Eric Dollard Introduction to Dielectric and Magnetic Discharges in Electrical Windings (complete OCR remake)

Eric Dollard Introduction to Dielectric and Magnetic Discharges in Electrical Windings (complete OCR remake)
I) INTRODUCTION TO DIELECTRIC & MAGNETIC
DISCHARGES IN ELECTRICAL WINDINGS
by Eric Dollard, ©1982
II) ELECTRICAL OSCILLATIONS IN ANTENNAE
AND INDUCTION COILS
by John Miller, 1919
PART I
INTRODUCTION TO DIELECTRIC & MAGNETIC DISCHARGES IN ELECTRICAL
WINDINGS
by Eric Dollard, ©1982
1. CAPACITANCE
2. CAPACITANCE INADEQUATELY EXPLAINED
3. LINES OF FORCE AS REPRESENTATION OF DIELECTRICITY
4. THE LAWS OF LINES OF FORCE
5. FARADAY'S LINES OF FORCE THEORY
6. PHYSICAL CHARACTERISTICS OF LINES OF FORCE
7. MASS ASSOCIATED WITH LINES OF FORCE IN MOTION
8. INDUCTANCE AS AN ANALOGY TO CAPACITANCE
9. MECHANISM OF STORING ENERGY MAGNETICALLY
10. THE LIMITS OF ZERO AND INFINITY
11. INSTANT ENERGY RELEASE AS INFINITY
12. ANOTHER FORM OF ENERGY APPEARS
13. ENERGY STORAGE SPATIALLY DIFFERENT THAN MAGNETIC ENERGY
STORAGE
14. VOTAGE IS TO DIELECTRICITY AS CURRENT IS TO MAGNETISM
15. AGAIN THE LIMITS OF ZERO AND INFINITY
16. INSTANT ENERGY RELEASE AS INFINITY
17. ENERGY RETURNS TO MAGNETIC FORM
18. CHARACTERISTIC IMPEDANCE AS A REPRESENTATION OF PULSATION OF
ENERGY
19. ENERGY INTO MATTER
20. MISCONCEPTION OF PRESENT THEORY OF CAPACITANCE
21. FREE SPACE INDUCTANCE IS INFINITE
22. WORK OF TESLA, STEINMETZ, AND FARADAY
23. QUESTION AS TO THE VELOCITY OF DIELECTRIC FLUX
APPENDIX I
0) Table of Units, Symbols & Dimensions
1) Table of Magnetic & Dielectric Relations
2) Table of Magnetic, Dielectric & Electronic Relations
PART II
ELECTRICAL OSCILLATIONS IN ANTENNAE & INDUCTION COILS
J.M. Miller
Proceedings, Institute of Radio Engineers. 1919
1) CAPACITANCE
The phenomena of capacitance is a type of electrical energy storage in the form of a field in
an enclosed space. This space is typically bounded by two parallel metallic plates or two metallic
foils on an interviening insulator or dielectric. A nearly infinite variety of more complex structures
can exhibit capacity, as long as a difference in electric potential exists between various areas of the
structure. The oscillating coil represents one possibility as to a capacitor of more complex form, and
will be presented here.
2) CAPACITANCE INADEQUATELY EXPLAINED
The perception of capacitance as used today is wholly inadequate for the proper
understanding of this effect. Steinmetz mentions this in his introductory book ''Electirc Discharges,
Waves and impulses''. To quote, ''Unfortunately, to a large extent in dealing with dielectric fields the
prehistoric conception of the electro- static charge (electron) on the conductor still exists, and by its
use destroys the analogy between the two components of the electric field, the magnetic and the
dielectric, and makes the consideration of dielectric fields unnecessarily complicated.''
3) LINES OF FORCE AS REPRESENTATION OF DIELECTRICITY
Steinmetz continues, ''There is obviously no more sense in thinking of the capacity current
as current which charges the conductor with a quantity of electricity, than there is of speaking of the
inductance voltage as charging the conductor with a quantity of magnetism. But the latter
conception, together with the notion of a quantity of magnetism, etc., has vanished since Faraday's
representation of the magnetic field by lines of force."
4) THE LAWS OF LINES OF FORCE
All the lines of magnetic force are closed upon themselves, all dielectric lines of force
terminate on conductors, but may form closed loops in electromagnetic radiation.
These represent the basic laws of lines of force. It can be seen from these laws that any line
of force cannot just end in Space.
5) FARADAY AND LINES OF FORCE THEORY
Farady felt strongly that action at a distance is not possible thru empty space, or in other
words, "matter cannot act where it is not." He considered space pervaided with lines of force.
Almost everyone is familiar with the patterns formed by iron filings around a magnet. These filings
act as numerous tiny compasses and orientate themselves along the lines of force existing around
the poles of the magnet. Experiment has indicated that a magnetic field does possess a fiberous
construct. By passing a coil of wire thru a strong magnetic field and listening to the coil output in
headphones, the experimenter will notice a scraping noise. J. J. Thompson performed further
experiments involving the ionization of gases that indicate the field is not continuous but fiberous
(Electricity and Matter, 1906).
6) PHYSICAL CHARACTERISTICS OF LINES OF FORCE
Consider the space between poles of a magnet or capacitor as full of lines of electric force.
See Fig. 1. These lines of force act as a quantity of stretched and mutually repellent springs. Anyone
who has pushed together the like poles of two magnets has felt this springy mass. Observe Fig. 2.
Notice the lines of force are more dense along AB in between poles, and that more lines on A are
facing B than are projecting outwards to infinity. Consider the effect of the lines of force on A.
These lines are in a state of tension and pull on A. Because more are pulling on A towards B than
those pulling on A away from B, we have the phenomena of physical attraction. Now observe Fig.
3. Notice now that the poles are like rather than unlike, more or all lines pull A away from B; the
phenomena of physical repulsion.
7) MASS ASSOCIATED WITH LINES OF FORCE IN MOTION
The line of force can be more clearly understood by representing it as a tube of force or a
long thin cylinder. Maxwell presented the idea that the tension of a tube of force is representative of
electric force (volts/inch), and in addition to this tension, there is a medium through which these
tubes pass. There exists a hydrostatic pressure against this media or ether. The value of this pressure
is one half the product of dielectric and magnetic density. Then there is a pressure at right angles to
an electric tube of force. If through the growth of a field the tubes of force spread sideways or in
width, the broadside drag through the medium represents the magnetic reaction to growth in
intensity of an electric current. However, if a tube of force is caused to move endwise, it will glide
through the medium with little or no drag as little surface is offered. This possibly explains why no
magnetic field is associated with certain experiments performed by Tesla involving the movement
of energy with no accompanying magnetic field.
8) INDUCTANCE AS AN ANALOGY TO CAPACITY
Much of the mystery surrounding the workings of capacity can be cleared by close
examination of inductance and how it can give rise to dielectric phenomena. Inductance represents
energy storage in space as a magnetic field. The lines of force orientate themselves in closed loops
surrounding the axis of current flow that has given rise to them. The larger the space between this
current and its images or reflections, the more energy that can be stored in the resulting field.
9) MECHANISM OF STORING ENERGY MAGNETICALLY
The process of pushing these lines or loops outward, causing them to stretch, represents
storing energy as in a rubber band. A given current strength will hold a loop of force at a given
distance from conductor passing current hence no energy movement. If the flow of current
increases, energy is absorbed by the field as the loops are then pushed outward at a corresponding
velocity. Because energy is in motion an E.M.F. must accompany the current flow in order for it to
represent power. The magnitude of this E.M.F. exactly corresponds to the velocity of the field. Then
if the current ceases changing in magnitude thereby becoming constant, no E.M.F. accompanies it,
as no power is being absorbed. However, if the current decreases it represents then a negative
velocity of field as the loops contract. Because the E.M.F. corresponds exactly to velocity it
reverses polarity and thereby reverses power so it now moves out of the field and into the current.
Since no power is required to maintain a field, only current, the static or stationary field, represents
stored energy.
10) THE LIMITS OF ZERO AND INFINITY
Many interesting features of inductance manifest themselves in the two limiting cases of
trapping the energy or releasing it instantly. Since the power supply driving the current has
resistance, when it is switched off the inductance drains its energy into this resistance that converts
it into the form of heat. We will assume a perfect inductor that has no self resistance. If we remove
the current supply by shorting the terminals of the inductor we have isolated it without interrupting
any current. Since the collapse of field produces E.M.F. this E.M.F. will tend to manifest. However,
a short circuit will not allow an E.M.F. to develop across it as it is zero resistance by definition. No
E.M.F. can combine with current to form power, therefore, the energy will remain in the field. Any
attempt to collapse forces increased current which pushes it right back out. This is one form of
storage of energy.
11) INSTANT ENERGY RELEASE AS INFINITY
Very interesting (and dangerous) phenomena manifest themselves when the current path is
interrupted, thereby causing infinite resistance to appear. In this case resistance is best represented
by its inverse, conductance. The conductance is then zero. Because the current vanished instantly
the field collapses at a velocity approaching that of light. As E.M.F. is directly released to velocity
of flux, it tends towards infinity. Very powerful effects are produced because the field is attempting
to maintain current by producing whatever E.M.F. required. If a considerable amount of energy
exists, say several kilowatt hours* (250 KWH for lightning stroke), the ensuing discharge can
produce most profound effects and can completely destroy inadequately protected apparatus.
* The energy utilized by an average household in the course of one day.
12) ANOTHER FORM OF ENERGY APPEARS
Through the rapid discharge of inductance a new force field appears that reduces the rate of
inductive E.M.F. formation. This field is also represented by lines of force but these are of a
different nature than those of magnetism. These lines of force are not a manifestation of current
