Download Transmission Line Model for Rectangular Waveguides

Transmission Line Model for Rectangular
Waveguides accurately incorporating Loss Effects
Konstantin Lomakin
[email protected]
Institute of Microwaves and Photonics Friedrich-Alexander-Universität Erlangen-Nürnberg
10 May 2017
Outline
✦
✦
✦
✦
✦
✦
Introduction
Modeling lossless TE10 Mode
Incorporating Loss Effects
Impact of Losses on the Phase Coefficient
Comparison to Simulation and Measurement
Conclusion
SPI-2017 Baveno, Italy
Konstantin Lomakin
Friedrich-Alexander Universität Erlangen-Nürnberg
10.05.2017
2
Introduction
Introduction
✦
Rectangular Waveguides (RWG) typically deployed e.g. in mm-wave or space applications
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Fundamental mode of RWG: TE10
✦
Inherently dispersive transmission line
✦
Only two loss-mechanisms: dielectric and conductor
✦
One typical modeling approach:
‣ Phase coefficient: solution of Maxwell’s equations
‣ Attenuation coefficient: perturbation method
✦
Perturbation method does not take into account any impact on phase coefficient
y
h
z
SPI-2017 Baveno, Italy
Konstantin Lomakin
Friedrich-Alexander Universität Erlangen-Nürnberg
w
x
10.05.2017
4
Current Distribution of the TE10 Mode
Transversal Field Components
⇣ ⇡x ⌘
Ey =
✦
jA10 ZF ⌦ sin
w
⇣ ⇡x ⌘
p
Hx = jA10 ⌦2 1 sin
w
✦
Longitudinal Component
Hz = A10 cos
⇣ ⇡x ⌘
A10 =
w
2Pin
p
whZF ⌦ ⌦2
1
⌦ = f /fc
Current density in conductive material: r ⇥ H = j!"E + J ' J
Distribution of surface currents on the RWG’s walls:
Jz,top
1 ˆ
= Hx e
Jx,top =
y
Jy
xn
H(xn ) / Ĥ(xn = 0)e
Jx,z
z
s
Jy,right =
1
1
Ĥz e
Ĥz e
y
h
y
h
x
w
x
SPI-2017 Baveno, Italy
Konstantin Lomakin
Friedrich-Alexander Universität Erlangen-Nürnberg
10.05.2017
5
Modeling lossless TE10 Mode
Modelling lossless TE10 Mode
3D model
Transmission line model
Z’
y
Hx
0
dzLo
Hz
h
w
z
ll
ZL,ll
!
2
=j =j
1 (f /fc )
c
ZF
=q
2
1 (f /fc )
c0
fc = p
"r 2w
SPI-2017 Baveno, Italy
dzC
0
Y’
Ey
x
q
00
Lo /dz
ll
=
p
0
0
Z Y =j
ZL,ll =
r
0
ll
X
0 =
Y
=j
s
s
0
Lo
L00o
! 2 L0o C 0
2
0
00
! Lo Lo
! 2 L00o C 0 1
µ0 w 2
Lo = µ C = " Lo =
⇡2
0
0
Konstantin Lomakin
Friedrich-Alexander Universität Erlangen-Nürnberg
00
10.05.2017
7
Incorporating Loss Effects
Transmission Line Model for lossy TE10 Mode
✦
Extending lossless model:
✦
Conductor losses due to longitudinal currents: R’
✦
Conductor losses due to transversal currents: R’’
✦
Dielectric losses in electric field:
0
0
G = !C tan
Il
dzL
0
dzR
0
00
L /dz
It
Model currents
✦
dzC
0
dzG
0
00
R /dz
Model holds as long as fields don’t degenerate dramatically
SPI-2017 Baveno, Italy
Konstantin Lomakin
Friedrich-Alexander Universität Erlangen-Nürnberg
10.05.2017
9
Deriving Model Currents
✦
Model currents are derived from field energies and Lo’ and Lo’’ in lossless case:
W m,x
W m,z
1
=
2
1
=
2
Z
Z
µHx2 dV
1 0
= Lo dzIl2
2
0
Lo = µ
µHz2 dV
1 00 2
=
Lo It
2dz
µ0 w 2
Lo =
⇡2
⇡
It = p dz
w 2
s
p
2Pin ⌦2
ZF ⌦
2Pin
p
Z F ⌦ ⌦2
1
1
00
Field distribution
Iz =
✦
j
Il = p
2
s
Z
w
0
Z
h+
h
2
Jz,top dydx = j
⇡
s
p
w 2Pin ⌦2
h
ZF ⌦
1
Model current does not explicitly scale with geometry (w,h) like physical current does!
