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The calculation of low-voltage fault currents with respect to time

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Title:
The calculation of low-voltage fault currents with respect to time
Creator:
Kilgore, W. Ronald
Publication Date:
Language:
English
Physical Description:
vi, 104 leaves : illustrations ; 29 cm

Subjects

Subjects / Keywords:
Electric power distribution ( lcsh )
Electric power systems ( lcsh )
Low voltage systems ( lcsh )
Electric power distribution ( fast )
Electric power systems ( fast )
Low voltage systems ( fast )
Genre:
bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

Notes

Bibliography:
Includes bibliographical references (leaves 103-104).
General Note:
Submitted in partial fulfillment of the requirements for the degree, Master of Science, Electrical Engineering.
General Note:
Department of Electrical Engineering
Statement of Responsibility:
by W. Ronald Kilgore.

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Source Institution:
|University of Colorado Denver
Holding Location:
|Auraria Library
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
37364469 ( OCLC )
ocm37364469
Classification:
LD1190.E54 1996m .K55 ( lcc )

Full Text
THE CALCULATION OF LOW-VOLTAGE
FAULT CURRENTS WITH RESPECT TO TIME
by
i W. Ronald Kilgore
B.S., Southern College of Technology, 1988
A thesis submitted to the
University of Colorado at Denver
in partial fulfillment
of the requirements for the degree of
Master of Science
Electrical Engineering
1996


Kilgore, W. Ronald (M.S., Electrical Engineering)
The Calculation of Low-Voltage Fault Currents with Respect to Time
Thesis directed by Professor Pankaj K Sen
ABSTRACT
This thesis explores the calculation of fault current in low-voltage (below 600 volts)
electric systems. A method is presented to calculate all practical circuit impedance
values by taking into account the change in resistance during a fault due to change in
conductor temperature. The purpose of the calculations is to present fault current
with respect to time rather than the maximum conceivable fault current that is
produced with standard calculation methods. Sample calculations and the source code
of a C++ computer program are provided to demonstrate the calculation methods
verified by experimental results
This abstract accurately represents the content of the candidates thesis. I recommend
its publication.
Signed
xv


CONTENTS
Chapter
1. Introduction...............................................................1
2. Calculation of Impedances .................................................4
2.1 Transformer Impedances...................................................4
2.1.1 Single-Phase Transformer ...............................................5
2.1.2 Single Transformer with a Center-Tapped Secondary......................7
2.1.3 Three-Phase Bank Connected Wye-Wye ....................................9
2.1.4 Three-Phase Bank Connected Delta-Wye................................. 11
2.1.5 Three-phase Bank Connected Wye-Delta (with a four-wire secondary) .... 13
2.2 Conductor Impedances................................................... 15
2.2.1 Resistance............................................................ 15
2.2.2 Capacitance...........................................................25
2.2.3 Inductance............................................................25
2.3 Impedance of Conductors in Conduit to a Ground Fault....................27
2.4 Resistance of Electric Arc..............................................28
3. Temperature Rise Calculations.............................................31
3.1 Calculation of Heat Produced in Conductor .............................31
3.2 Change of Conductor Temperature due to Heat Generated in Conductor .... 32
3.3 Calculation of Heating Efficiency ......................................33
3.4 Temperature Effect on Fault Calculations................................35
4. Calculation and Test/Measurement Verification ...........................36
5. Proposed Changes in the National Electrical Code ........................54
v


6. Proposed Product Changes...........................................56
7. Summary ...........................................................57
Appendix
A. Source Code for C++ Program ......................................58
B. Output of C++ Program.............................................87
C. Tabulation of Measured Fault Current..............................102
References........................................................... 103
vi


1. Introduction
Electric faults in low-voltage systems can be extremely costly to life and property. For
this reason, much effort has been spent on the calculation of available fault currents as
part of system design. The primary purpose of most fault calculations is to assure
proper protection coordination and equipment ratings. For this reason, most fault
calculations are interested in the maximum possible fault currents. Often, in the field
of failure analysis, the goal of fault calculations is not to find the maximum possible
fault current under the worst case scenario, but to find the probable fault current given
a set of circumstances. The methods commonly used to calculate available fault
current in electric systems make several very conservative assumptions (e.g. neglecting
the circuit resistance). Some methods do not neglect circuit resistance but assume the
circuit resistance is constant throughout the fault. This thesis examines the effects of
dynamic circuit resistance in low-voltage fault conditions.
The primary difficulty in determining probable fault currents is accurately determining
system impedances. In most fault-current calculations it is usually acceptable to use
conservatively low estimates of system impedances because the goal is to determine
the maximum possible fault current. One goal of this research is to locate and apply
calculation methods that will accurately depict the system impedances in order to
determine probable fault currents and compare results with common calculation
methods. The circuit impedances that will be examined include transformer
impedances with several common connection methods, circuit impedance, and fault
resistance (i.e., arc resistance).
1


The temperature of conductors during a short circuit can rise almost instantaneously
during fault conditions because circuit resistance is the largest part of the total
impedance and heat is generated within the conductors due to I2R losses. Because the
resistance of a conductor is directly proportional to the temperature of that conductor,
the temperature rise of the conductors greatly influences the total circuit impedance,
and therefore the total fault current. Also, as the resistance of the conductor
increases, the conductor heats more rapidly. This temperature rise effect on the time-
dependant fault current should be examined for proper protection coordination.
The temperature rise calculations will be modeled by calculating the heating efficiency
of the conductor. This heating efficiency is dependant on the fault duration and the
specific heat of the conductor material and cable insulation. Heat produced in the
cable is stored in the cable with less efficiency as time passes. This calculation method
has been successfully tested for fault durations of up to two seconds as part of this
thesis. The calculation method can be extrapolated beyond two seconds, but that has
not been tested. If the circuit has unusual thermal properties (i.e., immersed in water,
or gas insulated), special considerations must be made that are beyond the scope of
this research.
Presently the National Electrical Code (NEC) has no requirements for maximum
circuit lengths. The NEC does indicate minimum conductor sizes for branch circuits1;
however, the use of extension cords often decreases the conductor size, increases the
length, and limits the fault current to well below the instantaneous trip rating of the
circuit breaker. This thesis proposes changes to the NEC to include a maximum
circuit length for each conductor size protected by each circuit breaker rating and the
use of inline fuses in extension cords.
2


This thesis includes:
1. Method of calculation of equivalent impedances with several
different transformer connection methods.
2. Method of calculation of the impedances of electric circuits in most
typical installations.
3. Method of calculation of the impedances of electric conduits as
ground return circuit.
4. Method of calculation of instantaneous heating and associated
resistance change of insulated conductors during fault conditions.
5. Sample calculation of a faulted low-voltage circuit with laboratory
verification.
6. Proposed National Electrical Code and product design changes.
7. A C++ computer program to perform calculations presented in this
thesis.
The units presented in this thesis will be found to he inconsistent. This was done
purposely to accurately reproduce the calculation method of each reference. The
sample calculations were performed using a mathematical computation software that
automatically adjusts for units. The units are internally consistent in the sample
calculations.
3


2. Calculation of Impedances
Most impedances used in typical fault calculations are determined by looking for the
circuit device in standard tables. Due to the necessity of performing the calculations
described in this thesis by the use of a computer program, it is desirable to calculate all
cable impedances. This method will also enable more precise calculation of the
specific circuit in question rather than a general circumstance. The impedances applied
in the calculations will be applied in the ohmic values rather than per-unit values. This
is done because the faults examine will generally be only on the secondary of the
distribution transformer. The calculations could also be performed equally well in the
per-unit method.
2.1 Transformer Impedances
The name-plate impedance of most transformers is an adequate start to determine the
impedance of the transformer bank. When a transformer experiences a fault within or
electrically close to the secondary, the impedance of the transformer changes from the
steady-state value2. For purposes of this work, the fault current will be limited
primarily by the secondary cables. Short-circuit impedance of the transformers does -
not become a factor until the fault current reaches a level of 15 to 20 times the rated
current3. At 15 to 20 times the rated current, the transformer resistance value
increases due to winding temperature rise, and the reactance increases due to magnetic
forces causing winding movement. Because the current is greatly limited by the
secondary cables, the transformer impedance is not effected by faults deep within the
secondary distribution.
4


When two or three transformers are connected to form a three-phase bank, the system
must be modeled accommodating the particular connection scheme. The transformer
impedance must take into account the method chosen for connecting the transformer.
The connections that will be examined include:
1. Single-phase transformer
2. Single-phase transformer with a center-tapped Secondary
3. Three-phase bank connected Wye-Wye
4. Three-phase bank connected Delta-Wye
5. Three-phase bank connected Wye-Delta (with a four-wire secondary)
6. Three-phase bank connected Delta-Delta (with a four-wire secondary)
7. Three-phase bank connected Open Wye-Open Delta (with a four-wire
secondary)
2.1.1 Single-Phase T ransformer
The single transformer is handled using simple per-unit impedances using the name
plate data. The transformer is modeled as an impedance equal to the name plate
impedance as seen in Figure 2.1. If both resistive and reactive components are not
given, the impedance value can be assumed to be at an angle of between 75 and 80
(X/R ratio of 4 to 6). The larger the transformer, the greater the impedance angle:
5


Connection Diagram
Primary
(High Voltage)
Secondary
(Low Voltage)
3
Equivalent Impedance Model
Z=R+jX
Converted to
Ohmic values on
the secondary.
Figure 2.1
Model for single-phase transformer. Transformer impedance is that of the name-plate
converted to an ohmic value.
6


2.1.2 Single Transformer with a Center-Tapped Secondary
This connection is most commonly used for residential loads with a 120/240-volt
secondary. The neutral is derived by bonding the center tap to ground. The
determination of the transformer impedance to a ground fault must take into account
the half-winding impedance of the transformer. The impedance of the transformer to a
ground fault as seen in Figure 2.2 is calculated as follows:4
- 1.5 R{% + j 2.0 Xt % (2.1)
Where: R B is the resistive component of the nameplate per unit impedance
based on the transformer full winding KVA.
X 3 is the reactive component of the nameplate per unit impedance
based on the transformer full winding KVA.
The 1.5 and 2.0 multipliers found in equation 2.1 are rule of thumb values. Equation
2.1 is accurate for the majority of transformers that are wound in shell form If the
transformer is wound in core form the inductive multiplier is decreased from 2.0 to
1.15 due to the difference in winding geometry.
If the fault in question is between two energized conductors, the transformer is
modeled like the single-phase transformer in section 2.1.1 with the secondary voltage
equal to the line-to-line voltage.
7


Connection Diagram
Equivalent Impedance Model
\
% Z= Zhalf = 1.5Rt% + 2.0 Xt%
m Converted to
Ohmic values on
^ the secondary.
Figure 2.2
Schematic connections and equivalent circuit of single-phase transformer with
secondary center-tap grounded.
8


2.1.3 Three-Phase Bank Connected Wye-Wye
The three-phase bank of transformers connected Wye-Wye is modeled as one single-
phase transformer with a single-phase-to-ground fault as seen in Figure 2.3. If the
fault is phase-to-phase in nature, the transformer impedance must be doubled to
account for the two transformers involved in the fault. Also, care must be taken to use
the line-to-line voltage rather than the line-to-neutral voltage. Similar results can be
derived from the use of symmetrical components.
9


Connection Diagram
Equivalent Impedance Model
Z=R+jX for Line-to-Neutral
or Ground Faults.
Z=2R+j2X for Line-to-Line
faults.
Converted to Ohmic
values on the secondary.
Figure 2.3
Three-Phase Wye-Wye connected transformer with single-phase model.
10


2.1.4 Three-Phase Bank Connected Delta-Wye
The Delta-Wye transformer connection is modeled similar to the Wye-Wye
connection. The primary difference is that ground fault current will circulate within
the primary windings because there is neither a ground connection nor a neutral
conductor on the primary side of the transformer bank. The equivalent circuit model is
seen in Figure 2.4.
11