flow but of an electric compression or tension. This tension is termed voltage or potential
difference.
13) DIELECTRIC ENERGY STORAGE SPATIALLY DIFFERENT THAN MAGNETIC
ENERGY STORAGE
Unlike magnetism the energy is forced or compressed inwards rather than outwards.
Dielectric lines of force push inward into internal space and along axis, rather than pushed outward
broadside to axis as in the magnetic field. Because the lines are mutually repellent certain amounts
of broadside or transverse motion can be expected but the phenomena is basically longitudinal. This
gives rise to an interesting paradox that will be noticed with capacity. This is that the smaller the
space bounded by the conducting structure the more energy that can be stored. This is the exact
opposite of magnetism. With magnetism, the units volumes of energy can be thought of as working
in parallel but the unit volumes of energy in association with dielectricity can be thought of as
working in series.
14) VOLTAGE IS TO DIELECTRICITY AS CURRENT IS TO MAGNETISM
With inductance the reaction to change of field is the production of voltage. The current is
proportionate to the field strength only and not velocity of field. With capacity the field is produced
not by current but voltage. This voltage must be accompanied by current in order for power to exist.
The reaction of capacitance to change of applied force is the production of current. The current is
directly proportional to the velocity of field strength. When voltage increases a reaction current
flows into capacitance and thereby energy accumulates. If voltage does not change no current flows
and the capacitance stores the energy which produced the field. If the voltage decreases then the
reaction current reverses and energy flows out of the dielectric field.
As the voltage is withdrawn the compression within the bounded space is relieved. When the
energy is fully dissipated the lines of force vanish.
15) AGAIN THE LIMITS ZERO AND INFINITY
Because the power supply which provides charging voltage has internal conductance, after it
is switched off the current leaking through conductance drains the dielectric energy and converts it
to heat. We will assume a perfect capacitance having no leak conductance. If we completely
disconnect the voltage supply by open circuiting the terminals of the capacitor, no path for current
flow exists by definition of an open circuit. If the field tends to expand it will tend towards the
production of current. However, an open circuit will not allow the flow of current as it has zero
conductance. Then any attempt towards field expansion raises the voltage which pushes the field
back inwards. Therefore, energy will remain stored in the field. This energy can be drawn for use at
any time. This is another form of energy storage.
16) INSTANT ENERGY RELEASE AR INFINITY
Phenomena of enormous magnitude manifest themselves when the criteria for voltage or
potential difference is instantly disrupted, as with a short circuit. The effect is analogous with the
open circuit of inductive current. Because the forcing voltage is instantly withdrawn the field
explodes against the bounding conductors with a velocity that may exceed light. Because the
current is directly related to the velocity of field it jumps to infinity in its attempt to produce finite
voltage across zero resistance. If considerable energy had resided in the dielectric force field, again
let us say several K.W.H. the resulting explosion has almost inconceivable violence and can
vaporize a conductor of substantial thickness instantly. Dielectric discharges of great speed and
energy represent one of the most unpleasant experiences the electrical engineer encounters in
practice.
17) ENERGY RETURNS TO MAGNETIC FORM
The powerful currents produced by the sudden expansion of a dielectric field naturally give
rise to magnetic energy. The inertia of the magnetic field limits the rise of current to a realistic
value. The capacitance dumps all its energy back into the magnetic field and the whole process
starts over again. The inverse of the product of magnetic storage capacity and dielectric storage
capacity represents the frequency or pitch at which this energy interchange occurs. This pitch may
or may not contain overtones depending on the extent of conductors bounding the energies.
18) CHARACTERISTIC IMPEDANCE AS REPRESENTATION OF PULSATION OF
ENERGY FIELD
The ratio of magnetic storage ability to that of the dielectric is called the characteristic
impedance. This gives the ratio of maximum voltage to maximum current in the oscillatory
structure. However, as the magnetic energy storage is outward and the dielectric storage is inward
the total or double energy field pulsates in shape or size. The axis of this pulsation of force is the
impedance of the system displaying oscillations and pulsation occurs at the frequency of oscillation.
19) ENERGY INTO MATTER
As the voltage or impedance is increased the emphasis is on the inward flux. If the
impedance is high and rate of change is fast enough (perfect overtone series), it would seem
possible the compression of the energy would transform it into matter and the reconversion of this
matter into energy may or may not synchronize with the circle of oscillation. This is what may be
considered supercapacitance, that is, stable longterm conversion into matter.
20) MISCONCEPTIONS OF PRESENT THEORY OF CAPACITANCE
The misconception that capacitance is the result of accumulating electrons has seriously
distorted our view of dielectric phenomena. Also the theory of the velocity of light as a limit of
energy flow, while adequate for magnetic force and material velocity, limits our ability to visualize
or understand certain possibilities in electric phenomena. The true workings of free space
capacitance can be best illustrated by the following example. It has been previously stated that
dielectric lines of force must terminate on conductors. No line of force can end in space. If we take
any conductor and remove it to the most remote portion of the universe, no lines of force can extend
from this electrode to other conductors. It can have no free space capacity, regardless of the size of
the electrode, therefore it can store no energy. This indicates that the free space capacitance of an
object is the sum mutual capacity of it to all the conducting objects of the universe.
21) FREE SPACE INDUCTANCE IS INFINITE
Steinmetiz in his book on the general or unified behavior of electricity ''The Theory and
Calculation of Transient Electric Phenomena and Oscillation," points out that the inductance of any
unit length of an isolated filamentary conductor must be infinite. Because no image currents exist to
contain the magnetic field it can grow to infinite size. This large quantity of energy cannot be
quickly retrieved due to the finite velocity of propagation of the magnetic field. This gives a non
reactive or energy component to the inductance which is called electromagnetic radiation.
22) WORK OF TESLA, STEINMETZ AND FARADAY
In the aforementioned books of Steinmetz he develops some rather unique equations for
capacity. Tesla devoted an enormous portion of his efforts to dielectric phenomena and made
numerous remarkable discovers in this area. Much of this work is yet to be fully uncovered. It is my
contention that the phenomena of dielectricity is wide open for profound discovery. It is ironic that
we have abandoned the lines of force concept associated with a phenomena measured in the units
called farads after Farady, whose insight into forces and fields has led to the possibility of
visualization of the electrical phenomena.
23) QUESTION AS TO THE VELOCITY OF DIELECTRIC FLUX
It has been stated that all magnetic lines of force must be closed upon themselves, and that
all dielectric lines of force must terminate upon a conducting surface. It can be infered from these
two basic laws that no line of force can terminate in free space. This creates an interesting question
as to the state of dielectric flux lines before the field has had time to propagate to the neutral
conductor. During this time it would seem that the lines of force, not having reached the distant
neutral conductor would end in space at their advancing wave front. It could be concluded that
either the lines of force propagate instantly or always exist and are modified by the electric force, or
voltage. It is possible that additional or conjugate space exists within the same boundaries as
ordinary space. The properties of lines of force within this conjugate space may not obey the laws
of normally conceived space.
IMPORTANT REFERENCE MATERIAL
1. "Electricity and Matter," J. J. Thompson
New York, 1906, Scribner's Sons, and I904, Yale University
2. "Elementary Lectures on Electric Discharges, Waves, and Impulses and other Transients."
C. P. Steinmetz, second edition, 1914, McGraw-Hill
3. "Theory and Calculation of Transient Electric Phenomena and Oscillations," C. P.
Steinmetz, third edition, 1920, McGraw-Hill. Section III Transients in Space, Chapter VIII,
Velocity of Propagation of Electric Field
Table I
Magnetic Field
Dielectric Field
Magnetic flux:
φ = Li 108 lines of magnetic force.
Dielectric flux:
ψ = Ce lines of dielectric force.
Inductance voltage:
d  −8
di
e ' =n
10 =L
volts.
dt
dt
Capacity current:
d
di
i '=n
=C
amperes.
dt
dt
Magnetic energy:
L i2
joules.
w=
2
Dielectric energy:
C e2
joules.
w=
2
Magnetomotive force:
F = ni ampere turns.
Electromotive force:
e = volts.
Magnetizing force:
F
f=
ampere turns per cm.
l
Electrifying force or voltage gradient:
e
G=
volts per cm.
l
Magnetic-field intensity:
H = 4πf 10-1 lines of magnetic force per cm2.
Dielectric-field intensity:
G
2
K=
2 lines of dielectric force per cm .
4 v
Magnetic density:
B = μH lines of magnetic force per cm2.
Dielectric density:
D = kK lines of dielectric force per cm2.
Permeability: μ
Permittivity or specific capacity: κ
Magnetic flux:
φ = AB lines of magnetic force.
Dielectric flux:
ψ = AD lines of dielectric force.
v = 3 X 1010 = velocity of light
Table II
Magnetic Circuit
Dielectric Circuit
Magnetic Flux (magnetic current):
φ = lines of magnetic force.
Dielectric flux (dielectric current):
ψ = lines of dielectric force.
Magnetomotive force:
F = ni ampere turns.
Electromotive force:
e = volts.
Electric Circuit
Electric current:
i = electric current.
Voltage:
e = volts.
Permeance:
M=

