PULSE FORMING NETWORK INVESTIGATION
by
EDWARD GUY COOK, B.S. IN E.E.
A THESIS
IN
ELECTRICAL ENGINEERING
Submitted to the Graduate Faculty
of Texas Tech University in
Partial Fulfillment of
the Requirements for
the Degree of
MASTER OF SCIENCE
IN
ELECTRICAL ENGINEERING
t
j
Accepted
August, 1975
F\ n
tNio.n3
>-- n
Cop.
ACKNOWLEDGMENTS
I would like to thank Dr. T. R. Burkes for his
guidance and assistance during the development of this
project and during the writing of this thesis.
Special thanks are also extended to Drs. J. P. Craig
and S. S. Panwalker for serving on my committee and their
helpful suggestions.
I would like to express my appreciation to my fellow
graduate students in the High-Voltage Laboratory for their
comments.
Finally, I also thank ray wife Katherine for her
support and encouragement.
IX
TABLE OF CONTENTS
ACKNOWLEDGMENTS
ii
LIST OF TABLES
v
LIST OF FIGURES
vi
Chapter
I.
II.
INTRODUCTION
1
TRANSMISSION LINE THEORY AND LUMPED PARAMETER
TRANSMISSION LINES
4
Transmission Line Theory
5
Networks Derived From Transmission Lines . . 14
Guilleman's Theory
21
III.
DESIGN, CONSTRUCTION, AND EVALUATION OF A
PULSE FORMING NETWORK
Design of a PFN
Element Design and PFN Construction
Performance Evaluation of the PFN
IV.
MODIFICATION OF PULSE DURATION
The Use of Shielding to Reduce Inductance. .
Inductance
Eddy-Currents
Skin Effects
Inductor With Solid Cylindrical Conducting
Core
Inductor With a Tubular Conducting Core. . .
Magnetic Field Intensity Within a Coil . . .
V.
INDUCTANCE MEASUREMENTS AND EXPERIMENTAL
VERIFICATION OF REDUCING PFN PULSE DURATION
BY REDUCTION OF PFN INDUCTANCE
Calculated Inductance Values
Inductance Values for a Non-Uniform
Magnetic Field
Inductance Values for a Uniform
Magnetic Field
Measured Inductance and Pulse Duration
Values
Summary
...
111
28
28
29
35
41
43
45
51
55
59
73
85
98
99
100
102
108
113
VI.
CONCLUSIONS AND RECOMMENDATIONS FOR FURTHER
RESEARCH
115
REFERENCES
117
APPENDIX
119
A.
B.
C.
DERIVATION OF TYPE E NETWORK
VALUES OF JQÍZJ^^^) A N D J^ízj"^^^) FOR DETERMINING NORMALIZED COIL IMPEDANCE OF COIL
SURROUNDING A SOLID CYLINDRICiUi CORE
126
EXPRESSIONS FOR DETERMINING THE MAGNETIC
FIELD INTENSITY WITHIN A SOLENOID
133
IV
120
L I S T OF TABLES
Têible
III-l.
V-1.
V-2.
V-3.
Page
Measured Element Values
34
Calculated Inductance Values Assuming A
Non-Uniform Magnetic Field
103
Calculated Inductance Values Assuming A
Uniform Magnetic Field
107
Measured Inductance Values and Pulse
Durations for PFN
110
LIST OF FIGURES
Figure
II-l.
II-2.
Page
Schematic representation of distributed
inductance and capacitance of a lossless
transmission line
6
Coordinate system for lossless two conductor
transmission line
6
II-3.
Transmission line with resistive termination . 11
II-4.
Voltage-fed pulse generating circuit using
ideal transmission line
Current-fed pulse generating circuit using
ideal transmission line
Two terminal line-simulating network
consisting of an infinite number of
elements
II-5.
II-6.
II-7.
II-8.
II-9.
11-10.
11-11.
III-l.
13
15
17
Circuit representing the rational-fraction
expansion of the transmission line
impedance function
20
Circuit representing the rational-fraction
expansion of the transmission line
admittance function
20
Foann of voltage-fed network derived by
Fourier-series analysis of a specific
alternating-current waveform
23
Foster*s and Cauer's expansion of impedance
and admittance
25
Pulse forming network having equal
capacitance type E network
26
Definition of terms for determination of
mutual inductance between coaxial coils. . . 31
III-2.
Photograph of constructed PFN
33
III-3.
PFN and operational support equipment
36
II1-4.
Photograph of PFN and operational support
equipment
vi
38
Figure
III-5.
IV-1.
IV-2.
IV-3.
IV-4.
IV-5.
IV-6.
IV-7.
IV-8.
IV-9.
Page
Oscillogram of PFN generated voltage pulse
across linear resistive load
39
Determination of B at a point P due to a
loop current
46
Evaluation of magnetic flux for a single
loop
46
Determination of magnetic flux coupling in
loop a due to a current in loop b
48
Phase relationships of magnetic flux in a
coil
53
Cross-section of semi-infinite plate placed
in a magnetic field oriented in the zdirection
57
Orientation of the conducting cylinder in the
cylindrical coordinate system
61
Conducting cylinder surrounded by conducting
current sheath carrying a current I . . . .
65
Plots of normalized impedance for coil
surrounding solid cylindrical core
71
Conducting tube placed in a uniform axial
magnetic field
76
IV-10.
Plot of magnetic field intensity attenuation.
IV-11.
Impedance of a coil surrounding a conducting
tube
Definition of parameters for an elemental
loop in x-y plane
Definition of dimensions for coil having
finite length and finite thickness
rV-12.
IV-13.
rv-14.
V-l.
81
86
89
91
Coil and two sub-coils for detennination of
H^(z,0)
94
Normalized impedance of a coil surrounding
a conducting tube
104
Vll
Figure
Page
V-2.
Pulse duration as a function of core radius. . 109
V-3.
Oscillograms for 7.5 centimeter tube
111
V-4.
Oscillograms for 10.1 centimeter tube
112
Vlll
CHAPTER I
INTRODUCTION
High-power rectangular pulses have extensive applications in pulse modulated radar and pulse lasers.
Rectan-
gular pulses may be generated by many means, some of which
are either not capable of or not practical for the generation of high-power pulses.
Two of the more common methods
for developing high-power pulses are the hard-tube pulser
and the line-type pulser.
The hard-tube pulser is basically a vacuum-tube which
controls the power delivered to the load by modulation of
the grid or grids.
The line-type pulser is a network,
composed of passive elements (inductors and capacitors),
which delivers to the load all the energy stored in the
network in a time period predeteannined by the network
configuration.
The line-type pulser has several advantages over the
hard-tube pulser in terms of simplicity, size, weight, and
efficiency.
Generally, the line-type pulser does not need
as complex drive circuits for grid control as those required
by the hard-tube pulser.
The relatively large "on"
resistance (100 to 600 ohms) of the vacuum-tube and the
power dissipated make the hard-tube pulser much less
efficient than the line-type pulser.
The less intricate
circuits and the more efficient switches (thyratrons)
usually enable a line-type pulser to be smaller and
lighter in weight than a hard-tube pulser having comparable
output power levels.
Line-type pulsers are often lumped-parameter realizations of transmission lines.
Realizations for line-type
pulsers based on transmission line theory generally
require a large number of elements to generate an accurate
representation of a rectangular pulse.
Consequently,
techniques for designing pulse forming networks which
require a relatively small number of elements have been
derived.
These techniques are based on the synthesis of
a pulse having finite rise and fall times as opposed to
the zero rise and fall times for a rectangular pulse.
Line-type pulsers do have disadvantages and limitations.
The pulse generated by the line-type pulser is generally
not as rectangular as the pulse generated by the hard-tube
pulser.
The line-type generated pulse usually has a longer
rise and fall time and has oscillation on the top of the
pulse.
Overshoot on the leading and trailing edges is
coramon.
These distortions of the pulse shape may be reduced
by special network design.
Important disadvantages of the line-type pulser are
the difficulty of changing the impedance or the pulse
duration.
These factors are determined by the element
values.
The element values are not easily varied by
conventional methods.
In this thesis background theory and historically
important design techniques for the design of pulse forming
networks (PFNs) are presented.
A design technique for a
pulse forming network that is easily constructed and yet
generates a well-formed pulse is explained.
This design
technique is verified by a pulse forming network which is
designed, constructed, and tested.
A method for easily changing the PFN pulse duration
is also presented.
The pulse duration can be reduced by
decreasing the value of the network inductance.
An easily
applied technique for decreasing the inductance by placing
conducting solid cylinders or tubes of different diameters
inside a coil is derived.
This method for reducing the
pulse duration has effects on other PFN parameters and
these effects are discussed.
The accuracy of the method
is verified by actual application to a PFN.
Various
aluminum tubes of different diameters are positioned
within the PFN inductors.
Measured values of inductance
are compared with the values determined by the technique.
PFN perfonnance, with and without inductance modification,
is presented.
CHAPTER II
TRANSMISSION LINE THEORY AND LUMPED
PARAMETER TRANSMISSION LINES
The transmission line, composed of distributed
inductances and capacitances, has several interesting
characteristics.
Ideally, the transmission line can
transmit signals with no distortion, and theoretically
it can produce perfectly rectangular pulses into a matched
load.
In a near ideal transmission line, such as coaxial
cable, there are distinct limitations.
The major limitation
is the excessive length of cable required to generate a
pulse of even a few microseconds duration.
Several hundred
feet of cable, depending on the cable's propagation time,
are required to generate a one microsecond pulse.
Size limitations may be overcome by using a lumped
parameter transmission line.
A lumped parameter line
consists of discrete inductors and capacitors arranged
such that the characteristics of a transmission line are
approached.
The design of a lumped parameter line pennits
considerable flexibility in the choice of the pulse duration
time and the characteristic impedance.
A major disadvantage
of the lumped parameter line used for pulse generation is
its pulse shape.
parameter choice.
Rise and fall tiroes are limited by
Distortion due to oscillation during the
pulse and overshoot on the leading and trailing edges of the
pulse are common.
However, deviations from the ideal
pulse may be reduced by using special inductor-capacitor
configurations.
Transmission Line Theory
To better understand the characteristics needed by
lumped parameter lines to simulate ideal transmission line
operation, a brief review of relevant transmission line
theory and application is presented.
The two-conductor
transmission line may be described by means of four
parameters:
(a) l,
the distributed inductance in henrys
per unit length of line; (b) c, the distributed capacitance
in farads per unit length of line; (c) r, the distributed
resistance in ohms per unit length of line (the sum of the
resistances of the separate conductors per unit length of
line); (d) g, the distributed shunt conductance (leakage
from one conductor to the other through the insulation) in
siemens (mhos) per unit length of line.
Important and accurate results concerning the propagation properties for low-loss lines can be obtained by
assuming that r and g are zero.
This results in a lossless
2
transmission line as shown in Fig. II-l.
The lossless
line can be represented as shown in Fig. II-2 so that the
properties of the transmission line can be described more
clearly and a coordinate system can be estciblished.
!
I
I
i
i
I
I
I
I
i
i
i
i
i
I
I
I
I
i
i
i
i
I
I
I
I
I
_-i
I
I
I
I
I
I
1
Fig. II-l. Schematic representation of distributed
inductance and capacitance of a lossless transmission line
i(x,t)
•
+
v(x,t)
i(x,t)
í
x-Ax/2
Å,c
_
I
X
^
x+Ax/2
Fig. II-2. Coordinate system for a lossless two
conductor transmission line.
With the current and voltage polarities as established
in Fig. II-2, it is apparent that the voltage at x^ + Ax/2
differs from the voltage at x, - Ax/2 whenever the current
through the incremental inductance Aí, is changing.
The
wavelengths of the significant Fourier series terms which
forin the pulse are long compared to Ax and therefore the
current at x^ may be considered the average of the actual
current distribution over the interval from -Ax/2 to +Ax/2.
Thus as Ax approaches zero
3i(x^,t)
v(x^ + Ax/2,t) = v(x^ - Ax/2,t) - 2M
^
.
(II-l)
Similarly, the current at x, + Ax/2 is different from the
current at x, - Ax/2 due to voltage changes across the
incremental capacitance.
This expression as Ax approaches
zero is
8v(x,,t)
±{x^ + Ax/2,t) = i(x^ - Ax/2,t) - cAx
~
.
(II-2)
It can be assumed that the difference between the voltage
at X, + Ax/2 and x, - Ax/2 can be expressed in terms of
the first derivative of v with respect to x at x^.
yields
This
8
3i(x^,t)
-ilAx
9v(x^,t)"
= Ax
ãt
(II-3a)
3x
-^JP^X.
or
í ^i(x.t) ^ 8v(x,t)
3t
dx
(II-3b)
by dividing through by Ax and dropping subscripts since the
relationship holds for all points on the line.
Correspond-
ing assumptions and algebraic manipulations can be made
for the current to yield
-c
av(x,t)
8t
^
ai(x,t)
8x
(II-4)
By differentiating Eq. (II-3b) with respect to x and
Eq. (II-4) with respect to t, and noting that the order of
d/dx
and 9/9t on i or v is immaterial since the first and
second derivatives are continuous,
the following wave
equations can be derived:
a v(x,t)
3x2
=
ic
a^v(x,t)
at^
(II-5)
and
i!i%ti ^ ,^ a'i(x,t)
^
(,,.,j
Eqs. (II-5) and (II-6) are one-dimensional forms of
the wave equations.
The solutions of Eqs. (II-5) and (II-6)
are known to consist of waves that can travel at the
velocity \//%ô
in either direction, without change in
2
magnitude or foarm.
These solutions are
V = f^(TX-t) + f^^TX+t)
(II-7)
i = 1/ZQ rfj^(TX-t) - f^ÍTX+t)"! ;
(II-8)
and
where Z
= /Jí/c (characteristic impedance of an infinite
transmission line); T = / ô (inverse of the wave propagation
velocity); and f, and f^ represent any single-valued
function of the argument /Icx- t and /Ãac+ t respectively.
The discussion to this point has concerned transmission
lines of infinite length.
However, a practical transmission
line has a finite length and therefore has transition
points at its terminations.
Ohm's Law and Kirchhoff's Laws
10
must be obeyed at each transition point, and in order to
meet all requirements simultaneously, a set of voltage
and current waves departing on each of the lines joined
at the discontinuity is usually necessary.
For the purpose of examining the properties of
propagating voltages and currents at transition points,
one can assume that a voltage is propagating along a
transmission line toward a resistive termination as shown
in Fig. II-3. At any point along the line, the voltage
(v(x,t)) is the sum of the instantaneous voltage wave
propagating to the right (v (s,t)), and the instantaneous
voltage wave propagating to the left (v (x,t)).
Across the terminal resistance, Ohm's Law specifies
the voltage and current to be proportional to R . Combining
these conditions with Eqs. (II-7) and (II-8) yields an
expression for v (x,t) in terms of v (x,t)
R — Z
v^(x^,t) = v^(x^,t) ^^
^
r r
o r
^t + ^^
.
(II-9)
The expression (^'2^)/(R^+Z^), known as the reflection
coefficient, gives the magnitude and sign of any voltage
(and consequently the current) reflected from a resistance
due to an impedance mismatch at that termination.
Obviously,
if the transmission line is terminated by a resistance
equal to its characteristic impedance, the line behaves as
11
i(x,t)
+
v(x,t)
^
Fig. II-3.
Å,c
^
i(x,t)
Transmission line with resistive termination
12
an infinite line and there is no reflection.
For the
purpose of pulse generation, the terminations of interest
are the open-circuit and the short-circuit where the
reflection coefficients are positive unity and negative
unity respectively.
A pulse-generating circuit for the open-circuit
(voltage-fed) termination is shown in Fig. II-4 for a
transmission line having a characteristic impedance Z
and a one-way delay time T. The entire transmission line
is charged to an initial voltage V , and at time t=0, the
ideal switch is closed.
Since the transmission line appears
as a source with a series impedance Z , and since R, is
equal to Z , the current which flows when the switch
closes is V /2Z . A voltage of magnitude V /2 is developed
across R^.
Simultaneously, a voltage wave of magnitude
Li
-V /2 begins propagation down the transmission line toward
the open-circuit tennination.
At time T this propagating
voltage reaches the open-circuit termination at which time
the entire line has a potential equal to V /2. For the
open-circuit configuration, the reflection coefficient is
one and so a voltage of magnitude -V /2 is reflected from
the open-circuit teannination.
At time 2T, the propagating
voltage wave reaches the load R,.
reflections.
There are no further
Also at time 2T the entire line is discharged
and all the energy has been dissipated in R,. during the
13
+
t=o
v
RT
L
= Z
ZQ/ T
^P=°°
(open circuit)
o
Fig. I I - 4 . Voltage-fed pulse generating c i r c u i t
ideal transmission l i n e .
using
14
pulse of magnitude V^/2 and duration 2T.
A circuit for a pulse-generating network using a
transmission line terminated with a short circuit (currentfed) is shown in Fig. II-5.
For this circuit, a pulse is
generated when the switch is opened and the current I
interrupted.
V = I /2Z
is
At this time, a positive voltage of magnitude
is developed across R,. . This voltage propagates
to the short circuit termination where it is reflected
with a value of -v.
This voltage begins propagation back
toward the load reducing the voltage on the line to zero
as it progresses.
voltage I /2Z
The pulse generated across R,. has a
and duration 2T.
Networks Derived From Transmission Lines
The simulation of an ideal transmission line using
liamped parameters presents a problem which is stated by the
following:
From the mathematical point of view, physical
problems involving distributed parameters give rise
to partial differential equations, whereas lumpedparameter probleras give rise only to ordinary
differential equations. Inasmuch as the partial
differential equation for a physical problem may
usually be derived by taking the limit of a set of
ordinary differential equations as the number of
equations in the set approaches infinity (usually
referred to as Rayleigh's principle), it is clear
that a finite number can at best give only an
approximate answer to the distributed-parameter
problem, but that the degree of approximation improves
as the number of equations, and therefore the number
of physical elements, is increased.^
15
t=o
»©
R^
X = Zo
Zo'
,T
^
= 0
Fig. II-5. Current-fed pulse generating circuit using
ideal transmission line.
16
A network which represents a transmission line as
determined by Rayleigh's principle is shown in Fig. II-6.
Writing the current loop equations using Laplace techniques
yields a set of equations all in the form of Eq. (11-10),
except for the first and last loop;
i_,(s)
ir+i ^^^
- ^
+ (Ls + 2/Cs)i^(s) - -^J^
= 0
where r is the r^^ mesh.
(11-10)
These equations are recognized
as being difference equations having the general solution
irCs) = h/^
+ B^-^®
,
(11-11)
cosh 6 = 1 +
(LC/2)s2
.
(11-12)
where
A and B are arbitrary constants to be determined by
substitution into the first and last loop equations.
Assuming that the source impedance R^ is zero and the
switch is closed, currents flowing indefinitely and
representing the steady-state condition will occur.
If under these conditions, the values of the arbitrary
constants A and B are detearmined and placed in hyperbolic
form, the input impedance transfoann for the network can
be found by dividing the source voltage transform by the
17
Switch
n
•r+1
I
j Vi
Fig. II-6. Two terminal line-simulating network
consisting of an infinite number of elements (Rayleigh's
Principle).
18
expression for the input current.
This impedance expression
is dependent on the number of sections, n.
If the limit
as n approaches infinity is deterrained, the input irapedance
Z(s) of the network reduces to Z coth T S , the irapedance
o
^
function for an infinite lossless transmission line.
The obvious disadvantage of using a network based on
Rayleigh's principle is that the nuraber of sections needed
to siraulate a transraission line capable of producing a well
shaped pulse becorae prohibitively large.
To yield any
significant iraproveraent of the pulse shape for a finite
nuraber of sections, a large nuraber of sections are needed.
Another approach for developing networks which
siraulate a transraission line is to use the rational-fraction
expansion of the transmission line irapedance function,
Z
coth TS, or its inverse, the adraittance function.
The
rational-fraction expansion of the impedance function is
2ZoTs
Z(s) = Z^ST + j J - L - y ^ 2 —
•
(11-13)
S-ã— + 1
2 2^-^
TT n
The first term on the right hand side of the expression
is the impedance of a capacitance of value C^ = T/Z
C
N
is the total network capacitance.
where
The reraaining terms
represent the operational impedance of a series of parallel
inductor-capacitor corabinations.
The impedance of each
parallel corabination has the forra
L s
Zn "
2
^
1 + L C S^
n n
'
(11-14)
where
2Z T
2^
ir n
c
^n
-
"^ 2Z_
(11-15)
2
and L-^ is the total network inductance.
The network
having these values is shown in Fig. II-7.
A similar network may be derived by using the
rational-fraction expansion on the admittance function
Y(s) = 1/Z(s) = tanhTs/Z .