Connection Diagram
Z=R+jX for Line-to-Neutral
or Ground Faults.
Z=2R+j2X for Line-to-Line
faults.
Converted to Ohmic
values on the secondary.
Figure 2.4
Three-Phase Delta-Wye connected transformer with single-phase model.
12


2.1.5 Three-phase Bank Connected Wye-Delta (with a four-wire
secondary)
The delta connected secondary, seen in Figure 2.5, is often used for commercial three-
phase loads. One of the three transformers is center-tap grounded in order to provide
a secondary neutral for 120-volt loads. This provides for two lighting phases and one
power phase. The phase-to-phase faults are modeled like the single-phase fault in
section 2.1.1 with care taken to use the line-to-line voltage for the fault voltage.
The phase-to-neutral faults are modeled differently depending upon the faulted phase.
If the faulted phase is one of the two phases connected to the center-tapped
transformer (i.e., a lighting phase), the transformer bank is modeled using the half-
winding impedance discussed in section 2.1.2. If the faulted phase is the phase that is
not connected to the center-tapped transformer (i.e., the wild phase), the
transformer bank equivalent impedance is modeled as follows5:
Z + Z. ,,
Z = -!-----^ (2.2)
2
Where:
Z is the equivalent impedance of the transformer bank to a power
phase-to-ground fault.
Z, is the transformer nameplate impedance.
ZhalJ is the half-winding impedance of the center-tapped transformer.
13


Primary
(HV)
Connection Diagram
Secondary
(LV)
OOOOOOOOOOOOOOOQDOOOOOOr
Equivalent Impedance Model
Z =
Zt + Zhalf
Converted to Ohmic
values on the secondary.
Figure 2.5
Three-Phase Wye-Delta transformer with single-phase model.
14


Equation 2.2 is derived by observation that the fault current has two parallel paths in
the center-tapped transformer and the two power transformers. The total impedance
of each path is divided by two.
If the transformer bank is connected in an open wye-open delta, the system is modeled
identically except for the condition of the power phase-to-ground. In this condition
the sum of the half-winding and nameplate impedances are not divided by two because
there is no parallel path for the fault current.
2.2 Conductor Impedances
The secondary conductors from the service entrance conductors to the individual
branch-circuit conductors offer the greatest impedance to a bolted fault. For this
reason, the calculation of cable impedance is of paramount importance to proper fault
current calculations. For proper impedance calculation both the resistance and
reactance must be calculated.
2.2.1 Resistance
In order to calculate the total resistance the dc resistance must first be determined.
Then proper multipliers for skin and proximity effects must be calculated to determine
the ac resistance of a cable.
2.2.1.1 dc Resistance
The dc resistance of a conductor is calculated with the following formula:6
15


Where:
R is the resistance of the conductor (in ohms).
p is the resistivity of conductor (8.145 10"K------------------ for copper
and 1.254 x 10 K ^ x inch for aluminUm).
/ is the conductor length in feet,
a is the cross-sectional area of the conductor in inch2.
This dc resistance must then be adjusted to the conductor temperature with the
formula7:
^t2= ^/,0 + at^2~ *i)) (2.4)
Where: tj and t2 are conductor temperatures (in C)
R. is the conductor resistance at temperature tj (in ohms)
n
R. is the conductor resistance at temperature t2,C (in ohms)
a, is the temperature coefficient of resistance of the conductor
n
material at temperature tj (unitless)
The temperature coefficient of resistance (a) is dependent on temperature and must be


calculated for the conductor temperature. As the conductor temperature changes
during a fault, the coefficient of resistance must be recalculated using the following
equation:
a
h
1
1
+ (<2-'l)
(2.5)
Where: tl and t2 are the initial and final temperatures respectively (C).
a. is the resistance temperature coefficient at tx (unitless)
n
a. is the resistance temperature coefficient at t2 (unitless).
The previous equation assumes that the temperature coefficient of resistance is linear.
That assumption is valid in all but very cold temperatures.
2.2.1.2 ac Resistance
Due to the magnetic flux present due to alternating current flow, the ac resistance is
larger in magnitude than the dc resistance. This greater resistance is due skin and
proximity effects. The ac resistance is determined by first calculating the dc resistance,
and then multiplying this value by the skin and proximity effect multipliers (7^ and Ycp
respectively). The resistance change due to 60 Hz alternating current is greater for
larger conductors and can usually be ignored for conductors smaller than #6 AWG.8
2.2.1.2.1 Skin Effect
In a dc circuit the conductor can be modeled as a solid rod because the current density
is uniform throughout the cross-section of the conductor. Alternatively, the current in
17


an ac circuit actually travels largely along the circumference of the conductor. The
current density decreases from the surface of the conductor in inverse proportion to
the square root of the frequency. The greater concentration of current along the
circumference of the conductor lessens the effective cross-sectional area of the
conductor. Because the conductor resistance is inversely proportional to conductor
cross-sectional area, the ac resistance is greater than the dc resistance. The ratio of
to Rfc is called the skin-effect ratio (Y^).9 Skin effect is caused by magnetic lines of
flux internal to the conductor causing greater flux density toward the center of the
conductor. The increased flux density increases the inductive reactance of the center
portion of the conductor forcing the current to the outer conductor portion.10
The formula for the skin-effect ratio for a circular conductor is as follows:
Y=St
cs
nrJJnr)
IJfar)
(2.6)
Where is the skin-effect ratio
Si is the function indicating the real part of the argument
r is the radius of the conductor (meters)
JQ and Jj are Bessel functions of the first kind and of order zero and one
respectively
n =
\
-jyy-
P
(2.7)
18


where: 00 is the frequency (radians per second)
|i is the absolute permeability (4tt;x 10"A henrys per
meter if conductor is nonmagnetic, either copper
or aluminum)
p is the resistivity of the conductor material
(1.7241 x 10'* Q-meter for copper and
2.83 x 10"x Q-meter for aluminum)
2.2.1.2.2 Proximity Effect
When electric current flows in a
conductor an associated
magnetic field is formed around
the conductor. Two adjacent
conductors with identical current
flowing in opposite direction
form magnetic fields that attract
the current toward the adjacent
conductor. This is caused by the
vector addition of the magnetic
lines between the conductors
and cancellation on the external
sides of the conductors. This
effect causes additional inductive reactance in the external portion of the conductors
and an increased current density on the adjacent portion of the conductor (Figure 2.6).
19


The proximity effect also reduces the effective cross-sectional area of the conductor
thereby increasing the conductor resistance. This is calculated by determining a
multiplier (YC[) for the dc resistance of the conductor. Proximity effect for circular
conductors is calculated as follows:
Y F(x )
cp V P' s
1,18
F(xp) + 0.21
+ 0.312
( 2a\2
\
Where: Yf; is the proximity effect multiplier (no units),
a is the conductor diameter (inches).
s is the axial spacing between conductors (inches)
Ffej) is the function:
F(xp)=Bt
Xp/piXp)
2 J,(xp)
(2.8)
(2.9)
Where:
x =
p
\
izR
dc
(2.10)
A similar proximity effect takes place when a conductor is placed in a conductive
conduit. An increased resistance occurs because of the current that is induced in the
conduit. The resistance multiplier, Yp, for nonmagnetic conduit is determined as
follows:11
20


(2.11)
Where:
Y f(CI)
* 80 o
M
21
/ is the frequency (in hertz)
Cl is the conductor area (in circular inches)
p is the conductor resistivity
v £
Where:
1
p=e ve + 0
[a^+b*?- 2a*b *cos(nlAB)\ (2.12)
LAB is the angle formed between lines joining the
center of the pipe with the center of two
conductors (in degrees)
a and b are the distances between the center of the pipe
and the center of conductor the two conductors
(A and B) divided by the mean pipe radius
(unitless). See figure 2.7.
15.94 cor? __j
v =-----------x 10 1
P;
(2.13)
Where: r is the mean pipe radius (in inches)
t is the pipe thickness (in inches)
p. is the resistivity of the pipe material
(in microhm-inches)
21


22


A more straightforward simplified approximation for calculating the proximity effect in
a nonmagnetic pipe, Yp, is found in NEHER-McGRATH.12
Rdc is the dc resistance of the cable within the pipe (microhms per foot)
is the outside pipe diameter (inches)
/ is the frequency (Hz)
s = 0.578 D for close triangular spacing (inches)
D.- D
s = - for wide triangular spacing (inches)
where: D is the inside pipe diameter (inches)
The model introduced in NEHER-McGRATH for the pipe proximity effect, Yp, for -
magnetic pipes is also used:13
for a three conductor cable:
(2.14)
Where: Rs is the conduit resistance (microhms per foot)
Ds is the outside cable insulation diameter (inches)
1.545- 0.115 D
p
(2.15)
R
dc
23


for a single conductor in close triangular configuration (see figure 2.8):
Y =
p
0.895 0.115 D,
R
dc
(2.16)
for a single conductor cable in cradled configuration:
Y =
p
0.345 0.175 D
___ p
R
dc
(2.17)
24


Where all variables are identical to those used for the nonmagnetic pipe proximity
effect.
2.2.2 Capacitance
Typically the capacitance within a low-voltage system is so little that the effect is
neglected.14 The only time that capacitance of a low-voltage system is crucial is when
calculation of fault current availability on a system with an ungrounded transformer
secondary (i.e., a Delta-connected secondary) is necessary. These systems are
designed to limit fault current to a value below a level that would cause equipment
damage during a fault. The ground-fault current available for a fault on an
ungrounded secondary system is determined by the capacitance to ground of the
system. This design is usually employed only on systems that require constant
operation under all circumstances.15 Ungrounded secondary systems are beyond the
scope of this thesis; therefore, the capacitance of the electrical components will not he
examined.
2.2.3 Inductance
In order to calculate the inductance of a single-phase electric circuit a return path must
be specified. The return path may be a neutral conductor, the grounding conductor, or
an unintended return path. The neutral and grounding conductors can be
mathematically modeled by standard inductance formulas. Any unintended path would
have to be considered independently but general formulas may apply.
The inductive reactance of one of two cables not routed in conduit and overhead
distribution lines is determined by the following general equation:16
25


h=
+ 2 In
D '
iAv
x 10
-7
(2.18)
Z,j is the inductance of one of two conductors (in henrys per
D is the distance between the center of each of two conductors (in
ry, r2 are the radii of each conductor (in meters).
If the radius of each conductor is the same, the preceding equation will simplify to the
following:17
2.57 S
Where: A
meter). D
meters). n,r2
X1 = kx log
D
(2.19)
Where: is the positive sequence reactance of one of two conductors with
both self and mutual inductance accounted for.
kx is a constant that adjusts for conductors in proximity to magnetic
pipes. For conductors routed in steel pipes k .= 60.9. For
conductors in nonmagnetic pipes or free air k .= 52.9. The
magnetic constant yields 15% higher results.
Sis the geometric mean spacing between the conductors. For cables in
close triangular configuration S=D (the diameter of cables
including insulation). For the more typical cradled
configuration:
26


\ 2
(2.20)
Where Dp is the inside diameter of the pipe.
2.3 Impedance of Conductors in Conduit to a Ground Fault
The impedance of cables in conduit is greatly influenced by the return path of a ground
fault. If current returns on a neutral or ground conductor, the impedance can be
calculated by methods outlined in previous equations. However, if the current returns
along conduit, the impedance is dependant upon the degree of conduit magnetism and
the magnitude of the ground-fault current.
The primary model of impedance of a ground-fault circuit involving conduit-current
flow is that developed by FISHER18. This method calculates a multiplier (M) to the
positive sequence impedance of the conductors within the conduit that adjusts for the
coaxial-type currents in this situation and the magnetic circuit of the conduit. This
calculation is an approximation developed by empirical data only.
The basic equation for the model is the following:
(2.21)
27