.
4 F
Permittance or capacity:
C=
Inductance:
4  v2 
farads.
e
Conductance:
g=
i
mhos.
e
n2  −8 n  −8
L=
10 =
10
F
i
henry.
(Elastance ? ):
Reluctance:
2
Li F  −8
=
10 joules.
2
2
w=
Ce2 e 
joules.
=
2
2
Dielectric density:
Magnetic density:

B =
= μH lines per cm2.
A

D=
= κK lines per cm2.
A
Magnetizing force:
Dielectric gradient:
F
f=
ampere turns per cm.
l
G=
e
volts per cm.
l
Magnetic-field intensity:
H = .4πf
Dielectric field-intensity:
Permeability:
Permittivity or specific capacity:
K=
I=
i
= γG amperes per cm2.
A
Electric gradient:
G=
e
volts per cm.
l
Conductivity:
=
1 K
=
.
 D
I
mhos-cm.
G
2
Resistivity:
ρ=
1
G
=
ohms-cm.

I
Specific dielectric energy:
Specific magnetic energy:
per cm3.
Electric-current density:
(Elastivity ? ):
Reluctivity:
ρ=f/B
.4   f
w0 =
2
Electric power:
p = ri2 = ge2 = ei watts.
G
2
4 v
D
=
K
μ=B /H
e
ohms.
i
r=
Dielectric energy:
Magnetic energy:
w=
Resistance:
1
e
=
.
C 4 v 2 
F
R=
.