SPI-2017 Baveno, Italy
Konstantin Lomakin
Friedrich-Alexander Universität Erlangen-Nürnberg
10.05.2017
10
Modelling Conductor Losses
✦
Physical loss power inside conductive material gathered from current densities
✦
R’ and R’’, together with the model currents must yield the same loss power:
1
Z
Longitudinal currents
Jz2 dV
Field distribution
1
Z
= dzR
0
0
Il2
R =
2
h
Model
Transversal currents
2
Jx,y
dV
SPI-2017 Baveno, Italy
1 00 2
=
R It
dz
2w (w + 2h)
R =
h⇡ 2
Konstantin Lomakin
Friedrich-Alexander Universität Erlangen-Nürnberg
00
10.05.2017
11
Impact of Losses on the Phase
Coefficient
Additional Impact on Phase Coefficient
✦
Penetrating magnetic fields in conductors (skin effect) associated with:
✦
Current densities and conductor loss (taken into account by R’ and R’’)
✦
Magnetic field energy in conductive material: Inner Inductance
0
R
2
Li =
=
!
! h
00
00
R
2w (w + 2h)
Li =
=
!
!h⇡ 2
0
✦
0
0
0
00
00
00
L = Lo + Li
L = Lo + Li
Final equations for propagation coefficient and characteristic impedance:
s
✓
◆
1
0
0
= (R + j!L )
+
G
+
j!C
R00 + j!L00
s
✓
◆
1
0
0
0
0
Z = (R + j!L )/
+
G
+
j!C
R00 + j!L00
SPI-2017 Baveno, Italy
0
0
Konstantin Lomakin
Friedrich-Alexander Universität Erlangen-Nürnberg
10.05.2017
13
Comparison to Simulation and
Measurement
Simulation of RWG with different heights
✦
Finite conductivity, identical in all simulated hollow RWGs;
✦
Ideal smooth surfaces in simulation and proposed model; w = 4mm
✦
Continuous lines: proposed model; dashed: HFSS simulation;
✦
Full wave field solver and proposed model deliver almost identical responses
20
15
10
2
in 1/m
4
↵ in 1/m
h = 1 mm
h = 2 mm
h = 3 mm
Perturbation Method
h = 1 mm
h = 2 mm
h = 3 mm
5
0
37
37.5
SPI-2017 Baveno, Italy
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38.5
39
Frequency in GHz
39.5
40 37.3
37.35
37.4
37.45
Frequency in GHz
Konstantin Lomakin
Friedrich-Alexander Universität Erlangen-Nürnberg
0
37.5
10.05.2017
15
Measurement: WR10 Waveguide
✦
TRL calibration at waveguide flange
✦
Material: brass; Exact conductivity unknown
✦
Fabrication tolerances not exactly known
✦
Possible reason for apparently low conductivity: Surface Roughness
Estimation from phase coefficient: ~0.5 MS/m
Estimating w from phase coefficient: ~2.49 mm
20
Measurement
Proposed Model
/
0
15
10
1.2
↵ in 1/m
Measurement
Proposed Model
Perturbation Method
1.4
5
1
60
61
SPI-2017 Baveno, Italy
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63
64
Frequency in GHz
65
66
60
70
80
90
Frequency in GHz
Konstantin Lomakin
Friedrich-Alexander Universität Erlangen-Nürnberg
100
110
0
10.05.2017
16
Conclusion
Conclusion
✦
Transmission Line Model for RWG only requiring geometry and material parameters
✦
Analytical equations describing propagation characteristics with respect to losses
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Very efficient in terms of computation time
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Basic principle: Perturbation Method formulated in Transmission Line Model
✦
Inner inductance accounts for the impact of losses on the phase coefficient
✦
Model is easily extendable to include surface roughness effects
✦
Model potentially enables higher precision of waveguide measurements & calibration
SPI-2017 Baveno, Italy
Konstantin Lomakin
Friedrich-Alexander Universität Erlangen-Nürnberg
10.05.2017
18
Thank You very much for Your Attention