Where:
Zqj)U0 is the angle-phase line to housing impedance (ohms).
Zj is the positive sequence impedance of the conductor within the
conduit (ohms).
iy N
M= (2.22)
ix
Where: i is the RMS current within conductor/conduit
N and P are determined by table 2.1:
2.4 Resistance of Electric Arc
The resistance of a low-voltage electric arc has been the subject of a great many
studies. The primary model that will be used in this thesis is that of FISHER.19 This
simple model was developed using empirical data from many low-voltage arcing tests.
The resistance of the arc has been calculated with sufficient accuracy for low-voltage
arcs on 120-volt through 480-voh systems. The equation for the arc resistance is as
follows:
R
arc
25JL
i085
(2.23)
Where: Rarc is the resistance of the electric arc (in ohms).
L is the length of the arc (inches).
/ is the RMS current passing through the electric arc (amperes).
28


Table 2.1: Constants for calculation of ground fault impedance in conduit.
Cables in Alut nirnrm Conduit
Cable Size (AWG) Cables in Steel Conduit Copper Wire Aluminum Wire M
N P Minimum M Maximum M M Cable Size (AWG)
18 1.50 0.00 1.50 1.60 1.15 1.10 18
16 1.65 0.00 1.50 1.70 1.15 1.10 16
14 1.80 0.02 1.50 1.80 1.15 1.10 14
12 2.20 0.04 1.50 2.00 1.15 1.10 12
10 2.80 0.06 1.60 2.10 1.15 1.10 10
8 3.40 0.08 1.60 2.20 1.15 1.10 8
6 5.00 0.12 1.60 2.70 1.15 1.10 6
4 6.40 0.14 1.60 3.20 1.15 1.10 4
3 7.60 0.16 1.70 3.50 1.15 1.10 3
2 9.20 0.18 1.70 3.70 1.15 1.10 2
1 12.00 0.21 1.70 4.00 1.17 1.11 1
0 15.00 0.23 1.70 4.40 1.17 1.11 0
2/0 18.00 0.25 1.80 5.00 1.40 1.27 2/0
3/0 25.00 0.28 1.90 5.80 1.40 1.27 3/0
4/0 30.00 0.30 2.00 6.50 1.28 1.20 4/0
250 37.00 0.32 2.00 6.80 1.27 1.20 250
300 43.00 0.33 2.00 7.00 1.26 1.20 300
350 49.00 0.34 2.00 7.20 1.25 1.20 350
400 55.00 0.35 2.00 7.30 1.25 1.20 400
500 66.00 0.37 2.00 7.50 1.25 1.20 500
mo 80.00 ML- ml- 1 70 600
The length of the arc varies with each situation. For most self- extinguishing arcs, the
length is usually approximated between 0.1 and 0.2 inches. Arcing faults on 480/277-
volt systems are often not self-extinguishing and therefore the arc length can be up to
several inches. Each specific installation requires independent analysis and
assumptions.
Experimental results dictate that arcing faults on 120-volt systems are usually self-
29


extinguishing. In these situations an arcing fault will last from 0.5 to 15 cycles.20
Voltages at the 208-volt level and above are often not self extinguishing and can last
for several seconds or until the overcurrent protection operates.
30


3. Temperature Rise Calculations
3.1 Calculation of Heat Produced in Conductor
Conductor resistance is directly proportional to the temperature of the conductor
material. Because the conductor temperature during an electric fault can rise very
rapidly due to IeR losses, the temperature rise acts to increase the circuit resistance.
For this reason the temperature of the conductor must he determined before an
accurate determination of the circuit resistance can be calculated. The determination
of this temperature rise can be greatly simplified by the use of a model presented by
EICHORN.21 EICHORN has modeled the temperature rise in insulated conductors
for both adiabatic and nonadiabatic conditions.
To determine the instantaneous temperature rise of the cable first the heat generated in
the cable must be determined. This amount of generated heat is equivalent to the
following:
Q=I2Rt (3.1)
Where: Q is the heat generated in the conductor (joules)
/ is the current within the conductor (amperes)
R is the resistance of the conductor (ohms)
t is the time interval (seconds)
31


3.2 Change of Conductor Temperature due to Heat Generated in
Conductor
The value of heat generated (Q) previously calculated can be used to determine the
temperature rise by the following:
(3.2)
ffl V
Where: A T is the change in temperature for the time interval used to
calculate Q (Kelvin)
m is the mass of the conductor in which the heat is
generated (gram)
c is the specific heat constant:
c = 174 717 j e- for copper
lb C
c = 429 197 J0}1^ for aluminum
lbC
il ffQY- is the heating efficiency as calculated by the
EICHORN method (see section 3.3)
The mass of the conductor is determined by multiplying the volume of the conductor
by the density of the conductor material (0.32117 ^ for copper and
inch1
0.09765 for aluminum).
inch1
32


3.3 Calculation of Heating Efficiency
The heating efficiency of the conductor is the efficiency at which the heat produced in
the conductor is stored in the conductor. As the temperature increases during a short
circuit, heat is transferred to the cable insulation that is in direct contact with the
conductor. Because of this heat loss to the insulation all of the heat produced within
the conductor cannot be assumed to apply directly to temperature increases of the
conductor. To account for the heat loss to the insulation a heating efficiency factor
(y\heattn^ is applied. This factor is calculated as follows (see table 3.1):
T1
heating
1
1 etf(x)~ 1+2
(3.3)
I? n 2TtrJoX
Where: x Fyjt-------------
F is the contact factor that indicates the degree of thermal
contact between the conductor and the insulation
(usually 0.80 to 0.95)
t is the elapsed time into the fault (seconds)
r is the radius of the conductor (inches)
A is the insulation thermal conductivity
( 0.07 0.10 FTU_ for pyc insulation)
sec//1 F
a is the volumetric specific heat for the insulation
33


oc is the vohimetric specific heat for the conductor
q is the area of the conductor (square inches)
As seen from the equation, t\heating is dependant on the conductor and insulation
specifications, and elapsed time. For this reason i\heating can be calculated for the circuit
before any fault calculations are begun. Typical values of volumetric specific heat for
common conductor and insulation materials are shown in the following table:
Table 3.1: Volumetric Specific heat (a anda ) for various materials
Material Volumetric Specific Heat BTU \
{ inch1 F)
Copper 0.002948
Aluminum 0.022068
Polyvinyl Chloride (PVC) 0.014249
Polyethylene 0.01848
34


3.4 Temperature Effect on Fault Calculations
For each new temperature, the new temperature coefficient of resistance must be
calculated by the method outlined in equation 2.5. From this point in the calculation
the new resistance of each cable must be calculated due to the temperature rise. The
total fault current is then recalculated and additional heat is generated in the
conductor. The circular nature of these calculations continues until the fault can he
assumed to clear, or the temperature of the conductor exceeds the conductor melting
point of the metal (1981 F for copper and about 1200F for aluminum).
As can be expected, the change in resistance due to the instantaneous temperature rise
of the conductor can have a dramatic effect on the fault current with respect to time.
The analysis of this calculation includes comparison of the initial fault current and the
corresponding fault current decay with the time-current curves for the fuse or circuit
breaker protecting the circuit. If the ignition temperature for the insulation around the
smallest conductor within the circuit is exceeded before the overcurrent protective
device enters the operation region of the time-current curve, then a fire could result
from an electric fault on circuit in question.
35


4. Calculation and Test/Measurement Verification
A one-line diagram of the circuit that was used in the laboratory during verification of
this thesis is seen in figure 5.1. The measured values were obtained with a computer-
aided data acquisition system sampling the fault current at 2 kHz. The instantaneous
RMS value of the fault current was calculated and is presented in Appendix C. The
sample calculations were performed in MathCad (a registered trademark of Math Soft,
Cambridge, MA) and are seen as follows:
SOkVA 13.2 kV primary 208/120 V secondary 4% Impedance Pad-mounted transformer (r\y) jj 156 feet of #t AWGAI cable, In Al conduit, typeXHHW, 1/phase 10 feet of #18AWG lamp cord, PVC, no conduit
3 92 feet of500 kcmil Al a i_ cable In underground ZA/*sl PVC conduit, 1/phase, typeXHHW 137 feet of #12 AWG Copper conductor, 1/2 inch steel conduit, type THHN
Figure 4.1
One-line diagram of laboratory test circuit
36


Transformer Impedance:
ZpU := 4-%-Ej'85des Zpu = 0.00349 +0.03985j
V := 120-volt VA = 50000-volt amp
Z -V2
ZY:= Zv = 0.001 +0.01148j -ohm
x VA x
Number of segments:
Seg = 4 n = 1.. Seg -
Length of each segment :
lj = 92 ft 12 =156-6 ^ = 137-ft 14 = 10 ft
Diameter of conductor that makes up each segment:
dia^ := 708-mils dia^ = 289-mils dia^ = 80.8-mils dia^ = 40.3-mil
Insulation thickness for each conductor:
inSj := 95-mils ins^ = 40-mils in^ := 30-mils ins4 = 30-mils
Separation of conductors for inductance and proximity effect:
sep := dia + 2-ins
n n
Now die resistance of each segment is calculated at 20 degrees C:
p cu := 1.7241-10 6-ohmcm p aj := 2.8624-10 6-ohmcm
Pi = P al P2 := PJ
m Cl P cu P4 = P3
37


Sep
<------------->
Figure 4.2
Separation of conductors.
5.01264-105
a =
8.3521-104
6.52864-103
cmil
1.62409-103
a
3.43496-10
2.06154* 10~4
0.00159
0.00639
ohm
ft
The resistance of each segment is then calculated:
Rdc rdc
n n
R
dc total
:= XR
dc
Rdc =
Rdc_total = a31681ohm
38
0.00316
0.03216
0.21763
0.06386
ohm


The dc resistance must be adjusted for skin effect:
rad
a := 2-jc -60----
sec
p := 4-Jt 10 Permeability of free air
m
The Skin-effect ratio is seen in the following:
dia
The radius of each conductor.
90.96835 90.96835j
N :=

YCS :=
n
/
N -r Jn
n n U
K-r.)
2J
i I
Ycsn = RefYcsnj 1
N =
Yes =
90.96835 90.96835j
117.21259 - 117.21259j
117.21259 - 117.21259j
0.00926
2.58843-10
4.36028-10
-4
2.6983-10
-7
Yep is the adjustment factor for proximity effect to other conductors. This factor
is calculated for circuit segments that have the circuit conductor return path
rather than over conduit.
39


1.15676
0.47218
0.1701
0.08484
M o(x);= J o(x-j15) M i(x) =J i (x-j1:5)j2
0 0(x) = arg(M 0(x)) 0 ^x) := arg(M j(x))
M(x)
0 l(x) 0 0(x)
y%-m(xpJ-
/ dia
\Sq,n /
1.18
M/xp \ + 0.27
+ 0.312-
/dia
n
lsepJ
Ysp =
0.02543
7.23623-10"4
6.42297* 10-6
1.92572* 10"7
Now for the proximity effect contribution of a nonmagnetic pipe:
First the dc resistance of the pipe must be calculated. Segment #2 is the
only one that has a non-magnetic pipe.
^pipe^41 ODpipe:=415in t:=
pipe pipe
pipe''1
pipe -
it -
N.1
2 /
rs :=
Pal
pipe
40


. 2
apipe = 0-96015'in
r = 14.08442
pohm
ft
Ds = seP2
sm pipe
s := 1.55-in^ + 0.58-dia2
= n>
pipe
x
cp2
X
cp2
3-r
I
0.10901*%
Segment #3 has a magnetic conduit and the Xcp is
calculated as follows (assuming cradled arrangement):
a := 1.7 for steel conduit.
D_. = 0.7- in
P3
sep3 Dp3
0.34------- + 0.175----
xcp3
:= a-
m m
( ^3) (lohm
x
cp3
= 0.01823*%
cp
0
0.10901
0.01823
0
%
41