 G2
= fB / 2 10 joules w0 = 4  v 2 =
cm3.
-8
GD
joules per
2
Specific power:
p0 = ρI2 = G2 = GI watts per cm3.
Table of Units, Symbols, and Dimensions
Quantity
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
Length
Area
Volume
Mass
Time
Velocity
Acceleration
Force
Energy
Power
Charge
Dielectric constant of
free space
Dielectric constant
relative
Charge density
volume
surface
line
Electric intensity
Electric flux density
Electric flux
Electric potential
EMF
Capacitance
Current
Current density
Resistance
Resistivity
Conductance
Conductivity
Electric polarization
Electric susceptibility
Electric dipole moment
Electric energy density
Symbol
mks
Unit
Rationalized
Defining Equation
L
A
v
M, m
T, t
v
q
F
W
P
Q, q
m
m2
m3
kilogram
second
m / sec
m / sec2
newton
joule
watt
coulomb
ε0
ε
εr
farad / m
farad / m
numeric
ε0 = 1 / (μ0c2)
ρ
ρs
ρl
E
D
ψ
V
Vg
C
I, i
J
R
ρ
G
σ
P
xe
me
ωe
coulomb / m3
coulomb / m2
coulomb / m
volt / m
coulomb / m2
coulomb
volt
volt
farad
ampere
ampere / m2
ohm
ohm-m
mho
mho / m
coulomb / m2
farad / m
coulomb-m
joule / m3
ρ=Q/v
ρs = Q / A
ρl = Q / L
E=F/Q=-V/L
D = εE = ψ / A
ψ = DA
V = - EL
Vg = - dφ / dt
C=Q/V
I=Q/T
J=I/A
R=V/I
ρ = RA / L
G=1/R
σ=1/ρ=J/E
P = D – ε0E = ρL
xe = P / E = ε0 (εr -1)
me = QL
ωe = DE / 2
2
A=L
v = L3
v=L/T
a = L / T2
F = Ma
W = FL
P=W/T
F = Q2 / (4πε0L2)
εr = ε / ε0
Dimensional
Formula
Exponents of
L M T
Q
1
2
3
0
0
1
1
1
2
2
0
cgs emu
No. of emu
No. of mks
cgs esu
No. of esu
No. of mks
No. of esu
No. of emu
cm
cm2
cm3
gram
second
cm / sec
cm / sec2
dyne
erg
erg / sec
abcoulomb
102
104
106
103
1
102
102
105
107
107
10-1
cm
cm2
cm3
gram
second
cm / sec
cm / sec2
dyne
erg
erg / sec
statcoulomb
102
104
106
103
1
10%
102
105
107
107
10c
1
1
1
1
1
1
1
1
1
1
100c
1
4πc2 / 107
0
0
0
1
0
0
0
1
1
1
0
0
0
0
0
1
-1
-2
-2
-2
-3
0
0
0
0
0
0
0
0
0
0
0
1
-3 -1
-3 -1
0 0
2
2
0
2
2
0
-3
-2
-1
1
-2
0
2
2
-2
0
-2
2
3
-2
-3
-2
-3
1
-1
0
0
0
-2
0
0
-2
-2
2
-1
-1
-1
-1
1
1
0
2
0
-2
1 abcoulomb / cm3
1 abcoulomb / cm2
1 abcoulomb / cm
-1
abvolt / cm
1
1
-1
abvolt
-1
abvolt
2
abfarad
1
abampere
1 abampere / cm2
-2
abohm
-2
abohm-cm
2
abmho
2
abmho / cm
1 abcoulomb / cm2
2
1
0
erg / cm3
0
0
0
1
0
0
1
1
-1
0
0
1
1
-1
-1
0
-1
0
1
1
10-7
10-5
10-3
106
4π / 105
4π / 10
106
108
10-9
10-1
10-5
109
1011
10-9
10-11
10-5
1
statcoulomb / cm3
statcoulomb / cm2
statcoulomb / cm
statvolt / cm
statvolt
statvolt
statfarad
statampere
statampere / cm2
statohm
statohm-cm
statmho
statmho / cm
statcoulomb / cm2
1
statcoulomb-cm
erg / cm3
c / 105
c / 103
c / 10
104 / c
4πc / 103
4π10c
106 / c
106 / c
c2 / 105
10c
c / 103
105 / c2
107 / c2
c2 / 105
c2 / 107
c / 103
4πc2 / 107
103c
10
100c
100c
100c
1 / (100c)
100c
100c
1 / (100c)
1 / (100c)
(100c)2
100c
100c
1 / (100c)2
1 / (100c)2
(100c)2
(100c)2
100c
1
Quantity
Symbol
mks
Unit
Rationalized
Defining Equation
7
Dimensional
Formula
Exponents of
L M
T
No. of emu
No. of mks
cgs esu
No. of esu
No. of mks
No. of esu
No. of emu
105 / c2
105 / c2
1 / (100c)2
1 / (100c)2
108 / c
103
10
1 / (100c)
1
1
Q
34 Permeability of free space
35 Permeability
36
relative
37 Magnetic pole
μ0
μ
μr
p
henry / m
henry / m
numeric
weber
μ0 = 4π / 10
μ=B/H
μr = μ / μ0
p = A (B - B0)
1
1
0
2
1
1
0
1
0 -2
0 -2
0 0
-1 -1
38 Magnetic moment
39 Magnetic intensity
m
H
m = pL
H = U / L or F / p
3
-1
1
0
-1 -1
-1 1
40 Magnetic flux density
B
weber-m
ampere / m or
newton / weber
weber / m2
B = μH = φ / A
0
1
-1 -1
41
42
43
44
φ
U
F
M
weber
ampere
ampere
weber / m2
φ = BA = VgT
U = F = HL
F=I
M = B - B0 = m / L3
2
0
0
0
1
0
0
1
-1 -1
-1 1
-1 1
-1 -1
L
M
R
v
P
μ
Vg
P
ωm
xm
henry
henry
ampere / weber
meter / henry
weber / amp
henry / meter
volt
watts / m2
joule / m3
henry / m
Magnetic flux
Magnetic potential
MMF
Intensity of magnetization
cgs emu
107 / 4π
pole
= maxwell / 4π
pole-cm
oersted or
gilbert / cm
gauss or
maxwell / cm2
maxwell
gilbert
gilbert
pole / cm2 or
gauss / 4π
1
103 / 4π
1010 / 4π
4π / 103
104
108
4π / 10
4π / 10
104 / 4π
Inductance
45
46
47
48
49
50
51
52
53
54
self
mutual
Reluctance
Reluctivity
Permeance
Permittivity
EMF
Poynting's vector
Magnetic energy density
Magnetic susceptibility
μ0 = 4π / 107 henrys / m. For c = 2.998 X 108 meters / sec,
For c ~
3 X 108 meters / sec,
c2 = 8.988 X 1016 ~ 9 X 1016
L= φ/I
M = φ / I = W / I2
R=F /φ
v=1/μ
P=1/R
μ=1/v
Vg = -dφ / dt
P = EH
ωm = HB / 2
xm = M / H
= μ0 (μr - 1)
2 1 0 -2
2 1 0 -2
-2 -1 0 2
-1 -1 0 2
2 1 0 -2
1 1 0 -2
2 1 -2 -1
0 1 -3 0
-1 1 -2 0
1 1 0 -2
abhenry
abhenry
109
109
abvolt
abwatt / cm2
erg / cm3
henry / m
108
103
10
107 / 4π
ε0 = 1 / μ0c2 = 107 / (4πc2) = 8.854 X 10-12 farad / meter
ε0 ~ 1 / (36π109) farad / meter
statvolt
statwatt / cm2
erg / cm3
ELECTRICAL OSCILLATIONS IN ANTENNAS AND
INDUCTANCE COILS
By John M. Miller
CONTENTS
I. Introduction
II. Circuit with uniformly distributed inductance and capacity
III. The Antenna
1. Reactance of the aerial-ground portion
2. Natural frequencies of oscillation
(a) Loading coil in lead-in
(b) Condenser in lead-in
3. Effective resistance, inductance, and capacity
4. Equivalent circuit with lumped constants
5. Determination of static capacity and inductance
6. Determination of effective resistance, inductance, and capacity
IV. The Inductance coil
1. Reactance of the coil
2. Natural frequencies of oscillation
Condenser across the terminals
3. Equivalent circuit with lumped constants
I. INTRODUCTION
A modern radiotelegraphic antenna generally consists of two portions, a vertical portion or
"lead-in" and a horizontal portion or "aerial." At the lower end of the lead-in, coils or condensers or
both are inserted to modify the natural frequency of the electrical oscillations in the system. When
oscillating, the current throughout the entire lead-in is nearly constant and the inductances,
capacities, and resistances in this portion may be considered as localized or lumped. In the
horizontal portion, however, both the strength of the current and the voltage to earth vary from point
to point and the distribution of current and voltage varies with the frequency. The inductance,
capacity, and resistance of this portion must therefore be considered as distributed throughout its
extent and its effective inductance, capacity, and resistance will depend upon the frequency. On this
account the mathematical treatment of the oscillations of an antenna is not as simple as that which
applies to ordinary circuits in which all of the inductances and capacities may be considered as
lumped.
The theory of circuits having uniformly distributed electrical characteristics such as cables,
telephone lines, and transmission lines has been applied to antennas. The results of this theory
do not seem to have been clearly brought out; hazy and sometimes erroneous ideas appear to be
current in the literature, textbooks, and in the radio world in general so that the methods of
antenna measurements are on a dubious footing. It is hoped that this paper may clear up some of
these points. No attempt has been made to show how accurately this theory applies to actual
antennas.
FIG 1. - Antenna represented as a line with uniform
distribution of inductance and capacity
The aerial-ground portion of the antenna, or aerial for short (CD in Fig. 1), will be treated as
a line with uniformly distributed inductance, capacity, and resistance. As is common in the
treatment of radio circuits the resistance will be considered to be so low as not to affect the
frequency of the oscillations or the distribution of current and voltage. The lead-in, BC in Fig. 1 ,
will be considered to be free from inductance or capacity excepting as inductance coils or
condensers are inserted at A to modify the oscillations.
An inductance coil, particularly if a long single-layer solenoid, may also be treated from the
standpoint of the transmission-line theory. The theoretical results obtained furnish an interesting
explanation of certain well-known experimental results.
II. CIRCUIT WITH UNIFORMLY DISTRIBUTED INDUCTANCE AND CAPACITY
The theory, generally applicable to all circuits with uniformly distributed inductance and
capacity, will be developed for the case of two parallel wires. The wires (Fig. 2) are of length l and
of low resistance. The inductance per unit length L1, is defined by the flux of magnetic force
between the wires per unit of length that there would be if a steady current of 1 ampere were
flowing in opposite directions in the two wires. The capacity per unit length C1 is defined by the
charge that there would be on a unit length of one of the wires if a constant emf of 1 volt were
impressed between the wires. Further, the quantity L0 = l L1 would be the total inductance of the
circuit if the current flow were the same at all parts. This would be the case if a constant or slowly
FIG. 2
alternating voltage were applied at x = o and the far end (x = l) short-circuited. The quantity C0 = l
C1 would represent the total capacity between the wires if a constant or slowly alternating voltage
were applied at x =o and the far end were open.
Let it be assumed, without defining the condition of the circuit at x = l, that a sinusoidal emf
of periodicity ω = 2πf is impressed at x = o giving rise to a current of instantaneous value i at A and
a voltage between A and D equal to v. At B the current will be i
C will be v
i
dx and the voltage from B to
x
v
dx .
x
The voltage around the rectangle ABCD will be equal to the rate of decrease of the induction
through the rectangle, hence
v
v

dx −v =−  L 1 i dx 
x
t
v
i
=−L1
x
t
(1)
Further, the rate of increase of the charge q on the elementary length of wire AB will be equal to the
excess in the current flowing in at A over that flowing out at B.
Hence
q d
i
= C v dx =i−i
dx 
 t dt 1
x
−
i
v
=C 1
x
t
(2)
These equations (1) and (2) determine the propagation of the current and voltage waves along the
wires. In the case of sinusoidal waves, the expressions
v=cos  t  A cos   C 1 L1 xB sin   C 1 L1 x (3)
i=sin  t

