ABSTRACT Due to complexity of electromagnetic modeling, researchers and scientists always look for development of accurate and fast methods to extract the parameters of electronic transmission cables. These parameters play a critical role in various applications like electro surgery, fault location in cables and many more. In this paper, the modeling of stranded multiconductor cable in multilayered dielectric media is illustrated. We specifically determine the distributed parameters of two wire transmission lines in two-layered dielectric media. The effect of frequency change on the parameters is also discussed. This numerical analysis was successfully implemented for modeling of a cable which connects the hand piece of an electro surgical device to its generator. Comparison of the theoretical and the practically measured results is demonstrated with good agreements.

1. Analysis of Stranded Multi-Conductor Cable in Multilayered Dielectric Media
Jimmi James
www.linkedin.com/in/jimmi-james
April-2011
ABSTRACT
Due to complexity of electromagnetic modeling, researchers and
scientists always look for development of accurate and fast
methods to extract the parameters of electronic transmission
cables. These parameters play a critical role in various
applications like electro surgery, fault location in cables and many
more. In this paper, the modeling of stranded multiconductor
cable in multilayered dielectric media is illustrated. We
specifically determine the distributed parameters of two wire
transmission lines in two-layered dielectric media. The effect of
frequency change on the parameters is also discussed. This
numerical analysis was successfully implemented for modeling of
a cable which connects the hand piece of an electro surgical
device to its generator. Comparison of the theoretical and the
practically measured results is demonstrated with good
agreements.
Keywords: stranded conductors, multiconductor transmission
lines, capacitance per unit length, inductance per unit length, skin
effect, multilayered dielectric media.
1. INTRODUCTION
Electro Surgical Units are used for surgical cutting or to control
bleeding by causing coagulation at the surgical site. They deliver
high-frequency electrical currents and voltages through an active
electrode, causing desiccation, vaporization, or charring of the
target tissue.
The power delivered by the electrosurgical generator to the
handpiece electrode is highly critical, since any excess power
delivered to the electrode can lead to burns on patient’s body. The
modeling of the parameters of the cable can therefore be used to
keep a check on the power delivered to the electrode by the
generator. This process is based on the microcontroller-based self
calibration technique.
In this technique, the parameters of the cable are used to identify
the limiting values of the output current and voltage of the power
generator. These calibration values are programmed into a
microcontroller which keeps a check on the power output. During
device operation, the current and voltage levels are varied in such
a manner that the threshold limit of power set by the calibration
values is never crossed. Thus, the power reaching the handpiece is
always within safer limits, ensuring patient safety.
In the past, study and research has been done on different types of
wires, but with the consideration of a single dielectric or a solid
conductor. This paper deals with the numerical analysis of multi
stranded conductors. This makes this analysis more challenging.
Figure 1: Stranded Conductor
The electrical characteristics of a two-wire transmission line
depend primarily on the construction of the line. The two-wire
line acts like a long capacitor. The change of its capacitive
reactance is noticeable as the frequency applied to it is changed.
Since the long conductors have a magnetic field about them when
electrical energy is being passed through them, they also exhibit
the properties of inductance. The values of inductance and
capacitance presented depend on the various physical factors of
the wire. For example, the type of line used, the dielectric in the
line, and the length of the line must be considered.
The effects of the inductive and capacitive reactance of the line
depend on the frequency applied. Since no dielectric is perfect,
electrons manage to move from one conductor to the other
through the dielectric. This illustration of a two-wire transmission
line will be used throughout the discussion of transmission lines;
but, the principles presented apply to all transmission lines [1].
This paper deals with the numerical analysis of the electrical
parameters of a cable which connects the electrosurgical hand
piece with the generator, from the physical parameters of the
conductor and insulator. The conductors are multi-stranded and
are separated by two dielectric materials and this makes this
analysis a little complex.
Figure 2: Handpiece Cable

