transmission line model for rectangular waveguides ... may 2017 transmission line model for...
TRANSCRIPT
10 May 2017
Transmission Line Model for Rectangular Waveguides accurately incorporating
Loss Effects
Institute of Microwaves and Photonics Friedrich-Alexander-Universität Erlangen-Nürnberg
Konstantin Lomakin [email protected]
SPI-2017 Baveno, Italy Konstantin Lomakin 10.05.2017 Friedrich-Alexander Universität Erlangen-Nürnberg
Outline
✦ Introduction ✦ Modeling lossless TE10 Mode ✦ Incorporating Loss Effects ✦ Impact of Losses on the Phase Coefficient ✦ Comparison to Simulation and Measurement ✦ Conclusion
2
Introduction
SPI-2017 Baveno, Italy Konstantin Lomakin 10.05.2017 Friedrich-Alexander Universität Erlangen-Nürnberg
Introduction
✦ Rectangular Waveguides (RWG) typically deployed e.g. in mm-wave or space applications
✦ 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
4
y
x
z
w
h
SPI-2017 Baveno, Italy Konstantin Lomakin 10.05.2017 Friedrich-Alexander Universität Erlangen-Nürnberg
Current Distribution of the TE10 Mode
5
✦ Current density in conductive material:
✦ Distribution of surface currents on the RWG’s walls:
r⇥H = j!"E + J ' J
Jx,z
Jy
A10 =
s2Pin
whZF⌦p⌦2 � 1
Jz,top
=1
�H
x
e�y�h
�
Jx,top
=1
�H
z
e�y�h
�
Jy,right
=1
�H
z
e�x�w
�
⌦ = f/fc
x
y
z
H(xn) / H(xn = 0)e�x
n
�
Ey = �jA10ZF⌦ sin⇣⇡x
w
⌘
Hx
= jA10
p⌦2 � 1 sin
⇣⇡xw
⌘ Hz = A10 cos
⇣⇡xw
⌘
Transversal Field Components Longitudinal Component
Modeling lossless TE10 Mode
SPI-2017 Baveno, Italy Konstantin Lomakin 10.05.2017 Friedrich-Alexander Universität Erlangen-Nürnberg
Modelling lossless TE10 Mode
7
y
x
z
w
h
3D model Transmission line model
�ll = j� = j!
c
q1� (f/fc)
2
ZL,ll =ZFq
1� (f/fc)2
L0
o
= µ C0= " L
00
o
=µ0w2
⇡2
Hx
Hz
Ey
�ll
=pZ 0Y 0 = j�
ll
= j
s
!2L0o
C 0 � L0
o
L00o
ZL,ll
=
rX 0
Y 0 =
s!2L
0
o
L00
o
!2L00o
C 0 � 1
Z’
Y’
dzL0o L
00
o
/dz dzC0
fc =c0p"r2w
Incorporating Loss Effects
SPI-2017 Baveno, Italy Konstantin Lomakin 10.05.2017 Friedrich-Alexander Universität Erlangen-Nürnberg
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:
✦ Model holds as long as fields don’t degenerate dramatically
9
G0= !C
0tan �
Model currents
Il
dzL0
dzR0
L00/dz
It
R00/dz
dzC0
dzG0
SPI-2017 Baveno, Italy Konstantin Lomakin 10.05.2017 Friedrich-Alexander Universität Erlangen-Nürnberg
Deriving Model Currents
✦ Model currents are derived from field energies and Lo’ and Lo’’ in lossless case:
✦ Model current does not explicitly scale with geometry (w,h) like physical current does!
10
Wm,x
=1
2
ZµH2
x
dV =1
2L
0
o
dzI2l
Il =jp2
s2Pin
p⌦2 � 1
ZF⌦
It =⇡
wp2dz
s2Pin
ZF⌦p⌦2 � 1
Wm,z
=1
2
ZµH2
z
dV =1
2dzL
00
o
I2t
Field distributionI
z
=
Zw
0
Zh+�
h
J
z,top
dydx = j
2
⇡
sw
h
2Pin
p⌦2 � 1
Z
F
⌦
L0
o
= µ
L00
o
=µ0w2
⇡2
SPI-2017 Baveno, Italy Konstantin Lomakin 10.05.2017 Friedrich-Alexander Universität Erlangen-Nürnberg
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:
11
R0=
2
��h
1
�
ZJ2x,y
dV =1
dzR
00I2t
R00=
2w
h⇡2
(w + 2h)
��
1
�
ZJ2zdV = dzR
0I2l
Longitudinal currents
Transversal currents
Field distribution Model
Impact of Losses on the Phase Coefficient
SPI-2017 Baveno, Italy Konstantin Lomakin 10.05.2017 Friedrich-Alexander Universität Erlangen-Nürnberg
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
✦ Final equations for propagation coefficient and characteristic impedance:
13
L0
i =R
0
!=
2
!��h
L00
i =R
00
!=
2w
!h⇡2
(w + 2h)
��
L0= L
0
o
+ L0
i
L00 = L00
o
+ L00
i
� =
s
(R0 + j!L0)
✓1
R00 + j!L00 +G0 + j!C 0
◆
Z =
s
(R0 + j!L0)/
✓1
R00 + j!L00 +G0 + j!C 0
◆
Comparison to Simulation and Measurement
SPI-2017 Baveno, Italy Konstantin Lomakin 10.05.2017 Friedrich-Alexander Universität Erlangen-Nürnberg
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
15
37 37.5 38 38.5 39 39.5 400
2
4
Frequency in GHz
↵in
1/m
h = 1mmh = 2mmh = 3mm
37.3 37.35 37.4 37.45 37.50
5
10
15
20
Frequency in GHz
�in
1/m
h = 1mmh = 2mmh = 3mm
Perturbation Method
SPI-2017 Baveno, Italy Konstantin Lomakin 10.05.2017 Friedrich-Alexander Universität Erlangen-Nürnberg
Measurement: WR10 Waveguide
✦ TRL calibration at waveguide flange
✦ Material: brass; Exact conductivity unknown Estimation from phase coefficient: ~0.5 MS/m
✦ Fabrication tolerances not exactly known Estimating w from phase coefficient: ~2.49 mm
✦ Possible reason for apparently low conductivity: Surface Roughness
16
60 70 80 90 100 1100
5
10
15
20
Frequency in GHz
↵in
1/m
MeasurementProposed Model
60 61 62 63 64 65 66
1
1.2
1.4
Frequency in GHz
�/�
0
MeasurementProposed Model
Perturbation Method
Conclusion
SPI-2017 Baveno, Italy Konstantin Lomakin 10.05.2017 Friedrich-Alexander Universität Erlangen-Nürnberg
Conclusion
✦ Transmission Line Model for RWG only requiring geometry and material parameters
✦ Analytical equations describing propagation characteristics with respect to losses
✦ Very efficient in terms of computation time
✦ 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
18
Thank You very much for Your Attention