla, where
22
22
22
12
21
1111
laTT 11
TT
lb
and
111211 TSS
11
21
2221 1
T
T
SS
1122 T
11
12
1221
T
T
TTT