Rac_ = r dc_' (1 + Ycsn + YsPn + *q>rK
n n
Rac
0.00327
0.03223
0.21767
0.06386
ohm R ac totaj IR ac
Rac total =0-31703
The difference between ac and dc resistance can be seen as follows:
RaCj Rdc1
KdCj
ac3-Rdc3
= 3.46851 %
= 0.01931 %
R ac2 R dc2
dc
R
ac.
dc.
= 0.20726 %
= 4.62403*10~5
Rdc Rdc,
3 4
As seen in the previous calculation, the ac portion of the resistance can be
ignored for all circuit sized conductors (conductors smaller than #8 AWG).
The total ac resistance of the circuit is:
%
4
R ac tot ~ Yj Rac Rac tot=0-31703 'ohm
* -- n
n= 1
42


The total ac resistance can then be compared to the total dc resistance:
Rdctot = Y Rdc Rdctot = 0-31681-ohm
i i n
n= 1
^ ac tot dc tot
-dc tot
= 0.06891 -%
As seen the contribution of the ac portion of the
total circuit resistance is very minimal.
The inductance of each segment is calculated as follows:
k =52.9 =60.9 kv =60.9
1
*2
*3
S2==
1.26- sep
/ \ 2
sep2
DP ~sep2
P2 2j
kY := 52.9
X4
S3==
1.26-sep,
1 -
f sep3
S2 =0.46414 -in
S3= 0.17548 -in
Sj := sep1
S4 = sep4
Xn=kxH
2.57-S
dia
. polrnij
^ ift
X =
0.0025j
0.00585j
0.00623j
4.26336-10_4j
ohm
43


The X/R ratio of the circuit in question is seen as:
4
XoverR =
^ac tot
n= 1
XoverR= 0.0473257736
As seen in the preceeding equation, the X over R ratio is much less than 1.
For crude calculations the reactance can be neglected with similar answers
The initial fault current with all conductor temperatures at
20 deg.C is seen as the following:
The initial fault current has been calculated and now effects of change
in resistance as the conductor temperature rises during fault.
First the Temperature Coefficient of Resistance must be input:
^tot ^ ac tot+ ^ tot
Ztot = 0.31703 +0.015j -ohm
120- volt
i= 188.55823 *amp
l -
0.00403
a
degC
0.00393
3" degC
44


he specific heat of conductor material is as follows:
cal
0.092- c := 0.226
gm- degC
cl =cal c2:=cal
c3 = c cu C4 = C CU
cal
gmdegC
he density of the conductor material is used to calculate the mass
f the conductor as follows:
dencu = 8.87-
gm
cm
den^ := 0.09765-
lb
3
m
den1 := den ^
de^ := den al
der^ = den
cu
den4 :=dco
mass := a 1 den
n n n n
mass =
42.44218
11.99123
2.7013
0.04905
1
The thermal coefficient of resistance is as follows:
0.00403
degC
a
cu
0.00393
degC
The heating efficiency, as a function of short circuit time,
is calculated by the following method:
X fas := 0.08-
BTU
hr-ft -degF-ft
c ins
0.293
BTU
Ib-degF
45


a c c cu ^encu
F := 0.85
a ins c ins
1.35
62.247-
dia
2-71-
c ins^ ins
M = F-
n
m
n v
X :=M
n
n
240
sec
eff :=
n
1 erf(xn
eff=
0.9958
0.98977
0.96423
0.93052
46


The efficiency matrix can be set up before the calculation of the heating:
q = 1.. 480 dt =------sec
240
h i sec 500
eflj
: =
^3,1 efij
lv J efij
t + dt
q

1+2
mi'A
lq+l
\q+l
\q+l
n3,q+l
. ^4,q+ 1
M2'/q/
1 erf M,- t 1+2


1+2

m4.

i +,jr
1+2

47


The iterative calculation to determine the temperature rise, the resistance
increase, and the corresponding decrease of current is as follows:
q := 1.. 480 dt =
240
ll,l
2,1
3,1
l4,l
a
U
a
2,1
a
3,1
a4,l
RU
*2,1
*3.1
R
4,1
20- degC
20- degC
20- degC
20- degC
a al
aal
a
cu
a
cu
R
R
R
R
ac,
ac-
ac-,
ac.
sec
48


x,
dt-
T2,q-
masSj-Cj
MX,
mass/c
n.
'2,q
T3,q +
* w**
mas^'C3
:wx,
n
'3,q
dt-
4,q mass.-c. 4, 4 4
1, q+i
r
2, q+l
r3,q+l
r4,q+l
at,q+ 1
a2,q+l
3,q+l
X, 1
4,q + 1
^l,q+ 1
*2^+1
^3,q+ 1
R4,q+1
Vl .

masSj-Cj
1

/ 1 \ [*wx.l
(2,q/ + masSj-Cj '^.q
/ 1 \ \3,q/ + *wx, mas^*c3 *3*
i
/ 1 \ kq/ + *wx mass/c. 4 4 *U,q
\q 1+(l.q)- masSj-Cj \q
\q 1 + Kq) mas^Cj \q
V 1 + Kq) mas^Cj \q
V 1 + Kq)- *(/(%) mass/c. 4 4 j '\q
120-volt

49


he results of laboratory short-circuit tests on the identical circuit are read in from a data file a
ollows (See Appendix B for the tabulated results):
u = 1..100
imeas : =READ( data ) amp tmeas :=READ(time) sec
u u
*meas = 186.6312amp Ij = 188.55823amp
*i~*meas
----------- = 1.0219%
he results of the iterative calculation is seen in the following four graphs:
Time (Seconds)
Figure 4.3: Calculated Current vs. Time
50


(Amperes) Current (Amperes)
measu
Time (Seconds)
Figure 4.4: Measured Current vs. Time
51


As seen in the previous graph the measured value of fault current compares favorably to the
calculated value.
The temperature of the smallest conductor reaches over 300 deg. C in less than 2 seconds.
The circuit resistance has changed during the fault due to the conductor temperature change. T
total change is seen in the following calculation:
^finaltotal _Rl,480 + R5,480 + R3,480 + 4,480
Rfinal_total = 040846-ohm Rac_total =0.31703 ohm
Rac total Rfinal total ..
----=------------=---- =28.860?%
R dctotal
52


The 28.9% change in circuit resistance results in
the following change in total fault current:
Ij = 188.55823 amp
I4g0= 146.57978-amp
- =22.26286-%
>1
The 22.3% change in fault current can be very important in calculating proper circuit
protection and coordination of protective devices.
53


5. Proposed Changes in the National Electrical Code
Article 310 of the National Electrical Code (NEC) has provisions for minimum circuit-
breaker sizing for circuit size conductors.22 There is no provision indicating minimum
size of extension cord or maximum length of branch circuit conductors. This lack of
requirement can lead to situations where the maximum available fault current is below
the instantaneous trip level of the branch overcurrent protective device. This can lead
to a fault that lasts several seconds, heating the circuit conductors, and thereby
decreasing the fault current. As the fault current decreases circuit breaker tripping is
delayed even further.
The most common conductor insulation is PVC. When PVC reaches a temperature of
about 200 C the material begins releasing a flammable vapor that can be ignited with a
nearby open flame or electric arc. From the test conducted in the laboratory, the vapor
from the rapidly heating insulation is rapidly forced from the ends of the insulation.
Although at room temperatures PVC insulation will not sustain combustion, at
elevated temperatures the material usually will continue to bum freely due to the
increased heat supplied by the heating conductor.23 Although admittedly this is not a
typical situation, this problem can occur on circuits with the fault current limited
primarily by a single small conductor. The small conductor will heat rapidly while the
other larger conductors remain much cooler. The problem of short-circuit heating of
conductors can become much more apparent when the over-current protection of the
circuit is inadequate, fails to operate, or is defeated.
The NEC does contain a provision that states the circuit impedance shall not be of a
54


level that interferes with the proper operation of the over-current protection.24 The
NEC offers no guidelines or methods to insure this provision is met and most users of
the code do not perform analysis necessary to determine fault current levels. In order
to comply with this article of the NEC, an appendix could be referenced in which a
table is placed that outlines a method for calculating the maximum lengths of low-
voltage branch circuits given conductor sizes. The table could be as seen in Table 3.
Table #5.1: Unit resistance per 10 fee of circuit.
Conductor 15 A Circuit 20 A Circuit 30 A Circuit
Size Breaker Breaker Breaker
10 0.025 0.034 0.050
12 0.046 0.061 0.092
14 0.082 0.109 0.163
16 0.183 0.245 0.367
18 0326 0435 ... -usa
Table #3 is used by measuring the length of circuit necessary to feed desired load and
adding the number found in the table for each 10 feet of given circuit size. Care must
be exercised to include each cable within the circuit and expected appliance cords.
The sum total of all unit resistances must not be over 1.00. If the sum total exceeds
1.00, the fault current available at the load will likely he less than the instantaneous trip
region of the circuit breaker.
The values for Table #3 are obtained by calculating the single-phase, per-unit resis-
tance of the conductor on a 120-volt base. The current base is assumed to be 10 times
the circuit breaker rating, the level at which most circuit breakers enter the
instantaneous tripping region. The values of the resistance are then multiplied by 2 to
account for the return circuit.
55


6. Proposed Product Changes
The product design changes that result from the work surrounding this research are
primarily in the area of protection of extension cords. With current design practices,
nothing prevents the use of several extension cords in series, thereby greatly limiting
the fault current to well below the instantaneous trip level of a branch circuit breaker.
The branch circuit breaker is not designed to protect this situation.
A simple remedy for this problem is to insert a fuse in the male attachment plug on the
extension cord. This could protect the cord from continuous overload and fault
currents below the trip level of the branch circuit breaker. This fused plug design is
currently used in self-regulating heat cables and Christmas-tree lights. The attachment
plug for each device contains a fuse holder to hold a bayonet fuse. The same
arrangement could be installed on extension cords to limit the fault current and thereby
limit the short-circuit heating of the conductors. The fuse rating for this situation
should be a time-delay fuse. This will allow for starting of electric motor driven
appliances while protecting the cable from short-circuit heating.
56


7. Summary
Short circuit calculation methods that are common in engineering practice are very
useful but are overly conservative for predicting the response of low-voltage systems
under fault conditions. The two assumptions that lead to the greatest error are (1)
neglecting circuit resistance and (2) neglecting change in circuit resistance due to
temperature rise of faulted conductors. The calculation methods in this thesis address
both of these assumptions and present methods that more precisely calculate the
available fault current throughout the fault than commonly used methods. Some of the
applications of this analysis are as follows:
Calculation of the short circuit temperatures of faulted conductors in order to
determine possibility of conductor temperatures igniting conductor insulation.
Calculation of fault current with respect to time in order to determine proper
coordination of overcurrent protection devices.
Calculation of expected trip times for overcurrent devices.
The analysis has been compared very favorably to fault current measurements
completed in the laboratory.
57


