review of transmission lines & guiding structures
TRANSCRIPT
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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
' '
' '
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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
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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
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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
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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
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Line terminated on a resistive load
z = 0 z
ZL = R Z0
z = - l
ZL = R0L
0LL
ZR
ZR
Impedance, voltage and current
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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
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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
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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
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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
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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
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Coaxial Cable
Commercial productsflexible coaxial cables
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Commercial productssemi-rigid coaxial cables
Commercial productsspline coaxial cables
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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)
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EM field in the cable
Unimodal frequency region
ba
vfc
)11(
The first higher order mode is the TE11
For this mode it results:
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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
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rectangularwaveguide
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Geometry of the Rectangular Waveguide
Commercial Products
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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
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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
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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
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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).
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Field in Rectangular Waveguide TE10 Mode
xa
cosA)x(hz
xa
jBsen)x(hx
x
ajCsen)x(ey
Transverse Electric Field
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Longitudinal Electric Field
circularwaveguide
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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
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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
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Fundamental TE11 Mode
Electric field lines
Magnetic field lines
Electrical Circular Modes TE0m
TE01 mode TE02 mode
Electric field lines
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microstrip
Geometry of the Microstrip
metallic strip
dielectric substrate
ground plane
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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
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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 [
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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
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APPENDIX
wave impedance
good conductors
APPENDIX