Transmission Line Model for Rectangular Waveguides ... May 2017 Transmission Line Model for Rectangular Waveguides accurately incorporating Loss Effects Institute of Microwaves and

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]

mailto:[email protected]?subject=

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


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


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


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


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


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


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


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


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


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


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

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Thank You very much for Your Attention

Documents

Transmission Line Model for Rectangular Waveguides ... May 2017 Transmission Line Model for Rectangular Waveguides accurately incorporating Loss Effects Institute of Microwaves and