Appendix A: Source Code for Gt+ Program
^include
#include
^include
^include
#include
struct proj_data{
char eng[40];
char comp [40];
char proj_name[40];
char file_name[10];
double priavai;
int avaljnput;
int calc_per_sec;
};
struct. seg_data{
char ident[40];
double length ;
double dia;
double rad;
double temp;
double row;
double alpha;
double sep;
int path;
int mail;
int ins_matl;
int conduit;
double rac;
double x;
int set;
double mass;
double density;
double spec heat;
58


double insspecheat;
double volspecheat;
//THIS IS THE AREA WHERE ALL FUNCTIONS ARE LISTED,
GENERALLY IN THE ORDER IN
//WHICH THEY APPEAR.
int graph();
int in_window();
int mscm();
int bhiscm (int 1, int t, int r, int b);
int seginput(int);
int segin();
int rac(int);
int drac(int, double);
int x(int);
int calc();
double erf(double); //this calcs the error function for the value x
//int ini_struct();
int total(int);
int load(); //this loads saved system data
int mass(); //this calcs the mass of the conductor
int save(); //this saves the system data to a file,
int heat efiQ; //this sets the heating efficiency variable.
//THESE ARE THE GLOBAL VARIABLES:
double i[481]; //the array that holds the current
int maxn; //the number of segments currently in system
extern seg data seg[10]; //the structure name that holds the segment data
extern projdata info;
extern FILE *fptr;
float eff[10][481];
float v; INoltage at fault
const float pi=3.1415926;
int calculated=0; //This is used as a flag for calculation
59


void main (void)
{
clrscr(); // this clears the screen
max_n=0;
in_window(); // this function is the first input screen//
return;
}
in_window()
{
register int i;
register int j;
int s;
bhiscm(20,5,60,22);
// this section writes the message to the screen
gotoxy(5,3);
cputs(" 1. Input System Data.");
gotoxy(5,5);
cputs("2. Input Segment Data. ");
gotoxy(5,7);
cputsC'3. Load System Data from File.");
gotoxy(5,9);
cputs("4. Calculate Fault Current.");
gotoxy(5,ll);
cputs("5. Graph Results to Screen.");
gotoxy(5,13);
60


cputs("6. Save System Data to File.");
gotoxy(5,15);
cputs("7. Exit Program.");
gotoxy(5,18);
cputs("Make Your selection: (1-7)");
s = (getch());
switch(s){
case'l': inscm ();break;
case'2': segin();break;
case '3': load();break;
case'4': calc();break;
case'5': graph();break;
case'6': save();break;
case 7': _exit(0);
default: break; }
in_window();
return 0;}
int calc() {
const cmax_n=max_n;
int n; //This is the segment number
int cps; //calculations per sec. This will not be here in final ver.
int q;
double dt;
double dT; //This is the change in temperature of the conductor,
double denominator;
double numerator;
double k; //This is the factor in the heating eff. calc
FILE *fptr;
float temp[10][481];
double heating_eff[10,481];
61


//area of each segment conductor
double area[10];
double bot;
float time=0; //this is the running time for output file
// double a; //temp for area
// double radius; //temp for radius
// double vsh; //temp for seg[n].vol_spec_heat
// double ef; //temp for efficiency
calculated=l;
mass();
bluscm(15,10,65,17);
gotoxy(5,3);
cout"What is the fault voltage?";
gotoxy(5,5);
cinv; // TfflS IS A GLOBAL VARIABLE.
bluscm(25,12,55,17);
gotoxy(5,3);
cout"One Moment Please....";
gotoxy(5,5);
cout"Results written to OUT.TXT";
cps=480;
dt=2.0/cps;
//This calculates the initial fault current,
for (n=l;n temp [n] [ l]=seg[n] .temp;
rac(n);
x(n);
area[n]=pow(seg[n].dia/2,2)*pi;
};
total(l); //THIS FUNCTION CALCULATES THE FAULT CURRENT
FOR
//ITERATION #1
62


heat_efl(); //this function sets up the heating efficiency
//matrix
for (q=2;q for (n=l ;n numerator=dt*i[q- l]*i[q-1] *seg[n] ,rac*eff[n] [q];
denominator=seg[n] .mass* seg[n]. sp ec_heat;
dT=numerator/denominator;
seg[n] ,temp=seg[n] .temp+dT;
temp [n] [q]=seg[n] .temp;
drac(n,dT);

total(q); //This function will calc the fault current.

fptr=fop en( "out.txt"," w");
fprintf(fptr,"This program was written by W. Ronald Kilgore\n");
fprintf(fptr,"as a partial requirement for an MSEE degree\n");
fprintf(fptr,"from the University of Colorado at Denver.\n\n\n");
fprintf(fptr,"Project Name: %s.\n",info.proj_name);
fprintf(fptr,'"Engineer's Name: %s.\n",info, eng);
fprintf(fptr,"Comp any Name: %s.\n",info.comp);
fprintf(fptr,"____________________________\n");
for (n= 1 ;n fprintf(fptr," Segment %iname: %s.\n",n,seg[n].ident);
fj)rintf(fptr,"Segment Length: %f feet.\n",seg[n].length);
fprintf(fptr,"Segment Diameter: %f inches.\n",seg[n].dia);
fprintf(ij)tr, "Final Conductor Temp: %f DegC.\n",seg[n].temp);
fprintf(fptr,"Conductor Separation: %finches.\n",seg[n].sep);
if (seg[n].matl=l)
fprintf(fptr,"Conductor material is Copper.\n");
if (seg[n].matl=2)
fprintf(fpfr,"Conductor material is Aluminum.\n");
63


if (seg[n].conduit=l)
lprintf(fptr,"Segment is not in conduit.W);
else
if (seg[n].conduit=2)
fprintf(fptr,"Segment is routed in magnetic conduit.W);
if (seg[n].conduit=3)
fprintf(fptr,"Segment is routed in nonmagnetic conduit.W);
if (seg[n].path=l)
fprintf(iptr,"Conduit is return path.W);
else
fprintf(fptr,"Conduit is not in return path.W);
if (seg[n].ins_matl=l)
fprintf(fptr, "Insulation is PVC.W);
if (seg[n].ins_matl=2)
fprintf(fptr, "Insulation is PE.\n");
fprintf(fptr,"\nW);

fprintf(fptr,"Time (sec)\tl (amps)\t");
for (n=l;n fprintf(fptr,"Seg. %i\t",n);};
fprintf(fptr,"Seg. %iW,cmax_n);
for (q= 1 ;q fprintf(fptr,"'%f\t%f\t",time,i[q]);
time=time+dt;
for (n=l ;n ^)rintf(§)tr,"%f\t",temp[n] [q]);};
fprintf^tr, "%f\n",temp [cmaxn] [q]);
};
fclose(fptr);
return 1;
}
64


//THIS FUNCTION CALCULATES THE CHANGE IN RESISTANCE
AND ALPHA
drac(int n, double dT) {
seg[n].rac=seg[n].rac (l.+seg[n],alpha (dT));
seg[n]. alpha=l/(( l/seg[n]. alpha)+(dT));
return 1; }
total(int q) {
complex ztemp;
complex ztot=0;
float itemp;
const complex j=sqrt(complex (-1));
intn;
for (n=l;n ztemp =complex(seg[n].rac+ (seg[n].x)*j);
ztot=ztot+ztemp;
};
i[q]=v/(2 *(abs(ztot)));
return 1;
}
int x(int n) {
float k;
double arg;
//The next section adjusts X for magnetic conduit
if (seg[n]. conduit=2) {k=60.9;}
65


else
k=52.9;
arg=2.57* seg[n]. sep/seg[n]. dia;
seg[n].x=k (loglO(arg)) le-6 (seg[n],length);
return 1;
>
//This function calculates the initial dc resistance for each segment
int rac(int n) {
double area;
double r_20;
double alpha;
const double pi=3.1415926;
area=((pow(seg[n].dia/2,2) (pi)));
r_20=seg[n].row*seg[n].length/area;
if (seg [n] ,temp=20.) {
seg[n].rac=r_20;
return 1;}
else {
alpha=l/(( l/seg[n]. alpha)+(seg[n]. temp-20.));
seg[n].rac=r_20*(1.0+alpha*(seg[n].temp-20.0));

return 1;
}
mass() {
66


int n;
double num;
double den;
for (n=l;n seg[n].mass=seg[n], length* 12.0*pow(seg[n].dia/2,2)*3.141592*seg[n]. density;
};
return 1;
}
double erf(double x) {
int n; //this is the counter
int q;
double total; //this is the running total
double top; //this is the numerator
double bot;
double next;
double inside;
double fact=l;
int p;
total=0;
top=l;
inside=0;
for (n=l;n<50;n++) {
fact=l;
top=l;
for (q=n;q>0;q~) {
fact=fact*q;};
p=2*n+l;
top=pow(x,p);
bot=p*fact;
next=pow(-1 ,n)*top/bot;
67


// cout"top= "top;
inside=inside+next;

total=(2.0/1.7724538509)*(x+inside);
// cout"erf(x)=";
// couttotal;
// getch();
return total;
};
int heat_efl() {
const float e=2.71828183;
double bot;
double top;
double k;
double vsh;
double area[20];
double a;
double radius;
float test;
float ef;
int q;
int cps;
double t; //this is the elapsed time in the heating eff
double dt;
int n;
cps=480;
dt=2.0/cps;
for (q=l;q t=dt*q;
68


for (n=l;n<(max_n+l);n++){
radius=0.0254*seg[n]. dia/2;
area[n]=pow(radius,2)*pi;
k=0.85*sqit(t)*(2*pi*radius);
k=k*sqrt(seg[n] .insspecheat* 1.49036);
bot=area[n]*seg[n].vol_spec_heat;
k=k/bot;
eff[n] [q]=( 1.0/(k*k))*((pow(e,k*k)*( l-erf(k)))- l+(2*k)/sqr
t(pi));

};
return (1);};
69


/*
"BLUSCRN.CPP"
THIS FILE IS THE FUNCTION THAT DISPLAYS THE DIALOG WINDOW.
*/
#inchide
#include
int bluscm (int 1, int t, int r, int b)
{
int end;
register int i;
register int j;
int s;
int xmargin;
int left=l;
int top=t;
int right=r;
int bottom=b;
textmode(C80);
textbackground(BLACK);
clrscr();
textbackgroimd(BLUE);
textcolor(WHlTE);
70


window (left,top,right,bottom);
// this loop changes the background color to blue
for (i=l;i < 25; i++){
for(j=l J < 80j++){
gotoxy(j,i);
cprintf("");}}
return 10;
}
#include
#include
#include
#include
extern double i[480];
extern int calculated;
extern bhiscm (int 1, int t, int r, int b);
void graph ()
{
int driver,mode;
int max_x,max_y;
int left,right,top,bottom;
float max_i=0;
int count;
int temp;
int xl,x2,yl,y2;
float itemp;
71


char *yaxis[ll];
inty[ll];
float cnt;
char ychr[10];
if (calculated=0){
bhiscm(20,12,60,17);
gotoxy(5,3);
cout"You have to calculate the values";
gotoxy(5,5);
cout"before you graph them. Hit a key.";
getch();
return; }
for (count=l ;count< 100;count++) { //This loop finds the max I
if(i[count]>max_i) {
max_i=i[count];} }
//This loop sets max i to multiples of 50
for (count=0;coimt<100;count++){
temp=max_i+count;
if(temp%50=0) break;}
max_i=temp;
// This code initializes the graphics driver:
driver= DETECT;
mode= 0;
initgraph(&driver,&mode,"c:\\tc\\bgi");
72