This network, shown in Fig. II-8,
consists of n parallel sections of series inductancecapacitance combinations where the values of the inductors
are found to be \j/2
and the capacitor values are 8C^/
((2n-l)^TT^).
As the number of sections for either network approaches
infinity, the pulse shapes generated by each network become
20
n
/Tînnnrs
innnnrs
]\-
'N
n
^n "
2Z T
2LN
2_ 2_
2 2
ir n
C
n
=
2
2Z
TT n
Fig. II-7. Circuit representing the rational-fraction
expansion of the transmission line impedance function.
7
•3 Z
^i y
L
^n
- ^
2
^2 - j -
7
7
1
^3 -y
K
o
1
n
8C
N
C =
2 2
n
(2n-l) ir''
Fig. II-8. Circuit representing the rational-fraction
expansion of the transmission line admittance function.
21
identical.
However, under these conditions (n-^») , it can
be shown that there will be considerable overshoot at the
leading and following edges of the pulse (known as the
Gibbs phenomenon) . For a finite nuinber of sections, the
network of Fig. II-7 generates an excessive spike on the
leading edge due to the series capacitance and thus produces
a poor pulse.
The network of Fig. II-8 for a finite number
of sections, will generate a pulse having oscillation on
the top of the pulse and a leading
edge peak of lesser
magnitude than tha average of the pulse top.
Gui1lemin * s Theory
5
Guillemin
saw the problems that occurred when
lumped-parameter networks attempted to simulate the zero
rise and fall time of transmission line pulse generators
and realized that these problems could not be overcome
whenever lumped elements were involved.
He then argued
that since it was impossible to generate an ideal rectangular pulse by means of a lumped-parameter network, the
theoretical pulse that is chosen should intentionally have
finite rise and fall times.
Mathematically, this condition means that the
discontinuity in the pulse shape is eliminated and
that the Fourier series for the generated wave has
the necessary property of uniform connvergence
throughout the entire region. The property of
uniforin convergence insures that overshoots and
oscillation in the pulse can be reduced to any
desired degree by using a sufficient number of
sections.3
22
There are several pulse shapes that can reasonably be
used.
Two shapes Guillemin originally chose are a
trapezoidal alternating-current wave and an alternatingcurrent wave with a flat top and parabolic rise and fall.
The Fourier series for both of these waveforras has no
constant term and consists only of sine terms.
Each sine
term has a specific magnitude and frequency which can be
produced by a resonant series inductor-capacitor section.
The required pulse-generating network is developed by
placing these resonant L-C sections in parallel. A
closer approximation is developed by using more sections.
A network of this form is shown in Fig. II-9.
Networks of the form shown in Fig. II-9 are
inconvenient for practical use since the inductors have
appreciable distributed capacitance (due to the network
configuration) and the capacitors have different values.
Consequently, it is advantageous to devise equivalent
networks which avoid these problems.
There are many
possible equivalent networks since, from the mathematical
viewpoint, all networks having the same impedance function
are equivalent.
There are many mathematical operations
which yield such equivalent networks.
The three most useful and practical networks are
derived from expansion of the impedance function by
Foster's Theorem and expansion of both the impedance and
23
'n-l
n
rix
n
Fig. II-9. Form of voltage-fed network derived by
Fourier-series analysis of a specific alternating-current
wavefoann.
24
admittance function by Cauer's extension of Foster's
Theorem.
These networks are shown in Fig. 11-10.
Other equivalent networks may be developed by
utilizing combinations of the mathematical techniques
used to derive the networks in Fig. 11-10.
The most
important network developed using a combination of these
techniques, the type E network, is shown in Fig. 11-11.
The most significant features of this network are that all
capacitors are of identical value and that each shunt branch
consists of a capacitor and a negative inductance in series.
The ability to use identical capacitors is significant in
t e m s of cost and design simplicity, and negative inductances may be easily realized by mutual inductances between
the adjacent inductors.
This is a very practical network
form since all inductances including the mutual inductances
may be developed by winding coils on a single tubular form
and capacitors can simply be tapped in at the proper points.
A complete mathematical derivation of the network in
3
Fig. 11-11 is found in Appendix A.
The mutual inductances of the type E network are nearly
identical and the inductors have virtually the same values;
these elements may be made equal without significantly
altering the pulse shape.
The construction of a type E
network is thereby simplified to the following:
v
(a) a
continuous solenoid is wound such that the total inductance
25
"1
"2
_1
n
-nnnnn-.
][
il
'N
#
'n-l
(a) Foster's expansion of impedance function.
network.
^l
^3
— "5
•-nnnnrrs-^innnnr
—1
T-^nnnnnrv-^innnnrs
'n-2
n
(b) Cauer's expansion of impedance function.
network.
'n-3
i^
Type A
Type B
'n-l
j
•.. s
'n-4
n-2
1
(c) Cauer's expansion of admittance function.
network.
n
Type F
Fig. 11-10. Foster's and Cauer's expansion of impedance
and admittance function of network in Fig. II-9.
26
^nnrhpr—r-nnjinno-T
12
rnfTrmnn
n-l,n
T
Fig. 11-11. Pulse forming network having equal
capacitances. Type E network.
27
Lj, = "^^Q' (^) the total capacitance C.. = T/Z is divided
equally between the sections, and each capacitor is connected to a tap on the inductor such that all inductances
are identical except for the end inductances which improve
the pulse shape when their self-inductance is 20% to 30%
larger.
Z
represents the characteristic impedance and
T is the one-way delay time of the network.
shown in transmission line theory, Z
T = /L^C^.
As previously
= /Lu^/C
and
These formulas are accurate when used with
type E networks.
CHAPTER III
DESIGN, CONSTRUCTION, AND EVALUATION OF A
PULSE FORMING NETWORK
The design of a voltage-f ed pulse forming network satisfying specif ic requirements and performance needs is presented.
The pulse forming network has been built and tested.
Results
of measurements of the PFN's performance are presented.
Design of a PFN
Design requirements for the PFN are that the pulse duration (2T) be 20 microseconds and the characteristic impedance
be 15 ohms.
The desired physical network configuration is a
long solenoid with unif orm spacing between turns and with the
capacitors tapped in at the appropriate locations.
This con-
figuration, if properly designed, satisf ies the type E network's
requirement for the mutual inductances between adjacent inductors.
The PFN is chosen to have nine (9) sections.
The network capacitance and network inductance are determined f rom the relations L^ = Z T and Cj, = T/Z as def ined in
Chapter II. Thus, the total network inductance and total network capacitance have the values of 150 microhenries and 0.667
microfarads, respectively.
The values for the element cap-
acitances are f ound by dividing the total network capacitance
of C„ by the number of sections. This division yields an
N
element capacitance of 0.074 microfarads. The
28
29
mutual inductance is chosen to be 15 percent of the
internal inductance value and the end inductors are chosen
to be 20 percent larger than the internal inductor values.
The network inductance may be written as the sum of the
self-inductances plus the mutual inductances.
The self-
inductances of the inductors and the mutual inductances
may be written in terms of the element inductors.
The
relation for the network inductance is
Ljj = 7L + 2(1+.2)L + 2(8) (.15)L
= 11.8L
(Ill-la)
(Ill-lb)
The first term on the right-hand side of Eq. (Ill-la) is
the self-inductance of the internal inductances; the
second term is the self-inductance of the end inductors;
and the last term is the total mutual inductance.
Evaluation of Eq. (III-l) for L gives 12.71 microhenries.
The end inductors are 15.25 microhenries and all mutual
inductances are 1.9 microhenries.
Element Design and PFN Construction
Capacitors satisfying the design requirements are commercially available so the minimum size of the PFN is limited by the capacitor
30
dimensions.
The coil may be any length greater than or
equal to the minimum distance between the high-voltage
teanninals of adjacent capacitors a.s long as the coil
satisfies the design values for the inductance.
The coil inductance is determined by the number of
turns, the coil's length, and the coil's diameter.
The
self-inductance of a coil can be determined from the
formula
2 2
^ " (2.54H9r+10Jl)
"^icrohenries ,
(III-2)
where r is the coil radius in centimeters, n is the number
of turns, and £ is the coil length in centimeters.
This
forTTiula is accurate to within one percent for single layer
coils having a coil length greater than 80 percent of the
coil diameter.
The mutual inductance between two coils
c
can be determined by t h e foirmula
22
a A n,n^
M = 0.00986 —..
(Kj^k^ + K^^ "** ^^5^ microhenries, (III-3)
where all values are defined as shown in Fig. III-l.
For
the configuration used, the mutual inductance relation is
simplified since the coils which deteannine the mutual
inductance are identical.
In Eqs. (III-2) and (III-3), the
31
Coil 1
iCoil 2
a A n,n^
M = 0.00986 — A i
^^l^l''"^3^3''"S^5^ raicrohenries
A = radius of coil 1
a = radius of coil 2
2x = length of coil 1
2i = length of coil 2
D = axial distance between coil centers
n, and n^ = total number of tums on the coil 1 and 2, respectively
x , = E>-x
x^ = EH-z
r^ = //2~TT^
x, + A
= /x^ + A
x^
x^
^l " A72 r ^ - r ^
x.
K,. = - -TT
8
5
D
W
; k^ = a í- 3 -
^
K^ = ^
; k^ = 21
4x,
x.
3 -
l^
a
i^
a
4x,
3 -
All Measuronents are
in Centimeters
Fig. I I I - l . Definition of terras for determination of
mutual inductance between coaxial c o i l s .
32
values for the self-inductances and mutual inductances are
known.
So that the other unknown may be deterroined, the
coil radius is chosen to be 5.72 centimeters (2.25 inches)
for convenience.
To determine the coil length and the
number of turns Eqs. (III-2) and (III-3) may be solved
simultaneously.
The solution of the two equations for
the internal inductors specifies a coil 12.7 centimeters
(5 inches) long having 12.6 7 turns.
Solving the equations
for the inductance of the end inductors specifies coils
14 centimeters (5.5 inches) long having 14.37 turns. To
facilitate construction, the number of turns for the
inductors is rounded to the next larger integer value.
The internal inductors then have 13 turns, and the end
inductors have 15 turns.
All inductors are wound separately and are then
properly spaced on the coil form.
The inductors are
connected to .318 centimeter (-g- inch) copper bus bars which
are connected to the high-voltage terminals of the
capacitors.
The ground terminals of all capacitors are
connected by a common copper bus bar.
A photograph of the
finished PFN is shown in Fig. III-2.
The element values of the constructed PFN have been
measured and are listed in Table III-l.
The measured
values are not equal to the design values (the total
capacitance is 6.5 percent less than the design value and
33
Fig. III-2.
Photograph of constructed PFN
34
TABLE III-l
MEASURED ELEMENT VALUES
Capacitance
(Microfarads)
Self Inductance
(Microhenries)
Mutual Inductance
(Microhenries)
Cl=.0691
Ll=15.5
M^_2=2.2
C2=.0692
L2=12.5
M^.^^^.O
C3=.0684
L3=12.5
C4=.0681
L4=12.5
C5=.0683
L5=12.5
M, .=2.0
3-4
M. ^.=2.0
4-5
M^ ^=2.0
5-6
C6=.0720
L6=12.5
Mg_^=2.0
C7=.0690
L7=12.5
M^_g=2.0
C8=.0688
L8=12.5
Mg_^=2.0
C9=.0708
C^=.6237
L9=15.5
Sotal=^^ ^ ^ZM
=150.9
NOTE: Position of inductors and capacitors are
indicated in Fig. 11-11.
35
the total network inductance is 2 percent larger than the
design value).
Consequently, the values for the pulse
duration and characteristic impedance are slightly different.
Evaluating the pulse duration and characteristic impedance
for the measured values yields 19.5 microseconds and 15.7
ohms respectively.
The variation of these values from
the design values is 2.5 percent for the pulse duration
and 4.6 percent for the characteristic impedance.
Performance Evaluation of the PFN
An evaluation of the PFN's performance is determined
by placing the PFN into the circuit shown in Fig. III-3.
The circuit shown in Fig. III-3 consists of dc power supply,
a charging circuit, a switch, a triggering circuit, and
a linear resistive load.
The power supply used is capable
of supplying 7,5 kilowatts at a maximum voltage of 15
kilovolts.
The charging circuit serves to isolate the
power supply from the PFN during operation and yet allows
the PFN to be properly charged by the power supply.
The
switch is a hydrogen thyratron capable of switching 30
kilovolts.
The triggering circuit switches the thyratron
to the conducting state at desired times.
The linear
resistance is a copper sulfate-water solution.
The load voltage and load current are measured by
means of a 1000:1 compensated voltage divider and a
current transformer.
The outputs of the voltage divider
36
Charging C i r c u i t
PFN
nnnnnnrmmp
<^—nnnnnnr^
j
D.C. Power Supply -.=LrLjTrigger Circuit
Fig. III-3.
PFN and operational support equipment.
+
37
and the current transformer are observed by means of
an oscilloscope.
Fig. III-4 is a photograph showing
the PFN with its support equipment and its measuring
equipment.
The steps for generating a pulse with the PFN are
as follows:
(a) the PFN capacitors are charged to an
initial voltage; (b) the trigger circuit turns the
thyratron "on" and the thyratron becomes a short-circuit;
(c) the PFN discharges through the load resistance
generating a voltage pulse across the load; (d) after the
PFN completely discharges, the thyratron deionizes and
becomes an open-circuit; (e) the PFN capacitors begin to
recharge.
Oscillograms of the load voltage and current for
different values of initial capacitor voltages are shown
in Fig. III-5.
Examination of the oscillograms shows
that there is slight overshoot on the leading and trailing
edge of the pulse and very slight oscillation during the
pulse.
The pulse's magnitude varies linearly with the
initial voltage across the capacitors so the pulse shape
is always the same.
The pulse duration is observed to be
approximately 20 microseconds.
In summary, the constructed PFN meets specific design
requirements.
The measured values for the elements are
within ten percent of the design values.
The PFN parameters.
38
-p
c
<D
e
Qi
•H
P
cr
0)
4J
»4
O
Oi
0«
3
m
»0
<d
IM
o
X
M
o
-p
o
x:
I
•H
39
(a)
LíøBBáaBHBal
250v/cm
MHMMÍÍIIMMRMI
5 Ms/cm
(b)
500v/cra
HMHHRHHHH
MH
HI
m
in
HH H
H
HHHHHHHHH
5 ys/cra
I
500v/cm
20 ys Pulse
(No Time Scale)
Fig. III-5. Oscillograms of PFN generated voltage
pulse across linear resistive load.
40
pulse duration and characteristic impedance, as determined
from the measured element values, closely correspond to
design values.
Most of the variation of these PFN
parameters from the design values is due to the element
capacitors which are, on the average, 6.5 percent sraaller
than the design values.
The network inductance is within
2.0 percent of the design value. Measurements of the
voltage and current pulse generated by the PFN reveal that
the PFN operates as designed and within specifications.
CHAPTER IV
MODIFICATION OF PULSE DURATION
An inherent disadvantage of any pulse forming network
is the fixed pulse length.
The pulse duration is specified
by element values which are usually inflexible.
Often,
applications require high power pulses of different
durations, and consequently other methods for generating
pulses, such as hard-tube modulators, are usually employed.
PFNs have significant advantages over other methods
of pulse generation in terms of circuit simplicity and
higher efficiency.
If simple, easily-applied methods for
changing PFN pulse durations were available, applications
for PFNs and PFN usage would greatly increase.
To
deteannine a method of changing the pulse length of a PFN,
the expression for the pulse length is examined.
The
pulse duration for a type E network is defined by the
relation 2T = 2/L^C . As characterized by this equation,
the pulse duration can be reduced by decreasing the
value of the total network inductance, the total network
capacitance, or both.
For reasons to be explained later,
only reduction of the pulse length is to be considered.
A major advantage of the type E network is the
capability of using equal capacitances.
Since expenditures
for capacitors form the major portion of PFN costs, the
41
42
advantage of not changing capacitor values is evident.
The most viable alternative thus lies in changing the
network inductance.
Reducing the network inductance without changing the
relationships between the element inductances which aid in
the formation of the pulse shape presents problems.
Reducing the element inductances by methods such as the
removal of turns or the shorting of turns changes the
mutual inductances between the elements as well as
requiring additional switching and mechanical hardware.
Such methods also do not provide flexible or accurate
control of the inductance.
Changing the network inductance affects the PFN's
characteristic impedance considerably.
impedance Z
The characteristic
is determined from the expression Z = /L^/C^.
Comparison of this expression with the relation for the
pulse duration, 2 T = 2/L^C^,
shows that both Z
and T
vary at the same rate for an inductance change.
Changing Z
and T does not affect the energy storage
capabilities of the PFN since the network capacitance is
not changed.
However, since the energy stored in the PFN's
capacitors is constant (for a given voltage), the instantaneous power delivered to a matched load increases as the
inductance decreases.
43
The Use of Shielding to Reduce Inductance
Conducting shields are used in electronic circuits
when magnetic coupling between coils and the other elements
of the circuit needs to be minimized or eliminated.
Completely surrounding the inductor or coil with a conducting material having a closed electrical path reduces
the magnetic field outside the conductor.
This conductor
confines the magnetic field to the space within its
boundaries, and thus shields the external circuitry from
the coil's magnetic field.
The effectiveness of the
shield depends upon the excitation frequency, the shield's
conductivity, and the shield thickness.
The presence of the shield has two effects on the
7
impedance of the inductor which Welsby
follows:
summarizes as
(a) eddy-currents induced in the shield produce
an opposing field which lowers the inductance of the coil;
(b) the shield relies for its operation on absorption of
the field penetrating it, so energy is dissipated in the
shield with a resultant increase in the effective resistance of the winding.
The effect of the shield in reducing the inductance
of the coil suggests a practical method of reducing the
inductance of a PFN.
The inductance can be decreased
since eddy-currents, and therefore the opposing magnetic
field, increase as the distance between the core and the
44
shield decreases.
A useful modification of the normal
shield configuration is placement of the conductor inside
the inductor instead of outside.
This conductor configura-
tion has the same effect on the inductance as the external
conductor configuration, but the interior space of the
conductor instead of the external space is shielded from
the magnetic field.
The effectiveness of the conductor
still depends on its conductivity, its thickness, and the
frequency, but its effect on the inductance is even more
dramatic because the induced eddy-currents increase since
the magnetic flux density is greater inside the inductor
than outside the inductor.
The prospect of using internal
conductors is particularly attractive when the inductor is
a solenoid as for the type E PFN, since this allows the use
of cylindrical solid conductors or cylindrical conducting
tubes.
To deteannine completely the usefulness of this method
of reducing inductance, a careful evaluation of all the
effects is required.
This means that the effects which
determine eddy-currents and the field generated by eddycurrents must be determined.
include:
The areas to be investigated
(a) the relationship between the magnetic field
intensity and inductance; (b) the mechanism for eddycurrent generation and the field generated by eddy-currents;
(c) skin effect and skin depth and their effects on
45
eddy-currents; (d) the effect of a conducting solid
cylinder on an inductor; (e) the effect of a conducting
tube on an inductor; (f) the uniformity of the magnetic
field inside a solenoid.
Inductance
The inductance of a circuit relates the georaetry of
a circuit to the flux linkages produced by a current in
the circuit.
A derivation for the inductance for an
arbitrary circuit configuration begins with a single loop
and fundaraental raagnetic field relations.
The magnetic flux $ is given by
=í^
$ = <j> B-ds
S
,
(IV-1)
where B is the magnetic flux density.
For the single loop
shown in Fig. IV-1 the magnetic flux density at any point
p is
P^I r
B = 4Tr
°
áZxr
•'' dv
V
,
(IV-2)
r^
where I is the loop current, di^ is the incremental length
along the loop, and r, is the vector from d£ to the point p
For the arbitrary single loop shown in Fig. IV-2, $ is
evaluated to be
46
Fig. IV-1.
loop current.
Determination of B at a point P due to a
^
Fig. IV-2. Evaluation of magnetic flux for single loop
47
dl
* = ¥f (f
^b
(IV-3)
where r is the distance between a fixed point p on the
circuit and an element áZ of the same circuit. The double
—a
integral is explained as follows: (a) form the elementary
vector dí.a at a point p located at the position of another
element vector d£, of the same circuit; (b) sum the
—^D
elementary vectors corresponding to the element ái
around
a scalar
the circuit giving 6 (dÅ /r) at p; (c) then form the
^a
product of the vector with d^, at p; (d) sum the resultant
scalar quantities for all dÅ, 's around the circuit. 8 The
total flux linkage is $ = LI, where
'feí í
dJl
—^b
(IV-4)
is defined as the total self inductance of the circuit in
a linear homogeneous medium and depends only on the
geometry of the loop.