C1
 A sin   C 1 L1 x −B cos   C 1 L1 x (4)
L1
are solutions of the above equations as may be verified by substitution. The quantities A and B are
constants depending upon the terminal conditions. The velocity of propagation of the waves, at high
1
frequencies is V =
.
 C 1 L1
III. THE ANTENNA
1. REACTANCE OF THE AERIAL-GROUND PORTION
Applying equations (3) and (4) to the aerial of an antenna and assuming that x = 0 is the lead-in end
while x = l is the far end which is open, we may introduce the condition that the current is zero for
x = l. From (4)
A
=cot   C 1 L1 l (5)
B
Now the reactance of the aerial, which includes all of the antenna but the lead-in, is given by the
current and voltage at x = 0. These are. from (3), (4), and (5).
V 0 =A cos t=B cot  C 1 L1 l cos  t
i 0=−

C1
B sin  t
L1
The current leads the voltage when the cotangent is positive and lags when the cotangent is
negative. The reactance of the aerial, given by the ratio of the maximum values of v0 to i0, is
X =−

L1
cot   C 1 L1 l
C1
or in terms of C0 = lC1 and L0 = lL1
X =−

L0
cot  C 0 L 0 (6)
C0
or since
V=
1
 L1C1
X =−L1 V cot   C 1 L1 l as given by J. S. Stone.1
At low frequencies the reactance is negative and hence the aerial behaves as a capacity. At
1
the frequency f =
4  C 0 L0
FIG. 3 - Variation of the reactance of the aerial of an antenna with the frequency
1 J. S. Stone, Trans. Int. Congress, St. Louis, 8, p. 555; 1904.
the reactance becomes zero and beyond this frequency is positive or inductive up to the frequency
1
f=
at which the reactance becomes infinite. This variation of the aerial reactance with
2  C 0 L0
the frequency is shown by the cotangent curves in Fig. 3.
2. NATURAL FREQUENCIES OF OSCILLATION
Those frequencies at which the reactance of the aerial, as given by equation (6), becomes
equal to zero are the natural frequencies of oscillation of the antenna (or frequencies of resonance)
when the lead-in is of zero reactance. They are given in Fig. 3 by the points of intersection of the
cotangent curves with the axis of ordinates and by the equation
f=
m
; m=1, 3, 5, etc.
4  C 0 L0
The corresponding wave lengths are given by
=
V
l
4l
=
=
f f  C 0 L0 m
that is, 4/1, 4/3, 4/5, 4/7, etc., times the length of the aerial. If, however, the lead-in has a reactance
XX, the natural frequencies
FIG. 4 - Curves of aerial and loading coil reactance
of oscillation are determined by the condition that the total reactance of lead-in plus aerial shall be
zero; that is,
XX + X = 0
provided that the reactances are in series with the driving emf.
(a) Loading Coil in Lead-in. - The most important practical case is that in which an
inductance coil is inserted in the lead—in. If the coil has an inductance L, its reactance XL + ωL.
This is a positive reactance increasing linearly with the frequency and represented in Fig. 4 by a
solid line. Those frequencies at which the reactance of the coil is equal numerically but opposite in
sign to the reactance of the aerial, are the natural frequencies of oscillation of the loaded antenna
since the total reactance XL + X = 0. Graphically, these frequencies are determined by the
intersection of the straight line - XL = ωL (shown by a dash line in Fig. 4) with the cotangent curves
representing X. It is evident that the frequency is lowered by the insertion of the loading coil and
that the higher natural frequencies of oscillation are no longer integral multiples of the lowest
frequency.
The condition XL + X = 0, which determines the natural frequencies of oscillation, leads to
the equation
 L−
L
L0
  C 0 L0

L0
cot   C 0 L0=0 .
C0
TABLE 1 – Data for loaded antenna calculations
1
L
  C 0 L0
L 1 Difference,

per
cent
L
0
L0 3

0.0
.1
.2
.3
.4
1.571
1.429
1.314
1.220
1.142
1.732
1.519
1.369
1.257
1.168
10.3
6.3
4.2
3.0
2.3
.5
.6
.7
.8
.9
1.077
1.021
.973
.931
.894
1.095
1.035
.984
.939
.900
1.7
1.4
1.1
.9
.7
1.0
1.1
1.2
1.3
1.4
.860
.831
.804
.779
.757
.866
.835
.808
.782
.760
.7
.5
.5
.4
.4
1.5
1.6
1.7
1.8
1.9
.736
.717
.699
.683
.668
.739
.719
.701
.685
.669
.4
.3
.3
.3
.3
2.0
2.1
2.2
2.3
2.4
.653
.640
.627
.615
.604
.655
.641
.628
.616
.605
.3
.2
.2
.2
.2
2.5
2.6
2.7
2.8
2.9
.593
.583
.574
.564
.556
.594
.584
.574
.565
.556
.2
.2
.2
.1
.1
3.0
3.1
.547
.539
.548
.540
.1
.1

