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Cables and transmission lines attached to devices and complex systems may act as parasitic receiving antennas and can guide unwanted radiated electromagnetic disturbances into connected sensitive electronics like sensors or measurement units. In this talk, the basic field-to-wire coupling phenomena will be described. Analytical and numerical calculations will be explained and compared with each other.

- 1. Why the Wire is on Fire – Electromagnetic Field Coupling to Transmission Lines Mathias Magdowski Chair for Electromagnetic Compatibility Institute for Medical Engineering Faculty of Electrical Engineering and Information Technology Otto von Guericke University Magdeburg
- 2. Basic Concepts Sources of Coupling Applicability TL Equations BLT Equations Examples References Disclaimer Source: https://www.istockphoto.com/de/foto/strom%C3%BCbertragungst%C3%BCrme-mit-gl%C3%BChenden-dr%C3% A4hten-gm1439482487-479741079
- 3. Basic Concepts Sources of Coupling Applicability TL Equations BLT Equations Examples References Advice for future Distinguished Lecturer talks Source: https://imgflip.com/i/7z0zaa
- 5. Basic Concepts Sources of Coupling Applicability TL Equations BLT Equations Examples References Survey: Energy Flow in a Cable https: //strawpoll.de/49f8f46 A cable is used to transfer energy from a source (e. g. a generator) to a load (e. g. a light bulb). The current flows through the supply conductor, the load and the return conductor back to the source. Where does the flow of energy from the source to the load actually take place? 1. in the conductors 2. on the surface of the conductors 3. between the conductors 4. There is actually no energy flowing at all.
- 6. Basic Concepts Sources of Coupling Applicability TL Equations BLT Equations Examples References Poll Results Source: https://strawpoll.de/49f8f46
- 7. Basic Concepts Sources of Coupling Applicability TL Equations BLT Equations Examples References Analytical Methods Transmission Line Theory: ▶ well-known ▶ relatively simple ▶ but many simplifications and restrictions
- 8. Basic Concepts Sources of Coupling Applicability TL Equations BLT Equations Examples References Analytical Methods Transmission Line Theory: ▶ well-known ▶ relatively simple ▶ but many simplifications and restrictions Other methods: ▶ BLT equations (Baum, Liu & Tesche) ▶ TLST (Transmission Line Super Theory) ▶ Pertubation Theory
- 9. Basic Concepts Sources of Coupling Applicability TL Equations BLT Equations Examples References Solution with the Equivalent Circuit of the Line Z1 Ut1 L′ · ∆l Etan · ∆l L′ · ∆l Etan · ∆l Ut2 Z2 C′ · ∆l C′ · ∆l Figure: Equivalent circuit diagram of a line ▶ is based on the transmission line theory ▶ suitable for calculations in time and frequency domain ▶ extension: PEEC method (Partial Element Equivalent Circuit)
- 10. Possible Sources of Coupling
- 11. Basic Concepts Sources of Coupling Applicability TL Equations BLT Equations Examples References Possible Sources of Coupling ▶ fields from antennas and other sources of interference ▶ near fields ▶ far fields ▶ approximation by plane waves ▶ fields from other lines ▶ crosstalk ▶ direct, galvanic coupling of an interference current Source: https://pixabay.com/de/photos/ radaranlage-antennen-funkturm-1090106/
- 12. Basic Concepts Sources of Coupling Applicability TL Equations BLT Equations Examples References Crosstalk ▶ cross coupling between lines with small distance ▶ effect originally known from telephone systems, by which one could quietly listen to another conversation ▶ important on printed-circuit board level Source: https://pixabay.com/de/ photos/telefon-alt-holz-w%C3% A4hlscheibe-2663655/
- 13. Basic Concepts Sources of Coupling Applicability TL Equations BLT Equations Examples References When Cables do Crosstalk . . . I’m so lonely, lo-lo-lo-lo-lonely, you’re the one and only, to make my current return . . .
- 14. Basic Concepts Sources of Coupling Applicability TL Equations BLT Equations Examples References When Cables do Crosstalk . . . Okay, let’s look into a brighter future together.
- 15. Basic Concepts Sources of Coupling Applicability TL Equations BLT Equations Examples References When Cables do Crosstalk . . .
- 16. Basic Concepts Sources of Coupling Applicability TL Equations BLT Equations Examples References When Cables do Crosstalk . . .