2. Furthermore, actual circuits are made up of wires, not filaments of
negligible cross section and are wound in layers or channels of
rectangular cross section with insulating material between the
wires. The present work has for its purpose the simplification of
routine calculations of cable inductance, capacitance,
characteristic impedance and effect of frequency on these
parameters. For each case considered a simple working formula is
provided, which are practically verified. Errors in measurement of
the dimensions of existing apparatus will usually be the limiting
factors.
Figure 3: Internal view of the cable
In critical applications like Electro surgery, it becomes important
to use more than one insulating material for the conductors, to
increase the break down voltage of the wires. The power loss and
impedance matching become very important in such cases.
Theoretical approximations should be valid to yield results, which
are much closer to the practically measured values.
2. TRANSMISSION LINE MODEL
A two-wire transmission Line can be approximated by a
distributed parameter network with the circuit parameters
distributed throughout the line as shown in the figure.
Figure 4: Distributed model of transmission Line
• R is the resistance per unit length in  /m
• L is the inductance per unit length in H/m
• C is the capacitance between the conductors, per unit
length
• G is the conductance per unit length in S/m
3. PROBLEM STATEMENT
The parameter modeling of the cable is being done to keep a
check on the power delivered to the handpiece electrode. The
calculated values will act as calibration values programmed in the
microcontroller which controls the output power from the
generator.
We have a single phase two wire cable treated as a lossy
transmission line for which the parameters R, L, C, and G are
nonzero and are themselves functions of complex number and
change with frequency.
The given customized two wire cable consists of two stranded
conductors. Each conductor is insulated by a dielectric, and this
insulated pair is again surrounded by another dielectric which also
acts as a cover. The following parameters of the conductor and the
insulators are given:
Length of the conductor, l
Relative permeability of the conductor, r
Number of strands in each conductor, n
Radius of each strand, b
Thickness of Insulator, 1t
Thickness of Cover, 2t
Dielectric constant of Insulator, 1r
Dielectric constant of Cover, 2r
Resistance per Unit Length, R
Frequency of operation, f
We should determine the value of distributed parameters of this
transmission line. This infers that the following parameters of the
transmission line should be determined:
1) Inductance per unit length
2) Capacitance per unit length
3) Characteristic Impedance
4) Analysis of Skin effect on the parameters
The calculation of these parameters is important to maintain a
control on the output power, which is critical to ensure the safety
of the patient from surgical burns.
4. NUMERICAL RESULTS
For more accurate modeling of the transmission-line distributed
parameters, we must consider several important factors. Material
and structural make-ups, skin effect, and proximity effect, all play
a role. The mathematical analysis and formulae associated with
the transmission line parameters are discussed in this section.
4.1 Inductance Parameter
The inductance of multiple conductors, that is, a group of several
conductors joined in parallel, may be found by fundamental
circuit theory using the foregoing formulas for the self inductance
of a straight conductor and the mutual inductance of parallel
conductors [2].
Only in symmetrical arrangements where the contribution of each
conductor to the total impedance of the combination is the same,
will the resulting formula be simple.

3. Figure 5: A strand in a conductor
The following special cases include equal round wires of radius
‘b’ symmetrically placed.
The inductance of two wires, equal in length ‘l’, separated by a
distance ‘d’ between their centers and joined in parallel is given
by:













8
72
ln002.0
bd
l
lL Equation 1
L is in centimetres.
For the general case of ‘n’ equal round wires spaced uniformly on
a circle of radius ‘a’ and connected in parallel, the inductance is
given by:












 1
2
ln002.0
R
l
lL Equation 2
Where,
 nn
rnaR
1
1

Equation 3
Where ‘r’ is the geometric mean distance of the circular area of
radius ‘b’, given by:
4
1
lnln  br Equation 4
→If a conductor is composed of several strands twisted together,
it is of interest to inquire how the stranding affects the inductance.
The rigorous solution of the problem, especially when the number
of strands is large leads to results that are far from simple, because
in general, the impedance of any individual wire strand does not
involve the same mutual inductances as all the other strands.
Equation 3 however enables a survey of the numerical relations
that hold good in case of a symmetrical arrangement [2].
Two cases are of interest:
4.1.1 Strands of same radius
When the strands used for different arrangements have all the
same cross-sectional radius ‘b’,
The equation 2 will be used for strands in contact, so that they are
equally spaced on a circle of radius,
n
b
a

sin
 Equation 5
The inductance of a single wire is given by,













4
32
ln002.0 '
b
l
lL Equation 6
And by comparing the above result with the result given by
Equation 2, the equivalent radius b’ of a single wire giving the
same inductance can be found.
Note: The inductance of the stranded wire is less than that of a
single strand.
4.1.2 Constant cross section of conductor
If radii nb of the strands are decreased as the number of strands
is increased, to keep the cross section of the conductor the same,
i.e.
22
bnn   Equation 7
The value of 'b
b
for larger values of ‘n’ would be near to unity.
A good approximation to the inductance of a solid stranded
conductor will be to find the inductance of the solid wire having
the same cross-sectional radius. This furnishes the upper limit of
the inductance of the stranded conductor.
The equivalent inductance of the two wires is given by,










 
'2
cosh10
b
d
L r
eq

 Equation 8
Summarizing the above steps to determine inductance:
(1). Determine self inductance of a single conductor with ‘n’
strands
(2). Determine the equivalent radius b’ of a single strand having
the same inductance
(3). Treating each conductor as one with single strand, calculate
the equivalent inductance
a
b
Strand
Conductor

4. 4.2 Capacitance Parameter
Now considering b’ as the radius of the single stranded conductor,
the capacitance per unit length can be calculated as follows.
Figure 6 : Parallel cylindrical wires of radius R
Considering two cylindrical conductors of radius ‘a’, with their
centers separated by distance ‘d’. The capacitance per unit length
has been modelled and calculated [5-8]. It is given by,








'
1
cosh
b
d
C
 (F/m) Equation 9
Often the cylinders are wires and it is appropriate to approximate
this result for large ratios of
b
d .
Thus, the capacitance per unit length is approximately,
Equation 10
In case of two dielectrics, with dielectric constants 1r and 2r
respectively, the equivalent capacitance between the two
conductors can be determined by following the lines of the
analysis of a parallel plate capacitor separated by two dielectrics.
The capacitance between the conductors is given by:













'
1
2'
2
1
021
12
lnln
b
d
b
d
C
rr
rr

 Equation 11
Where,
)(2 1
'
1 tbd 
)(2 2
'
2 tbd 
)(2 21
'
ttbd 
1t and 2t represent the thickness of the dielectric material used
for the Insulator and the Cover.
Equation 11 gives the capacitance per unit length between the two
conductors. For transmission line modeling, the capacitance is
defined between the conductor and the neutral [11]. This is shown
in Figure 6.
Figure 7: Capacitance between two conductors
Therefore, the value of capacitance per unit length is given from
Figure 6 as,













'
1
2'
2
1
021
12
lnln
2
2
b
d
b
d
CC
rr
rr
eq

 Equation 12
Figure 8: Parallel conductors separated by multiple dielectrics
The equivalent dielectric constant of a material which separates
the two conductors, having a thickness of 1t or 2t , such that the
capacitance between the conductors is the same as eqC
, can be
determined as follows:







'
0
ln
b
d
C r
eq
 Equation 13
0
'
ln









b
d
Ceq
r
Equation 14
The equivalent dielectric constant r can be calculated from the
above expression.
4.3 Skin Effect
Skin effect reduces current density at distances away from the
surface. This is true for solid or stranded conductors [9]. Skin
effect can be reduced by using stranded rather than solid wire.
This increases the effective surface area of the wire for a given
gauge. Change in frequency has a very less effect on the
capacitance parameter of the cable, whereas the resistance and
inductance parameters show variation.







