Why the Wire is on Fire - Electromagnetic Field Coupling to Transmission Lines

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

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

Advice for future Distinguished Lecturer talks
Source: https://imgflip.com/i/7z0zaa

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.

Poll Results
Source: https://strawpoll.de/49f8f46

Analytical Methods
Transmission Line Theory:
▶ well-known
▶ relatively simple
▶ but many simplifications and restrictions

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

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)

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/

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/

When Cables do Crosstalk . . .
I’m so lonely, lo-lo-lo-lo-lonely, you’re the one
and only, to make my current return . . .

Okay, let’s look into a brighter future together.

Applicability of Transmission Line Theory

Cross Section Dimensions and Wavelength
Conditions:
▶ cross-sectional dimensions must be small compared to the wavelength
▶ h ≪ λ
▶ k · h ≪ 1

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

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)

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

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

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)

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

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

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

Formulation of the Transmission Line Equations

Geometry
x
y
z
h
−h
d0
l
k
E
H
Figure: Geometry of a double-wire line and an incident plane wave

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

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

Poll Results
Source: https://strawpoll.de/frayr29

Calculation with the Help of the BLT Equations

Finite Line
Z1
Us
Z2
Is
0 zs l z
Figure: Finite line with a lumped voltage and current source

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

Not to be Confused With:

Not to be Confused With:
Figure: BLT sandwich made with the ingredients bacon, lettuce and tomato

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)

Advantages:
▶ simpler, more compact representation
▶ each matrix has a certain meaning
▶ modularization of the equation

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

Exemplary Transmission Line
Dimensions:
▶ line length l = 30 m
▶ conductor spacing s = 20 cm
▶ conductor diameter d0 = 3 mm

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

Load impedances:
▶ load at the beginning Z1 = Zc
2 = 293.4 Ω
▶ load at the end Z2 = Zc
2 = 293.4 Ω

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

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

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

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

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

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

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

Twisted-Pair Cable
Figure: Bifilar helix with 5 twists as a NEC model

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)

Explanation of the Observed Behavior
Colors: external field, forward conductor, return conductor
Low-frequency case:
Coupling along the line cancels.

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.

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.

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

Thank you very much for your attention!
https://twitter.com/MarkusRidderbu8/status/
1523708966039351297
Are there any questions?

Why the Wire is on Fire - Electromagnetic Field Coupling to Transmission Lines

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