GIFT OF

^^*"*^

OSCILLATING -CURRENT
CIRCUITS

AN EXTENSION OF THE THEORY OF GENERALIZED

ANGULAR VELOCITIES, WITH APPLICATIONS TO

THE COUPLED CIRCUIT AND THE ARTIFICIAL

TRANSMISSION LINE

V. BUSH

ABSTRACT

OF
A THESIS

SUBMITTED TO THE FACULTY OF THE
MASSACHUSETTS INSTITUTE OF TECH-
NOLOGY IN PART FULFILMENT OF
THE REQUIREMENTS FOR THE DEGREE
OF DOCTOR OF ENGINEERING

JUNE, 1916

OSCILLATING-CURRENT
CIRCUITS

AN EXTENSION OF THE THEORY OF GENERALIZED

ANGULAR VELOCITIES, WITH APPLICATIONS TO

THE COUPLED CIRCUIT AND THE ARTIFICIAL

TRANSMISSION LINE

V. BUSH

ABSTRACT

A THESIS

SUBMITTED TO THE FACULTY OF THE
MASSACHUSETTS INSTITUTE OF TECH-
NOLOGY IN PART FULFILMENT OF
THE REQUIREMENTS FOR THE DEGREE
OF DOCTOR OF ENGINEERING

JUNE, 1916

CONTENTS.

PAGE

LIST OF SYMBOLS EMPLOYED . . 4

INTRODUCTION 5

THE COUPLED CIRCUIT 8

APPLICATION TO THE ARTIFICIAL LINE 1 1

SUGGESTIONS FOR A CONTINUATION OF THE WORK 14

SUMMARY 14

333554

LIST OF SYMBOLS EMPLOYED IN THESIS.

i The instantaneous oscillating current in a branch of a network am-
peres

n A generalized angular velocity of oscillation hyperbolic radians per
second Z

Z Generalized impedance ohms Z

E Initial potential volts

Naperian base 2.718 ....

twi 2 Roots of the equation Z =o, hyperbolic radians per second Z

C Total capacitance farads

L Total self inductance henries

M Total mutual inductance henries

R Total resistance ohms

Constants of the primary and secondary of a coupled circuit are dis-
tinguished by subscripts

a, /3, 7, 5, 77 Coefficients of the equation Z = o for the coupled circuit

q Correction to be applied to the absolute values of the free angular
velocities of a resistanceless coupled circuit to obtain the absolute
values of the angular velocities of the complete circuit. numeric

p A correction to be added and subtracted to - to obtain the decrements

4
of the complete coupled circuit. hyp. rad. per sec.

s, t Sum and difference respectively of the squares of the angular velocities
of the resistanceless coupled circuit.

/hyp. rarl \2

>. rad.V
sec. /

\ sec

j The pure imaginary, V i

A A generalized amplitude of current oscillation amperes Z
m Number of sections of an artificial line
h Auxiliary constant numeric

.Z This sign appended to the units of an equation denotes that the expres-
sion contains, in general, complex quantities

OSCILLATING-CURRENT CIRCUITS.

INTRODUCTION.

Heaviside,* and since then several others,! have shown that for the free
oscillations of a network the generalized impedance, formed from the con-
stants of the network and the complex angular velocity of oscillation, is
zero for any complete circuit. This principle enables the frequencies and
decrements of the free oscillations of a network to be readily found. There
is a similar principle which enables the finding of the amplitudes of free
oscillation at the several frequencies, which is also in Heaviside, derived
from a series of theorems concerning the distribution of energy during
subsidence. It is the purpose of the thesis, of which this is an abstract, to
demonstrate the application of this second principle to practical engineering
problems.

The principle may be stated as follows: If Z is the generalized impedance
of a branch of the network initially containing a store of energy, corre-
sponding to the initial voltage E, and if n is the complex angular velocity
of oscillation, so that Z= /(), then the first order term in the Taylor ex-

j rj

pansion of Z, namely, n , will be of the nature of an impulsive impedance ;

d n

and the oscillatory current will be of the form :

E nt ,

amperes Z

--^ dz

n
dn

where the summation extends over the roots i, n^, - - of the equation Z =o.

It will be convenient to call the expression n the "threshold im-

d n
pedance."*

The equation, as given, applies to the current in the branch initially
charged, where the generalized and threshold impedances are formed for
that branch.