'
ln
b
d
C


5. 4.3.1 Effective Resistance
As frequency increases, the depth of penetration into adjacent
conductive surfaces decreases for the boundary currents
associated with electromagnetic waves. This results in the
confinement of the voltage and the current waves at the boundary
of the transmission line, thus making the transmission line lossier.
The skin effect causes the effective resistance of the conductor to
increase with the frequency of the current. The skin depth is
calculated as:



2
 Equation 15
r 0
Where,
ρ is the resistivity of the conductor
ω is the angular frequency of the current
μ is the absolute magnetic permeability of the conductor
For long cylindrical conductors such as wires, with diameter ‘D’
larger compared to δ, the resistance is approximately that of a
hollow tube with wall thickness δ carrying direct current. The
effective resistance is approximately,
 









2D
R
  





D
 
Equation 16
Where,
 is the length of the conductor.
D is the diameter of the conductor.
The final approximation above is accurate if D >> δ.
4.3.2 Inductance per unit length
The expression obtained for the values of inductance of two
parallel conductors at high frequencies is as follows:






 
d
D
L 1
cosh

 Equation17
Where,
d -> diameter of the conductors
D -> distance between the conductors
The equivalent radius “R” of the conducting region is calculated
as follows:
Total area of conductor = Number of strands x Area of each strand
 NrR
Where,
r -> radius of each strand
N->Number of Strands
4.4 Characteristic Impedance
Reference [8] gives us the expression for the characteristic
impedance of the pair of conductors, which is given by:






 '0 ln
120
b
d
Z
r
Equation 18
Where
‘d’ is the distance of separation between the two conductors
(centre to centre)
b’ is the radius of conductor
r is the dielectric constant of the material separating the
conductors.
The value of r can be calculated from Equation 9 above.
5. RESULTS
A comparative study of the theoretical and the practically
measured values of the cable parameters are discussed in this
section. The practical measurements were made with an LCR
meter at different operational frequencies. The model of the LCR
meter used was HP 4263B.The cable used for measurement has
copper conductors and the two dielectrics as Polyethylene (PE)
and Poly Vinyl Chloride (PVC). The open circuit capacitance and
the short circuit inductance values have been tabulated as under.
A transmission line parameter calculator has also been developed
using macros and can be implemented in various applications
involving transmission lines. Basic inputs like geometry of
conductors, dielectric constants for the multiple layers etc. need to
be fed and the transmission line parameters are obtained as the
output.
The specifications of the cable used for measurement are as
follows:
Table 1: Specifications
S.
No
Parameter Value Unit
1 l 3600 mm
2 n 41
3 2b 0.08 mm
4 1t 0.15 mm
5 2t 0.23 mm
6 1r 2.5
7 2r 4
8 R 93.25 Ω/km
9 f 100000 Hz
10 r 1