When two loops are involved, the total flux through
either loop is a function of the currents in both loops
and the geometrical arrangement of the loops with respect
to each other.
Referring to Fig. IV-3, the magnetic flux
48
Fig. IV-3. Determination of magnetic flux coupling in
loop a due to current in loop b.
49
*^ab P^o^^^c^^ i^ ^ loop b and linking loop a is
$
= t ^^
ab - r
^b'^a
^a
(IV-5)
'
where —
ds^
a is an element of area on the surface Sa bounded
by loop a, and B, is the magnetic flux density at a point
on S
due to the current I, .
Replacing B, by the curl of
the vector potential (B = VxA) and applying Stokes Theorem
yields
$
ab
= f ^'^,
=f
^o^
(IV-6a)
(IV-6b)
. djla = M^b^b
where
«a. = îf f J,
d£a -d—^D
il,
—
(IV-6C)
M , is a quantity depending only on the geometry of the
ab
two loops, which when multiplied by the current in loop b
gives the flux linking loop a.
An evaluation of the flux linking loop b due to the
current in loop a yields an expression for M,
which is
identical to the expression for M , since it is symmetrical
I t^AAw
5 É;|> ii ^ .>«.'J»|fisY
50
with respect to the indices a and b.
The mutual induct-
ance is positive if a positive current in loop a produces
a positive flux in loop b.
The total inductance of an
inductor is the sum of the self inductances of its turns
plus twice the sum of the mutual inductances between
tums.
The inductance can also be determined in terms of
the magnetic flux density and the current.
From the
relation L = $/1 and Eq. (IV-1), the inductance can be
evaluated as
<j>
L = -^-=
B'ds
.
(IV-7a)
When B is the average flux density, the relation reduces
to
B
X AREA
L = ZÊZe
^
(IV-7b)
These expressions are particularly useful when the change
of inductance for a change of flux density is needed.
Both Eq. (IV-7a) and Eq. (IV-7b) are expressions for the
inductance of a single loop.
When the coil has more than
one turn, multiply the inductance value in Eq. (IV-7b) by
n
where n is the number of turns of the coil.
51
Eddy-Currents
The eddy-currents generated in a conductor when the
conductor is placed in a magnetic field alter the magnetic
field.
To determine the exact effect on the magnetic
field, the generation of eddy-currents and the magnetic
field produced by eddy-currents are investigated.
When a material is placed in a time-varying raagnetic
field, a voltage is induced in the material.
voltage is described by Faraday's Law.
The induced
In integral form
Faraday's Law is
f
E-d£ = -II .
C
"~
(IV-8)
^^
Eq. (IV-8) states that the total induced voltage around
the surface of the material is equal to the negative
time rate of change of the total magnetic flux contained
within the area of the material.
If the material in the magnetic field is a conductor,
currents will flow in the conductor.
The magnitudes of
the currents are determined by the magnitude of the
induced voltage and the conductivity of the conductor
as they are related by the general form of Ohm's Law.
In
a good conductor displacement currents are negligible and,
J = OE
,
(IV-9)
52
where J is the current density and o is the conductivity.
The paths which the currents follow are determined by the
shape of the conductor and the frequency with which the
field is changing.
The conductor shape having the most potential for
application with a solenoid is the circular cross section.
This allows the choice of conductors to be either a solid
cylinder or a tube.
Further analysis will be concerned
only with these two conductor forms.
When a tube or solid cylinder is placed in a unifoann
time-varying magnetic field with the field parallel to
the conductor axis, the induced currents flow in concentric
circular paths in a plane perpendicular to the axis if the
9
conductor is homogeneous.
The current changes in
cunplitude and phase angle with increased depth of penetration into the conductor.
Assume that a uniform magnetic field is produced by
a long solenoid having a constant sinusoidal current
excitation as shown in Fig. IV-4a.
The voltage across the
solenoid is 90 degrees out of time phase with the solenoid
primary magnetic field H , which is proportional to and in
Cr
phase with the magnetic flux density.
When the tube or
solid cylinder is placed inside the solenoid as shown in
Fig. IV-4b, the currents induced in the conductor produce
9
a secondary field H .
In the case of nonmagnetic
53
)iE.
90
excitation_
current
r
$
(a) Relation of raagnetic flux and coil voltage without
conducting cylinder.
E„ = E + E
T
p
s
excitation
current
(b) Relation of magnetic flux and coil voltage in a
coil with a cylindrical conducting core.
Fig. IV-4.
Phase relationships of magnetic flux in a coil
54
conductors, H
opposes the priraary field.
The degree to
which the secondary magnetic field opposes the primary
field is determined by the conductor conductivity, conductor size, and the frequency of the primary field.
As
the conductivity or the frequency increases, the angle
between H
and H
approaches 180 degrees.
In both Fig. IV-4a and Fig. IV-4b the coil resistance
of the coil is assumed to be negligible and the secondary
magnetic field in the turns of the coil is neglected.
The
vector sum of the primary and secondary fields produces a
total field.
Since the magnetic field has changed, the
voltage induced into the conductor changes.
the conductor current changes.
As a result
In the steady state the
conductor current will finally attain a value which is
consistent with the final existing magnetic field.
The impedance of the coil at a particular frequency
can be determined by the ratio of the coil phasor voltage
to the coil phasor current.
Examining the phase relation-
ships of the coil voltage and current in Fig. IV-4b, it is
observed that the impedance of the coil decreases since E^
has a smaller magnitude and I is held constant.
This is
the effect desired since it means that the inductance has
decreased.
The phase angle between the coil voltage and
coil current is no longer 90 degrees which indicates that
the coil impedance is no longer totally inductive but now
55
includes a
reflected resistance.
The core resistance is
reflected to the coil in a sirailar manner as the resistance of the secondary winding of a transformer reflects
into the primary winding.
The degree to which the
inductance is decreased will depend largely on the coupling
between the coil and the conducting core.
Increased
coupling increases the secondary field and decreases the
coil inductance.
Skin Effects
The skin effect phenomenon is one result of the flow
of eddy-currents in a conductor.
The skin effect causes
the currents to be concentrated on the conductor surface
nearest the excitation coils or other sources of the
magnetic field.
Due to the skin effects, the magnitude of
the eddy-currents decreases almost exponentially with the
depth of penetration into the conductor.
As the penetration
depth increases, the phase angle of the eddy-currents
becomes increasingly lagging with respect to the surface
current.
The skin effect increases with increased operating
frequency, conductor conductivity, and magnetic permeability
The skin effect can be explained in several different
9
but related ways.
One explanation shows that at any
depth the eddy-currents generate raagnetic fields at greater
depths which oppose and reduce the primary magnetic field
causing a decrease in current as depth increases.
The
56
skin effect can also be considered a result of energy
absorption from the electromagnetic wave as it penetrates
the conductor.
As an approach for deteannining the skin effect,
assume that a conductor plate, extending infinitely into
the x-direction and the z-direction and having a finite
thickness 2b in the y-direction, is placed in a magnetic
field having a component only in the z-direction.
The
magnetic field is further constrained to be a spatial
function of y alone and to vary sinusoidally in time.
This
configuration is shown in Fig. IV-5. Under these conditions
eddy-currents are induced which flow only in the x-direction.
Using Maxwell's equations and Ohm's Law,
VxE = - If
and
J = aE
,
(IV-10)
gives
3J
8H
OM^ -3t
^
3y^ = "^o
.
(IV-11)
For a good conductor
VxH = J
.
(IV-12)
For the configuration shown in Fig. rv-5, Eq. (IV-12) is
57
Hsl
X
-P
Hs2
-^— X
-b
Fig. IV-5. Cross-section of semi-infinite plate placed
in a magnetic field oriented in the z-direction.
58
3H
-97 = Ix
•
(IV-13)
Differentiating Eq. (IV-13) with respect to y and substituting into Eq. (IV-11) yields
3 ^H
3H
-—^=ou^-^
.
(IV-14)
Eq. (IV-14) is known as the diffusion equation and has the
identical form as the diffusion equation for the diffusion
of heat through matter.
Since H
varies sinusoidally, the substitution of juj
for —• into Eqs. (IV-11), (IV-13), and (IV-14) is permissibOe
This yields
3J
- ^ = Í^ou^^
^
= J^
^ \^ = a^H^
2
^2
—z
ay
2
= a H^
;
;
(IV-15)
(IV-16)
;
where a = (l+j)/ô and 6 =
(IV-17)
/2/U)0M^.
Eq. (IV-17) has the
59
general solution
-z " ^l®"^ "^ K^e-^y
,
(IV-18)
where K^ and K^ are coefficients that depend on the system
in which the plate is situated.
The evaluation of Eq.
(IV-18) is straightforward and it shows that the magnetic
field intensity is attenuated and delayed at greater depths
in the conductor.
The derivative of H
with respect to y
gives the current density.
The value 6 is called the standard depth of penetration
or often just the skin depth.
At the depth equal to 6,
the eddy-currents decrease to approximately 1/e times the
value at the surface.
Thus, the magnetic field and eddy-
currents exist below the skin depth.
When y = 6/ their
magnitudes are about 37 percent of the surface values and
are just under two percent when y = 46.
Inductor With Solid Cylindrical
Conducting Core9
A conducting core has two effects on the impedance of
the inductor:
(a) the inductance is reduced; (b) the
effective series resistance is increased.
The degree to
which these effects occur depends considerably upon the
conductor's characteristics and the frequency.
For this dis-
cussion the material is assumed to be isotropic, linear, and
60
homogeneous.
Other assumptions made are that the inductor
has a steady-state sinusoidal excitation waveform, no
charge accumulation occurs, and that the magnetic field
is uniform in the z-direction.
is discussed later.
The validity of the latter
For convenience, the axis of the con-
ducting cylinder coincides with the z-axis of the cylindrical coordinate system as shown in Fig. IV-6.
The magnetic field intensity is determined at all
points in the conducting core.
To accomplish this, the
required wave equation for magnetic wave propagation in
the material is derived.
This equation is then solved in
cylindrical coordinates.
The solution consists of Bessel
functions which may be evaluated from the boundary
conditions.
The solution for the magnetic field intensity
is used to determine the magnetic flux $ which, in turn,
is used to evaluate the coil impedance.
Beginning with Maxwell's field equations and using
sinusoidal steady-state field relations yields
VxH = (a+ja)e)E
,
(IV-19)
and
3H
VxE = - P Q J= = -Jt^yQH
.
(IV-20)
61
r
î
1
^^^^^^
Z '^
^
•N^
•p(:",z,e)
>
'v.^^y^fc^^J^
"
1
Fig. IV-6. Orientation of the conducting cylinder in
cylindrical coordinate system.
62
Taking the curl of Eq. (IV-19) and substituting Eq. (IV-20)
for VxE which then occurs on the right-hand side of the
resulting equation yields
VxVxH + -j (a+ja)e)(jop H
,
(IV-21)
which by using vector identities reduces to
V^H = j(a+ja)e)y wH
,
(IV-22)
V^H = jom
,
(IV-23)
or
H
where o is much greater than coe (a good approximation in
a good conductor).
In cylindrical coordinates (see Fig.
IV-6)
2
1 3 . 3H,
1 8^H
3^H
Since the solution is independent of 6 and z, Eq. (IV-24)
reduces to
^2„ _ ^ ^z . 1 !5z
^ ^ " ":r2"
3r
r 3r
.
(IV-25)
63
Substituting Eq. (IV-23) for V^H, Eq. (IV-25) can then be
written to yield the wave equation for magnetic wave
propagation in a good conductor.
a^H
8H
dr
where
k^ = (iJU a
.
(IV-27)
At this point, for convenience, vector notation will
be dropped from the field vectors since they are only rdependent.
The notation will only be used when necessary
to prevent confusion.
Eq. (IV-26) is recognized as being a form of Bessel's
differential equation which has a solution of the foann
H
= AJ^(krj^/^) + BK^(krj-'-/^)
z
o
o
= Al^^krj-"-/^) + BK^(krj-''/^)
(IV-28a)
.
(IV-28b)
where J is a Bessel function of the first kind and order
o
zero,' and I o and K o are modified Bessel functions of the
first and second kind respectively, both of zero order.
64
A and B are arbitrary constants determined by boundary
conditions and j is / ^ .
The constant B can be evaluated since the modified
Bessel function K (krj1/2
' ) increases without limit as r
approaches zero.
This condition is not realizable with
the given boundary conditions since the magnetic field
intensity would approach infinity at the center of the
cylinder.
Consequently the constant B must be zero.
Eq. (IV-28b) then reduces to
H = AI (krj-"-/^) = A(ber kr + jbei kr)
z
o
,
(IV-29)
where ber kr and bei kr are the Kelvin Bessel real and
imaginary functions respectively, with argument kr.
The constant A can be solved by evaluating Eq. (IV-29)
for r = a, the cylinder radius in centimeters (seeFig. IV-7) ,
and H
= H , where H is the magnetic f ield intensity inside the
solenoid before the conductor is inserted.
A = ber ka !°jbei ka
This yields
'
'^^-^O'
which when substituted into Eq. (IV-29) yields
H
= H
z
ber kr + jbei kr
o ber ka + 3bei ka
^
^^^,3
65
Conducting
Cylinder
a and b in raeters
Current
Sheath
Fig. IV-7. Conducting cylinder surrounded by conducting
current sheath carrying the current I .
66
Eq. (IV-31) describes the magnetic field intensity at any
point inside the cylinder.
Given the magnetic field intensity, the flux and its
effect on the inductance of the surrounding coil can be
determined.
The inductance is equal to the total flux
divided by the coil current and multiplied by the number
of turns in the coil.
The total flux is equal to the sum
of the flux within the cylinder and the flux within the
air-gap between the cylinder and the coil.
The total flux
in the cylinder can be determined by multiplying Eq. (IV-31)
by 2\I Trrdr and integrating from r = 0 to r = a.
The total
flux is then
^T = ^iry^
^^ H J^ (ber kr + jbei kr) ,
0 o (ber ka + jbei ka)
+ 27rp^ I H^rdr
a
.
(IV-32)
Eq. (IV-32) can be evaluated by using the formulas
í
z ber(kz) dz = fbei' kz
,
(IV-33a)
í
z bei(kz) dz =^ber* kz
,
(IV-33b)
and
67
where
I^(kziV2)-J
dz [_
= kber kz + jkbei
kz
(IV-34)
Evaluation of Eq. (rV-32) then yields
$-, =
2Try aH
o o bei
k
ber
1
ka - jber
ka + jbei
ka
ka
+ y^TTH^OD^-a^)
o o
(IV-35)
To determine the core's total effect on the coil, the
impedance of the coil is needed.
The impedance is found
by dividing the phasor voltage across the coil by the
phasor current through the coil.
The induced coil voltage
may be determined by applying Faraday's Law.
This is
(IV-36)
^i =
= - Ht tc - ' — " • dt '
where e. is the total induced voltage for a single-turn
loop.
Applying Eq. (IV-36) to Eq. (IV-35) yields
e^
xj
2a
= -oíTTy^H^
^
ber
ber
ka + jbei ka
ka + jbei ka
janry^H^(b^-a^) .
(IV-37)
For the analysis purposes, it is convenient to assume
that coil is infinitely long, has a single turn, and has
68
a current I^ amperes/meter.
For this configuration the
magnetic field intensity can be determined by applying
Ampere's circuital law and is found to be H = I . 8 Subo
s
stituting I for H in Eq. (IV-37) and dividing both sides
by I
results in an expression for the coil impedance Z.
Z
= (jayira'
ka
ber
ber
ka + jbei
ka + jbei
ka
ka
2
+ j(i)\iT\ {h
2
-a )
(IV-38)
T h e e x p r e s s i o n m a y b e n o r m a l i z e d b y d i v i d i n g the impedance
of t h e coil w i t h o u t t h e c o n d u c t o r into Eq. ( I V - 3 8 ) .
A s s u m i n g that t h e c o i l h a s n e g l i g i b l e r e s i s t a n c e and
c a p a c i t a n c e , t h e coil impedance, w i t h o u t the c o n d u c t o r , is
d e t e r m i n e d by letting the radius of the core shrink t o zero
For these
conditions
Z = joay TTb
(IV-39)
= jwL
w h e r e u)L is the r e a c t a n c e o f the coil w i t h o u t the c o n d u c t o r
o
D i v i d i n g Eq. (IV-38) b y Eq. (IV-39) g i v e s the n o r m a l i z e d
coil
impedance.
oúL
2 íber
ka Iber
ka + jbei
ka + jbei
ka
ka
2 2
+ j b -a
(IV-40)
69
The value of k is /a)y a.
The value for the argument
ka simplifies to
(IV-41)
ka - a/ojp a
Evaluation of Eq. (IV-40) is simplified by using the
relations
3/2
J^(kaj ^ ) = ber ka + jbei ka
,
(IV-42)
and
J Q ' (kaj^/^) = j"^/2
(j^g^- j^^ ^ jj^g^' j^^j
g-JTr/4j^^j^^j3/2j ^ j^^^' j^^ ^ jbei' ka
(IV-43a)
(IV-43b)
Eq. (IV-40) then becomes
Z
caL
o
a'
,2
b
_2^
ka
re-^^/S^(kaj^/2j:
+ j
,2-a 2
b
(IV-44)
jQ(kaj^/2)
Tabulated values for JQ and J^ consisting of their
magnitude and phase angles are found in many sources and
70
1
for convenience are listed in Appendix B.
A plot of
2 2
Eq. (IV-44) for several values of a /b as a function
of increasing ka is shown in Fig. IV-8.
The normalized
inductance is the imaginary part of Eq. (IV-44) and the
normalized resistance is the real part of Eq. (IV-44).
As seen from Fig. IV-8, the normalized inductance, at
high frequencies, may be reduced to a neglible value as
the ratio of the radii approaches one.
The previous relations for the impedance are defined
for a long, single-turn coil having a current I
per meter.
amperes
For a coil having n turns with current I
in
each turn
I„ = nl„
s
n
.
(IV-45)
The induced voltage for such a coil is n times as large
as derived in Eq. (IV-37) and the new impedance is
2
ne.
n e.
-,
Z = T^ = -H' = ^ Z
^
^n
^s
.
(IV-46)
The impedance of a shorter coil may be approximated
by noting that if there are n turns per meter, the induced
voltage is ne• where e. is the induced voltage for a
single-turn coil, one meter long carrying a current I .
O
If the coil has n turns and length l,
the induced coil
71
ka = a/u)ai
_L_
L
a)L
Fig. IV-8. Plots of normalized impedance for coil
surrounding solid cylindrical core.
72
v o l t a g e is p r o p o r t i o n a l to n/l.
7
T h e coil impedance Z
« n^Z
^n "
is
(IV-47)
~
If t h e c o n d u c t o r is a m a g n e t i c
raaterial,
i.e., h a s a
p e r m e a b i l i t y rauch g r e a t e r than that of a i r , the impedance
o f t h e coil is g r e a t l y changed.
D e f i n i n g the p e r m e a b i l i t y
as
(IV-48)
y = P^ii^
where y
is t h e r e l a t i v e p e r m e a b i l i t y of the m a g n e t i c
material and y
is t h e p e r m e a b i l i t y of free s p a c e , a n d
9
s u b s t i t u t i n g into E q . (IV-38) y i e l d s
Z = a^yira'
I 2 íber ka + jbei
\ka. [ber ka + jbei
ka
ka
+ ja)y TTb 2-a2
N o r m a l i z i n g t h i s e q u a t i o n w i t h a)y irb
Z = u
a)L^
2
2 íber
^ ka (ber
^7
ka + jbei
k a + jbei
= a)L
ka
ka
(IV-49)
yields
u2 2
+ j b -a
(IV-50)
73
A plot of Eq. (rV-50) has the same shape as Fig. IV-8.
The significant difference is that at small values of
ka (k = /a)y9) the normalized inductance a)L/a)L
than one.
is greater
However, as ka gets large the normalized
inductance approaches the value determined by j íb^-a^'
—=—
lb
In other words, the magnetic material increases the
inductance at low frequencies, but as the frequency
increases theraagneticfield generated by the eddy-currents
negates the effect of the magnetic material and the
inductance decreases.
Because the norraalized inductance
varies considerably as the frequency changes and at
large frequencies has a value less than unity, the duration
of short pulses cannot be increased using conducting
magnetic materials.
Inductor With a Tubular Conducting Core
A tubular conducting core and a solid cylindrical core
affect the coil impedance in a very similar manner.
The
coil inductance decreases and the effective coil resistance
increases.
The degree to which the inductance is decreased
is dependent upon the dicimeter, conductivity, and thickness
of the tube.
The magnetic field intensity within the tube is
decreased by the opposing magnetic field generated by the
induced eddy-current flow around the tube.