- 17. Applicability of Transmission Line Theory
- 18. Basic Concepts Sources of Coupling Applicability TL Equations BLT Equations Examples References Cross Section Dimensions and Wavelength Conditions: ▶ cross-sectional dimensions must be small compared to the wavelength ▶ h ≪ λ ▶ k · h ≪ 1
- 19. Basic Concepts Sources of Coupling Applicability TL Equations BLT Equations Examples References Cross Section Dimensions and Wavelength Conditions: ▶ cross-sectional dimensions must be small compared to the wavelength ▶ h ≪ λ ▶ k · h ≪ 1 Reason: ▶ mathematical simplifications ▶ sin(kh) ≈ kh ▶ cos(kh) ≈ 1 − (kh)2 2 ▶ ejkh ≈ 1
- 20. Basic Concepts Sources of Coupling Applicability TL Equations BLT Equations Examples References Single Lines and Multiconductor Lines Single line: ▶ forward and return conductor ▶ double-wire line in free space ▶ single-wire line above a ground plane (that serves as return conductor)
- 21. Basic Concepts Sources of Coupling Applicability TL Equations BLT Equations Examples References Single Lines and Multiconductor Lines Single line: ▶ forward and return conductor ▶ double-wire line in free space ▶ single-wire line above a ground plane (that serves as return conductor) Multiconductor line: ▶ n forward conductors and 1 return conductor yield n + 1 conductors ▶ transmission line theory becomes more elaborate ▶ elegant description with supermatrices
- 22. Basic Concepts Sources of Coupling Applicability TL Equations BLT Equations Examples References Common-Mode and Differential-Mode Current I1 Icm/2 Idm I2 Common-mode current: ▶ leads to radiation (antenna mode) ▶ vanishes at the end of symmetrical circuits ▶ transfers via stray capacitances to other conductors
- 23. Basic Concepts Sources of Coupling Applicability TL Equations BLT Equations Examples References Common-Mode and Differential-Mode Current I1 Icm/2 Idm I2 Common-mode current: ▶ leads to radiation (antenna mode) ▶ vanishes at the end of symmetrical circuits ▶ transfers via stray capacitances to other conductors Differential-mode current: ▶ only very few radiation ▶ calculable by means of transmission line theory (transmission line mode)
- 24. Basic Concepts Sources of Coupling Applicability TL Equations BLT Equations Examples References Common-Mode and Differential-Mode Current Conversion: I1 = Iam + Itl I2 = Iam − Itl (1) Iam = I1 + I2 2 Itl = I1 − I2 2 (2) I1 Iam Itl I2 Figure: Total current as superposition of two modal currents
- 25. Basic Concepts Sources of Coupling Applicability TL Equations BLT Equations Examples References Uniform and Non-Uniform Lines Uniform line ▶ constant cross-sectional dimensions along the line ▶ location independent parameters like per-unit-length capacitance, per-unit-length inductance, propagation constant and characteristic impedance ▶ simple description
- 26. Basic Concepts Sources of Coupling Applicability TL Equations BLT Equations Examples References Uniform and Non-Uniform Lines Uniform line ▶ constant cross-sectional dimensions along the line ▶ location independent parameters like per-unit-length capacitance, per-unit-length inductance, propagation constant and characteristic impedance ▶ simple description Non-uniform line: ▶ changing cross-sectional dimensions along the line ▶ location-dependent parameter like C′ and L′ ▶ more complicated description
- 27. Formulation of the Transmission Line Equations
- 28. Basic Concepts Sources of Coupling Applicability TL Equations BLT Equations Examples References Geometry x y z h −h d0 l k E H Figure: Geometry of a double-wire line and an incident plane wave
- 29. Basic Concepts Sources of Coupling Applicability TL Equations BLT Equations Examples References Taylor Formulation of the Transmission Line Equations dUt(z) dz + jωL′ It (z) = jω h Z −h Bi y dx (3a) dIt(z) dz + jωC′ Ut (z) = −jωC′ h Z −h Ei x dx (3b) ▶ coupled differential equations of 1st order ▶ description of the exciting wave by: ▶ normal to the line incident B-field ▶ transverse to the line incident E-field ▶ solution are forward and backward traveling waves
