1
Review of Transmission lines & Guiding structures
1) Theory of Transmission Lines
Telegraph equationsTransmission line parametersSmith chart
2) Guiding Structures
Coaxial cableWaveguideMicrostrip
Theory of transmission lines (eg coaxial cable)
I(z)
V(z)
Z’ d z I(z + d z)
V(z + d z)Y’ d z
dz z
a
bln
'2''C
0
V
Q
C '= Parallel capacity per unit of length (F / m)
L = Series inductance per unit length (H / m)
G '= Parallel conductance per unit of length (S / m)
R '= Series resistance per unit of length (Ω / m)
Z '= R' + jL ‘ Impedance per unit of length (Ω / m)
Y '= G ’+ j C' Admittance per unit length (S / m)
a
bln
2
''L
0
I
ψ
' '
' '
2
Theory of transmission lines(coaxial cable)
=====
tan'C
a
bln
''2
a
bln
2
V
'I'G D
0
c
CS
2R
=
tan = ''/'
D = dielectric Conductivity
C = metal conductivity
1
2𝑅′𝐼0
2 =1
2𝑅𝑒
𝑙1+𝑙2𝐸𝑡 ×𝐻𝑡
∗ ∙ 𝑛0 𝑑𝑙 = 1
2𝑅𝑆
𝑙1+𝑙2𝐻𝑡
2 𝑑𝑙
𝑅′ =𝑅𝑆 𝑙1+𝑙2
𝐻𝑡2 𝑑𝑙
𝑙1𝐻𝑡 𝑑𝑙
2= 𝑅𝑆 𝑏+𝑎
2𝜋 𝑎𝑏
Telegraph equations
z'Z
dz
zdI
V
z'Y
dz
zdV
I
zz eez VVV zz
C
eeZ
z
VVI
1
'Cj'G'Lj'R'Y'Zj
''
''
'
'
CjG
LjR
Y
ZjZZZ CjCrC
Direct and reflected voltage wave Direct and reflected current wave
Propagation constant
(attenuation and phase)
Characteristic impedance
Secondary parameters of the line
3
Coaxial cable
low losses ''''
2
1CLjZG
Z
RC
C
n
fcv
smc
v
cCL
rrr
rr
rr
000
8
00
/103
''≈
(phase speed)
'
2
1
4
12
2
1'
2
1≈
CCC
S
C Zab
ab
Z
ab
abR
Z
R(m-1)
(m-1)
n = index of refraction
Coaxial cable
a
b
a
b
C
LZ
r
C ln60
ln2
1
'
'
Typical value for cables and microstrips ZC = Z0= 50
]dB[l686.8eP
Plog10
P
Plog10A
l2IN
IN
OUT
INdB
=== Attenuation
]m/dB[686.8A m/dB =for l = 1m
4
zz
zz
ee
eeZ
z
zzZ
VV
VV
I
V0
z2
z
z
ee
ez
V
V
V
V
0
0
ZzZ
ZzZz
z
zZzZ
1
10
Impedance along the line
reflection coefficient along the line
Transmission lines terminated on loads
Z0
z = - l z = 0 z
ZL
j
L
LL e
ZZ
ZZ
0
0
LLL
L0L jXR
-1
1ZZ +=+
=
ljZlZ
ljZlZZlZ
L
L
sincos
sincos-
0
00
lZlZ
lZlZZlZ
L
L
sinhcosh
sinhcosh-
0
00
( ) lj2-l2-jl2-L eel- +==
For a "long" line with losses it results ( ) 0l- =
with losses without losses
5
Line terminated on a matched load
z = 0 z
ZL=Z0 Z0
z = - l
ZL = Z00
ZZ
ZZ
0L
0LL
Impedance, voltage and current
V(z)
I(z)
z z=0
z
R(z)
z=0
Z0
6
Line terminated on a resistive load
z = 0 z
ZL = R Z0
z = - l
ZL = R0L
0LL
ZR
ZR
Impedance, voltage and current
7
Line terminated on a short
ItanjZIZ 0
z = 0 z
ZL = 0 Z0
z = - l
ZL = 01
Z0
Z0
0
0L
Impedance, voltage and current
Series resonance
Parallel resonance
8
Resonant circuits
Short circuited Line
For l < lambda / 4
Leq = X(-l) / = (Z0 / ) tan l