The derivation
74
presented in this section determines the magnetic field
intensity within the tube as a function of the magnetic
field intensity outside the tube.
The variation of the
magnetic field intensity within the tube walls as a
function of penetration depth into the conductor is not
determined.
For purposes of determining the total magnetic
flux ($) inside the coil, the magnetic flux density within
the tube walls is assumed to be equal to the raagnetic
flux density inside the tube.
This assumption is valid
for a thin-walled tube since the cross-sectional area of the
tube is small compared to the cross-sectional area of the
coil.
The evaluation of the magnetic field intensity within
the tube begins with the wave equation obtained in the
previous section for magnetic wave propagation in a good
conductor (Eq. rv-23).
The solution of the wave equation
is, again, in terms of modified Bessel functions I
and
K . The solution is determined by the boundary conditions
o
on the inner and outer surfaces of the tube.
The magnetic
field intensity on the inner wall of the tube, and
consequently the magnetic field intensity within the tube,
is solved in terms of the magnetic field intensity outside
the tube.
Assume that a conducting tube is placed in a uniform
axial magnetic field H^ (such as the field produced inside
75
a long coil) with the tube axis parallel to the field
axis as shown in Fig. IV-9.
As in the previous section,
vector notation will be dropped (except where needed for
clarity) from the field vectors since they are only rdependent.
The magnetic field intensity inside the tube wall,
H^, is in the axial (z) direction and is described by
the wave equation
2
dr
2
where k
= a)ay
for a nonmagnetic material, and r is the
radial distance from the axis (cylindrical coordinates).
The solution of Eq. (IV-51) has the form
Ho = AI (krj-"-/^) + BK^(krj^/^)
2
O
,
(IV-52)
O
where I and K are modified Bessel functions of the first
o
o
and second kind respectively, both of zero order.
The tangential components of the magnetic field
intensity must be continuous across the boundary of the
conductor.
For the outer surface of the tube r = b.
H^ = H^ = AI^(kbj^/^) + BK^(kbj^/^)
,
(IV-53)
76
Fig. IV-9.
magnetic field.
Conducting tube placed in a uniform axial
77
and on the inner surface of the tube r = a.
H^ = H^ = AI (kaj-"-/^) + BK (kaj-^-^^)
±
z
o
o
.
H^ is the magnetic field intensity inside the tube.
(IV-54)
In
Eq. (IV-53) and (IV-54) there are three unknowns, A, B,
and H-, so a third equation is needed.
Applying Faraday's
Law to the path r = a gives
{
I2'— ^
dt
B^-ds
(IV-55)
or
^iraE^ = -j(A)y H^ira'
(IV-56)
Using Eq. (IV-56) and the general expression for Ohm's Law,
(IV-57)
I2 = ^^2
yields
-]k a „
" 2
^l
Í2
r=a
(IV-58)
78
The curl of H^ is
(IV-59)
VxH^ = (a+ja)e)E2 " ^-2 ^ -2
r=a
since in a metal a>>a)e. The curl of H^ is independent of
í6 and z and is dependent only on r.
VxH^ =
3H2
3r
(IV-60)
Combining Eqs. (IV-58), (IV-59), and (IV-60) yields
^"2 , -ik^a H
-5F 2 ~ ^l
(IV-61)
r=a
Applying Eq. (IV-61) to Eq. (IV-52) yields the third
equation
«1 =
kaj 1/2
["«• (kaj^/^) + BK^'(kaj^/^)' ] •
where the primes denote first order derivatives.
(IV-62)
Applying
one of the recurrence formulas for modified Bessel
functions"^^ to Eq. (IV-62) yields
79
«1 =
kaj 1/2
AI^(kaj^/^) - BK^^kaj-"-/^)
(IV-63)
Subtracting Eq. (IV-62) from Eq. (IV-54) and applying the
appropriate recurrence formulas yields
= Al^^kaj-^/^) + BK^^kaj-"-/^)
(IV-64)
From Eq. (IV-64), A can be found in terms of B.
Sub-
stituting the value for A into Eq. (IV-53) yields an
expression from which B and consequently A can be determined in terms of H . Substituting the value of A and
B into Eq. (IV-54) yields
K2(kaj^/2)lQ(kaj^/2) - I^(kaj^/^)K^(kaj^/^)
«1 = %
lQ(kbj^/^)K2(kaj^/^) - KQ(kbj^/^)l2(kaj^/^)
(IV-65)
The numerator may be reduced using recurrence formulas for
K
and I~.
This results in a numerator which is a constant
times the Wronskian of IQ and K^.
reduces to another constant.
to
The Wronskian, in turn,
Eq. (IV-65) finally reduces
80
-F-1
2H
lQ(kbj^/2)K2(kaj^/2) - KQ0dDJ^2jj^(j^jV2j
^ - . 3k
, 2 a2 . ^
(IV-66)
Eq. (IV-66) may be further simplified if both a and b are
much greater than the conductor's skin depth at the frequency of interest.
For these conditions, the modified
Bessel functions may be replaced by the first two terms
of their asymptotic expansions.
The asymptotic expansions
may be found in several sources.11 '12 If, in addition, a
is approximately equal to b, as for a thin wall tube,
other simplifications can be performed.
The resulting
expression, taking into consideration all simplifications.
is
-1-1
«1 =
cosh k(b-a) j"^^ + icaj-^^sinh ka>-a)j"^'^^
a H
(IV-67)
The ratio of H^ to H for tubes of different thicknesses
1
o
and different conductivities as plotted in Figs. rv-lOA
and IV-IOB.
The total flux within the tube will be H, times the
area of the tube since H, is independent of r.
The total
flux with the coil is equal to the sum of the flux in the
tube plus the flux in the gap between the tube and the coil
81
1.0
Copper Tube Inside Radius
.1 Meter
H,
H.
10,000
a)(Radians)
Fig. IV-IOA.
attenuation.
Plot of magnetic field intensity
100,000
82
1.0
Aluminiin Tube Inside Radius
.1 Meter
H.
H
1 0 , 0 0 0 l o o V o 000
c
a) (Radians)
Fig. rV-lOB.
attenuation.
Plot of magnetic field intensity
83
For a tube placed with its axis lying on the axis of the
coil, the total flux is
$„
=
-r-1
/1"
H y TTb' cosh k(b-a)j-'-/^ + ikaj-'-/^sinh k(b-a) j-^-^^
o'^o
+ 2Try
(IV-68)
H rdr
^ib
^
where C is the radius of the coil.
-V|
$_ = H y TT
T
o^o
This simplifies to
-1
oosh k(b-a) j-^^ + ^j-''/^sinh k(b-a) j-*-/^
(IV-69)
+ (C^-b^)
To determine the core's total effect on the coil,
the impedance of the coil is needed.
The impedance is the
ratio of the phasor voltage across the coil to the phasor
current through the coil.
The voltage may be found by
applying Faraday's Law, Eq. (rv-36).
By assuming, as in
the previous section, that the coil is infinitely long, has
a single turn, and has a current I
amperes/meter, H
is
84
found to be equal to I .
By using Faraday's Law and H
=
I , the impedance of the single turn coil is found to be
-1
Z = ja)y TT b ^ / |
cosh k(b-a) j-^/^ + Ícaj-^-Z^sinh k^b-a^j"'-/^
(C^-b^)
(IV-70)
A s in the previous section, E q s . (IV-46) and
(IV-47) apply
to coils having more than a single turn or a finite length
By letting b approach zero, the impedance of the coil
w i t h the tube is found to be
Z = ja)y TTC
-' o
(IV-71)
= ja)L_
-* o
This is assuming the coil has negligible resistance and
capacitance.
Dividing Eq.
(IV-70) by (J^L , the coil
impedance with the tube is normalized to the coil
impedance
w i t h o u t the tube.
Z
^o
-1
^ .b^ / | ícosh k(b-a) j-'-/^ + ^j-'^^sinh k(b-a) j-"^^
" C^
2
2
(IV-72)
85
The norraalized irapedance may be evaluated for
selected values of a, b, and c as k varies over a wide
range of frequencies.
Impedance plots for various values
of tube thickness are shown in Figs. IV-lla and rv-llb.
Magnetic Field Intensity Within a Coil
The preceding derivations for the impedance of a coil
having a conducting core assume that the magnetic field
intensity within the coil is uniform everywhere within the
coil.
This condition exists for an infinitely long coil,
but does not exist for a coil of less than infinite length
If the magnitude of the magnetic field intensity varies
considerably within the coil, the expressions for the coil
impedance derived in the previous two sections may need to
be altered to compensate for the variation.
Consequently,
an accurate technique for determining the magnetic field
intensity everywhere within a coil of any dimensions is
necessary.
To evaluate the magnetic field intensity in a short
solenoid, this section begins with the evaluation of the
magnetic field intensity on the central axis of a single
loop.
The axial magnetic field intensity for a finite .
length, finite thickness coil is determined, and finally,
a series expansion for the magnetic field intensity off
the central axis is presented.
86
T u b e W a l l s 5 mm T h i c k
1.0
3=kb
k b = b/a)ay
.9
C o i l Radius . 1 Meter
.8
.7
L_
.6
.5
.4
.3
.2
.1
0
.1
.4
.5
.6
u) L
Fig. IV-llA.
conducting tube.
Impedance of a coil surrounding a
.7
87
1.0
T u b e W a l l s 2 ram T h i c k
Coil R a d i u s .1 M e t e r
.9
.8
.7
_I^
L_
kb=8
.6
.5 ,
kb=10
.4
.3 ,
a)L
Fig. IV-llB.
tube.
Impedance of a coil surround a conducting
88
The axial magnetic field intensity for the single
loop shown in Fig. IV-12 is determined from the BiotSavart Law to be
2(a +z ) '
where I is the loop current in amperes, a is the loop
radius, and z is the distance from the loop center to the
point on the axis where the magnetic field intensity is
being determined.
If for convenience a and z are expressed
in centimeters instead of meters, Eq. (IV-73) is written
2
H^(z,0) =
j;^ ^ ./o
^
200(a'^+z^)-^^'^
.
(IV-74)
The axial field at the center of the loop (at z = 0) is•
«o-
H = -rkc^Z
o
200a
'
(IV-75)
The axial magnetic field intensity may then be written in
terms of H^ as,
3
H (z,0) = H^ — ^ ^n -./o
2
° (a^+z^)^/^
.
(IV-76)
89
- ^
Hz
->- z
Fig. IV-12. Definition of parameters for an elemental
loop in x-y plane.
90
An expression for the axial magnetic field intensity
at the center of a coil of length 2b and thickness a^-a,
can be determined by integrating Eq. (IV-74)
H
o
1
200
= H (0,0) = T T ^ J
z' ' '
í^2 ^^
ave J
'a ^ -b
•••
^
2^2.5/2 ^^^^ '
(a +z )
(IV-77)
where a,,
X a^,
c. and b are defined in Fig. IV-13, and Ja ve is
the average current density of the coil and is determined
from
ave
^^
^b^a^-a..)
.
(rv-78)
N is the number of turns in the coil, and I is the current
in amperes in each turn of the coil.
Integration of
Eq. (rV-77) yields the expression
HQ = ^ave'TM' ai(""h-l I - sinh-1 \)
where a is a^/a, and 3 is b/aj^.
F(a,3) = Y ^
the value of H
.
(l^J-19)
If F(a,3) is defined as
(sinh"-^ I - sinh"^ i)
,
(IV-80)
simplifies to
H = J
a,F(a,3)
o
ave 1
.
(IV-81)
91
2b
C o i l Windings
11
H
-+^
Coil Windings
a =
3 =
All Dimensions in Centimeters
Fig. IV-13. Definition of dimensions for coil having
finite length and finite thickness.
92
The axial magnetic field intensity H (z,0) at any
point along the axis of a coil can be determined by noting
from symmetry that the end field (the axial magnetic
field intensity at the end of a coil) is one-half of the
magnetic field intensity generated by two such coils
extending in opposite directions.13 In other words, the
end field of a coil is one-half the central magnetic field
intensity H
of a coil twice its length.
At any point z along its axis, the coil may be
divided into two sub-coils:
one sub-coil extending to the
left of z and one sub-coil extending to the right of z.
The magnetic field intensity at z, H (z,0), is equal to the
sum of the end fields of the sub-coils.
As explained in
the previous paragraph, H (z,0) is also equal to one-half
z
the sum of the central fields of two coils which are
twice the length of the sub-coils.
In teinns of Eq. (IV-81)
this results in
F(a,3+^) + F(a,3-;^]
(IV-82)
«zí^'°^ = ^ave^l
and since H^ = 'Jave^l^ ^^^'^^
F ( a r 3 + ^ ) + F ( a , 3--
H^(z,0) = H^
2F(a,3)
^ , "
(IV-83)
93
The coil and the two sub-coils used for the deterraination of
H^^z^O), as specified inEqs. (IV-82) and (IV-83) , are represented in Fig. IV-14.
When the f ield point 2 lies beyond the end of the coil
(z>3) / the expression-F (a,^3) should be used instead of
^l
F(a,3-~) in Eqs (IV-82) and (IV-83) .
The magnetic f ield intensity at points inside the coil,
other than on the axis, cannot be represented in a closed
form solution. 14 However, by imposing an area of convergence upon the solution, the magnetic f ield intensity can
be written as a power series involving Legendre polynomials
and coefficients derived from the coefficients of a Taylor
series expansion of H (z,0). 13 '15 The power series expansion
converges within a sphere having as its radius the distance
f rom any point on the coil axis to the nearest point of the coil
For points inside the coil, the radius of convergence is
the coil radius.
The power series may be written
Hz (r,e)
E E„
' = H^
o n=0
n
n
P^(u)
(IV-84)
and
oo
H^(r,0)
H
Z E„
o n=0 ^
Pn (u)
^'
(IV-85)
94
2b
H^(z,0)
^
(a) Coil for which H (z,0) is determined
2(b+z)-
Hol
(b) Sub-coil 2(b+z)
1^2 ( b - z ) ^
H
o2
(c) Sub-coil 2(b-z)
«z^^'°^ = 2 («ol^ %2^
Fig. IV-14.
of H^(z,0).
z
Coil and two sub-coils for determination
95
where H^ and H^ are the magnetic field intensities in the
z- and radial directions for a point specified by the
spherical coordinates r and 6.
P (u) and P '(u) are
n
n
Legendre polynomials found in Appendix C.
The E
co-
efficients are determined by the derivatives of H (z,0)
evaluated at z = 0.
, , d^H^(z,0)
E„ = •— -^
h.
^
"o ''' dz^
Evaluation of E
(IV-86)
is simplified by substituting Eq. (IV-83)
for H (z,0), Eq. (IV-83).
=n = IF
•
TBHT
This yields
£i f ("'6^§ + F(a,B4j •
The terms of E have F(a,3) in the denominator.
n
(IV-87)
The
expressions of Eqs. (IV-82) and (rv-83) have the term H
as part of the series, and as described by Eq. (IV-81),
H
includes the expression F(a,3).
Consequently, E
can
be multiplied by F(a,3) to yield an expression more
easily evaluated.
If the origin for the sphere of convergence is located
on the coil axis at the midplane of symmetry, the series
expansion will consist only of even terms.
96
H^(r,e) = H^ 1 + B^P^M
z
o
+ E^P^^u)
J
V
(IV-88)
^ave^l F + FE^P^(u)
Hj^(r,0) = H^ 0 + E^P^' (u)
(IV-89)
+ E^P^'íu)
(IV-90)
.^.4
J
a,
ave 1 FE^P^ (u)
+ FE^P^ (u)
V.
J
(rv-91)
FE
is equal to F(a,3)E . Expressions for F(a,3) and FE
n
"
n
are given in Appendix C. Values of F, FE^, FE., and FE^
97
are tabulated in Appendix C for various values of a and 3.
FE8 is not listed since its contribution to the magnetic
field intensity is negligible.
be determined by dividing FE
Specific values of E
may
by F(a,3).
The magnetic field intensity within the coil, varies
as a function of geometry and the number of turns in the
coil.
The magnetic field intensity becomes more uniform
as the coil length increases with respect to the coil
diameter and also as the coil's total number of turns
increases.
The magnetic field intensity within the coil
is strongest in the immediate vicinity of the coil windings
and decreases as the distance from the coil's windings
increases.
The coil inductance for a given geometry depends upon
the magnetic field intensity within the coil.
Consequently,
the relations describing the coil impedance in the preceding
two sections are accurate only when the actual magnetic
field intensity is used.
CHAPTER V
INDUCTANCE MEASUREMENTS AND EXPERIMENTAL
VERIFICATION OF REDUCING PFN PULSE
DURATION BY REDUCTION OF
PFN INDUCTANCE
Reduction of the PFN pulse duration can be accomplished
by reducing the network inductance.
Theoretically, reduction
of the network inductance is easily achieved by inserting
a conducting solid cylinder or a conducting tube into the
inductor.
The inductance may be reduced to any degree by
allowing the conductor diaraeter to approach the inductor
diameter.
In this chapter the relations presented in
Chapter IV for the impedance of a coil surrounding a tube
are applied to the inductors of the nine-section, 20
microsecond PFN described in Chapter III.
Examination of the graphs for normalized impedance
(Fig. IV-8, Fig. IV-lla, and Fig. IV-llb) shows that there
is no significant advantage in choosing a conducting
solid cylinder over a conducting tube.
considerations tubing should be used.
In view of weight
For experimental
purposes, aluminum tubes of different diameters are used
to reduce the PFN inductance.
The aluminum tubes have
tube wall thicknesses of 0.125 centimeters and outside
diameters of 7.5 centimeters (approximately three inches)
and 10.1 centimeters (approximately four inches).
For
each of the two tubes the element inductances and the
98
99
network inductances are calculated from Eqs. (IV-70) and
(IV-72).
The calculated values are compared with the
measured values.
For each tube, the PFN pulse duration
is determined from the calculated network inductance and
compared with the measured pulse duration.
The network
capacitance is constant for all conditions.
The impedance relations derived in Chapter IV are
determined by assuming that the magnetic field intensity
is uniform and is generated by an infinitely long, one
turn per meter solenoid.
However, the magnetic field
generated by a finite length solenoid having several turns
per meter is not perfectly uniform.
Consequently the
impedance relations of Chapter IV must be used carefully
and only when their application gives accurate results.
Calculated Inductance Values
The values for the end inductors, the internal
inductors, and the network inductance are calculated by
two methods.
The first raethod assuraes that the magnetic
field intensity within the solenoid is not uniform.
This
method uses the magnetic field intensity as determined by
the power series expansion presented in Chapter IV to
evaluate the inductance values.
The second method assumes
that the magnetic field intensity within the solenoid is
uniform and uses the plots for the normalized impedances
to determine the inductance values.
The values calculated
100
by both methods are compared to the measured values.
Inductance Values for a NonUniform Magnetic Field
lî the magnetic field is not uniform, the impedance of
a coil surrounding a core can be determined from a variation
of Eq.
(IV-70).
For a coil having n turns and an excitation
current of one ampere, this equation is:
b ^ / ^ (cosh k OD-a) j-^/^+l/^kaj-"-/^ sinh k (b-a) ^^"^ ) '^
Z = ia)y 7m H
-• o ' z
+ (cW)
(V-1)
where |H | is the average magnitude of the magnetic field
intensity for a coil having a specified geometry, n turns,
and an excitation current of one ampere.
Since the excita-
tion frequency is large, the first term in brackets is
negligible and Eq.
(V-1) reduces to
Z = jcoy^TTnlH^I (c^-b^)
The right-hand side of Eq.
.
(V-2) is purely imaginary and
consequently the term is only inductive.
side of Eq.
Z = ja)L
(V-2)
The left-hand
(V-2) may be written
.
(V-3)
101
Combining E q s .
(V-2) and
L = yQTrn|H^|
(c^-b^)
(V-3)
yields
.
(V-4)
The magnetic field intensity inside the coil may be
determined by the power series expansion presented in the
preceding chapter.
The magnetic field intensity is deter-
mined only for points in the volume between the core and
the coil.
As an example the power series expansion is applied
to a coil having the dimensions of the constructed PFN's
internal inductors.
The magnetic field intensity is
evaluated all points within the sphere of convergence.
The magnetic field intensity at the origin (on the coil
axis at the midplane of symmetry) is .006349 amperes per
centimeters.
From the value at the origin, the magnetic
field intensity increases 11 percent in the radial direction
and decreases 25 percent in the axial direction.
With the coreraaterialsplaced inside the solenoid,
the average magnetic field intensity between the tubes and
the coil is determined.
For the 7.5 centimeter tube the
average magnetic field intensity between the core and
the coil is approximately .00651 amperes per centimeters.