6. 5.1.1 Inductance
Table 2: Comparision of Inductance Values
Frequency
(kHz)
Measured Value
(uH/m)
Theoretical Value
(uH/m)
100 0.597 0.603
1 0.504 0.469
5.1.2 Capacitance
Table 3: Comparison of Capacitance Values
Frequency
(kHz)
Measured Value
(pF/m)
Theoretical Value
(pF/m)
100 49.166 43.392
1 48.721 43.392
If we consider the presence of only a single dielectric, say Poly
ethylene (PE) with a dielectric constant value of 2.5 and thickness
0.15mm. Theoretical calculations yield a capacitance value of
241.48pF for a 3.6m long cable. The effect of adding an
additional layer of dielectric over the existing layer can be
observed. Theoretical calculations show that, the capacitance of
the cable reduces and the practically measured values also show
the same trend.
5.1.3 Effective Resistance
Table 4: Comparison of Resistance Values
Frequency
(kHz)
Measured Value
(Ω/m)
Theoretical Value
(Ω/m)
100 0.175 0.171
1 0.0182 0.0171
5.1.4 Characteristic Impedance
Table 5: Characteristic impedance
Measured Value
(Ω)
Theoretical Value
(Ω)
97.62 109.83
It can be observed from the above results that the theoretical
values stand very close in agreement with the practically measured
values.
6. APPLICATIONS
In critical applications like electro surgery, these parameters play
an important role in controlling the output power of the surgical
device. This is based on the microcontroller-based self calibration
technique. This ensures the safety of the patient by preventing
unintended burns in patients [9].
Wiring is pervasive from private, commercial and military aircraft
to the space shuttle, modest homes to massive skyscrapers,
communication networks for business, entertainment, data
collection, and control of critical systems, over land power lines
to nuclear reactors and power plants, trains, warning systems, and
switching stations to ships, dockyards, cranes, and autonomous
loading systems, and even down to the “simple” family car.
As these buried wires age, they may begin to crack and fray or
their connectors may break, corrode, leak or be damaged in
careless maintenance. Detecting and locating these faults is
extremely important. The capacitance and inductance can be used
to measure the length of wires and the distance to faults.
The capacitance value can be used to measure and detect the
presence of a dielectric material or human, humidity and water
content micro imaging, position measurement, angular position
and angular speed measurement, liquid level, pressure and
temperature measurements [3].
7. CONCLUSION
The mathematical modeling of the electrical parameters of a two-
wire cable is demonstrated in this paper. Comparative study of the
theoretical results and the measured values showed close
agreement. The current analysis is being successfully
implemented in critical application of electro surgery for output
power control and can also be implemented in many more
applications as discussed. The effect of frequency change on the
parameters is also discussed.
These observations can help in making the correct choice of a
cable for an application. Further work is in progress towards the
extension of the analysis to a bundle of more than two conductors
and the inclusion of the parasitic effects on the parameter values.
The effect of transmission lines can also be included in
applications like Automotive Wire Harness Simulation and
Design, for interconnect modeling of the on-chip clock
distribution for high-performance microprocessors to reduce on-
chip interconnect delays (VLSI), for characterizing cable
parameters for cable manufacturers, for analysis and design of
medical instruments and many more.
Moreover, the applications can also include modeling and
simulation of high voltage transmission lines with frequency
dependent parameters.
The development of a transmission line parameter calculator
based on the research done is one of the achievements.
8. REFERENCES
[1] “Principles of Transmission Lines”,
www.tpub.com/content/neets/14182_118.htm
[2] Frederick W. Grover, Inductance calculations: working
formulas and tables
[3] You Chung, Nirmal N. Amarnath, Cynthia M. Furse and
JohnMahoney,” Capacitance and Inductance Sensors for

7. Location of Open and Short-Circuited Wires”, University of
Utah
[4] Transmission Line Fundamentals, e-book on mit.edu.
http://web.mit.edu/6.013_book/www/chapter14
[5] H. E. Green, “A Simplified derivation of the capacitance of a
two-wire transmission line,” IEEE Transactions on
Microwave Theory and Techniques, v 47, pp 365-366, Mar.
1999.
[6] B. B. Wadell, Transmission Line design handbook, Artech
House, Norwood, MA, 1991.
[7] W. H. Hayt, Engineering Electromagnetics, 5th edition,
McGraw-Hill Book Co. 1989.
[8] B. Whitfield Griffith, “Radio-Electronic Transmission
Fundamentals”, Atlanta, Ga.: Noble Pub. Co. 2000
[9] Electrosurgery for the Skin. Barry L. Hainer M.D., Richard
B. Usatine, M.D., American Family Physician (Journal of the
American Academy of Family Physicians), 2002 Oct :1259-
66.
[10] The Quest Group,” Cable Design, Theory vs Evidence”,2006
[11] “Power System Analysis”, Electrical Engineering,
Webcourse
http://nptel.iitm.ac.in/courses/Webcourse-contents/IIT-
KANPUR/power-system/ui/Course_home-1.htm