- 30. Basic Concepts Sources of Coupling Applicability TL Equations BLT Equations Examples References Survey: Coupling into Cables https: //strawpoll.de/frayr29 What causes more current and voltage to be coupled into a line? ▶ greater distance between conductors ▶ longer conductors ▶ thicker conductors ▶ load mismatches at the ends of the conductors
- 31. Basic Concepts Sources of Coupling Applicability TL Equations BLT Equations Examples References Poll Results Source: https://strawpoll.de/frayr29
- 32. Calculation with the Help of the BLT Equations
- 33. Basic Concepts Sources of Coupling Applicability TL Equations BLT Equations Examples References Finite Line Z1 Us Z2 Is 0 zs l z Figure: Finite line with a lumped voltage and current source
- 34. Basic Concepts Sources of Coupling Applicability TL Equations BLT Equations Examples References Origin of the BLT equations ▶ after Carl E. Baum, Tom K. Liu and Frederick M. Tesche ▶ developed 1978 at Kirtland Air Force Base in Albuquerque, New Mexico, USA (a) C. E. Baum (b) T. K. Liu (c) F. M. Tesche Figure: Developers of the BLT equations
- 35. Basic Concepts Sources of Coupling Applicability TL Equations BLT Equations Examples References Not to be Confused With:
- 36. Basic Concepts Sources of Coupling Applicability TL Equations BLT Equations Examples References Not to be Confused With: Figure: BLT sandwich made with the ingredients bacon, lettuce and tomato
- 37. Basic Concepts Sources of Coupling Applicability TL Equations BLT Equations Examples References BLT Equations for Lumped Sources IL1 IL2 = 1 Zc · (1 − A1) 0 0 (1 − A2) · −A1 ejkl ejkl −A2 −1 · ejkzs 2 (Us + ZcIs) − ejk(l−zs) 2 (Us − ZcIs) # (4) UL1 UL2 = (1 + A1) 0 0 (1 + A2) · −A1 ejkl ejkl −A2 −1 · ejkzs 2 (Us + ZcIs) − ejk(l−zs) 2 (Us − ZcIs) # (5)
- 38. Basic Concepts Sources of Coupling Applicability TL Equations BLT Equations Examples References BLT Equations for Lumped Sources Advantages: ▶ simpler, more compact representation ▶ each matrix has a certain meaning ▶ modularization of the equation
- 39. Basic Concepts Sources of Coupling Applicability TL Equations BLT Equations Examples References BLT Equations for Lumped Sources Advantages: ▶ simpler, more compact representation ▶ each matrix has a certain meaning ▶ modularization of the equation Meaning of each matrix: ▶ 1st matrix indicates whether a current or voltage is to be calculated ▶ 2nd matrix contains the line resonances ▶ 3rd matrix contains the sources for excitation of the line
- 41. Basic Concepts Sources of Coupling Applicability TL Equations BLT Equations Examples References Exemplary Transmission Line Dimensions: ▶ line length l = 30 m ▶ conductor spacing s = 20 cm ▶ conductor diameter d0 = 3 mm
- 42. Basic Concepts Sources of Coupling Applicability TL Equations BLT Equations Examples References Exemplary Transmission Line Dimensions: ▶ line length l = 30 m ▶ conductor spacing s = 20 cm ▶ conductor diameter d0 = 3 mm Transmission line parameters: ▶ per-unit-length capacitance C′ = πε 1 arcosh s d0 = 5.685 pF m ▶ per-unit-length inductance L′ = µ π · arcosh s d0 = 1.957 µH m ▶ characteristic impedance Zc = q L′ C′ = 586.7 Ω
- 43. Basic Concepts Sources of Coupling Applicability TL Equations BLT Equations Examples References Exemplary Transmission Line Load impedances: ▶ load at the beginning Z1 = Zc 2 = 293.4 Ω ▶ load at the end Z2 = Zc 2 = 293.4 Ω
- 44. Basic Concepts Sources of Coupling Applicability TL Equations BLT Equations Examples References Exemplary Transmission Line Load impedances: ▶ load at the beginning Z1 = Zc 2 = 293.4 Ω ▶ load at the end Z2 = Zc 2 = 293.4 Ω Exciting wave: ▶ polar angle ϑ = 60◦ ▶ azimuth angle φ = 180◦ ▶ polarization angle α = 180◦ ▶ phase angle β = 0 ▶ amplitude E0 = 1 V m ▶ frequency f = 20 MHz