// This gets the max values of x and y and stores them.
max_x=(getmaxx());
max_y=(getmaxy());
// Set colors and font:
setbkcolor(BLUE);
setcolor(WHITE);
settextstyle( SMALL_FONT,HORIZ_DIR, 5 );
// This code draws the outside box of the graph:
left=max_x/10;
top= max_y/10;
right=max_x-(max_x/10);
bottom=max_y-(max_y/9);
rectangle (left,top,right,bottom);
outtextxy(left,bottom+10,"0.0");
outtextxy((right-left)*0.1+left ,bottom+10,"0.2");
outtextxy((right-left)*0.2+left ,bottom+10,"0.4");
outtextxy((right-left)*0.3+left ,bottom+10,"0.6");
outtextxy((right-left)*0.4+left ,bottom+10,"0.8");
outtextxy((right-left)*0.5+left ,bottom+10," 1.0");
outtextxy((right-left)*0.6+left ,bottom+10,'T.2");
outtextxy((right-left)*0.7+left ,bottom+10,"1.4");
outtextxy((right-left)*0.8+left ,bottom+10," 1.6");
outtextxy((right-left)*0.9+left ,bottom+10," 1.8");
outtextxy(right,bottom+l 0,"2.0");
// This will label the y-axis:
xl=left*0.6;
for (count=10;count>-l;count~){
y[coimt]=bottom-(bottom-top)*(count)/10-(bottom-top)/50;}
73


for (count=0;count yl=(count)*(max_i)/10;
itoa(yl,ychr,10);
outtextxy(xl,y[count],ychr);}
// This code lables all titles:
settextstyle(SANS_SERIF_FONT,HORIZ_DIR, 1);
outtextxy(max_x/2-50,bottom+30,"Time in Seconds");
settextstyle(SAN S_SERIF_FONT,HORIZDIR,2);
outtextxy(max_x/2-95 ,top/10, "Fault Current vs. Time");
settextstyle( SAN SSERIFFONT, VERTDIR, 1);
outtextxy(left*. l,max_y/2-75,"Current in Amperes");
// This changes the active window to inside the graph for
// ease of use
setviewport(left,top,right,bottom, 1);
max_x=right-left;
max_y=bottom-top;
setcolor(CYAN);
//Drawing vertical grids:
line(max_x*0.10, 0, max_x*0.1, max_y);
line(max_x*0.20, 0, max_x*0.2, maxjy);
line(max_x*0.30, 0, max_x*0.3, maxjy);
74


line(max_x*0.40, 0, max_x*0.4, max_y);
line(max_x*0.50, 0, max_x*0.5, max_y);
line(max_x*0.60, 0, max_x*0.6, max_y);
line(max_x*0.70, 0, max_x*0.7, max_y);
line(max_x*0.80, 0, max_x*0.8, max_y);
line(max_x*0.90, 0, max_x*0.9, max_y);
//Drawing horizontal grids
line(0,max_y*0.10, max_x, max_y*. 1);
line(0,max_y*0.20, max_x, max_y*.2);
line(0,maxjy*0.30, max_x, max_y*.3);
line(0,max_y*0.40, max_x, max_y*.4);
line(0,max_y*0.50, max_x, max_y*.5);
line(0,max_y*0.60, max_x, max_y*.6);
line(0,max_y*0.70, max_x, max_y*.7);
line(0,max_y*0.80, max_x, max_y*.8);
line(0,max_y*0.90, max_x, max_y*.9);
setcolor(LIGHTRED);
for (count=l;connt<240;count-H-){
cnt=count*2;
xl=((max_x)*cnt/480.);
y 1 =max_y-(max_y*i[cnt]/max_i);
x2=((max_x)*(cnt+1)/480.);
y2=max__y-(max_y*i[cnt+l]/max_i);
Iine(xl,yl,x2,y2); }
getch();
75


restorecrtmode;
return;
}
/*
"DMPUSCRN.CPP"
THIS FILE HANDLES THE INPUT OF DATA.
*1
#include
^include
#include
#include
#include
struct proj_data{
char eng[40];
char comp [40];
char proj_name[40];
char file_name[10];
double priaval;
int avalinput;
int calc_per_sec;
};
struct seg_data{
char ident[40];
double length;
76


double dia;
double rad;
double temp;
double row;
double alpha;
double sep;
int path;
int matl;
int insmatl;
int conduit;
double rac;
double x;
int set;
double mass;
double density;
double specheat;
double insspecheat;
double volspecheat;
};
struct projdata info;
struct seg_data seg[20];
int bluscm(int, int, int, int);
int seginput(int);
extern int max_n;
int inscm ()
{
FILE *fptr;
char end;
register int i;
register int j;
int s;
int xmargin;
77


bluscm(10,5,70,25);
xmargin=8;
gotoxy(20,2);
cputs("Input project data:'1);
gotoxy(xmargin,4);
cputs("Engmeer:");
gotoxy(xmargin,6);
cputs("Company:");
gotoxy(xmargin,8);
cputs("Project Name:");
gotoxy(xmargm, 10);
cputsC'DOS File name for data:");
gotoxy(xmargin, 12);
cputs("Primary Fault Current Available:");
gotoxy(xmargin, 14);
cputs("Primary Fault Current in MVA or Amperes?");
gotoxy(xmargin, 16);
cputs("How many Calculations Per Cycle? (1-20)");
gotoxy(xmargin, 18);
cputs("Is above information correct? (Y/N)");
/* The next code puts the cursor at each new place that it needs to be
to get next input*/
gotoxy(xmargin+l 1,4);
gets(info.eng);
78


gotoxy(xmargm+10,6);
get s(info. comp);
gotoxy(xmargin+15,8);
gets(info.proj_name);
gotoxy(xniargin+24,10);
gets(info.file_name);
gotoxy(xmargin+34,12);
cininfo .priaval;
gotoxy(xmargin+41,14);
cminfo. avalinput;
gotoxy(xmargin+40,16);
cminfo. calc_per_sec;
gotoxy(xmargin+45,18);
end = (getchQ);
if (end=y) {
return 1;
};
if(end=rY,){
return 1;};
inscm ();
return 10;
}
79


segin(){
strcpy(seg[0].ident,"fake name");
bluscm(25,10,55,15);
intn;
int insmatl;
cout"Highest Seg="max_n;
gotoxy(5,2);
cputs("Which Segment to edit?");
gotoxy(5,7);
cinn;
if(n>max_n){
max_n=n;}
seginput(n);
return 1;
}
seginput(int n) {
int xmargin=8;
int ins matl;
char end;
double thickness;
80


// this section writes the message to the screen
bluscm(10,1,70,24);
gotoxy(20,2);
cout" Segment #"n" of "max_n;
gotoxy(xmargin,4);
cputs("Segment Name:");
gotoxy(xmargin,6);
cputs( "Length (feet):");
gotoxy(xmargin, 8);
cputs("Conductor Diameter (inches):");
gotoxy(xmargin, 10);
cputs("Conductor Material (l=Cu, 2=A1):);
gotoxy(xmargin, 12);
cputs("Insulation Thickness (mils):");
gotoxy(xmargin, 14);
cputs("Insulation Material (1=PVC, 2=PE):");
gotoxy(xmargin, 16);
cputs("Circuit in conduit? (l=no,");
gotoxy(xmargin, 17);
cputs("2=magnetic, 3=nonmagnetic):");
gotoxy(xmargin, 19);
cputs("Conduit in return circuit? (l=Yes, 2=No)");
gotoxy(xmargin,21);
cputs("Conductor Temp (deg. C.):");
gotoxy(xmargin,23);
cputs("Is above information correct? (Y/N)");
/* The next code puts the cursor at each new place that it needs to be
to get next input*/
gotoxy(xmargin+15,4);
81


gets(seg[n] .ident);
gotoxy(xmargin+16,6);
cinseg[n].length;
gotoxy(xmargin+3 0,8);
cinseg[n].dia;
gotoxy(xmargin+34,10);
cinseg[n] .matl;
gotoxy(xmargin+30,12);
cinthickness;
seg[n]. sep=seg[n]. dia+2*thickness/1000;
gotoxy(xmargin+41,14);
cinseg[n].ins_matl;
gotoxy(xmargin+3 0,17);
cinseg[n]. conduit;
gotoxy(xmargin+45,19);
cinseg[n].path;
gotoxy(xmargin+26,21);
cinseg[n].temp;
gotoxy(xmargin+38,23);
end=(getch());
seg[n].set=l; // this will be the flag that tell me if
// the segment has been entered.
/* The next code sets the coefficient of resistivity
for the conductor material.*/
82


if (seg[n].matl=l){
seg[n].row=8.14535e-6; //TfflS IS FOR COPPER
seg[n].alpha=0.00393; // /DEGC
seg[n].density=0.32117;//#/INA3
seg[n]. spec_heat=l 74.717; //JOULE/#DEGC
seg[n].vol_spec_heat=3.4166*pow(10.,6.);}
// seg[n].ins_spec_heat=T.65129*pow(10.,6.);}
else if (seg[n].matl=2){
seg[n].row=1.3523 le-5; //TfflS IS FOR ALUMINUM
seg[n], alpha=0.00403;
seg[n]. density=0.09765;
seg[n].vol_spec_heat=3.4166*pow(10.,6.);
//seg[n].ins_spec_heat=1.65129*pow(10.,6.);
seg[n].spec_heat=429.197; }
else{
bluscm(15,5,65,19);
gotoxy(5,5);
cout"The value you entered for Material";
gotoxy(5,7);
cout"is out of bounds. Try again.(Hit a Key)";
getch();
seginput(n);
// THE FOLLOWING SETS THE VOLUMETRIC SPECIFIC
HEATS FOR
// BOTH THE CONDUCTOR AND INSULATION
if (seg[n].ins_matl=l){
seg[n].ins_spec_heat=1.65129*pow(10.,6.);}
//BTU/INA3 *DEGF
else if (ins_matl=2){
seg[n].ins_spec_heat=l.65129*pow( 10.,6.); }
else{
bluscm(15,5,65,19);
gotoxy(5,5);
83


cout"The value you entered for insulation material"
gotoxy(5,7);
cout"is out ofbounds. Try again.(Hit a Key)";
getch();
seginput(n);
};
if (end=y) {
return 1;
};
ifCend^^K
return 1;
};
seginput(n);
return 1;
}
save () {
FILE *fptr;
if ((fjptr=fopen(info.file_name,"wb"))=NULL) {
bluscm(10,12,70,17);
gotoxy(5,3);
cout"I couldn't save the file and I think it's your fault! !\n";
gotoxy(5,5);
cout"You may need to input system data.";
getch();
return 1;};
84


fwrite(&info, sizeof (info), 1,lptr);
fwrite(&max_n,sizeof (maxn), 1, fptr);
fwrite(&seg, sizeof(seg), 1, fptr);
fclose(fptr);
bluscm(15,12,65,17);
gotoxy(5,3);
cout"Input data has been written to "info.file_name".";
return 1;
}
intload() {
char *filename;
FILE *fptr;
bluscm( 15,5,65,20);
gotoxy(5,5);
cputs("What DOS file name is the data stored as?");
gotoxy(5,7);
cinfilename;
if((fptr=fop en(filename, "rb" ))=NULL) {
bluscm(20,5,60,20);
gotoxy(5,5);
cout"Can't open file "filename".";
getch();
return 1;
fread(4rinfo,sizeof(info), l,fptr);
fread(&max_n,sizeof(max_n), 1 ,fptr);
fread(&seg,sizeof(seg), l,fptr);
fclose(fptr);
85


bhiscm(25,5,55,20);
gotoxy(5,5);
cout"Loaded data for:
gotoxy(5,7);
coutinfo.proj_name;
gotoxy(5,10);
cout"Press any key...";
getch();
return 1;
}
86


Appendix B: Output of C++ Program
This program was written by W. Ronald Kilgore
as a partial requirement for an MSEE degree
from the University of Colorado at Denver.
Project Name: Master's Thesis: UCD.
Engineer's Name: W. Ronald Kilgore, P.E..
Company Name: Kilgore Engineering, Inc..
Segment 1 name: Transformer to Meter.
Segment Length: 92.000000 feet.
Segment Diameter: 0.708000 inches.
Final Conductor Temp: 20.008894 DegC.
Conductor Separation: 0.898000 inches.
Conductor material is Aluminum.
Segment is routed in nonmagnetic conduit.
Conduit is not in return path.
Insulation is PVC.
Segment 2 name: Meter to Panel.
Segment Length: 156.000000 feet.
Segment Diameter: 0.289000 inches.
Final Conductor Temp: 20.296771 DegC.
Conductor Separation: 0.379000 inches.
Conductor material is Aluminum.
Segment is routed in magnetic conduit.
Conduit is not in return path.
Insulation is PVC.
Segment 3 name: Panel to Receptacle.
Segment Length: 137.000000 feet.
Segment Diameter: 0.080800 inches.
87