Substituting this value for the magnetic field intensity
in Eq. (V-4) yields
102
L = (4TT X 10"^) (Tí) (13) (.00651) (6.35^ - 3.75^) = 8.77 H.
(V-5)
The calculation of the magnetic field intensity is
repeated for the other core diameter and for the end
inductors.
This method does not provide for the determina-
tion of mutual inductances.
However, it is reasonable to
assume that the mutual inductances are decreased to the
same degree as the element inductances.
The calculated values for the end inductors and the
internal inductors are listed in Table V-1 for the two
tube cores.
A value for the network inductance is deter-
mined by summing the values of all element inductances and
the mutual inductances as described in the preceding
paragraph.
The value determined for the network inductance
and the value for the pulse duration calculated frora the
network inductance are also listed in Table V-1.
Inductance Values for a
Uniform Magnetic Field
For a uniform magnetic field, the normalized inductance may be determined from Eq. (IV-72).
Plots of this
equation for each of the two core matericú-s are presented
in Fig. V-1. To determine at which points on these curves
the normalized impedance is accurate, the coil's excitation
frequencies are needed.
103
TABLE V-1
CALCULATED INDUCTANCE VALUES ASSUMING A
NON-UNIFORM MAGNETIC FIELD
7.5 cm
Core
%Deviation
10.0 cm
Core
%Deviation
10.27 yH
+2.70%
6.14 yH
+ 2.39%
Internal Inductors
8.77 yH
+3.17%
5.18 yH
+ 3.60%
Network Inductance
96.46 yH
+0.50%
53.60 yH
+14.00%
Pulse Duration
15.51 ys
-5.40%
11.56 ys
+ 1.40%
End Inductors
104
2
3
Tube Thickness .00125 m
Coil Diameter .0635 m
kb = b/a)ay
8=kb
_L_
L_
10=kb
.4
R
.5
.6
.7
a)L.
Fig. V-1. Normalized impedance of a coil surrounding
a conducting tube.
105
The frequencies of the coil's excitation may be determined from the natural frequencies of the network.
The
natural frequencies of the network are determined by the
impedance function of the type E network.
The impedance
function is given inAppendixA and for convenience is
repeated for this discussion.
"^
2
n
(L
C
s
"
._, ^
n n ^ + 1)
Z(s) =
^"•^"^"" ^
n
n
j
z
c
s
n
(L
C
s''
+
.
_
_
,
^
n
.
_^
^
m
m
•j — ± , .J , • » .
^ ^ X , <3 , . . .
i=j omitted
(V-6)
1)
The frequencies of the currents through the coil are of
interest and these frequencies are specified by shortcircuit natural frequencies.17 The short-circuit natural
frequencies are the zeros of the right-hand side of
Eq. (V-6).
The natural frequencies for the element
inductances and capacitances required to generate a 20
microsecond pulse consists of a fundamental at 25 kH , and
its harmonics.
For a frequency of 25 kH , the kb term of Fig. V-1 is
99.1 and 133.5 for the 7.5 centimeter tube and the 10.1
centimeter tube, respectively.
For these values of kb,
the points on the curves are on the vertical axis.
Con-
sequently, the inductance of each coil is determined by
106
multiplying the value specified by the intersection of the
curve and the vertical axis by the value of the coil inductance without the core.
The inductance values for the end
inductors, the internal inductors, and the network inductance are listed in Table V-2. The pulse durations calculated
from the network inductances are also listed in Table V-2.
Examination of the network inductance value for the
5.05 centimeter tube in Table V-2 shows that it is 20 percent
larger than the calculated value.
However, the sum of the
individual element values yields a value which is within 7
percent of the measured values.
The difference in the sum
of the element values and the value in Table V-2 can be
attributed to the mutual inductance.
This suggests that
the mutual inductance between the PFN's element inductors
is reduced to a negligible value as the ratio of the core
to coil radii approaches one.
Experimental verification
of this suggestion is obtained when measurement of the
mutual inductance is attempted.
For the PFN solenoid
surrounding the 5.05 centimeter tube the mutual inductance
between adjacent inductors is unmeasurable with the
universal bridge.
For the purpose of determining the PFN pulse duration,
the use of the normalized impedance plots is inconvenient
since the value for the pulse duration cannot be directly
ascertained.
al to ÆJT .
The PFN pulse duration is directly proportionFor an inductor with a conducting core, the
107
TABLE V-2
CALCULATED INDUCTANCE VALUES ASSUMING A
UNIFORM MAGNETIC FIELD
No
Core
Nbrmalized Iirpedance
1.0
7.5 cm
Core
%Deviation
.65
10.1 on
Core
%Deviation
.37
End Inductors
15.5 yH
10.10 yH
+1.0%
5.74 yH
- 4.30%
Intemal Inductors
12.5 yH
8.13 yH
-4.4%
4.63 yH
- 7.40%
Network Inductance
153.0 yH
99.50 yH
+3.6%
56.60 yH
+20.40%
22.0 ys
15.75 ys
-4.0%
11.88 ys
+ 4.25%
Pulse Duration
108
value of L^ is the product of the coil inductance without
2 2
a core and the value 1 - b /c , where c is the coil's outside diaraeter in meters and b is the core's outside diameter
measured in meters.
The values of 1 - b^/c^ are deterrained
by examination of the relation for the normalized impedance
(Eq. (IV-72)) and are the points at which the norraalized
impedance curves intersect with the vertical axis. The
ratio of the pulse duration without a conducting core to
the pulse duration with a core can be plotted against the
ratio of the coil radius to the core radius.
This curve
is shown in Fig. V-2. From this curve the pulse duration
of a PFN can be determined if the pulse duration without
a core and the coil and core radii are known.
Measured Inductance and Pulse
Duration Values
Inductance measurements for each core material were
made with a universal bridge.
For each case the values
of the end inductors, the internal inductors, and the
network inductance were measured.
The measured values
are listed in Table V-3.
The oscillograms in Fig. V-3 and Fig. V-4 show the
durations of the pulse generated by the PFN for the 7.5
centimeter tube and the 10.1 centimeter tube, respectively.
In both Fig. V-3 and Fig. V-4, the pulse duration for the
PFN without the tube is shown for the purpose of comparison.
The values for the pulse duration are listed in Table V-3.
109
.6
T withoutoore
T with core
Radius of Coil
Fig. V-2.
P u l s e d u r a t i o n a s a f u n c t i o n of c o r e r a d i u s
110
TABLE V-3
MEASURED INDUCTANCE VALUES AND PULSE
DURATIONS FOR PFN
No
Core
7 . 5 cm
Aluminum Core
1 0 . 1 cm
Aluminim Core
End Inductors
1 5 . .5 yH
10.00 yH
6.00 yH
Internal Inductors
1 2 , . 5 yH
8.50 yH
5.00 yH
Network Inductance
1 5 3 , .0 yH
96.00 yH
47.00 yH
2 2 . .0 y s
16.40 ys
11.40 ys
Pulse Duration*
*Measured from oscillograms.
111
500v/cra
5 ys/cm
(a) PFN pulse (without tube)
500v/an
•MBH aaBaB
5 ys/cra
í
(b) PFN pulse (with tube)
SOOv/cmlJIIJlf]
2 ys/cra
(c) PFN pulse (with tube)
Fig.
V-3.
Oscillograms for 7.5 centimeter tube
112
500v/cra
5 ys/cra
(a) PFN pulse (without tube)
500v/cra
5 ys/cm
(b) PFN pulse (with tube)
500v/cm
2 ys/cm
(c) PFN pulse (with tube)
Fiq. V-4.
Oscillograms for 10.1 centiraeter tube
113
In Tables V-1 and V-2 the differences in the calculated
values and the measured values are listed under the heading
"% Deviation."
For convenience, the values under this
heading are listed with + and - prefixes to indicate whether
the calculated values are larger or smaller than the
measured values.
Summary
The pulse durations calculated by assuming that the
magnetic field is uniform are within five percent of the
measured values.
Because the results are so accurate, the
assumption that the magnetic field is uniforra appears
valid.
For small values of core radii, the mutual induct-
ances between adjacent inductors make the field sufficiently
uniform for the normalized impedance relations to be
accurate.
As the ratio of the core radius to the coil
radius approaches unity, the mutual inductances become
negligible.
However, the lack of mutual inductances for
larger core radii is not as critical since the raagnetic
field is adequately uniform in the immediate vicinity of
the coil's turns.
Since the relations for the magnetic
field yield accurate results, a plot from which the pulse
duration can easily be determined by generated.
This
plot is shown in Fig. V-3.
Accurate values for the element inductances may be
determined if a precise value for the average magnetic
114
field intensity is determined.
However, determination of
the element inductance frora the value of the average
magnetic field intensity is much more rigorous than using
the normalized impedance curves.
Consequently, there is
no advantage in using power series expansion to determine
inductance values except in cases of very short solenoids
where the magnetic field is non-uniform.
CHAPTER IV
CONCLUSIONS AND RECOMMENDATIONS
FOR FURTHER RESEARCH
To give the PFN more versatility, a raethod for easily
changing the pulse duration has been demonstrated.
The
method involves placing a conducting tube inside the PFN
inductors.
Eddy-currents induced into the conducting tube
generate a magnetic field which opposes the magnetic field
generated by the PFN inductors.
Since the inductance of
a coil is proportional to the value of the average magnetic
field within the coil, the inductance decreases.
The method for reducing the PFN pulse duration is
verified experimentally for tubes of different diameters.
The calculated values are within five percent of the
measured values.
To reduce calculations, a curve from
which the pulse duration may be determined for a given
ratio of core radius to coil radius has been developed.
The PFN used to verify the method of reducing the
pulse duration was a nine-section, 20 microsecond, 15 ohm
characteristic impedance, type E pulse forming network.
This PFN was designed, constructed, and tested.
The design
formulas derived by Guillemin and presented in Chapter II
are applied to the PFN design.
The PFN' is constructed
according to design requirements, and thus verify the
design techniques.
The PFN is capable of storing 124.7
115
116
joules of energy and can deliver a 6.4 megawatt pulse of
20 microsecond duration to a matched load.
A suggestion for additional research is the development
of a method for reducing the pulse duration linearly without requiring the use of many conducting cores of different
radii.
The use of a single core whose radius could be
varied mechanically would give added versatility to the
PFN.
Since the PFN's characteristic impedance is reduced
at a rate proportional to the rate at which the pulse
duration decreases, a method for continually matching the
load to the PFN would also need to be developed.
Another
area for additional research is the development of a method
for modifying the pulse shape by using core materials
having different shapes.
For example, conical or funnel
shaped cores may be used.
A through examination of the
effect on mutual inductances due to a conducting core is
also suggested.
REFERENCES
1.
Magnusson, P. C.
Propaqation.
1970.
Transmission Lines and Wave
2nd ed. Boston: Allyn and Bacon,
2.
Johnson, W. C. Transmission Lines and Networks.
New York: McGraw-Hill Book Company, Inc, 1950.
3.
Glasoe, G. N. and Lebacqz, J. F., eds. Pulse Generators
2nd ed. New York: Dover Publications, Inc, 1965.
4.
Whittaker, E. T. and Watson, G. N. A Course of Modern
Analysis. (Ara. ed.) New York: The MacMillan
Company, 1946.
5.
Guillemin, E. A. "A Historical Account of the Development of a Design Procedure for Pulse-Forming
Networks." Radiation Laboratory Report Number 43.
(October, 1944).
6.
Terman, F. E. Radio Engineers' Handbook.
McGraw-Hill Book Company, Inc, 1943.
7.
Welsby, V. G. The Theory and Design of Inductance
Coils. 2nd ed. London: John Wiley and Sons,
I n c , 1960.
8.
Corson, R. D. and Lorrain, P. Introduction to Magnetic
Fields and Waves. San Francisco: W. H. Freeman
and Co., 1962.
9.
Libby, H. L. Introduction to Electromagnetic Nondestructive Test Methods. New York: WileyInterscience, 1971.
New York:
10.
Stoll, R. L. The Analysis of Eddy Currents.
CÍarendon Press, 1974.
11.
McLachlan, N. W. Bessel Functions for Engineers.
2nd ed. London: Oxford University Press, 1961.
12.
Tranter, C. J. Bessel Functions with Some Physical
Applications. New York: Hart Publishing Company,
Inc, 1968.
13.
Montgomery, D. B. Solenoid Magnet Design.
Wiley-Interscience, 1969.
117
Oxford:
New York:
118
14.
Parkinson, D. H. and Mulhall, B. E. The Generation of
High Magnetic Fields. New York: Plenum Press,
1967.
15.
Garret, M. W. "The Method of Zonal Harmonics" in High
Magnetic Fields. Edited by Henry Kolm and others.
Cambridge, Massachusetts: M.I.T. Press and John
Wiley and Sons, I n c , 1962.
16.
Guillemin, E. A. Synthesis of Passive Networks.
New York: John Wiley and Sons, I n c , 1957.
APPENDIX
A.
DERIVATION OF TYPE E NETWORK
B.
VALUES OF JQ(ZJ ^^) AND J^(zj^^^) FOR DETERMINATION OF
NORMALIZED COIL IMPEDANCE OF COIL SURROUNDING A SOLID
CYLINDRICAL CORE
C.
EXPRESSIONS FOR DETERMINING THE MAGNETIC FIELD INTENSITY
WITHIN A SOLENOID
119
APPENDIX A
DERIVATION OF TYPE E NETWORK
The type E network is of special importance in pulse
forming networks because of its construction ease and
minimal cost.
The ease of construction is due to the
inductances being in the, form of a long solenoid having
mutual inductances and capacitors tapped on at the appropriate points.
The minimal cost is due to all the capacitors
being of equal value.
Derivation of the type E network
begins with the network derived by a continued-fraction
expansion of the impedance function which Guillemin derived
when he represented the Fourier coefficients of his desired
pulse as a set of parallel L-C series resonant sections
(Eq. A-1) .
The network of Fig. A-1 is changed to a network having
identical capacitances.
The derived network may be expected
to have inductances in series with the capacitive legs to
compensate for the changed capacitance.
If the capacitance
is increased the inductance will be negative and vice-versa.
The network that is desired is shown in Fig. A-2. The
negative inductances are desired since they can be realized
as mutual inductances.
The capacitors are of equal value
all being C = C /n where C
is the total network capacitance
and n is the number of sections.
The values of the inductors
and their mutuals are unknowns to be determined.
120
121
Z T
n
L =
n
nirb.
b
Cn
Tb
= —VTTZ
s -v
N
values specified by the choice of the alternating
current waveform.
Fig. A - 1 . Form of voltage-fed network derived by
Fourier-series analysis of a specified-alternating-current
waveform. Type C network.
n
n-l,n'^
T ^T
Fig. A - 2 .
Type E network.
122
The impedance of Fig. A-1 is
Z(s) =
J
n
2
n
(L C s"^ + 1)
n n
j-1,3,. . .
. (A-1)
n
n
~
l
C s
n
(L C s + 1)
•*-/"J# • • .
1= 1 , 3 , . . .
i=j omitted
In Eq. A-1, it is noted that, if an impedance L,s is
subtracted from Z(s) so that
Zj^(s) = Z(s) - Lj^s
,
(A-2)
a zero of Z,(s) appears-that is, the series combination of
L, 2 and C corresponds to a zero of Z, (s) or to a pole of
Y, (s) = 1/Z, (s).
The admittance of the series combination
2
of L,2 and C is Cs/(L^2Cs + D • The poles of Y^(s)
corresponding to the L,^ and C resonant section must be
given by
s = tl/^-h^^C)
= ±s^
,
(A-3)
and Y,(s) can be expressed in the form
^i<=' = i ^ ^ i î l : +^2<^'
'
<*-"'
123
where ^^(s) i s the reraainder admittance function r e g u l a r
at ± s , .
The constanct a, and a^ are found by algebra
a^ = lim (s-s^)Y^(s) = 1 im
s->-s
s-^s
s-s
Z(s) - L,s
(A-5)
and using L'Hospitals rule
(A-6)
a,
1 = _d_ Z (s) - L^s
ds
Z (s^) - L^
s=s.
Likewise a^ = l/(z'(-s,) - L,). Since z'(s) is a function
2
of s , as may be seen by differentiating Eq. A-1, a^^a^^a.
Thus Y.. (s) can be expressed as
^l( = ' = s ^-s.
^
(A-7)
^^2< = '
The first term on the right hand side of Eq. A-7 must be
the admittance of L^ ^ ^^^ C in series so
Cs
L^jCs + 1
12
s2 +
L^jC
2as
2
s -s.2
(A-8)
124
Eq. A-8 gives two equations for determining the unknowns s.
and L^2=
.3^2 = l/2a = (Z' (s^) - L^)/2
(A-9)
and
1/L^2^ = -s^'
(A-10)
2
where s^ is a root of Z(s) - L^s; thus L^ = Z(s,)/s .
From Eq. A-1 it is evident that the roots of Z(s) - L, s = 0
2
are all of the form s. , and that there are n such roots.
2
The root s, is found by eliminating L,^ between Eqs. 9 and
10 which gives
1
-s.
Z(s^)
(A-11)
C
The value of s, is specified since it is the only unknown.
Then
(A-12)
^l "" Z(s^)/Sj^
and
1*^2 " "^/CSi
= 1/2 (Z (s^) - L )
(A-13)
125
^12 ^^ negative for all cases where s,
is positive.
The foregoing procedure determines L,^ and L, and
reduces the degree of Z (s) by 2.
The whole process may be
repeated on the remainder function Z^^s) = l/Y^^s), where
Y^^s) is defined by Eq. 7, and L^ and L^^ may be determined
A new remainder function Z-(s) is left and the whole
process may be repeated again and again until all roots are
exhausted.