For l < lambda / 12
Leq = Z0l / = Z0l / c = L'l
9
Line terminated on an open circuit
ljYlY tan0
z = 0 z
Z0
z = - l
ZL =
ZL = 1Z
Z
0
0L
ljZlZ cot0
Impedance, voltage and current
Series Resonance
Parallel Resonance
10
Line terminated on an open circuit
For l < lambda / 4
Ceq = B(-l)/ = (Y0 / ) tan l
For l < lambda / 12
Ceq = Y0 l / c = C'l
SWR
-1
1
V
VSWR
MIN
MAX +==
Standing Wave Ratio
(SWR) 1
1-WR
SWR
S
Matched load ZL = Z0 = 0 SWR = 1
Short Circuit ZL = 0 = 1 SWR =
Open Circuit ZL = = 1 SWR =
with no
losses
11
Impedance Smith Chart
CC 0
0.5
1.0
2.0
-0.5
-1.0
-2.0
0.5 1.0 2.0
P
CM CA
ZL = 100 + j50
Inductive Region
Capacitive Region
Load as a function of the frequency
Ideal Inductance Ideal Capacity
12
Coaxial Cable
Commercial productsflexible coaxial cables
13
Commercial productssemi-rigid coaxial cables
Commercial productsspline coaxial cables
14
EM field in the cable
00
0tt
00
tt
20
212
r
1
a
bln
V1ze
1h
rr
1
a
bln
Ve
C)rln(
a
bln
VC)rln(
a
bln
),r(
(r0,0,z0)
15
EM field in the cable
Unimodal frequency region
ba
vfc
)11(
The first higher order mode is the TE11
For this mode it results:
16
Cable RG 58/U
Unimodal band
a = 0.44 mm b = 1.46 mm (2a=0.035 inches, 2b=0.116 inches)
ft(TEM) = 0.0 GHz
ft(11) = v/((a+b)) = (3108/2.1)/(3.14 1.9 10-3) = 34.7 GHz
Characteristic impedance
506.49
44.0
46.1ln
21.2
120
a
bln
2Z0
Cable RG 58/U
Wire attenuation (copper)
3
7
99
10016.0107.5368
101.2102
g8
f =1 GHz
040.0
44.0
46.1ln
46.1
10
44.0
10
10016.0
a
bln
b
1
a
1
g8
33
3c
]dB[6,105,30686,8040.0 ft100c AdB
1 foot = 30,5 cm
100 ft = 30,5 m
17
rectangularwaveguide
18
Geometry of the Rectangular Waveguide
Commercial Products
19
Study of the EM Field in the Waveguide
Purely transverse electric field (TE): EZ = 0
Purely transverse magnetic field (TM): HZ = 0
TE Mode Transverse Solution (Rectangular Waveguide)
2
22
2
222t
b
n
a
mk
yb
ncosx
a
mcosA)y,x(hz
m = 0, 1, .. n = 0, 1, ..
the case m = n = 0 null field is excluded
eigenfunction
eigenvalue
20
EM field Longitudinal Solution
z
fz
1zzz
1tjzzjk
1e
dt
dzv0dzdt
ztsenePeePIm)t,z(Z
Progressive wave
Regressive wave
z
fz
1zzz
2tjzzjk
2e
dt
dzv0dzdt
ztsenePeePIm)t,z(Z
constant phasephase
velocity
phase
velocity
Phase Constant
2c
2tk
20
2
2
2c
z 111
22z
2t
2 kkk
2
222 1
c
czzz jk
(lossless waveguide)
For KZ = 0 we set: = C for which
For > C
Longitudinal transversal
f
fc
Condition of separability
21
Dispersion Curves
k
Unimodal Region
2
22
2
22
cb
n
a
m
2
cf
a
cff
a2
cff )0,2(c2c)0,1(c1c
2
22
2
222c
2t
b
n
a
mk
m = 0, 1, .. n = 0, 1, ..