1
L 1

L0 3
Difference,
per cent
3.2
3.3
3.4
0.532
.524
.517
0.532
.525
.518
0.1
.1
.1
3.5
3.6
3.7
3.8
3.9
.510
.504
.4977
.4916
.4859
.511
.504
.4979
.4919
.4860
.1
.0
.0
.0
.0
4.0
4.5
5.0
5.5
6.0
.4801
.4548
.4330
.4141
.3974
.4804
.0
.4549
.0
.4330
.0
........................ ........................
........................ ........................
6.5
7.0
7.5
8.0
8.5
.3826
.3693
.3574
.3465
.3366
........................
........................
........................
........................
........................
........................
........................
........................
........................
........................
9.0
9.5
10.0
11.0
12.0
.3275
.3189
.3111
.2972
.2850
........................
........................
........................
........................
........................
........................
........................
........................
........................
........................
13.0
14.0
15.0
16.0
17.0
.2741
.2644
.2556
.2476
.2402
........................
........................
........................
........................
........................
........................
........................
........................
........................
........................
18.0
19.0
20.0
.2338
.2277
.2219
........................ ........................
........................ ........................
........................ ........................
or
cot   C 0 L0
  C 0 L0
=
L
(8)
L0
This equation has been given by Guyau2 and L. Cohen3. It determines the periodicity ω and hence
the frequency and wave length of the possible natural modes of oscillation when the distributed
capacity and inductance of the aerial and the inductance of the loading coil are known. This
equation can not, however, be solved directly; it may be solved graphically, as shown in Fig. 4, or a
table may be prepared indirectly which gives the values of   C 0 L0 for diferent values of
L
from which then ω, f, λ may be determined. The second column of Table 1 gives these values
L0
for the lowest natural frequency of oscillation, which is of the major importance practically.
FIG. 5 - Curves of aerial and series condenser reactance
(b) Condenser in Lead-in. - At times in practice a condenser is inserted in the lead-in. If the
1
capacity ofthe condenser is C, its reactance is X 0=−
. This reactance is shown in Fig. 5 by
C
the hyperbola drawn in solid line. The intersection of the negative of this curve (drawn in dash line)
with the cotangent curves representing X gives the frequencies for which X0 + X = 0, and hence
the natural frequencies of oscilltion of the antenna. The frequencies are increased (the wave length
decreased) by the insertion of the condenser and the oscillations of higher frequencies are not
integral multiples of the lowest.
The condition X0 + X = 0 is expressed by the equation
−
tan   C 0 L0
2 A. Guyau, Lumiere Electrique, 15, p. 13; 1911.
3 L. Cohen, Electrical World, 65, p. 286; 1915
 C 0 L 0
=
C
(9)
C0
which has also been given by Guyau. Equation (9) may be solved graphically as above or a table
C
similar to Table 1 may be prepared giving   C 0 L0 for different values of
. More
C0
complicated circuits may be solved in a similar manner.
3. EFFECTIVE RESISTANCE, INDUCTANCE, AND CAPACITY
In the following the most important practical case of a loading coil in the lead-in and the
natural oscillation of lowest frequency alone will be considered. The problem is to replace the
antenna
FIG. 6 - (a) Antenna with loading coil; (b) artificial antenna
with lumped constant: to represent antenna in (a)
of Fig. 6, (a), which has a loading coil L in the lead-in and an aerial with distributed characteristics,
by a circuit (Fig. 6, (b)) consisting of the inductance L in series with lumped resistance Re,
inductance Le, capacity Ce, which are equivalent to the aerial. It is necessary, however, to state how
these effective values are to be defined.
In practice the quantities which are of importance in an antenna are the resonant wave length
or frequency and the current at the current maximum. The quantities Le and Ce are therefore defined
as those which will give the circuit (b) the same resonant frequency as the antenna in (a). Further
the three quantities Le, Ce, and Re must be such that the current in (b) will be the same as the
maximum in the antenna for the same applied emf whether undamped or damped with any
decrement. These conditions determine Le, Ce, and Re uniquely at any given frequency and are the
proper values for an artihcial antenna which is to represent an actual antenna at a particular
frequency. In the two circuits the corresponding maxima of magnetic energies and electrostatic
energies and the dissipation of energy will be the same.
Zenneck4 has shown how these effective values of inductance capacity and resistance can be
computed when the current and voltage distributions are known. Thus, if at any point x on the
oscillator the current i and the voltage v are given by
i = I f (x); v = V φ (x)
where I is the value of the current at the current loop and V the maximum voltage, then the
differential equation of the oscillation is
4 Zenneck, Wireless Telegraphy (translated by A. E. Selig), Note 40, p. 430.
I
2
2
∫ C 1  x dx 
I
 I
2
R
f

x

dx

L f  x 2 dx
=0
∫
1
2∫ 1
t
t
∫ C 1  x 2 dx
where the integrals are taken over the whole oscillator. If we write
2
Re =∫ R1 f  x  dx (10)
L e=∫ L1 f  x2 dx (11)
C e=
∫ C 1  x dx 
∫ C 1  x2 dx
2
(12)
the equation becomes
2
Re
I
 I I
 Le 2  =0
t
Ce
t
which is the diierential equation of oscillation of a simple circuit with lumped resistance,
inductance, and capacity of values Re, Le, and Ce and in which the current is the same as the
maximum in the distributed case. In order to evaluate these quantities, it is necessary only to
determine f (x) and phi (x); that is, the functions which specify the distribution of current and
voltage on the oscillator. In this connection it will be assumed that the resistance is not of
importance in determining these distributions.
At the far end of the aerial the current is zero; that is, for x = l; il = 0. From equations (3)
and (4) for x = l
v l =cos  t Acos   C 1 L 1 lB sin   C 1 L1 l ,
i l =sin t

C1
 Asin   C 1 L1 l−B cos   C 1 L1 l ;
L1
and since il = 0,
A sin   C 1 L1 l=B cos   C 1 L 1 l .
From (3), then, we obtain
v =v l cos   C 1 L1 l−  C 1 L1 x .
Hence
 x=cos  C 1 L1 l−  C 1 L1 x  .
Now for x = 0 from (4) we obtain
i 0=−B


C1
C1
sin t=−A
tan   C 1 L1 l sin t ,
L1
L1
whence
i=i 0
sin  C 1 L1 l−  C 1 L1 x
sin  C 1 L1 l
and
f  x =
sin   C 1 L1 l−  C 1 L1 x 
sin   C 1 L1 l
.
We can now evaluate the expressions (10), (11), and (12). From (10)
sin 3   C 1 L1 l−  C 1 L1 x dx
Re =∫0 R1
sin2   C 1 L1 l
l
l sin 2   C 1 L1 l
[ −
]
sin   C 1 L 1 l 2
4   C 1 L1
R1
=
=
2
R0
cot   C 0 L0
1
[ 2
−
] (13)
2 sin   C 0 L0
  C 0 L0
and from (11) which contains the same form of integral
L e=
L0
cot   C 0 L0
1
[ 2
−
] (14)
2 sin   C 0 L0
  C 0 L0
and from (12)
l
C e=
2
∫0 C 1 cos   C 1 L1 l−  C 1 L1 x dx 
l
∫0 C 1 cos2  C 1 L1 l−  C 1 L1 x dx
C 21
sin 2   C 1 L1 l
2
  C 1 L 1
=
l sin 2 C 1 L1 l
C 1 

2
4   C 1 L1
=
C0
1
2
  C 0 L0 cot   C 0 L0
 C 0 L0
(15)
[

]
2
2
2 sin   C 0 L0
The expressions (14) and (15) should lead to the same value for the reactance X of the aerial
as obtained before. It is readily shown that
X = Le −

L
1
=− 0 cot   C 0 L 0
C e
C0
agreeing with equation (6).
It is of interest to investigate the values of these quantities at very low frequencies (omega =
0), frequently called the static values, and those corresponding to the natural frequency of the
unloaded anterma or the so-called fundamental of the antenna. Substituting w=o in (13), (14), and
(15), and evaluating the indeterminant which enters in the iirst two cases, we obtain for the lowfrequency values
Re =
L e=
R0
3
L0
(16)
3
C e =C 0
At low frequencies the current is a maximum at the lead-in end of the aerial and falls off linearly to
zero at the far end. The effective resistance and inductance are one-third of the values which would
obtain if the current were the same throughout. The voltage is, however, the same at all points and
hence the effective capacity is the capacity per unit length times the length, or C0.
At the fundamental of the anterma, the reactance X of equation (6) becomes equal to zero

and hence   C 0 L0=
. Substituting this value in (13), (14), and (15),
2
Re =
R0
2
L e=
L0
(17)
2
C e=
8
C0 .
2
Hence, in going from low frequencies up to that of the fundamental of the antenna, the resistance
(neglecting radiation and skin effect) and the inductance (neglecting skin effect) increase by 50 per
2
L and
cent, the capacity, however, decreases by about 20 per cent. The incorrect values
 0
2
C have been frequently given and commonly used as the values of the effective inductance
 0
L
and capacity of the antenna at its fundamental. These lead also to the incorrect value L e= 0 for
2
the low-frequency inductance.5
5 These values are given by J.H. Morecroft in Pro. I.R.E., 5, p. 389; 1917. It may be shown that they lead to correct
values for the reactance of the aerial and hence to correct values of frequency, as was verified by the experiments.
They are not, however, the values which would be correct for an artificial antenna in which the current must be
equal to the maximum in the actual antenna and in which the energies must also be equal to those in the antenna.
The resistance values given by Prof. Morecroft agree with these requirements and with the values obtained here.
The values for other frequencies may be obtained by substitution in (13), (14), (15). If the
value L of the loading coil in the lead-in is given, the quantity   C 0 L0 is directly obtained from
Table 1.
4. EQUIVALENT CIRCUIT WITH LUMPED CONSTANTS
In so far as the frequency or wave length is concerned, the aerial of the antenna may be
considered to have constant values of inductance and capacity and the values of frequency or wave
length for different loading coils can be computed with slight error using the simple formula
applicable to circuits with lumped inductance and capacity. The values of inductance and capacity
L
ascribed to the aerial are the static or low frequency; that is, 0 for the inductance and C0, for the
3
L
capacity. The total inductance in case the loading coil has a value L will be L 0 and the
3
frequency is given by
f=
1
2   L