APPENDIX B
VALUES OF Jo(zj-^/^) AND J^^zj^^^) FOR DETERMINATION OF
NORMALIZED COIL IMPEDANCE OF COIL SURROUNDING
A SOLID CYLINDRICAL CORE
126
127
TABLE B-1
VALUES OF Jo(zj"^'^^)
Jo(2J^^^) = M^(z)e^®^^^ = ber z + jbei z
M^(z)
e^(z)
0.00
0.05
0.10
0.15
0.20
1. 000
1. 000
1. 000
1. 000
1. 000
0. 00°
0. 04
0. 14
0. 32
0. 57
0.25
0.30
0.35
0.40
0.45
1. 000
1. 000
1. 000
1. 000
1.,001
0. 90°
1. 29
1. 75
2. 29
2. 90
0.50
0.55
0.60
0.65
0.70
1..001
1.,001
1,,002
1.,003
1..004
3. 580
4. 33
5. 15
6.,04
7.,01
0.75
0.80
0.85
0.90
0.95
1..005
1..006
1..008
1..010
1..013
8,,04°
9..14
10,.31
11..55
12.,86
1.00
1.05
1.10
1.15
1.20
1,.016
1,.019
1,.023
1..027
1..032
14,.23°
15,.66
17..16
18..72
20..34
1.25
1.30
1.35
1.40
1.45
1..038
1,.044
1,.051
1,.059
1,.067
22..02°
23,.75
25 .54
27 .37
29 .26
1.50
1.55
1.60
1.65
1.70
1,.077
1,.087
1..098
1..111
1..124
31 .19°
33 .16
35 .17
37 .22
39 .30
128
TABLE B-1—Continued
MQ(Z)
e^^z)
1. 75
1.80
1. 85
1.90
1.95
1.139
1.154
1.171
1.189
1.208
41.41°
43.54
45.70
47.88
2. 00
2. 10
2. 20
2. 30
2. 40
1.229
1.274
1.325
1.381
1.443
52.29°
56.74
61.22
65.71
70.19
2. 50
2. 60
2. 70
2.,80
2.,90
1.511
1.586
1.666
1.754
1.849
74.65°
79.09
83.50
87.87
92.21
3..00
3.,10
3..20
3.,30
3,.40
1.950
2.059
2.176
2.301
2.434
96.52°
100.79
105.03
109.25
113.43
3,.50
3,.60
3..70
3.,80
3..90
2.576
2.728
2.889
3.061
3.244
117.60°
121.75
125.87
129.99
134.10
4..00
4..50
5,.00
5..50
6..00
3.439
4.618
6.231
8.447
,
1.150x10-^
138.19°
158.59
178.93
199.38
219.62
7,.00
8..00
9 .00
10 .00
11 .00
2.155x10^4.082x10^7.796x10^
1.498x10^
2.895x10"^
260.29°
300.92
341.52
22.10
62.66
12 .00
14 .00
16 .00
18 .00
20 .00
5.618x10^
2.137x10::
8.217x10^
3.185x10^
1.242x10
103.22°
184.32
265.40
346.46
67.52
129
TABLE B - 1 — C o n t i n u e d
z
25.00
30.00
35.00
40.00
45.00
M (z)
o
3.809x10^
1.192xl0q
3.786x10^
1.215x10:-^
3.929x10
e (z)
o
270.15°
112.75
315.75
157.94
0.53
130
TABLE B-2
.3/2
VALUES OF J^(zj^ )
J^(zj^/^) = M^(z)e^^^^^ = ber z + jbei
M^(z)
e^(z)
0.00
0.05
0.10
0.15
0.20
0.0000
0.0250
0.0500
0.0750
0.1000
135.00°
135.02
135.07
135.16
135.29
0.25
0.30
0.35
0.40
0.45
0.1250
0.1500
0.1750
0.2000
0.2250
135.45°
135.64
135.88
136.15
136.45
0.50
0.55
0.60
0.65
0.70
0.2500
0.2751
0.3001
0.3252
0.3502
136.79°
137.17
137.58
138.03
138.51
0.75
0.80
0.85
0.90
0.95
0.3753
0.4004
0.4256
0.4508
0.4760
139.03°
139.58
140.17
140.80
141.46
1.00
1.05
1.10
1.15
1.20
0.5013
0.5267
0.5521
0.5776
0.6032
142.16°
142.89
143.66
144.46
145.29
1.25
1.30
1.35
1.40
1.45
0.6290
0.6548
0.6808
0.7070
0.7333
146.17°
147.07
148.02
148.99
150.00
1.50
1.55
1.60
1.65
1.70
0.7598
0.7866
0.8136
0.8408
0.8684
151.04°
152.12
153.23
154.38
155.55
131
TABLE B - 2 — C o n t i n u e d
M^(z)
e^(z)
1.75
1.80
1.85
1.90
1.95
0. 8962
0. 9244
0. 9530
0. 9819
1. 011
156. 76°
158. 00
159. 27
160. 57
161. 90
2.00
2.05
2.10
2.15
2.20
1. 041
1. 072
1. 102
1. 134
1. 166
163. 27°
164. 66
166. 08
167. 53
169. 00
2.25
2.30
2.35
2.40
2.45
1. 199
1.,232
1.,266
1.,301
1,.337
170. 50°
172. 03
173. 58
175. 16
176. 76
2.50
2.55
2.60
2.65
2.70
1..374
1..411
1..450
1.,489
1..530
178. 39°
180. 03
181. 70
183.,39
185.,10
2.80
2.90
3.00
3.10
3.20
1.,615
1,.705
1,.800
1,.901
2..009
188..57°
192..11
195,,71
199,.37
203,.08
3.30
3.40
3.50
3.60
3.70
2..124
2..246
2,.376
2,.515
2,.664
206..83°
210..62
214,.44
218,.30
222,.17
3.80
4.00
4.25
4.50
5.00
2,.823
3,.173
3,.681
4..378
5 .809
226 .07°
233 .90
243 .77
253 .67
273 .55
5.50
6.00
6.50
7.00
7.50
7 .925
,
1 .085x107
1 .490xl0f
2 .050x10^
2 .827x10-^
293 .48°
313 .45
333 .46
353 .51
373 .59
132
TABLE B-2—Continued
M^(z)
e^(z)
8.00
9.00
10.00
11.00
12.00
3.907x10^7.497x10^
1.447x10^
2.804x10^
5.456x10"^
13.69°
73.96
114.28
154.63
195.02
14.00
16.00
18.00
20.00
25.00
2.084x10^
8.038x10^
3.123x10^
1.220x10^
3.755x10
275.84°
356.72
77.63
158.57
00.98
30.00
35.00
40.00
45.00
1.178xl0q
3.748xlOÎ^,
1.204x107^
3.899x10 '^
203.45°
45.94
248.46
90.98
APPENDIX C
EXPRESSIONS FOR DETERMINING THE MAGNETIC
FIELD INTENSITY WITHIN A SOLENOID
The magnetic field intensity at points inside the
coil can be written as a power series involving Legendre
polynomials and coefficients derived from the coefficients
of a Taylor series expansion of H (z,0).
The power series
expansion converges within a sphere, the radius of which
is the distance from any point on the coil axis to the
nearest point of the coil.
For points inside the coil, the
radius of convergence is the coil radius. The power
series may be written
n
00
H^(r,e) = H
n=0
n
P^(u)
(C-1)
Pn (u)
(C-2)
and
H^(r,e) = H
where H
and H
£»
n
oo
n=0
n
are the magnetic field intensities in the
^
z- and radial directions for a point specified by the
spherical coordinates r and e.
Legendre polynomials Pj^(u)
and P1(u)
are listed in Table C-1.
n
133
The En coefficients
134
TABLE C-1
EVEN-ORDER LEGENDRE POLYNOMIALS AND
THEIR FIRST DERIVATIVES
u = cos e
u' = sin e
Po u) = 1
Po (u) = 0
^2 u) = l/2(3u^-l)
^2 (u) = l/2(6u)u'
^
u) = l/^^^^u'^-^Ou^+^)
^4 (u) = l/8(140u^-60u)u'
^6 u) = l/16(231u^-315u^+105u^-5)
^6 (u) = l/16(1386u^-1260u^+210u)u'
^8 u) = l/128(6,435u®-12,012u^+6,930u^-l,260u^+35)
^8 (u) = 1/128(51,480u^-72,072u^+27,720u"^-2,520u)u
135
are d e t e r m i n e d by the d e r i v a t i v e s of H (z,0) evaluated at
£
z=0.
If the o r i g i n for the sphere of convergence is
l o c a t e d on the coil axis at the midplane of
symmetry,
the s e r i e s e x p a n s i o n w i l l consist only of even t e r m s .
T h e p o w e r s e r i e s then simplifies to
«z^-'^>
= ^ave^i F(a,B) + FE^P^^u)
r r. \
^
(C-3)
and
H^(r,e) = J
a,
r
'
ave 1 F E ^ P ^ Í u )
+ FE^P^^u)
(C-4)
where FE
is e q u a l to F ( a , 3 ) E .
E x p r e s s i o n s for F(a,$) and
F E ^ a r e listed in T a b l e C - 2 .
n
F ( a , 3 ) and FE
are g e o m e t r y dependent terms w h i c h are
d e t e r m i n e d by the coil length, coil t h i c k n e s s , and coil
diameter.
F o r d i f f e r e n t v a l u e s of a and 3, as defined in
F i g . I V - 1 3 , F ( a , 3 ) , FE^r F E ^ , and FEg are listed in
T a b l e s C-3 t h r o u g h C - 1 6 .
136
TABLE C-2
EXPRESSIONS FOR F(a, 3) AND FE
n
^1 =
F(a,3) = i
1+3
2
S =
3^
a
C
=
^3-^2^32
1+3^
6'
C.
-4 = ^ 2 ^ 3 2
(sinh"^ I - sinh"^ ^)
FE - A Ji. fr V 2
^ 3/2.
^ 2 - ã0 23 ^^1
" ^3 >
FE - - L
^ C^-^/^^^ + 3^2 + 15^2^) - ^3^/^(2 + 30^+ 15C^^)
^ 4 -100773
243 *-
- I
1 C^^/^(8 + 12^2 + 15^2^ - 70^2^ + 3150^^)
^6
100 «.««5
2403^
- ^3*^/^(8 + 12C^ + 150^^ - 700^^ + 315C^^)
"^8 ^ ^ 0 0 3 ^ 7 0^^'^ (16 + 24^2 + 30^2^ + ^^C^"^ + Bl^C^"^
- 2079^2^ + 3003^2^)
0^^^ (16 + 24C4 + 30C^^ + 35C^^ + 315C^^
2079C^^ + 300X4^)
137
TABLES C-3 THROUGH C-16
COMPUTER TABLES GENERATING THE TERT^S
F(a,3), FE2, FE4, AND FEe FOR
VARIOUS VALUES OF a AND 3
138
TABLE
Al.PHA
BETA
U02
O.IO
0.20
0.30
0.40
O.50
0.60
0.70
O.fiO
0.90
1.00
1.10
1.20
1.30
1.40
l.bO
1.60
1.70
1.80
1.90
i.o;?
I.Ú2
1.02
1.02
1.02
1.02
1.ÍJ2
1.02
1.J2
1.02
1.02
1.02
1.Û2
1.02
1.02
1.02
1.02
1.02
1..J2
1.02
1.02
1.02
1.02
1.02
1.02
1.02
1.02
1.02
1.02
1.02
1.02
1.02
1.02
1.02
1.02
1.02
1.02
1.02
1.02
Z.OO
2.20
2.40
2.c>0
2.H0
3.;J0
3.20
3.40
3.60
3.80
4.00
4.20
4.40
4.60
4.80
5.00
5.20
5.40
5.60
5.8 0
6.00
F(a ,3 )
0.0000197
0.0000389
0.0UÛO569
0.0000736
0.J00Û387
0 . 0 0 0 1021
0 . 0 0 0 1 139
O.OO') 12 42
O.OuOl331
0.0001407
0.0001473
0.0001530
0.0001579
0.0001622
0.0001659
0 . 0 0 0 1691
0.0001719
0 . 0 0 0 1744
0.0001766
0.0001785
0 . 0 0 0 1 8 18
0.0001843
0 . 0 0 0 186^
0.0001881
0 . 0 0 0 1 89b
0.0001907
0.0001917
0.0001926
0.0001933
0.0001939
0.0001944
0.0UO1949
O.U00195<t
0.0001957
0.000196U
0.0001963
0.0001966
0.0001968
0.0001970
0.0001972
C-
3
Ffc2
-0.00002d4
-0.0000529
-0.uOOu707
-0.0000809
-0.0000842
-0.00J0821
-0.0000764
-0.0000689
-0.0000608
-0.0000528
-0.00)0453
-0.0000387
-0.00J0329
-0.0000279
-0.0000237
-0.0000202
-0.0000172
-0.0000lt7
-0.0000126
-0.0000108
-o.ooooosi
-0.000C061
-0.00000^+7
-0.0000037
-J.00v)002 9
-0.0000023
- 0 . 0 0 0 0019
- 0 . 0 0 0 0015
-0.0000012
-O.OOJOOIO
-0.0000009
-0.J0O0OU7
-0.O00O006
-0.0000005
-0.O0J00O4
-0.0000004
-0.00UUOO3
- 0 . 0 0 0 0003
-0.000000 3
-0.00000)2
rE4
0.0000338
0.0000569
0 . 1000646
0.0000586
0.0000^48
0.000C291
0..JOO0154
0.0000052
-0.0000014
-0.0000051
-O.0OO0C68
-0.J000072
-0.JOOJ069
-0.0000063
-0.0000055
-0.0000047
-0.0000040
-0.0000033
-0.0000028
-0.0000023
-0.0000016
-O.OOOOOll
-0.0000008
-0.0000006
-0.000)004
-0.0000003
-0.0000002
-0.0000002
-0.0000001
-0.0000001
-0.0000001
-Û.OOOOOOI
-0.0000000
-0.0000000
- 0 . 0 0 0 )000
-0.0000000
-0.0000000
-O.OOÔ ) 0 0 0
-O.OOOÔOOO
-0.0000000
FE6
-0.0000358
-0.0000539
-0.0000466
-0.0000261
-0.0000*05 7
0.0000073
0.0000123
0.0000121
0.0000096
0.0000067
0.0000042
0.0000023
O.OOOOOll
0.0000004
0.0000000
-0.0000002
-0.0000003
-0.0000003
-0.0000003
-0.0000003
-0.0000002
-0.0000001
-0.0000001
-0.0000001
-0.0000000
-0.0000000
-O.OOOOJOO
-0.0000000
-0.0000000
-0.0000000
-0.0000000
-0.0000000
-0.0000000
-0.0000000
-0.0000000
-0.0000000
-0.0000000
-0.0000000
-0.0000000
-0.0000000
139
TAcil.t; C- 4
ALPHA
BtTA
F(a,3)
ftP.
1.04
1.04
1.04
1.04
1.04
1.04
1.04
1.04
1 .0^
1.04
1.04
1.04
1.04
0.10
0.20
0.30
0.^0
0.50
0.60
0.70
0.80
0.90
1.00
1.10
1.20
1.30
1.40
1.50
1.60
1.70
1.80
1.90
2.00
2.20
2.40
2.60
2.80
3.00
3.20
3.40
3.60
3.80
4.00
4.20
4.40
4.60
4.80
5.00
5.20
5.40
5.60
5.80
6.00
0.00003 90
0.*)0 00770
0.0001 129
0.0001460
0.0001761
0.0002028
0.0002263
0.0002469
O.0J026^7
0.000280J
0.000293J
0.00 03048
0.0003147
0.0003233
0.000 3 308
0.0003373
0.0003430
0.0003480
0.000352t
0.0003563
0.0U03629
0.0J03681
0.000372^
0.0003758
0.000378/
0.0003811
0.000 3832
0.0003849
U.JO03863
0.0003876
0.0003887
0.0003 397
0.00 03 905
0.0003912
0.0003919
Û.0U03925
0.0003931
0.0003935
0.0003939
U.0J03943
-0.000 055 2
-0.0001030
-0.0001379
-0.J0U153?
-0.0001651
-0.0001614
-0.00015J8
-0.0001364
-0.00ul2o6
-0.0001050
-0.00U0904
-0.000U773
-0.0ÛJ0659
-0.0000560
-0.0000477
-0.O0JO4U6
-0.0000346
-0.00UÛ296
-0.00o02í)4
-0.0000219
-0.0000164
-0.0000124
-0.0000096
-0.000 00 74
-0.00C00t39
-0.00JO04 7
-0.0000038
-0.0000031
-0.ÛOOu025
-0.0000021
-0.0000017
-0.000 0015
-0.0000012
-O.OOJOOll
-0.000 0009
l.O^
1.04
1.04
1.0-+
1.04
1.04
1.04
1.04
1.04
1.04
1.04
1.04
1.04
1.04
1.04
1.04
1.04
1.04
l.Ot
1.04
1.04
1.04
1.04
1.04
l.Ot
1.04
1.04
-o.ooo uoa
-0.0000007
-O.0OO0J06
-0.0000005
-0.0000004
FE4
0.000 0644
0.0001090
0.0001243
0.J0O1136
0.0000876
0.0000576
0.0000311
0.0000113
-0.J000018
-0.0000093
-0.0000128
-0.0000138
-0.0000134
-0.0000122
-0.0000103
-0.0000093
-0.0000079
-0.0000066
- 0 . OOOJ055
-0.0000046
-U.0000032
-0.0000022
-0.0000016
-0.000 0011
-0.0000008
-0.0000006
-0.0000004
-0.0000003
-0.0000UO2
-0.0C00002
-O.OOOJOOl
-O.OUOOOOl
-0.0000001
-0.0000001
-0.0000001
-0.0000000
-O.OOOOOOO
-0.0000000
-O.OOOJOOO
-0.0000000
FF6
-0.0000676
-0.0001015
-0.0000889
-0.0000508
-0.0000122
0.0000128
0.0000230
0.0000231
0.OJJO186
0.0000131
0.0000083
0.0000047
0.0000024
0.0000009
0.0000001
-0.0000003
-0.0000005
-0.00000O6
-0.000JOO5
-0.0000005
-0.0000004
-0.0000003
-0.0000002
-o.ooooooi
-0.0000001
-0.0000001
-O.OOÛJOOO
-0.0000000
-0.0000000
-0.0000000
-O.OUOJOOO
-0.0000000
-0.0000000
-0.0000000
-O.COOOOOO
-O.ÛOOOOÛO
-0.0000000
-0.0000000
-0.0000000
-0.0000000
140
TARLt?
ALPHA
HFTA
F(a,3 )
1.06
1.06
1.06
1.06
1.06
1.06
1.06
1.06
1 .06
1.06
1.0 6
1.06
1.0 6
1.06
1.06
1.06
1.06
1• 06
1.06
1.06
1.06
1.06
1.06
1.06
1.06
1 .06
1.06
1.06
1.06
1.06
1.06
1.06
1.06
1.06
1.06
1 .06
1.06
1.06
1.06
1.06
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.1)0
1. 10
1.20
1.30
1.40
1.50
1.60
1.70
1 .80
1.90
2.00
2.20
2.40
2.60
2.80
3.00
3.?0
3.40
3.t»0
3.80
4.00
4.20
4.40
4.6 0
4.8 0
5.00
5.20
5.4 0
5.00
5.8 0
6.00
0. J00058Û
0.000U44
0.0)01678
0.000217Z
0.J002621
0.0003020
0.0003373
0.U0U36'U
0.C003948
C-
b
FL2
-0.000^806
-0.0001504
-J.JJJ2018
-0.0002321
-u.0uu2428
-0.000233 1
-0.0002232
-0.JOO2 02H
-0.0001795
0. )VJ0418O
-O.OOOl56f5
0.0004380 -0.0001351
0.0004553 -0.0001158
O.00O4703 -U.0JJO989
0.0 004 33 3 -0.j0J0d42
Û.0JJ4 946 -0.000 0 718
0.0005045
-0.00J0612
0.0005132 -0.00)0523
O.JO0 5208 -0.00 J04^8
0.0005275
-0.Û0JÛ335
0.00^)5334 -U.00J033 1
0.0005^34 -0.0000248
0.000551^
-0.0000189
0.00 0 55 78 -O.OO )01^5
-0.0000113
0.0005631
-J.OO0U089
0.0005675
-0.0000071
J.OOJ5711
-0.0000057
0.0005742
0.0005768 -0.JO)0047
-0.00 )0038
0.0005791
0.00 0581J -0.0000032
0.0 )05827 -0.0000027
-0.0000022
0.0005Í42
J . 0 J 0 5 3 5 5 -0.0») J JJ19
-0.00J0016
O.0JO5 366
0.0005877 -0.00J00Í4
0.UJ0538O
-U.JOJ0ul2
0.0)0589^ -O.OOJOOiO
J.Ut) 05901 -J.0JJ0Û09
0.0005908
-0.0000008
0.O0O5914 -0.JJJOJ0 7
Fh
FE6
0.00 0 0S*26
J.00J1567
0.00U1795
O.OOJ 1651
0.0ÚJ1285
0.00J085tj
0.00 )J4 7l
0.JUJ0180
-O.0OJOJ13
-J.JUJ0126
-0.O0JO181
-0.0000199
-0.0)00195
-0.0000179
-0.00J0159
-0.0000137
-O.ÛUJ0117
-0.JUJ0099
-0.0000083
-û.0000069
-0.0000048
-0.0000034
-0.000JÛ24
-0.0000017
-0.000J012
-0.0000009
-0.0000006
- 0 . JOO )005
-0.0000004
-0.00OJO03
-0.0000002
-0.0000914
-0.UU01438
-0.0001272
-0.000J742
-J.0000l9t
0.00001o9
0.O0J0322
0.J0003 3 1
0.0000270
0.0u0ol92
0.0000123
0.0000072
0.0000037
0.0000015
J.O0OJJJ3
-0.0Û00U04
-0.0(J00002
- 0 . JOOOOOl
-Û.OÛOOOOI
-J.JOOOOOl
-0.0000001
-0.0000001
-0.0000000
-0.0000000
-O.OOOJOOO
-0.0UJ)JJ7
-0.00JJÛO8
-0.0000008
-0.O0OJ0O7
-0.0000005
-J.OOJJJ04
-0.0000003
-0.0000002
-0.000JJOl
-0.0000001
-O.OÛOJOOl
-O.OOOOJJO
-0.0000000
-J.OJOOOoO
-0.0000000
-0.0000000
-O.OOOOJJO
-0.0000000
-O.OOOOJJJ
-0.0000000
-0.0000000
-J.OOOJOOO
-0.0000000
-0.JJOOOJO
141
TM-ILÊ
AL PH'V
BETA
F(a,3)
1.08
l.OB
1.08
1.08
1.08
1.08
1.08
1.08
1.08
1.08
1.08
1.08
1.08
1.08
1 .08
1.08
1.08
1.08
1.0 8
1.08
1.08
1.08
1.08
1.08
1.08
1.08
1.08
1.08
1.08
l.OS
1.08
1. 08
1.08
O.IO
0.20
0. 30
0.4U
0.50
0.6 0
0.70
0.80
0.90
1.00
1. 10
1.20
1.30
1.40
1.50
1.60
1. 70
1.80
1.90
2.00
2.20
2.^0
2.60
0.J000766
0.000 15 1 l
0.0002218
0.000 28 7 3
0.0003467
0.0003999
1 .») 8
1.08
1.08
l .08
1.08
1.08
1.08
Z.ãi)
3.00
3.20
3.40
3.60
3.80
4.00
4.20
-•.40
4.60
4.8 0
5.00
5.20
5.^0
5.6 0
5.80
6.00
C-
6
FL2
-J.0001045
-0.0001953
-J.0002626
-O.OOJ3Û27
-0.0003175
-0.0OU3123
0 . 0 J 0 <+ + 6 8 -0.000 29 35
0.0004378
-0.0002670
O.Oo0 52 3t) -0.0002373
0.00 0 554'j -0.0002C7Î3
O.OUJ5813 -0.0001795
O.U')06 J'+o -J. 000 15-» 2
-0.0001319
0.00J624
0.Ju0 642 2 -0.0001126
0.0006574 -0.0000960
0.000 670 7 -O.0OOOÔ20
0.0)06824 -O.OOOU /02
0.00 0692 7 -0.0000601
0.000 7017 -O.OO)05lt
0.0007097 -0.0000446
0.0007233 -0.O0JO335
0. )0()7340 -0.000 02 3-+
0.0U07428 -0.00001 ^6
U.UOJ7499 -O.0OJ0153
U.000 7559
-0.0000121
0. 00)7608 -0.0OUC096
0.00 0 76 50 -J.00J0078
0.000 7685
-0.0ÛU0063
O.00U7716
-J.OOJ0052
0.0007/42 -0.0000043
0. U(»u77b5 -0.0OJ0036
O.OOJ7785
-U.O0JO03 0
O.000 7d0 3 -0.0u00026
0.1)007818 -0.000 0022
0.000 6^2
-0.0000019
0.0007845 -0.0000016
0.0007856
-J.0000014
0.000 7866 -0.0000012
0.U007874 -O.OOoOOll
0.0007882 -0.0000009
FE4
FF6
0.J001176
0.0C02003
0.0002306
0.0002134
0.0001674
0.000 1127
0.0000633
0.0000254
-0.OOOJOOl
-0.0000152
-0.0000228
-0.00)0255
-0.0000252
-0.00J0233
-0.0000208
-0.000018 )
-0.000015^
-0.0000130
-0.0000110
-0.0000092
-0.0000064
-0.0000045
-0.0000032
-0.000)022
-0.0000016
-0.0000012
-0.0000009
-0.0000006
-0.000 0005
-0.0000004
-0.0OOJ0O3
-0.0000002
-0.0000002
-0.JOJOOOl
-O.OCOOOOl
-O.OOOJOOl
-0.0001192
-0.0001314
-0.0001620
- 0 . 0O0962
-0.0000271
0.0000196
0.0000401
0.0000420
0.0000348
0.0000251
0.UO0O163
J.0000096
0.0000051
0.0000022
0.0000005
-0.0003004
-0.0000008
-0.0000010
-0.000001J
-0.0000009
-0.0000007
-0.0000005
-0.0000003
-0.0000002
-0.0000002
-0.0000001
-0.0000001
-0.0000001
-0.0000000
-0.0000000
-0.0000000
-0.0000000
-0.0000000
-0.0000000
-0.0000000
-0
.0000000
-O.OOODOOO
-O.OOOOOOO
-0.0000000
-0.0000000
- O . O O J )001
-0.0000001
-O.uOJOOOO
-0.0000000
142
TAHLE C- 7
ÂLPHA
BÉTA
I.IO
1.10
1.10
1.10
i.lO
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
1.10
1.20
1.30
1.40
1.50
1.60
1.70
1.80
1.90
2.00
2.20
2.40
2.60
2.80
3.00
3.20
3.40
3.60
3.80
4.00
4.20
4.40
4.60
4.80
5.00
5.20
5.40
5.0 0
5.80
6.00
1. 10
1.10
1.10
I.IO
I.IO
1.10
1.10
1.10
1. 10
1.10
1.10
1.10
1.10
1.10
1.10
1.10
I.IO
1.10
1.10
1.10
1.10
1.10
1.10
1.10
1.10
1.10
1.10
1.10
1.10
1.10
1.10
I.IO
1.10
1.10
1.10
F ta f3 )
0.0000949
0.0001872
0.0002749
0.0003562
0.000^301
0.0004963
0.0005549
0.000 6062
0.0006509
0.0006897
0.0007234
0.0007526
0.000 7 780
0.0008000
0.0008192
0.0008360
0.0008508
0.J008638
0.0008752
0.0008854
0.0009025
0.0009161
0.0009272
0.C009363
0.0009438
0.0009501
0.0009555
0.0009600
0.0009639
0.0009672
0.0009701
0.0009727
0.0009749
0,0009769
0.00 0 9786
0.0009802
0.00098 16
0.0009829
0.0009839
O.UUU9850
FE2
-0.0001272
-0.0002380
-0.0003205
-0.0003703
-0.0003894
-0.0003840
-0.0003619
-0.0003301
-0.0002942
-0.0002579
-0.0002235
-O.0OU1924
-U.0001648
-0.0001409
-0.0001204
-0.0001030
-0.0000882
-0.0000757
-0.0000651
-0.0000562
-0.0000422
-0.0000322
-0.00O02-+8
-0.0000194
-0.00U0153
-0.0000122
-0.0000099
-0.0000080
-0.0000066
-0.0000055
-0.0000046
-0.0000038
-0.0U00033
-0.0000028
-0.O0JÛ024
-0.0000020
-0.0000018
-O.ÛUJ0015
-0.0000013
-0.000001?