with a > 2b
Unimodal region 1.25 fc1 < f < 0.95 fc2
1c
22
Example WR-90 waveuide
a = 22.86 mm b = 10.16 mm
(a=0.9 inches, b=0.4 inches)
ft(10) = c/2a = 3108/(2 22.86 10-3) = 6.56 GHz
ft(20) = c/a = 3108/(22.86 10-3) = 13.12 GHz
Banda unimodale
6.56 1.25 = 8.2 GHz 13.12 0.95 = 12.4 GHz
RF/microwave SystemsDenominations Frequency Interval GHz (109 Hz)
HF 0.003 - 0.030
VHF 0.030 - 0.300
P (Previous) UHF 0.300 - 1.000
L (Long) Band 1.0 - 2.0
S (Short) Band 2.0 - 4.0
C (Compromise) Band 4.0 - 8.0
X (Cross*) Band 8.0 - 12.0
Ku ( Under K) Band 12.0 - 18.0
K Band 18.0 - 26.5
Ka (Above K) Band Ka 26.5 - 40.0
Q Band 40.0 - 50.0
V Band 50.0 - 75.00
millimeter 40.0 - 300.0
Terahertz > 300.0
*Used in WW II for fire control, X for cross (as in crosshair).
23
Field in Rectangular Waveguide TE10 Mode
xa
cosA)x(hz
xa
jBsen)x(hx
x
ajCsen)x(ey
Transverse Electric Field
24
Longitudinal Electric Field
circularwaveguide
25
Geometry of the Circular Waveguide
TE Modes
ncosra
CJ,rh
nsinr
ra
J
nk
kC,rh
ncosra
Jak
kC,rh
0,re
ncosra
Jak
jC,re
nsinr
ra
J
nk
jC,re
]m,n[nz
]m,n[n
2]m,n[t
]n,m[z
]m,n[n
]m,n[
2]m,n[t
]n,m[zr
z
]m,n[n
]m,n[
2]n,m[t
]m,n[n
2]n,m[t
r
m-th zeros of the
n-th Bessel function derivative
m,n′
a
vf
mn
nmc
,
,,2
Cutoff frequency
Bessel function of the first kind
26
0,rh
ncosra
Jak
jC,rh
nsinr
ra
J
nk
jC,rh
ncosra
CJ,re
nsinr
ra
J
nk
kC,re
ncosra
Jak
kC,re
)m,n(n
]m,n[
2)m,n(t
c
)m,n(n
2)m,n(t
cr
)m,n(nz
)m,n(n
2)m,n(t
)n,m(z
)m,n(n
)m,n(
2)m,n(t
)n,m(zr
TM Modes
m,n
a
vf
mn
nmc
,
,,2
m-th zeros of the
n-th Bessel function
Cutoff frequency
Mode Spectrum
27
Fundamental TE11 Mode
Electric field lines
Magnetic field lines
Electrical Circular Modes TE0m
TE01 mode TE02 mode
Electric field lines
28
microstrip
Geometry of the Microstrip
metallic strip
dielectric substrate
ground plane
29
Quasi TEM Mode
Substrates
material
surface
finish (m)
104.tan
(10 GHz) r
Thermal
conductivity
(W/cm2/°C)
Alumina 99 % 0.25 1 - 2 10 0.37
Alumina 96 % 20 6 9 0.28
Alumina 85 % 50 15 8 0.20
Sapphire 0.025 0.7 9.4 0.4
Glass 0.025 20 5 0.01
Polyolefin 1 1 2.3 0.001
Duroid (Roger) 0.75-8.75 5-60 2-10 0.0026
Quartz 0.025 1 3.8 0.01
Beryllium 1.25 1 6.6 2.5
GaAs (high-res) 0.025 6 13 0.3
Silicio (high-res) 0.025 10-100 12 0.9
Air (dry) - 0 1 0.00024
30
Analysis Equations
'0
'
effC
Cε
effeff0c
effeff
CC
Cc
ZZ
'
0
0 1
Characteristic Impedance (t = 0)
0
50
100
150
200
250
300
0.1 1 10 w/h
r
1
2 3 4 6
16
10
Z0 [
31
Phase Constant
f
propagazione in aria
Modo dominante
modo quasi-TEM
propagazione nel dielettrico
modi di ordine
superiore
/c0
0
r
c
dielectric
propagationdominant
mode
quasi-TEM
mode
air
propagation
higher
order
modes
Higher Order Modes
x
y
x
y weff
a) b)
weff
eff
eff10c
w2
/cTEf
eff
eff20c
w
/cTEf
32
APPENDIX
wave impedance
good conductors
APPENDIX
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