(18)
L0
C 0
3

L0
C 0 (19)
3
or the wave length in meters by
=1884  L
where the inductance is expressed in microhenrys and the capacity in microfarads. The accuracy
with which this formula gives the wave length can be determined by comparison with the exact
formula (8). In the second column of Table 1 are given the values of   C 0 L0 for different
L
values of
as computed by formula (8). Formula (18) may be written in the form
L0
1
  C 0 L0 =
L 1 so that the values of   C 0 L0 which are proportional to the frequency,

L0 3
may readily be computed from this formula also. These values are given in the third column and the
per cent differences in the fourth column of Table 1. It is seen that formula (18) gives values for the
frequency which are correct to less than 1 per cent excepting when very close to the fundamental of
the antenna; that is, for very small values of L. Under these conditions the simple formula leads to
values of the frequency which are too high. Hence to the degree of accuracy shown, which is amply
L
sufficient in most practical cases, the aerial can be represented by its static inductance 0 with its
3
static capacity C0, in series, and the frequency of oscillation with a loading coil L in the lead-in can
be computed by the ordinary formula applicable to circuits with lumped constants.

In an article by L. Cohen,6 which has been copied in several other publications, it was stated
that the use of the simple wave length formula would lead to very large errors when applied to the
antenna with distributed constants. The large errors found by Cohen are due to his having used the
L
value L0, for the inductance of the aerial, instead of 0 , in applying the simple formula.
3
6 See footnote 3.
5. DETERMINATION OF STATIC CAPACITY AND INDUCTANCE
In applying formula (8) to calculate the frequency of a loaded antenna, a knowledge of the
L
quantities of L0, and C0 is required. In applying formula (18), 0 and C0, are required. Hence
3
either formula. may be used if the static capacity and inductance values are known. We will call
L
these values simply the capacity Ca and inductance La of the antenna. Hence C a =C 0 , L a= 0
3
and the wave length from (19) is given by
=1884   LL a C a (20)
where inductance is expressed in microhenrys and capacity in microfarads, as before.
The capacity and inductance of the antenna are then readily determined experimentally by
the familiar method of inserting, one after the other, two loading coils of known values L1 and L2
in the lead-in and determining the frequency of oscillation or wave length for each. From the
observed wave lengths λ1 and λ2 and known values of the inserted inductances, the inductance of the
antenna is given by
L a=
L1 22−L 2 12
21− 22
(21)
and the capacity of the antenna from either
1=1884  L 1L a C a
2=1884   L2 La C a (22)
using, preferably, the equation corresponding to the larger valued coil. This assumes that formula
(20) holds exactly.
As an example, let us assume that the antenna has L0 = 50 gomicrohenrys and C0 = 0.001
microfarad and that we insert two coils of 50 and 150 microhenrys and determine the wave lengths,
experimentally. We know from formula (8) and Table 1 that the wave lengths would be found to be
491 and 771 m. From the observed wave lengths and known inductances, the value of La would be
found by (21) to be
La = 17.8 microhenrys
and from (22)
Ca = 0.000999 microfarad.
L0
. This accuracy
3
would ordinarily be sufficient. We can, however, by a second approximation, derive from the
experimental data a more accurate value of La. For, the observed value of La furnishes rough values
L1
L2
of
and
, which in this example come out 0.96 and 2.88, respectively. But Table 1 gives
L0
L0
Ca is very close to the assumed value C0 but La differs by 7 per cent from
L
and shows that this formula gives a 0.7
L0
L
=0.96 but no appreciable difference
per cent shorter wave length than 491 m (or 488 m) for
L0
L
=2.88 . Recomputing La, using 488 and 771 m, gives
for
L0
the per cent error of formula (20) for diierent values of
La = 0.0168
which is practically identical with the assumed
L0
.
3
6. DETERMINATION OF EFFECTIVE RESISTANCE, INDUCTANCE, AND CAPACITY
When a source of undamped oscillations in a primary circuit induces current in a secondary
tuned circuit, the current in the secondary, for a given emf, depends only upon the resistance of the
secondary circuit. When damped oscillations are supplied by the source in the primary, the current
in the secondary, for a given emf and primary decrement, depends upon the decrement of the
secondary; that is, upon the resistance and ratio of capacity to inductance. The higher the decrement
of the primary circuit relative to the decrement of the secondary the more strongly does the crurent
in the secondary depend upon its own decrement. This is evident from the expression for the current
I in the secondary circuit.
P=
N E 20
4f R2 '  I 
'


where δ' is the decrement of the primary, δ that of the secondary, R the resistance of the secondary, f
the frequency, E0 the maximum value of the emf impressed on the primary, and N the wave-train
frequency.
These facts suggest a method of determining the effective resistance, inductance, and
capacity of an antenna at a given frequency in which all of the measurements are made at one
frequency and which does not require any alteration of the antenna circuit whatsoever. The
experimental circuits are arranged as shown in Fig. 7, where S represents a coil in the primary
circuit which may be thrown either into the circuit of a source of undamped or of damped
oscillations. The coil L is the loading coil of the antenna, which may be thrown over to the
measuring circuit containing a variable inductance L', a variable condenser C', and variable
resistance R'. The condenser C' should be resistance free and shielded, the shielded terminal being
connected to the ground side. First, the undamped source is tuned to the antenna, and then the L'C'
circuit tuned to the source. The resistance R' is then varied until the current is the same in the two
positions. The resistance of the L'C' circuit is then equal to Re, the eifective resistance of the aerialground portion of the antenna and L'C' = LeCe. Next, the damped source is tuned to the antenna and
the change in current noted when the connection is thrown over to the L'C' circuit. If the current
increases, the value of C' is greater than Ce, and vice versa. By varying both L' and C',
FIG. 7 - Circuits for determining the effective resistance, inductance,
and capacity of an antenna
keeping the tuning and R' unchanged, the current can be adjusted to the same value in both
C ' Ce
=
positions. Then, since L'C' = LeCe and
, the value of C' gives Ce and that of L' gives Le.
L' Le
Large changes in the variometer setting may result in appreciable changes in its resistance so that
the measurement should be repeated after the approximate values have been found. To eliminate the
resistance of the variometer in determining Re, the variometer is shorted and, using undamped
oscillations, the resonance current is adjusted to equality in the two positions by varying R'. Then R'
= Re. The measurement requires steady sources of feebly damped and strongly damped current. The
former is readily obtained by using a vacuum-tube generator. A resonance transformer and
magnesium spark gap operating at a low-spark frequency serve very satisfactorily for the latter
source, or a single source of which the damping can be varied will suffice. An accuracy of 1 per
cent is not difficult to obtain.
IV. THE INDUCTANCE COIL
The transmission-line theory can also be applied to the treatment of the effects of distributed
capacity in inductance coils. In Fig. 8, (a) , is represented a single-layer solenoid connected to a
variable condenser C. A and B are the terminals of the coil, D the middle, and the condensers drawn
in dotted lines are supposed to represent the capacities between the different parts of the coil. In Fig.
8, (b), the same coil is represented as a line with uniformly distributed inductance and capacity.
FIG. 8 - Inductance coil represented as a line with uniform
distribution of inductonce and capacity
These assumptions are admittedly rough but are somewhat justified by the known similarity of the
oscillations in long solenoids to those in a simple antenna.
1. REACTANCE OF THE COIL
Using the same notation as before, an expression for the reactance of the coil, regarded from
the terminals AB (x = 0) will be determined considering the line as closed at the far end D (x = l).
Equations (3) and (4) will again be applied, taking account of the new terminal condition: that is,
for x = l; v = 0. Hence
A cos   C 1 L1 l=−B sin   C 1 L1 l
and for x = 0
v 0 =A cos t =−B tan   C 1 L1 cos t
i 0=−