Final Conductor Temp: 37.085621 DegC.
Conductor Separation: 0.140800 inches.
Conductor material is Copper.
Segment is routed in magnetic conduit.
Conduit is not in return path.
Insulation is PVC.
Segment 4 name: Zip Cord.
Segment Length: 10.000000 feet.
Segment Diameter: 0.040300 inches.
Final Conductor Temp: 324.624092 DegC.
Conductor Separation: 0.100300 inches.
Conductor material is Copper.
Segment is not in conduit.
Conduit is not in return path.
Insulation is PVC.
Time (sec)
0.000000
0.004167
0.008333
0.012500
0.016667
0.020833
0.025000
0.029167
0.033333
0.037500
0.041667
0.045833
0.050000
0.054167
0.058333
0.062500
0.066667
0.070833
I (amps)
189.221479
189.038273
188.858745
188.682234
188.508318
188.336704
188.167166
187.999529
187.833651
187.669414
187.506715
187.345469
187.185600
187.027040
186.869730
186.713618
186.558654
186.404794
Seg. 1
20.000000
20.000025
20.000051
20.000076
20.000103
20.000128
20.000153
20.000179
20.000204
20.000229
20.000254
20.000278
20.000303
20.000328
20.000355
20.000380
20.000404
20.000427
Seg. 2
20.000000
20.000919
20.001833
20.002743
20.003649
20.004551
20.005449
20.006346
20.007238
20.008127
20.009014
20.009897
20.010778
20.011658
20.012533
20.013405
20.014277
20.015144
Seg. 3
20.000000
20.065210
20.129585
20.193253
20.256296
20.318777
20.380739
20.442219
20.503246
20.563845
20.624039
20.683844
20.743279
20.802357
20.861090
20.919493
20.977573
21.035343
Seg. 4
20.000000
21.002895
21.986345
22.954220
23.908955
24.852253
25.785391
26.709364
27.624973
28.532885
29.433655
30.327768
31.215635
32.097626
32.974056
33.845219
34.711369
35.572739
88


Time (sec)
0.075000
0.079167
0.083333
0.087500
0.091667
0.095833
0.100000
0.104167
0.108333
0.112500
0.116667
0.120833
0.125000
0.129167
0.133333
0.137500
0.141667
0.145833
0.150000
0.154167
0.158333
0.162500
0.166667
0.170833
0.175000
0.179167
0.183333
0.187500
0.191667
0.195833
0.200000
0.204167
0.208333
0.212500
0.216667
0.220833
I (amps)
186.252000
186.100233
185.949460
185.799650
185.650774
185.502803
185.355714
185.209482
185.064085
184.919502
184.775713
184.632699
184.490443
184.348928
184.208137
184.068056
183.928670
183.789965
183.651928
183.514546
183.377807
183.241700
183.106214
182.971337
182.837059
182.703371
182.570263
182.437725
182.305749
182.174326
182.043449
181.913108
181.783296
181.654005
181.525230
181.396961
Seg. 1
20.000452
20.000477
20.000502
20.000526
20.000551
20.000576
20.000599
20.000624
20.000648
20.000673
20.000696
20.000721
20.000746
20.000769
20.000793
20.000816
20.000841
20.000864
20.000889
20.000912
20.000937
20.000959
20.000982
20.001007
20.001030
20.001053
20.001078
20.001101
20.001123
20.001148
20.001171
20.001194
20.001217
20.001240
20.001263
20.001287
Seg. 2
20.016010
20.016872
20.017735
20.018593
20.019449
20.020302
20.021154
20.022003
20.022852
20.023697
20.024540
20.025381
20.026220
20.027058
20.027893
20.028725
20.029556
20.030386
20.031214
20.032038
20.032862
20.033684
20.034504
20.035320
20.036137
20.036951
20.037764
20.038574
20.039383
20.040192
20.040997
20.041800
20.042603
20.043404
20.044201
20.044998
Seg. 3
21.092812
21.149986
21.206875
21.263485
21.319824
21.375896
21.431709
21.487270
21.542582
21.597649
21.652479
21.707073
21.761440
21.815580
21.869497
21.923199
21.976685
22.029961
22.083031
22.135895
22.188559
22.241024
22.293293
22.345371
22.397261
22.448961
22.500477
22.551811
22.602966
22.653942
22.704744
22.755373
22.805830
22.856117
22.906239
22.956196
Seg. 4
36.429546
37.281979
38.130219
38.974426
39.814754
40.651340
41.484314
42.313801
43.139912
43.962753
44.782421
45.599018
46.412621
47.223320
48.031197
48.836319
49.638760
50.438587
51.235867
52.030659
52.823017
53.613003
54.400669
55.186062
55.969231
56.750225
57.529087
58.305862
59.080589
59.853306
60.624058
61.392872
62.159790
62.924847
63.688068
64.449493
89


Time (sec)
0.225000
0.229167
0.233333
0.237500
0.241667
0.245833
0.250000
0.254167
0.258333
0.262500
0.266667
0.270833
0.275000
0.279167
0.283333
0.287500
0.291667
0.295833
0.300000
0.304167
0.308333
0.312500
0.316667
0.320833
0.325000
0.329167
0.333333
0.337500
0.341667
0.345833
0.350000
0.354167
0.358333
0.362500
0.366666
0.370833
I (amps)
181.269193
181.141918
181.015131
180.888825
180.762994
180.637632
180.512733
180.388292
180.264303
180.140761
180.017661
179.894997
179.772765
179.650961
179.529578
179.408613
179.288061
179.167918
179.048180
178.928842
178.809900
178.691351
178.573190
178.455414
178.338018
178.221000
178.104355
177.988080
177.872172
177.756628
177.641443
177.526615
177.412140
177.298016
177.184239
177.070806
Seg. 1
20.001310
20.001333
20.001356
20.001379
20.001402
20.001425
20.001448
20.001471
20.001493
20.001516
20.001537
20.001560
20.001583
20.001606
20.001629
20.001652
20.001673
20.001696
20.001719
20.001741
20.001762
20.001785
20.001808
20.001829
20.001852
20.001875
20.001896
20.001919
20.001940
20.001963
20.001984
20.002007
20.002028
20.002050
20.002071
20.002094
Seg. 2
20.045794
20.046587
20.047380
20.048170
20.048960
20.049747
20.050533
20.051317
20.052099
20.052881
20.053659
20.054438
20.055216
20.055990
20.056763
20.057535
20.058306
20.059074
20.059843
20.060608
20.061373
20.062138
20.062899
20.063660
20.064419
20.065176
20.065931
20.066687
20.067440
20.068192
20.068943
20.069693
20.070440
20.071186
20.071932
20.072676
Seg. 3
23.005987
23.055618
23.105089
23.154402
23.203560
23.252562
23.301411
23.350109
23.398657
23.447056
23.495308
23.543413
23.591375
23.639194
23.686872
23.734411
23.781809
23.829071
23.876196
23.923185
23.970041
24.016764
24.063354
24.109816
24.156145
24.202349
24.248425
24.294373
24.340197
24.385895
24.431471
24.476925
24.522257
24.567469
24.612562
24.657536
Seg. 4
65.209152
65.967064
66.723274
67.477798
68.230667
68.981903
69.731537
70.479591
71.226089
71.971054
72.714508
73.456467
74.196968
74.936012
75.673630
76.409843
77.144661
77.878105
78.610199
79.340958
80.070396
80.798531
81.525383
82.250961
82.975281
83.698357
84.420212
85.140854
85.860291
86.578552
87.295639
88.011566
88.726341
89.439995
90.152519
90.863930
90


Time (sec)
0.375000
0.379166
0.383333
0.387500
0.391666
0.395833
0.400000
0.404166
0.408333
0.412500
0.416666
0.420833
0.425000
0.429166
0.433333
0.437500
0.441666
0.445833
0.450000
0.454166
0.458333
0.462500
0.466666
0.470833
0.475000
0.479166
0.483333
0.487500
0.491666
0.495833
0.500000
0.504166
0.508333
0.512500
0.516666
0.520833
I (amps)
176.957715
176.844962
176.732545
176.620460
176.508706
176.397279
176.286176
176.175396
176.064935
175.954792
175.844962
175.735445
175.626237
175.517337
175.408742
175.300449
175.192457
175.084763
174.977365
174.870261
174.763448
174.656926
174.550690
174.444741
174.339075
174.233690
174.128585
174.023758
173.919207
173.814930
173.710925
173.607190
173.503724
173.400525
173.297590
173.194919
Seg. 1
20.002115
20.002138
20.002159
20.002180
20.002203
20.002224
20.002245
20.002268
20.002289
20.002310
20.002331
20.002354
20.002375
20.002396
20.002417
20.002438
20.002460
20.002481
20.002502
20.002523
20.002544
20.002565
20.002586
20.002607
20.002628
20.002649
20.002670
20.002691
20.002712
20.002733
20.002754
20.002775
20.002796
20.002817
20.002838
20.002859
Seg. 2
20.073420
20.074160
20.074900
20.075638
20.076376
20.077112
20.077847
20.078581
20.079313
20.080044
20.080774
20.081503
20.082230
20.082956
20.083681
20.084404
20.085127
20.085848
20.086567
20.087286
20.088003
20.088720
20.089434
20.090149
20.090860
20.091572
20.092283
20.092993
20.093700
20.094408
20.095114
20.095818
20.096521
20.097223
20.097925
20.098625
Seg. 3
24.702391
24.747129
24.791754
24.836262
24.880655
24.924934
24.969103
25.013159
25.057104
25.100939
25.144663
25.188278
25.231787
25.275188
25.318481
25.361671
25.404755
25.447733
25.490608
25.533381
25.576050
25.618618
25.661083
25.703449
25.745716
25.787884
25.829952
25.871922
25.913795
25.955572
25.997252
26.038836
26.080326
26.121719
26.163021
26.204227
Seg. 4
91.574249
92.283478
92.991638
93.698723
94.404762
95.109756
95.813713
96.516647
97.218567
97.919487
98.619408
99.318344
100.016304
100.713295
101.409325
102.104408
102.798553
103.491760
104.184036
104.875404
105.565857
106.255409
106.944061
107.631836
108.318718
109.004738
109.689888
110.374176
111.057617
111.740204
112.421959
113.102882
113.782974
114.462242
115.140709
115.818359
91


Time (sec)
0.525000
0.529166
0.533333
0.537500
0.541666
0.545833
0.550000
0.554166
0.558333
0.562500
0.566666
0.570833
0.575000
0.579166
0.583333
0.587500
0.591666
0.595833
0.600000
0.604166
0.608333
0.612500
0.616666
0.620833
0.625000
0.629166
0.633333
0.637500
0.641666
0.645833
0.650000
0.654166
0.658333
0.662500
0.666666
0.670833
I (amps)
173.092510
172.990360
172.888469
172.786835
172.685456
172.584330
172.483456
172.382832
172.282458
172.182330
172.082449
171.982812
171.883418
171.784265
171.685352
171.586678
171.488241
171.390040
171.292074
171.194341
171.096839
170.999569
170.902527
170.805713
170.709126
170.612765
170.516628
170.420713
170.325021
170.229549
170.134296
170.039262
169.944444
169.849843
169.755456
169.661283
Seg. 1
20.002880
20.002899
20.002920
20.002941
20.002962
20.002983
20.003002
20.003023
20.003044
20.003065
20.003084
20.003105
20.003126
20.003145
20.003166
20.003187
20.003206
20.003227
20.003246
20.003267
20.003286
20.003307
20.003328
20.003347
20.003368
20.003387
20.003408
20.003428
20.003447
20.003468
20.003487
20.003508
20.003527
20.003548
20.003567
20.003586
Seg. 2
20.099323
20.100021
20.100718
20.101414
20.102108
20.102802
20.103495
20.104185
20.104876
20.105564
20.106253
20.106939
20.107624
20.108309
20.108994
20.109676
20.110357
20.111038
20.111717
20.112396
20.113073
20.113750
20.114426
20.115099
20.115772
20.116446
20.117117
20.117786
20.118456
20.119125
20.119791
20.120459
20.121124
20.121788
20.122452
20.123114
Seg. 3
26.245342
26.286366
26.327295
26.368135
26.408884
26.449541
26.490110
26.530590
26.570980
26.611282
26.651497
26.691626
26.731665
26.771620
26.811489
26.851273
26.890970
26.930584
26.970114
27.009560
27.048923
27.088203
27.127401
27.166517
27.205553
27.244507
27.283379
27.322172
27.360886
27.399519
27.438074
27.476551
27.514948
27.553268
27.591511
27.629677
Seg. 4
116.495209
117.171257
117.846519
118.520996
119.194695
119.867622
120.539772
121.211166
121.881805
122.551682
123.220818
123.889214
124.556862
125.223785
125.889977
126.555443
127.220200
127.884232
128.547562
129.210190
129.872101
130.533325
131.193863
131.853699
132.512863
133.171341
133.829147
134.486282
135.142746
135.798553
136.453705
137.108185
137.762024
138.415207
139.067749
139.719635
92