FE4
0.0001411
0.0002404
0.0002779
0.0002587
0.0002046
0.0001392
0.0000795
0.0000333
0.0000018
-0.0000171
-0.0000269
-0.0000305
-0.0000305
-0.0000284
-0.0000254
-0.0000222
-U.0000190
-0.0000161
-0.0000136
-0.0000114
-0.0000080
-0.G0O0O56
-0«0000040
-0.0000028
-0.0000020
-0.0000015
-0.0000011
-0.0000008
-0.0000006
-0.0000005
-0.0000004
-0.0000003
-0.C000002
-0.0000002
-0.0000001
-0.0000001
-O.OUOJOOl
-0.0000001
-0.0000001
- 0 . OOOJOOl
FE6
-0.0001391
-0.0002148
-0.0001935
-0.0001168
-0.0000350
0.0000213
0.0000468
0.0000500
0.0000420
0.00U0306
0.0000202
0.0000121
0.0000065
0.0000029
0.0000008
-0.0000004
-0.0000009
-0.0Û00012
-0.0000012
-0.0000011
-0.0000009
-0.0000OU6
-0.0000004
-0.0000003
-0.0000002
-0.0000001
-0.0000001
-0.0000001
-0.0000000
-0.0000000
-0.0000000
-0.0000000
-0.0000000
-0.0000000
-0.0000000
-0.0000000
-0.0000000
-0.0000000
-0.0000000
-0.0000000
143
TAf^Lt
ALPHA
bÊ'^A
F í a t3 )
1.20
1.20
U20
1.20
1.20
1.20
1.20
1.20
1 .20
1.20
1.20
1.20
1.20
1.20
1.20
1.20
1.20
1.20
1.20
1.20
1.20
1.20
1.20
1.20
1.20
Í.2Q
1.20
1.20
1.20
1.20
1.20
1.20
1.20
1.20
1.20
1.20
1.20
U20
1.20
1.20
0.10
0.20
0.30
0.^0
0.5 0
0.6 0
0.70
0.80
0.90
1.00
l.lo
1.20
1.30
1.40
1.50
1.60
1.70
1.80
1.90
2.00
2.20
2.40
2.60
2.80
3.00
3.20
3.40
3.60
3.80
4.00
4^20
4.40
4.60
4^80
5^00
5.20
5.40
5^60
5^80
6.00
0.0001316
0.0003587
0.000527-+
0^UUU6849
0.0008290
0 . U 0 09 5 9 )
0.0010749
0 . 0 0 11774
0 . 0 0 126 73
0.001346 0
O^OJ14147
0.001A746
0.0J152/U
0 . 0 0 15 728
0.0016128
0.0016481
0.0016791
O.OO17064
O.UJl73)7
O^0017522
0.001788/
0.0018179
0.0018417
0.0018613
0.0018776
0.0018912
0.001902 7
0.0019125
0.0019210
0.0019282
0.0019346
0.0019402
0.0019450
0 . 0 0 1 9^+9 3
0.0019532
3.001^566
0.0019597
0.0019624
0 . 0 0 19648
0 . 0 0 1 9 6 72
C-
t
hb2
- 0 . 0 0022-+•+
-0.0004217
- O . O O Û 5 72:>
- 0 .0006675
- 0 . 0 0 0 70*^9
- J . 0 0 0 70d6
-0.ÛÛÛ6761
-U.O0U6243
-O.OUU5630
-0.000499 1
-0.0OO-»372
- 0 . 0 0 0 3 79 9
-U.ÛUU3284
-0.0002831
-0.0002437
-o.ooo20<:8
-0.0001807
-0.U0U1559
-0.0001348
-0.0001169
- 0 . 0 0 0 0 8 85
-Û.0000678
-U.OU00526
-0.00J0412
-0.0000327
- J . O O )0262
-0.00J0212
-0.OUJ0173
- 0 . 0 0 0 0142
-0.0000118
- 0 . 0 0 0 009 9
-0.0000083
-0.00UOO71
-0.00JUU60
-0.0000052
-0.0OÔ0045
-0.0000039
-0.0000034
-J.0000029
-0.0000026
I-L4
0.J002309
0.J003982
0.00046^2
0.í>004^66
0.0003649
0.0(»0 2 598
0.0C015<^2
0.0000778
0.0000194
-0.0000180
- 0 . J0iJO392
-0.0000490
-0.UU00515
-0.0000497
-O.OU00457
-0.J000407
-0.0000355
- 0 . )0J0306
-0.0000261
-0.0000221
-0.JO00158
-0.0000113
-0.0000080
-0.0000058
-0.0000042
-0.0000031
-0.0000023
-0.0000017
-0.0000013
-0.0000010
-0.0000008
-0.C000006
-0.0000005
-0.J000004
-0.0000003
-0.0000002
-0.0C0J0O2
-0.00OJOO2
-0.0000001
-0.0000001
FE6
-0.0002106
-0.0003359
-0.U00313O
-0.0002016
-0.0000748
0.0000188
0.000066B
0.0000787
0.0000703
0.0000541
0.0000376
0.0000240
0.0000141
0.0000073
0.0000031
0.0000005
-0.0000009
-O.u0uu015
-0.0000018
-0.0000018
-0.O00J015
-0.0000011
-0.00000O8
-0.0000006
-0.0000004
-0.00000O3
-0.0000002
-0.0000001
-0.0000001
-0.0000001
-0.0000000
-0.0000000
-O.OOOOOOO
-O.OOOJOOO
-0.0000000
-0.0000000
-0.0000000
-0.0000000
-0.0000000
-0.0000000
144
T A R I L- C - 9
ALPHA
3ETA
F(a,3)
FF2
FE4
FF6
1.30
1.30
1.30
1.30
1 .30
1.30
1 .30
1.30
1.30
1.30
1.30
1.30
1.30
1.30
1.30
1.30
1.30
1.30
1.30
1.30
1.30
1.30
1.3J
1.30
1.30
1.30
1.30
1.30
1.30
1.30
1.30
1.30
1.30
1.30
1.30
1.30
1.30
1.30
1.30
1.30
0.10
0.2O
0.30
0.40
0.50
0.6 0
0.70
0.80
0.90
1.00
1.10
1.20
1.30
1.40
1.50
1.6 0
1.70
l.PO
1.90
2.00
2.20
2.40
2.60
2.3 0
3.U0
3.2 0
3.40
3.60
3.80
4.00
4.20
4.40
4.6 0
4.80
5.00
5.20
5.40
5.60
5.80
6. 00
0.00 026 13
0.UO05168
0.000 75 09
0. 000^3^)8
0.JO12005
0.0013919
0.(^015637
0.00 17165
0.0018517
U.O0197J8
0.002 0 754
0.OU21672
0.002 24 79
0.0J23187
0.0 J23811
0.00 24 36 1
0.0024347
O.002527H
0.J02 5661
0.0026002
0.0026581
0.OJ2 7 )4 8
0.00 2 7430
0.J02 7744
0.00 2 8006
0.002 8226
0.0J28-rl3
0.00 285 /2
O.C02 8/09
0.0028327
0.002 89 3 0
-0.000 3002
0.0002905
0.J005049
0.0006012
0.0005840
0.OUO4891
0.0003603
0.000232^
0.0001252
0.0000453
- 0 . CUO )085
-0.0000410
-0.0002543
-0.0004067
-0.0003877
-0.0002610
-0.0001098
0.0000074
0.0000 724
0.0000935
J.0000880
0.0000708
0.0000515
0.00003^5
0.0000214
0.0000122
0.0000060
0.0000022
-0.0000001
-0.0000013
-O.OOOJ019
-0.0000021
-0.0000019
-O.U0OJO15
-0.0000011
-0.000)008
-0.000J006
-0.0000004
-0.0000003
-0.0000002
-O.OOOJOOl
-O.OOOOOOl
-0.0000001
-O.OOOJOol
-0.0000000
-O.OOOOOOO
-0.0000000
-0.0000000
-0.0000000
-0.0000000
-O.OJ056O3
0.00 29100
0.0029170
-0.0007731
-0.0009038
-O.OOOS753
-0.0009834
-0.3009^82
-0.0008849
-VÍ.0U08063
-0.0007221
-0.0006386
-O.00O5599
-0.000488 1
-0.0004239
-0.000 36 7 5
-0.0003134
-ÍJ.00J2759
-0.0002394
-0.0002080
-0.0001811
-0.0001382
-0.000 106 6
-0.0000830
-0.00)0654
- 0 . JOi)0521
-0.0000418
-0.00 003^0
-0.0000278
-0.00J0229
-U.0»j)0l9l
-0.0000160
-J.OOJ0135
-0.00JU114
-O.0OJÚ098
0^0J29233
- 0 . oojoot3'+
0^00292H9
^•0029339
0.0029384
0.0029424
n.0029-+61
-0.0000072
-C.0OJ0JO3
-0.00J0055
-0.00J0Û48
-0.00.) 004 2
0.0029021
-o.v)ooo5ao
-0.0000646
-0.J00J647
-0.JOOJ6 10
-0.0000555
-0.0U0U493
-0.0000430
-0.0000372
-0.J000319
-0.0000232
- ). J000168
-0.0000121
-0.JOO )088
-0.O000065
-0.0000048
- 0 . )000036
-Û.0OOÛO27
-0.i)000021
-0.0000016
-0.0000012
-0.JOOOOIO
-0.0000008
-0.0000006
-0.J000005
-0.0000004
-0.0000003
-Û.000J003
-0.0000002
-0.J000002
-0.0000000
-0.0000000
145
TAiue
ALPHA
BcTA
Fía»3)
1.^0
1.40
1.40
1.40
1.40
1 . 40
1.40
1.40
1.40
1.40
1.40
1.40
1.40
1.40
1.40
U40
1.40
1.40
1.40
1.40
1 . +0
1.40
1.40
1.40
1.40
0.10
0.20
0.30
0.40
0.50
0.J0U3353
0.0006 5 34
0.0009779
0.00 12/40
J . 6 ()
J.Oi) 17982
0.0020241
0.002226^
0.0024065
0.0025661
0.0027071
0.0028316
0.0029416
0.0030386
0.0031244
J.J03 2J04
0.0032678
u.0033278
0.0033813
0.00342 91
0.0 03t>lJ4
0.00 3 5764
0.0036 304
0.00 3 675 2
0.(037125
O.0!)37^-»0
0.0037707
1 . -+0
1.40
1.40
U40
U40
1.40
1.40
1.40
1.40
1.40
l . 40
1.40
1.40
1.4»)
1.40
0.70
O.BO
0.^)0
1.00
I.IO
1.20
1.30
1.40
1.50
1.60
1.70
1.80
1.90
2.00
2.20
2.4 0
2.6 0
2.80
3.00
3.20
3.40
3.6 0
3.8u
4.00
-+.20
4.40
4.60
4.80
5.00
5.20
5.40
5.60
5.80
6.00
O.UI)1543J
(•».00 3 7 9 3 5
J.00 3 8 132
0.0038302
O.í)03845o
0.0038581
0.0J3 86 95
0.0038797
0.00 3 88 87
0.<J 0 3 8 9 6 8
0.0039040
0.003 9105
0.00 39164
0.0039217
C-IO
Ft-2
- 0 . JOu'i6u5
-U.0006821
-0.0009356
-0.001106 8
-0.0011967
-J.J012167
-0.001183 7
-O.OOl1149
-0.0010252
-0.0009263
-0.JOU8264
-O.00O73Û5
-0.0006^17
-0.0005614
-0.0004899
-O.OO0*+2 71
-0.0003722
-0.OOÛ3246
-0.0002834
-J.00u2479
-0.000190 7
-0.0001480
-0.0001160
-0.0000918
-0.0000733
-0.0OJJ592
-0.000048 1
-0.00003^5
-0.0000327
-0.00002 72
-J.0OJ0229
-0.00JU193
-0.0JJU164
-U.0000140
-0.000012 1
- 0 . v)00ul04
-0.000009 0
-0.000007^
-0.0000069
-0.000 006 1
Ft4
0.0003314
0.0005789
0.J006961
0.0006856
0.0005852
0.0004426
0.0002967
0.00)1710
0.0000743
0.CoOOO71
-u.JOJ0358
-0,0000601
-0.0000714
-0.0000743
-0.J0J0719
- 0 . )JJ0668
-0.0000603
-0.J000534
-0.0000468
-0.00004 06
-0.0000301
-0.0000220
-0.0 0OO162
-0.0000119
-O.0 00O88
- 0 . )0J0o66
-0.0000049
-0.000003 7
-0.0000029
-0.0000022
-0.OUOJOl7
-0.0000014
-0.0000011
-O.CC00009
-0.O0J0OO7
-0.0000006
-0.0000004
-0.U00U004
-0.0000003
-0.0000003
«^^6
-0.00027^2
-0.0004496
-0.000-+355
-0.0003022
-0.0001*378
-0.0000060
0.0000710
0.0001000
0.0000983
0.0000820
0.0000617
0.0000430
0.000 02 79
0.0000169
0.O000J93
0.0000U43
0.0000012
-0.0000007
-0.0000016
-0.0000021
-0.00o0o21
-0.0000018
-0.00OJ013
-0.0000010
-0.O00O0O7
-0.0000005
-0.0000004
-0.0000003
-0.0000002
-0.0000001
-0.0000001
-0.0000001
-O.OOOJOol
-O.OOOOOJO
-0.0000000
-J.0000000
-0.0000000
-O.OOOOOuO
-0.0000000
-O.OOOOOuO
146
TAhLt
Aí PH(^. BtTA
1.50
1.50
1.50
1.50
1.5u
1.50
1.50
1.50
1.50
1.50
1.50
1.50
1.50
1.50
1.50
1.50
1.50
1.50
1.50
1.50
1.50
1.50
1.50
1.50
1.50
1.50
1.50
1.50
1.5u
1.50
1.50
1.50
1.50
1.50
1.50
1.50
1.50
1.5í)
1.50
1.50
0.10
0.2 0
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
1.10
1.20
1.30
1.40
1.50
1.60
1.70
1.80
U9 0
2.00
2.20
2.40
2.60
2.80
3.00
3.20
3.40
3.60
3.80
4.00
-».?0
4.40
4.60
4.80
5.00
5.20
5.40
5.60
5.80
6.00
F(a ,3 )
C-11
FP2
0.0004041
-O.o0u4092
0.0008001 r-u. J0U7762
O.COllBOo
-U.0010688
0.0015400 -0.0012709
0.00 16741 -O.OU13827
0.0021806
-0.0014158
0.00245 39 -0.00138 79
0.00 2 7096 -0.uOr3176
0.u02fi34u -0.001221^
0.0031339 -U.UU11124
0.0033116
-U.0010001
0.003 46 9 3 -0.0008907
0 . U 0 3 6 0') 1-0.0007880
0.0037332 -0.0006941
0.003843^+ -0.0OU6096
-0.00053^5
0.U0394U
0.00402^7 -0.0004685
0.004 1066 -0.0004106
0.0041762 -0.0003602
0.0042387 -û.0003164
0. J0-+34 5^ -0.O0O2453
-0.0001916
0.004^323
0.00 450 3 8 -U.0001510
0.0J456 31 -0.0001201
0.0046129 -0.0000964
O.U0465'+8 -0.0000780
-0.0000637
0.0046905
0.004 7211 -O.O0JU524
0.00 474 74 -0.0000^34
0.004 7 7 03 -Û.00J0363
0.UJ479O3 -O.O0J03O5
0^0048078 -0.0000258
0.0048232 -0.00<J0220
0.0048369 -0.0000188
0.004 8491 -0.0000162
0.00'+8oOO -J.U000140
0.0048698
-0.00J0122
0.0048786 -0.OOJO106
0.0048 86 5 -0.0000093
0.004893/ -0.0000082
FE4
FE6
0.0003604
0. )006316
0.0007649
0.0007615
0.00065^)6
0.0005092
0.J003518
0.0002130
0.0001040
0.J0002 56
-0.0000262
- 0 . )000572
-0.0Û0O734
- 0 ^ 00C794
-0.OU0U791
-^•0000749
-0.0000688
-0.000 0618
-0.0000547
-0.0000480
-Û.00JU362
-0.0000270
-0.J000200
-0.0000149
-0.0000111
- 0 . )U0J0 83
-0.U00J063
-0,0000048
-0.0000037
-0.0000029
- 0 . JOOJ022
-0.0000018
-0.0000014
-0.0000011
-0.0000009
-0.0000007
-0.0000006
-0.0000005
-0.0000004
-0.0000003
-0.0002861
-0.0004763
-0.0004666
-0.0003308
-0.0001592
-0.0000186
0.00O06O5
0.000101/
0.0001)37
0.000089 1
0.0000689
J.0000495
0.0000333
0.0000211
0.0000124
0.0U000O4
0.0000026
0.0000003
-0.0000011
-0.0O00O18
-0.0000022
-0.0000019
-O.00O0015
-0.0000011
-0.0000008
-0.0000006
-0.0000004
-0.0000003
-0.0000002
-0.0000002
-0.0000001
-0.0000001
-0.0000001
-0.0000001
-0.0000000
-0.0000000
-0.0000000
-O.OûûOOOO
-O^0000000
-O^OOOOOOO
147
TAr^Lt
C-12
ALPHA
BETA
F (a ,3 )
Fr2
1.60
1.60
1.60
1.60
1 .60
1.60
1.60
1.60
1.6 0
1 .oO
1.60
1.60
1.6 0
1.60
1.60
1.60
1.60
1 .60
1.60
1 .60
1.60
1.60
1.60
1.60
1.6 0
1.6 0
1.60
1.60
1.60
1 .oO
1.60
1.60
1.60
1.60
1 .60
1.60
1.60
1.60
1 .60
1 .60
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.^0
l.uO
1.10
1.20
1.30
1.40
1.50
1.60
1.70
1.80
1.90
2.00
2.20
2.40
2.60
2.80
3.00
3.20
3.40
3.60
3.80
4.00
4.20
4.40
4.60
4.R0
5 . JO
5.20
5.40
5.60
5.80
6.00
0 . 0 0 04 6 85
0 . 0 0 0 9 2 81
0^0013707
0.0017899
0.0021311
0.0025417
0 . 0 0 2 8 706
0.0031683
0.0034362
O.OJ36761
0.0038904
0.0J40815
0.0042518
0.0044036
-0.0004492
-0.0008536
- 0 . 0 0 1 1792
-U.0O14U83