- 45. Basic Concepts Sources of Coupling Applicability TL Equations BLT Equations Examples References Geometry of the Exemplary Transmission Line x y z h −h d0 l Z1 Z2 k E ϑ . Figure: Geometry of the double-wire line and the incident wave
- 46. Basic Concepts Sources of Coupling Applicability TL Equations BLT Equations Examples References Magnitude of the Current Along the Line 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 0 2 4 6 8 10 Location along the line (in m) |I|/E 0 (in mA m V ) I1 in the forward conductor I2 in the return conductor Itl from decomposition
- 47. Basic Concepts Sources of Coupling Applicability TL Equations BLT Equations Examples References Magnitude of the Current Along the Line 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 0 0.2 0.4 0.6 0.8 1 Location along the line (in m) |I|/E 0 (in mA m V ) Itl from NEC Itl from transmission line theory
- 48. Basic Concepts Sources of Coupling Applicability TL Equations BLT Equations Examples References Magnitude of the Current at the End of the Line 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3 0 0.1 0.2 0.3 0.4 0.5 0.6 k · h |I|/E 0 (in mA m V ) IL2 from BLT equations IL2 from NEC
- 49. Basic Concepts Sources of Coupling Applicability TL Equations BLT Equations Examples References Calculation Examples as Octave Online Buckets Current along the line: https://octav.onl/I_along_tl Current at the end of the line: https://octav.onl/I_l_total_f
- 50. Basic Concepts Sources of Coupling Applicability TL Equations BLT Equations Examples References Outlook Other options: ▶ simulation of transient coupling ▶ single-wire lines over a ground plane ▶ twisted pair lines ▶ shielded lines and coaxial cables ▶ multiconductor lines ▶ transmission line networks ▶ statistics for different incident directions and polarizations
- 51. Basic Concepts Sources of Coupling Applicability TL Equations BLT Equations Examples References Twisted-Pair Cable Figure: Bifilar helix with 5 twists as a NEC model
- 52. Basic Concepts Sources of Coupling Applicability TL Equations BLT Equations Examples References Solution for the Twisted-Pair Cable 107 108 10−5 10−4 10−3 10−2 10−1 100 Frequency, f (in Hz) |I|/E 0 (in mA m V ) twisted (NEC) twisted (BLT) untwisted (BLT)
- 53. Basic Concepts Sources of Coupling Applicability TL Equations BLT Equations Examples References Explanation of the Observed Behavior Colors: external field, forward conductor, return conductor Low-frequency case: Coupling along the line cancels.
- 54. Basic Concepts Sources of Coupling Applicability TL Equations BLT Equations Examples References Explanation of the Observed Behavior Colors: external field, forward conductor, return conductor Low-frequency case: Coupling along the line cancels. High-frequency case: Coupling along the line adds up.
- 55. Basic Concepts Sources of Coupling Applicability TL Equations BLT Equations Examples References For Further Reading I R. W. P. King, Transmission-Line Theory. New York: McGraw-Hill Book Company, 1955. C. R. Paul, Analysis of Multiconductor Transmission Lines, ser. Wiley series in microwave and optical engineering. New York, USA: John Wiley Sons, 1994. C. D. Taylor, R. S. Satterwhite, and C. W. Harrison, “The response of a terminated two-wire transmission line excited by a nonuniform electromagnetic field,” IEEE Transactions on Antennas and Propagation, vol. 13, no. 6, pp. 987 – 989, Nov. 1965.
- 56. Basic Concepts Sources of Coupling Applicability TL Equations BLT Equations Examples References For Further Reading II C. E. Baum, T. K. Liu, and F. M. Tesche, “On the analysis of general multiconductor transmission line networks,” Air Force Weapons Laboratory, Kirtland Air Force Base, Albuquerque, NM, USA, Interaction Note 350, Nov. 1978, eMP 3-39. [Online]. Available: http://ece-research.unm.edu/summa/notes/In/0350.pdf
- 57. Basic Concepts Sources of Coupling Applicability TL Equations BLT Equations Examples References Thank you very much for your attention! https://twitter.com/MarkusRidderbu8/status/ 1523708966039351297 Are there any questions?