C1
B sin  t
L1
which gives for the reactance of the coil regarded from the terminals A B,
X '=

L1
tan   C 1 L1 l
C1
or
X '=

L0
tan   C 0 L0 (23)
C0
2. NATURAL FREQUENCIES OF OSCILLATION
At low frequencies the reactance of the coil is very small and positive but increases with

increasing frequency and becomes infinite when   C 0 L0=
. This represents the lowest
2
frequency of natural oscillation of the coil when the terminals are open. Above this frequency the
reactance is highly negative, approaching zero at the frequency   C 0 L0= . In this range of
frequencies the coil behaves as a condenser and would require an inductance across the terminals to
form a resonant circuit. At the frequency   C 0 L0= the coil will oscillate with its terminals
short-circuited. As the frequency is still further increased the reactance again becomes increasingly
positive.
Condenser Across the Terminals. - The natural frequencies of oscillation of the coil when
connected to a condenser C are given by the condition that the total reactance of the circuit shall be
zero.
X' + X0 = 0
From this we have

L0
1
tan   C 0 L0 =
C0
C
or
cot   C 0 L0
  C 0 L0
=
C
(24)
C0
This expression is the same as (8) obtained in the case of the loaded antenna, excepting that
C
C0
L
and shows that the frequency is decreased and wave
L0
length increased by increasing the capacity across the coil in a manner entirely similar to the
decrease in frequency produced by inserting loading coils in the antenna lead-in.
occurs on the right-hand side instead of
3. EQUIVALENT CIRCUIT WITH LUMPED CONSTANTS
It is of interest to investigate the effective values of inductance and capacity of the coil at
very low frequencies.
Expanding the tangent in equation (23) into a series we find
X '= L0 1
2 C 0 L 0
......
3
and neglecting higher-power terms this may be written
3

 C0
X '=
3
 L 0−
C 0
 L0 −
C0
, which shows that at low
3
C0
frequencies the coil may be regarded as an inductance L0, with a capacity
across the terminals
3
and therefore in parallel with the external condenser C. Since at low frequencies the current is
uniform throughout the whole coil, it is self-evident that its inductance should be L0.
This is the reactance of an inductance L0 in parallel with a capacity
Now, the similarity between equations (24) and (8) shows that, just as accurately as in the
similar case of the loaded antenna, the frequency of oscillation of a coil with any capacity C across
the terminals is given by the formula
f=
1

2  L 0 C
C0

3
This, however, is also the expression for the frequency of a coil of pure inductance L0 with a
C0
capacity
across its terminals and which is in parallel with an external capacity C. Therefore,
3
in so far as frequency relations are concerned, an inductance coil with distributed capacity is closely
equivalent at any frequency to a pure inductance, equal to the low-frequency inductance (neglecting
skin effect), with a constant capacity across its terminals. This is a well-known result of
experiment7, at least in the case of single-layer solenoids which, considering the changes in current
and voltage distribution in the coil with changing frequency, is not otherwise self-evident.
WASHINGTON, March 9, 1918.
7 G.W.O. Howe, Proc. Phys. Soc. London, 21 p. 251, 1912; F.A. Kolster, Proc. Inst. Radio Eng., 1, p. 19, 1913; J.C.
Hubbard, Phys. Rev., 9, p. 529, 1917.
FURTHER DISCUSSION* ON
"ELECTRICAL OSCILLATIONS IN ANTENNAS AND
INDUCTION COILS" BY JOHN M. MILLER
By
John H. Morecroft
I was glad to see an article by Dr. Miller on the subject of oscillations in coils and antennas
because of my own interest in the subject, and also because of the able manner in which Dr. Miller
handles material of this kind. The paper is well worth studying.
I was somewhat startled, however, to find out from the author that I was in error in some of
the material presented in my paper in the Proceedings of The Institute of Radio Engineers for
December, 1917, especially as I had at the time I wrote my paper thought along similar lines as does
Dr. Miller in his treatment of the subject: this is shown by my treatment of the antenna resistance.
As to what the effective inductance and capacity of an antenna are when it is oscillating in
its fundamental mode is, it seems to me, a matter of viewpoint. Dr. Miller concedes that my
treatment leads to correct predictions of the behavior of the antenna and I concede the same to him;
it is a question, therefore, as to which treatment is the more logical.
From the author’s deductions we must conclude that at quarter wave length oscillations
L e=
L0
(1)
2
and
C e=
8
C 0 (2)
2

The value of L really comes from a consideration of the magnetic energy in the antenna
keeping the current in the artificial antenna the same as the maximum value it had in the actual
antenna, and then selecting the capacity of suitable value to give the artificial antenna the same
natural period as the actual antenna. This method of procedure will, as the author states, give an
artificial antenna having the·same natural frequency, magnetic energy, and electrostatic energy, as
the actual antenna, keeping the current in the artificial antenna the same as the maximum current in
the actual antenna.
But suppose he had attacked the problem from the viewpoint of electrostatic energy instead
of electromagnetic energy, and that he had obtained thc constants of his artificial antenna to satisfy
these conditions (which are just as fundamental and reasonable as those he did satisfy); same
natural frequency, same magnetic energy, same electrostatic energy and the same voltage across the
condenser of the artificial as the maximum voltage in the actual antenna. He would then have
obtained the relations
L e=
and
*Received by the Editor, June 26, 1919.
8
L 0 (3)
2
C e=
C0
2
(4)
Now equations (3) and (4) are just as correct as are (1) and (2) and moreover the artificial
antenna built with the constants given in (3) and (4) would duplicate the actual antenna just as well
as the one built according to the relations given in (1) and (2).
I had these two possibilities in mind when writing in my original article "as the electrostatic
energy is a function of the potential curve and the magnetic energy is the same function of the
current curve, and both these curves have the same shape, it is logical, and so on." Needless to say, I
still consider it logical, and after reading this discussion I am sure Dr. Miller will see my reasons for
so thinking.
When applying the theory of uniform lines to coils I think a very large error is made at once,
which vitiates very largely any conclusions reached. The L and C of the coil, per centimeter length,
are by no means uniform, a necessary condition in the theory of uniform lines; in a long solenoid
the L per centimeter near the center of the coil is nearly twice as great as the L per centimeter at the
ends, a fact which follows from elementary theory, and one which has been verified in our
laboratory by measuring the wave length of a high frequency wave traveling along such a solenoid.
The wave length is much shorter in the center of the coil than it is near the ends. What the capacity
per centimeter of a solenoid is has never been measured, I think, but it is undoubtedly greater in the
center of the coil than near the ends.
The conclusions he reaches from his equation (22) that even at its natural frequency the L of
the coil may be regarded as equal to the low frequency value of L is valuable in so far as it enables
one better to predict the behavior of the coil, but it should be kept in mind that really the value of L
of the coil, when defined as does the author in the first part of his paper in terms of magnetic energy
and maximum current in the coil at the high frequency, is very much less than it is at the low
frequency.
One point on which I differ very materially with the author is the question of the reactance
of a coil and condenser, connected in parallel, and excited by a frequency the same as the natural
frequency of the circuit. The author gives the reactance as infinity at this frequency, whereas it is
actually zero. When the impressed frequency is slightly higher than resonant frequency there is a
high capacitive reaction and at a frequency slightly lower than resonant frequency there is a high
inductive reaction, but at the resonant frequency the reactance of the circuit is zero. The resistance
of the circuit becomes infinite at this frequency, if the coil and condenser have no resistance, but for
any value of coil resistance, the reactance of the combination is zero at resonant frequency.
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