0.0003607
0.0UJ6699
-0.0O154J2
0.00J7176
0.0005630
U.00u3981
0.000250t
O^OJ^5 389
0.0046597
0 . 0 0 4 76 7 /
0.0048644
0 . 0 0 4 95 11
O.OJ5029 1
0 . 0 0 5 16 29
^•0052723
0.0053o27
0^u05438J
0.0055013
0.0055548
J^00 5 600^
0.0056395
O.OJ56733
0 . J 0 5 ^027
0 . 0 0 5 72 34
0.O(j5 751íJ
0 . 0 0 5 7 709
0 . 0 0 5 78 86
0 . 0 0 58043
0.0053183
0 . 0 0 5 83 l u
0.0058423
0 . 0 0 5 85 26
0.0058619
-Û.0Û158o5
-0.0015654
-0.0014963
-0.0013968
-0.0012812
-0.0011599
-0.0010401
-0.0009263
-0.0008210
-J.0007253
-0.0006396
-J.0U05635
-0.0004964
-0.0004374
-0.JO03859
-0.0003014
-J.OOU2 3 70
- 0 . 0 0 0 1 8 79
-0.U0015J1
-Û.0001210
-0.0000983
-O.UOJ08J5
-0.00u06o^
-0.00u0552
-O.0OJ0462
-0.0000389
-0.OUO0330
- 0 . 0 0 0 02 8 1
-O.OJ )0241
-0.0000208
-0.0000180
-0.0000156
-O.OOJ013 7
-O.O0J0120
-0.0000106
C -r
0.0C08158
0 . ) 0 0 8188
0.0001320
0 . J 0 J 0 4 50
•0.0000141
0.JOJ0512
•0.0000719
•0.0000812
•0.J0J0831
•0.J0J0803
•0.0000749
•0.0000683
•0.0000612
•0.0000542
-0.0000416
0.JU00315
•0.0000236
0.00)0178
•0.0C0Û134
•O.JOJOIOI
• 0 . J 0 0 0 0 77
•0.0000059
•0.0U0U046
•0.0000036
•0.0000028
•0.0000022
•0.0000013
0.Û00J014
O.OÛOOOU
•0.0000009
0.00)0008
•0.0000006
0 . J 0 0 0 0 05
0.0000004
FE6
-0.0002980
-0.UJ04936
-0.0004872
-0.0003507
-0.0001753
-0.0000293
O.OOOO0I2
0.0001009
0.0001061
0.0000932
0.0000738
0.00005^+2
0.000037O
0.0000246
O.O0JJ152
0.0000086
0.00000-+1
J.OOJJOl3
-O.OJOJJ04
-0.0u0JOl4
-0.0000021
-0.J00u020
-0.0000016
-0.0uJ)013
-0.0000009
-0.0000007
-O.0OO0OO5
-0.0000004
-0.0000003
-0.0000002
-O.OOOOJOI
-0.00000Jl
-0.0000001
-0.0000001
-0.0000000
-0.0000000
-0.0000000
-0.0000000
-0.0000000
-0.0000000
148
TABLE C-13
ALPHA
BETA
1.70
1.70
1.70
1.70
1.70
1.70
1.70
1.7J
1.70
1.70
1.70
1.70
1.70
1.70
1.70
1.70
1.70
1.70
1.70
1.70
1.70
1.70
1.70
1.70
1.70
1.70
1.70
1. 70
1.70
1.70
1.70
1.70
1.70
1.70
1. 70
1.70
1.70
1.70
1.70
1.70
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
i.OO
UIO
1.20
1.30
1.40
1.50
1.60
1.70
1.80
1.90
2.00
2.20
2.40
2.60
2.80
3.00
3.20
3.40
3.60
3.80
4.00
4.20
4.40
4.60
4.80
5.00
5.20
5.40
5.60
5.80
6.00
F(a »3)
0.0005290
0.00104 84
0.0015496
0.0020256
0.0024712
0.0028835
0.0032613
0.0036047
0.0039151
0.0041945
0.0044452
0.0046697
J.00487J7
0.0050506
0.0052116
0.0053558
0.0054853
0.JJ56U15
0.0057062
0.0058005
0.00596 29
0.0060964
0.00620 70
0.0062996
0.00637/5
0.006-+-+35
0.0065001
0.0065486
0.U065906
0.00662/2
0.0066592
0.00668/3
0.0067122
0.0067342
0.006 75 39
0.0067715
0.0067873
U.0068015
0.0068144
0.0068261
FE2
-0.0004823
-0.0009182
-0.0012717
-0.0015243
-0.0016744
-0.0017336
-0.0017200
-0.00165^0
-0.0015536
-0.0014340
-0.0013064
-0.0011788
-0.0010561
-0.00J9415
-0.000 83o4
-0.0007415
-0.0006565
-0.0005810
-0.0005143
-0.0004555
-0.0003585
-0.0002338
-0.0002262
-0.0001317
-0.O0U1470
-0.0001199
-0.00J0985
-0.0OJ0815
-0.00)0679
-0.00J0570
-0.0000481
-0.0000408
-O.00J0349
-0.0000299
-0.0000258
-0.0000224
-0.0000195
-0.001)01 70
-0.0000150
-0.0000132
Ft4
0.0C03958
0.0006981
0.0008539
0.0008627
0.000 7630
0.0006064
0.0004369
0.0002829
0.0001578
0.0000641
-O.JOOOÛIO
-0.0000432
-0.J000680
-0.0000805
-0.000J847
-0.0000835
-0.0000791
- 0 . 1000730
-0.0000662
-0.0000593
-0.0000463
-0.0000355
-0.JÛ00270
-0.0000205
-0.0000156
-0.0000119
-0.0000091
- 0 . J00U070
-0.0000055
-0.0U0U043
-0.0000034
-0.0000027
-0.0000021
-0.0000017
-0.0000014
-0.0000011
-0.0000009
-0.J000008
-0.00Û0006
-0.0000005
FE6
-0.0003060
-0.0005052
-0.0005012
-0.0003648
-0.00013/4
-0.0000381
0.0000560
0.0000989
0.0001067
0.000095"+
0.0000768
0.0000576
0.0000408
0.0000275
0.UÛ00175
0.0000105
0.0000056
0.0000024
0.0000004
-0.0000009
-0.0000019
-0.0000019
-0.0000017
-0.0000013
-0.0000010
-0.0000008
-0.0000006
-0.0000004
-0.0000003
-0.0O0JOO2
-0.0000002
-0.0000001
-0.0000001
-0.0000001
-O.ÛOOOOOI
-0.0000000
-0.0000000
-O.ûOOOOOO
-0.0000000
-O.OOJOOOO
149
TAKLc
AIPHA
t^ZTA
1.80
U8u
1 . 80
1 . 80
1.80
1.80
1 .80
1.80
1.80
1.80
1.80
1.80
1 . 80
1 .80
1.80
1.80
1 .80
1.80
1.80
1.30
1.80
1.80
1.80
U80
1.80
1.80
1 .80
1.80
U80
1.80
1.80
1.80
1.80
1.80
1.80
1.80
l .80
1.80
1.80
1.80
O.IO
0.20
0.30
0.40
0.5 0
0.60
0.70
0.80
0 . 9 tj
1.00
1.10
1.20
1.30
1.40
1.50
1.60
1.7 J
l.í<0
1.90
2.00
2.20
2.40
2 . í. 0
2.P0
3.00
3.20
3.^+0
3.6 0
3.80
4.00
4.20
4.4U
4.60
4.80
5.00
5.20
5.40
5.60
5.80
6.00
F (a ,3 )
0.0005361
O.OJ116 20
0.0017186
0.0)22^85
0.0027460
U . 0 0 3 2 079
0 . J J 3 6 3 27
0 . 0 0 4 02u5
0.00^+3725
0.0046907
0.0049774
0.0052353
0.0054671
O.005o753
0.0U38624
0.0)60306
tJ.0U6l>^?l
0.0J631B6
0 . JJ6^-»17
0.00655J>1
0.JJ67455
0.,)J690^4
0 . 0 0 7 03 66
0 . 0 0 71-+ /5
0 . J J 7 2412
O.Oû/3209
0.0073892
0.007448Û
0.0U74989
0 . 0 0 75433
0.00/5822
0.JU76165
0.0076468
0 . 0 0 76 738
0.0076977
0.0077192
0 . 0 0 7 7386
0.0077560
0.J077718
0.0J77861
C-i4
fb2
Ft4
Fr6
- V ) . 0 U J 5 1 Jl
0 . 0 O J 4 073
0.0007193
0.0008829
0«0008966
0*0007C)89
0.J006415
0.0004692
•0.000^020
•0.0005121
^•0005108
•O^ 0 0 0 3 74 8
-u.0009724
-0.0013499
-U.UO16230
-0.0017896
-0.00186J9
-0.0018554
-0.0017934
-0.0016938
-0.0015722
-0.0014404
-0.0013069
-û.00117/4
-0.001055?
-0.0 00 942 3
-0.0008395
-0.000/468
-0.UU066 39
-v). 0 00 5 902
-0.0005248
-0.0004161
-J.OOU3313
-0.0002657
-0.00021H4
-0.0001743
-0.0001427
-0.0001176
-0.0000976
-0.0000815
-0.00006 35
-o.o 0u5ao
-O.OOU0493
-0.0000422
-0.0000363
-0.0000313
-0.0000272
-O.00J0237
-0.0000207
-0.00)0182
-0.0000161
C.0003110
0 . 0 0 0 1809
0.0000822
0.J000122
0.0000342
•0.0000626
0. )0007ao
•0.00JÛ844
•0.0OOU849
0.0000817
•0.0000763
0.0000699
•0.J0J0632
•0.0000502
•0.0000390
•0.0000300
^•0000230
•0.0000177
•Û.0000136
0.JUJ0105
•0.0000082
0.0000064
•0.0000050
•0.000J040
•0.00O0O32
• 0 . )U00025
•0.JOOJ020
-0.0000017
•0.0000013
•0.0000011
-0.0000009
•0.0000008
-0.0000006
0^000l963
^•U000451
0.0u003l4
0.0000966
0.00U1062
0.0000964
0.JU00787
0.0000598
0.000043 1
0.00JJ297
0.00001^5
0.0000121
0.0000069
0.0000034
O.OOOJJl1
•0.0000003
•0.0000016
•0.0000018
•0.0000017
•0.0000014
•0.0000011
-0.0OUJOO8
•0.0000006
-0.0000005
•0.0000003
-0.0000003
-0.000J002
-O.OOOOOOl
-0.0000001
-0.0000001
-0.0000001
-0.0000001
-0.0000000
-0.0000000
-J.OOOOOUÛ
-0.0000000
150
TABLt C-15
ALPHA
3ETA
1.90
1.90
1.90
1.90
1.90
1.90
1.90
1.90
1.90
1.90
1.90
1.90
1.90
1.90
1.90
1.90
1.90
1.90
1.90
1.90
1.90
1.90
1.90
1.90
1.90
1.90
1.90
1.90
U90
1.90
1.90
1.90
1.90
1.90
1.90
1.90
1.90
1.90
1.90
1.90
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
1.10
1.20
1.30
1.40
1.50
1.6 0
1.70
1.80
1.90
2.00
2.20
2.40
2.60
2.80
3.00
3.20
3.40
3.60
3.80
4.00
4.20
4.40
4.60
4.80
5.00
5.20
5.40
5.60
5.80
6.00
F(a ,3)
0.0006401
0.0012695
0.0018 788
0.0024598
0.003007U
0.0035165
0.0039867
0.0044175
0.0048100
0.0051662
0.0054885
0.0057796
0.0060421
0.0062788
0.0064922
0.0066848
0.0068537
0.0070159
0.0071582
0.0072872
0.0075108
0.0076964
O.OJ7Ô514
0.0079819
O.OJ80924
0.0081866
0.00826/6
O.OU83374
0.0083980
0.0084509
0.0084973
0.0085333
0.00 85746
0.0086063
0.0086356
0.0J86614
0.0086846
0.0087055
0.0087245
0.0J87417
Fe2
-0.0005337
-0.001U185
-0.0U14166
-0.0017076
-0.0018890
-0.0019717
-0.0019741
-0.0019169
-0.0018192
-O.U016970
-0.0015627
-0.0014251
-0.0012903
-0.0011622
-0.0010428
-0.00J9333
-0.0008340
-0.0007446
-0.0006645
-0.0005932
-0.0004736
-0.0003797
-0.0003060
-0.0002482
-0.0002026
-0.00016o5
-0.0001377
-0.0001146
-0.0000960
-0.0000809
-0.0000686
-0.0000584
-U.00JU5U1
-0.0000431
-0.0000373
-0.0000324
-0.0000283
-0.0000248
-0.0000218
-0.0000192
FÊ4
0.0004156
0.0007356
0.J0O9053
0.0009231
0.000827-»
0.0006700
0.0004961
0.0003351
0.0002014
0.0000989
0.0000251
-0.0000248
-0.0000563
-0.0000743
-0.0000828
-0.0000849
-0.0000828
-0.0000784
-0.0000726
-0.0000662
-0.0000534
-0.J000421
-0.0000328
-0.0000254
-0.0000196
-0.0000152
-0.0000118
-0.0000092
-0.0000073
-0.0000057
-0.0000046
-0.0000036
-0.0000029
-0.0000024
-0.J000019
-0.0000016
-0.0000013
-O.OÚOOOU
-0.0000009
-0.0COO007
FE6
-0.0003099
-0.0005178
-0.0005175
-0.0003820
-0.0002030
-0.0000507
0.0000474
0.0000942
0.0001053
0.0000966
0.0000796
0.000J613
0.0000448
0.0000314
0.0000211
0.0000135
0.0000081
0.0000044
0.0000019
0.0000003
-0.0000013
-0.0000017
-0.0000016
-0.0000014
-O.OOOOOll
-0.0000009
-0.0000007
-0.0000005
-0.0000004
-0.0000003
-0.0000002
-0.0000002
-0.0000001
-O.OOOOOOl
-0.0000001
-0.0000001
-0.0000000
-O.OOOOOOu
-0.0000000
-0.0000000
151
TARLt
ALPHA
BhTA
F (a ,3 )
2 . JO
2.0«)
2.0U
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.0 0
2.00
2.00
2.00
2.0»)
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.J0
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
0 . 10
0.20
0.30
0.4J
0.50
0.6 0
0.7 0
0.80
0.9J
UOO
0 . 0 0 0 6 9 13
0.0J1371O
0.0020308
u . 0 0 2 6 6 08
0.0032554
0.0038106
0.0043246
0.0J4 7J/1
0.0)52291
0.0056226
0.0059799
0 . ('06 30 37
0.0065968
J . U J 6 36 20
0.0071020
0 . 0 0 73l4^í
0.0075159
0.0076942
O.J) 78561
0.OJ80032
0.0082592
0.0084725
0 . 0 0 86 5 14
0.«JJ8tí025
0.0Jd9308
U . O J 9 0 4 06
0 . )J91350
0.009216 7
0.()«í928 /7
0 . 0 0 9 349 8
0.0094043
O.OJ^H^^^
0.0094953
0 . 0 0 9 53 3 3
0.00^^5672
0.0095977
0 . 0 0 9 62 52
0.0096499
U.0096723
0.0096^27
U lu
1.2 0
1.3U
1 .^0
U50
1.6 0
1.70
1.80
1 .90
2.00
2.20
2.4J
2.60
2.80
3.00
3.20
3.40
3.6 0
3.8U
4.00
4.20
-».40
4.6 0
4.8 0
5.00
5.2)
5.40
5 . 6 (J
5.80
6.00
C-lo
ft2
-U.uOu55 3 8
-0.0010580
-0.00147^9
-U.OU17807
-0.0019753
-0.0U2U686
-0.0020788
-O.00202o7
-u.OU19316
-O.OO1809 9
-u.0016743
-0.0015339
-0.0U13952
-0.0012623
-0.0011377
-O.0OÍ022/
-J.00U917/
-0.0008226
- J . U U 0 7 3 70
-u.0006603
-O.OOU53 0 7
- 0 . 0 0042 3 1
-0.0003469
-0.0002627
-0.0002313
-0.0001912
-0.0001536
-0.0001325
-0.JU01112
-O.0000940
-0.00JC798
-O.OUOÛ631
-0.0000565
-0.00)050-»
-0.0000437
-0.UOJO380
-O.OOJ0332
-0.00)0291
-0.0000256
-0.0000226
Ff 4
FE6
0 . JUJ4228
0 . 0 0 0 74 81
0.00J9227
0.00J9440
0.0003502
0.JJ06933
C.0005135
0.U0J3556
0 . 0 0 0 219«+
0.0001140
0.0000372
•0.0000155
-0.0000496
-O.U0OO698
-0.0000rt02
-0.J0Ú0838
•0.0000830
-0.0000794
- 0 . J 0 0 0742
•0.0000683
-0.0000559
•0.00OÛ446
-0.0000351
-0.0000275
-0.0000215
•0.0000168
•0.0000131
-0.0000103
-0.0OOUU81
-0.0000065
-0.0000052
•0.0000041
-0.J000033
•0.000002 7
•0.0000022
-0.J0J0018
•0.0000015
•0.0000012
0.JoOOOlO
•0.0000009
• 0 . 0 0 0 3 )20
0.0005215
•0.0005223
•0.0003872
•0.0002081
•0.0000550
0.0000441
0 . 0 0 j J 9 2u
0.O0J1U41
0.000096 3
0.0000801
0.0000622
0.0ÚJ0459
0.00O0326
0.0000223
0.000Jl'+6
0.0000091
0.000OO52
0.O000J26
0.0000008
-O.OOOJOIO
-0.0000015
-0.0000016
-0.0)00014
-O.UOOOOll
-0.0000009
-0.000000/
- 0 . 0 0 0 0 005
-0.0000004
-0.0000003
-O.OOOJ 0 0 2
-0.0000002
-0.0000001
-O.OOOJOOl
-0.0000001
-0.0000001
-O.OOOJOOl
-0.0000000
-0.0000000
-0.0000000