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

4. Basic Calculation Concepts

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

40. Calculation Examples

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?