Transmission Line Model for Rectangular Waveguides accurately incorporating Loss Effects

Transcription

1 Transmission Line Model for Rectangular Waveguides accurately incorporating Loss Effects Konstantin Lomakin Institute of Microwaves and Photonics Friedrich-Alexander-Universität Erlangen-Nürnberg 10 May 2017

2 Outline Introduction Modeling lossless TE 10 Mode Incorporating Loss Effects Impact of Losses on the Phase Coefficient Comparison to Simulation and Measurement Conclusion 2

3 Introduction

4 Introduction Rectangular Waveguides (RWG) typically deployed e.g. in mm-wave or space applications Fundamental mode of RWG: TE 10 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 y Perturbation method does not take into account any impact on phase coefficient h z w x 4

5 Current Distribution of the TE 10 Mode Transversal Field Components x E y = ja 10 Z F sin w p H x = ja 10 2 x 1 sin w Current density in conductive material: Longitudinal Component H z = A 10 cos x w A 10 = r H = j!"e + J ' J s = f/f c 2P in whz F p 2 1 Distribution of surface currents on the RWG s walls: H(x n ) / Ĥ(x n = 0)e x n J x,z J z,top = 1 Hx ˆ e y h y z x 5 J y J x,top = 1 Ĥ z e y J y,right = 1 Ĥ z e x h w

6 Modeling lossless TE 10 Mode

7 Modelling lossless TE 10 Mode y 3D model H x Transmission line model Z z h ll = j Z L,ll = w = j! c Z F q 1 (f/f c ) 2 f c = c 0 p "r 2w x q 1 (f/f c ) 2 H z E y dzl 0 o ll = p Z 0 Y 0 Z L,ll = = j ll = j r X 0 Y 0 = L 00 o /dz dzc 0 s s! 2 L 0 oc 0 L 0 o L 00 o! 2 L 0 ol 00 o! 2 L 00 o C 0 1 L 0 o = µ C 0 = " L 00 o = µ 0w 2 2 Y 7

8 Incorporating Loss Effects

9 Transmission Line Model for lossy TE 10 Mode Extending lossless model: Conductor losses due to longitudinal currents: R Conductor losses due to transversal currents: R Dielectric losses in electric field: G 0 =!C 0 tan I l dzl 0 dzr 0 L 00 /dz I t dzc 0 dzg 0 Model currents R 00 /dz Model holds as long as fields don t degenerate dramatically 9

10 Deriving Model Currents Model currents are derived from field energies and L o and L o in lossless case: W m,x = 1 2 W m,z = 1 2 Z Z Field distribution s µhxdv 2 = 1 2 L0 odzil 2 I l = p j p 2P in Z F µh 2 z dv = 1 2dz L00 o I 2 t I z = Z w 0 Z h+ h L 0 o = µ L 00 o = µ 0w 2 2 I t = J z,top dydx = j 2 s w p 2 dz 2P in Z F p 2 1 Model current does not explicitly scale with geometry (w,h) like physical current does! s w h 2P in p 2 1 Z F 10

11 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: Longitudinal currents 1 Z J 2 z dv =dzr 0 I 2 l R 0 = 2 h Field distribution Model Transversal currents 1 Z J 2 x,ydv = 1 dz R00 I 2 t R 00 = 2w h 2 (w +2h) 11

12 Impact of Losses on the Phase Coefficient

13 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 L 0 i = R0! = 2! h L 00 i = R00! = 2w!h 2 (w +2h) Final equations for propagation coefficient and characteristic impedance: = Z = L 0 = L 0 o + L 0 i L 00 = L 00 o + L 00 i s 1 (R 0 + j!l 0 ) + G 0 + j!c 0 R 00 + j!l 00 s 1 (R 0 + j!l 0 )/ + G 0 + j!c 0 R 00 + j!l 00 13

14 Comparison to Simulation and Measurement

15 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 in 1/m 4 2 h =1mm h =2mm h =3mm h =1mm h =2mm h =3mm Perturbation Method in 1/m Frequency in GHz Frequency in GHz 15

16 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 1.4 Measurement Proposed Model Perturbation Method Measurement Proposed Model / in 1/m Frequency in GHz Frequency in GHz 0 16

17 Conclusion

18 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

19 Thank You very much for Your Attention