Time (sec)
0.675000
0.679166
0.683333
0.687500
0.691666
0.695833
0.700000
0.704166
0.708333
0.712499
0.716666
0.720833
0.724999
0.729166
0.733333
0.737499
0.741666
0.745833
0.749999
0.754166
0.758333
0.762499
0.766666
0.770833
0.774999
0.779166
0.783333
0.787499
0.791666
0.795833
0.799999
0.804166
0.808333
0.812499
0.816666
0.820833
I (amps)
169.567322
169.473573
169.380034
169.286704
169.193582
169.100667
169.007959
168.915455
168.823156
168.731059
168.639164
168.547470
168.455976
168.364681
168.273584
168.182684
168.091980
168.001471
167.911156
167.821034
167.731105
167.641367
167.551820
167.462462
167.373293
167.284312
167.195518
167.106909
167.018486
166.930248
166.842193
166.754321
166.666630
166.579121
166.491792
166.404642
Seg. 1
20.003607
20.003626
20.003645
20.003666
20.003685
20.003704
20.003723
20.003744
20.003763
20.003782
20.003801
20.003822
20.003841
20.003860
20.003880
20.003899
20.003918
20.003939
20.003958
20.003977
20.003996
20.004015
20.004034
20.004053
20.004072
20.004091
20.004110
20.004129
20.004148
20.004168
20.004187
20.004206
20.004225
20.004244
20.004263
20.004282
Seg. 2
20.123775
20.124435
20.125095
20.125753
20.126411
20.127068
20.127724
20.128378
20.129030
20.129684
20.130335
20.130987
20.131636
20.132286
20.132933
20.133581
20.134228
20.134872
20.135517
20.136160
20.136803
20.137444
20.138084
20.138725
20.139364
20.140001
20.140638
20.141275
20.141911
20.142546
20.143179
20.143812
20.144444
20.145075
20.145704
20.146334
Seg. 3
27.667767
27.705778
27.743715
27.781576
27.819363
27.857073
27.894711
27.932272
27.969761
28.007175
28.044518
28.081787
28.118984
28.156109
28.193161
28.230143
28.267054
28.303892
28.340662
28.377361
28.413990
28.450550
28.487040
28.523460
28.559813
28.596098
28.632315
28.668463
28.704544
28.740557
28.776505
28.812386
28.848200
28.883947
28.919630
28.955248
Seg. 4
140.370895
141.021530
141.671524
142.320892
142.969635
143.617752
144.265259
144.912155
145.558441
146.204102
146.849182
147.493637
148.137512
148.780777
149.423462
150.065552
150.707062
151.347977
151.988312
152.628067
153.267258
153.905869
154.543915
155.181396
155.818298
16.454651
157.090439
157.725677
158.360367
158.994492
159.628067
160.261108
160.893600
161.525543
162.156952
162.787827
93


Time (sec)
0.824999
0.829166
0.833333
0.837499
0.841666
0.845833
0.849999
0.854166
0.858333
0.862499
0.866666
0.870833
0.874999
0.879166
0.883333
0.887499
0.891666
0.895833
0.899999
0.904166
0.908333
0.912499
0.916666
0.920833
0.924999
0.929166
0.933333
0.937499
0.941666
0.945833
0.949999
0.954166
0.958333
0.962499
0.966666
0.970833
I (amps)
166.317671
166.230878
166.144262
166.057822
165.971558
165.885469
165.799553
165.713811
165.628241
165.542843
165.457615
165.372559
165.287671
165.202953
165.118402
165.034019
164.949803
164.865752
164.781867
164.698147
164.614590
164.531197
164.447966
164.364898
164.281990
164.199243
164.116656
164.034229
163.951960
163.869849
163.787895
163.706098
163.624457
163.542972
163.461642
163.380466
Seg. 1
20.004301
20.004320
20.004339
20.004356
20.004375
20.004395
20.004414
20.004433
20.004452
20.004469
20.004488
20.004507
20.004526
20.004543
20.004562
20.004581
20.004601
20.004618
20.004637
20.004656
20.004673
20.004692
20.004711
20.004728
20.004747
20.004766
20.004784
20.004803
20.004820
20.004839
20.004858
20.004875
20.004894
20.004911
20.004930
20.004948
Seg. 2
20.146963
20.147591
20.148216
20.148842
20.149467
20.150091
20.150715
20.151337
20.151958
20.152580
20.153200
20.153818
20.154436
20.155054
20.155670
20.156286
20.156902
20.157516
20.158129
20.158741
20.159353
20.159964
20.160574
20.161184
20.161793
20.162399
20.163006
20.163612
20.164217
20.164822
20.165426
20.166029
20.166632
20.167233
20.167833
20.168432
Seg. 3
28.990799
29.026287
29.061708
29.097067
29.132362
29.167591
29.202759
29.237862
29.272902
29.307880
29.342794
29.377647
29.412439
29.447168
29.481834
29.516441
29.550987
29.585472
29.619896
29.654259
29.688562
29.722807
29.756990
29.791117
29.825182
29.859188
29.893137
29.927027
29.960857
29.994633
30.028347
30.062006
30.095608
30.129150
30.162638
30.196068
Seg. 4
163.418167
164.047974
164.677246
165.306000
165.934219
166.561920
167.189102
167.815765
168.441910
169.067535
169.692657
170.317261
170.941360
171.564957
172.188049
172.810638
173.432724
174.054321
174.675415
175.296005
175.916122
176.535736
177.154877
177.773514
178.391678
179.009354
179.626541
180.243256
180.859497
181.475266
182.090561
182.705383
183.319733
183.933609
184.547012
185.159973
94


Time (sec)
0.974999
0.979166
0.983333
0.987499
0.991666
0.995833
0.999999
1.004166
1.008333
1.012499
1.016666
1.020833
1.024999
1.029166
1.033333
1.037500
1.041666
1.045833
1.050000
1.054167
1.058333
1.062500
1.066667
1.070833
1.075000
1.079167
1.083334
1.087500
1.091667
1.095834
1.100001
1.104167
1.108334
1.112501
1.116667
1.120834
I (amps)
163.299444
163.218574
163.137858
163.057293
162.976879
162.896616
162.816504
162.736540
162.656726
162.577060
162.497543
162.418172
162.338948
162.259870
162.180938
162.102151
162.023509
161.945011
161.866656
161.788444
161.710374
161.632446
161.554660
161.477014
161.399509
161.322144
161.244918
161.167831
161.090882
161.014071
160.937397
160.860860
160.784459
160.708194
160.632065
160.556070
Seg. 1
20.004967
20.004984
20.005003
20.005020
20.005039
20.005056
20.005074
20.005093
20.005110
20.005129
20.005146
20.005163
20.005182
20.005199
20.005219
20.005236
20.005253
20.005272
20.005289
20.005306
20.005323
20.005342
20.005360
20.005377
20.005394
20.005413
20.005430
20.005447
20.005465
20.005484
20.005501
20.005518
20.005535
20.005552
20.005569
20.005589
Seg. 2
20.169031
20.169630
20.170227
20.170824
20.171421
20.172016
20.172609
20.173204
20.173796
20.174389
20.174980
20.175571
20.176161
20.176750
20.177338
20.177925
20.178513
20.179098
20.179684
20.180269
20.180853
20.181437
20.182018
20.182600
20.183182
20.183762
20.184341
20.184921
20.185499
20.186077
20.186653
20.187229
20.187805
20.188379
20.188953
20.189528
Seg. 3
30.229443
30.262760
30.296022
30.329227
30.362377
30.395472
30.428513
30.461496
30.494427
30.527302
30.560122
30.592888
30.625601
30.658260
30.690865
30.723415
30.755915
30.788361
30.820753
30.853092
30.885380
30.917614
30.949799
30.981928
31.014008
31.046036
31.078012
31.109938
31.141811
31.173635
31.205408
31.237129
31.268801
31.300423
31.331995
31.363518
Seg. 4
185.772446
186.384476
186.996033
187.607147
188.217789
188.827972
189.437714
190.046997
190.655838
191.264221
191.872162
192.479660
193.086700
193.693314
194.299469
194.905197
195.510483
196.115326
196.719742
197.323715
197.927261
198.530365
199.133057
199.735306
200.337128
200.938538
201.539505
202.140060
202.740189
203.339905
203.939194
204.538055
205.136520
205.734558
206.332184
206.929398
95


Time (sec)
1.125001
1.129168
1.133334
1.137501
1.141668
1.145834
1.150001
1.154168
1.158335
1.162501
1.166668
1.170835
1.175002
1.179168
1.183335
1.187502
1.191668
1.195835
1.200002
1.204169
1.208335
1.212502
1.216669
1.220835
1.225002
1.229169
1.233336
1.237502
1.241669
1.245836
1.250003
1.254169
1.258336
1.262503
1.266669
1.270836
I (amps)
160.480210
160.404484
160.328891
160.253431
160.178104
160.102909
160.027845
159.952913
159.878111
159.803440
159.728898
159.654486
159.580202
159.506047
159.432020
159.358121
159.284349
159.210703
159.137184
159.063791
158.990523
158.917380
158.844362
158.771468
158.698698
158.626051
158.553527
158.481126
158.408847
158.336689
158.264653
158.192738
158.120944
158.049269
157.977715
157.906280
Seg. 1
20.005606
20.005623
20.005640
20.005657
20.005674
20.005692
20.005709
20.005726
20.005743
20.005760
20.005777
20.005796
20.005814
20.005831
20.005848
20.005865
20.005880
20.005898
20.005915
20.005932
20.005949
20.005966
20.005983
20.006001
20.006018
20.006035
20.006052
20.006069
20.006086
20.006102
20.006119
20.006136
20.006153
20.006170
20.006187
20.006203
Seg. 2
20.190100
20.190672
20.191242
20.191813
20.192383
20.192951
20.193520
20.194088
20.194654
20.195221
20.195787
20.196352
20.196917
20.197481
20.198044
20.198606
20.199167
20.199728
20.200289
20.200848
20.201406
20.201965
20.202522
20.203079
20.203636
20.204191
20.204746
20.205301
20.205854
20.206408
20.206961
20.207512
20.208063
20.208614
20.209164
20.209713
Seg. 3
31.394989
31.426413
31.457787
31.489111
31.520386
31.551613
31.582790
31.613920
31.645000
31.676035
31.707020
31.737957
31.768847
31.799688
31.830482
31.861231
31.891932
31.922585
31.953192
31.983753
32.014267
32.044735
32.075157
32.105530
32.135860
32.166145
32.196384
32.226578
32.256725
32.286827
32.316883
32.346897
32.376865
32.406792
32.436668
32.466503
Seg. 4
207.526199
208.122589
208.718567
209.314133
209.909302
210.504059
211.098419
211.692368
212.285919
212.879059
213.471817
214.064163
214.656113
215.247665
215.838821
216.429596
217.019974
217.609955
218.199539
218.788742
219.377548
219.965973
220.554016
221.141663
221.728928
222.315811
222.902313
223.488434
224.074173
224.659531
225.244522
225.829117
226.413345
226.997208
227.580688
228.163788
96


Full Text
This thesis for the Master of Science
degree by
W. Ronald Kilgore
has been approved
by
William R. Roemish
Miloje S. Radenkovic
e-1 %
Date