The discussion of the application of this principle to various typical net-
works has indicated the truth of the following additional propositions
which will be found useful in attacking particular problems:

* Heaviside. Electrical Papers, Electromagnetic Theory, Vol. II.
t Campbell, Proc. AIEE, 1911;

Kennelly, Proc. IRE, 1915;

Eccles & Makower, Electrician, 1915.

(1) In determining the amplitude of oscillation at some point of the net-
work distant from the branch initially charged, the generalized impedances
of the elements combine in the manner of simple resistances. Upon com-
bining with the generalized impedance of an element, each term of a cur-
rent or voltage expression is combined with the generalized impedance of
the element formed for the free angular velocity of the term considered.

(2) When several stores of energy are simultaneously discharged they
may be considered separately and the results added.

(3) In order to ensure that the correct free angular velocities be ob-
tained, the generalized impedance should be formed for the branch under
examination; as in special cases certain free angular velocities may be
absent in particular branches of the network.

(4) The threshold impedance is formed always from the generalized
impedance which considers the initially charged element as the main
branch.

(5) The sudden application of a steady electromotive force may be
treated as the inverse of the discharge from the final state attained.

(6) The sudden application of an alternating electromotive force may be
treated in similar manner, the unbalanced stores of energy being in this
case the differences between the initial stores of energy in the network, and
the energies at the same points of the network corresponding in the steady
state to the point of the voltage wave at which it was suddenly applied.

The method of applying the threshold impedance is shown by various
examples. One of these, the series circuit containing resistance, inductance,
and capacitance is included here for illustration.

Resistance
f\ o/ims.

Inductance
L_ henries.

C farads.

P/g. / . 5imf)le Series Osci/latin<j Circuit.

In this circuit (see fig. i) the generalized impedance will be:

Z = R+Ln+ ohms Z

Equaling to zero and solving for n, we obtain the free angular velocities:

_ R . // R\* r

71% % / I - I -

*L V \ 2 L/ LC

hyp- ra d.

sec.

The threshold impedance is:

n =Ln ohms Z

dn Cn

If now we consider the condenser as discharging through the circuit from
an initial voltage E, the current will be:

n=nz E ni ,

i= ^ e amperes Z

n^x n ^
H dn

E nit i E n%t /

t = e -j- e amperes Z.

i _ i

which, with the values of n\ and n* given above, is the complete oscillatory
solution. This expression may be reduced to the usual form by inserting
the values of i and n^. There will, of course, be three cases according as
the quantity under the radical is positive, zero, or negative. For the third
case the expression becomes upon reducing:

amperes
LC \2L'

which is the solution obtained by the usual methods. The solutions for
the other cases may be obtained by similar reductions.

It will be noted that this method of solving the circuit is much more con-
cise and direct than is the method of determining the constants of integra-
tion in the differential equation solution, in accordance with the boundary
conditions. It is also convenient to retain all three cases in the single
expression.

If we wish the oscillatory voltage across, for instance, the reactor in this
circuit, we may obtain it by multiplying the oscillatory current by the
generalized impedance of the reactor, and treat the current terms separately,
thus:

nit , ELnz n 2 t ,

- e + volts Z

i i

Cn\ Cn 2

and this expression may also be reduced by inserting the values of n\ and n^.

THE COUPLED CIRCUIT.

The coupled circuit has been thoroughly solved by the method of differ-
ential equations.* These solutions have been discussed from the point of
view of the applications of this circuit, particularly to radio work. Many
approximate solutions have been obtained for the case of the free discharge
of the primary condenser, either by neglecting the effects of resistance, or
the reaction of the secondary upon the primary, or in some similar way.
The complete exact solution has been generally avoided, principally because
of its complication. The resistance operator method, or the method of
generalized angular velocities, has also been applied to this circuit in as far
as the frequencies and decrements are concerned.! This method gives the
same equation for the determination of the free angular velocities as does
the differential equation solution, namely:

numerc

where n is the complex angular velocity, and the constants are those shown
on fig. 2.

Fi<j.i. JndvctiVely Counted Circuit.

* Bjerkness, Wied. Ann. 55, 1895; Oberbeck, Wied. Ann. 55, 1895; Domalys & Kolacek,
Wied. Ann. 57, 1896; Wien, Wied. Ann. 61, 1897; Rayleigh, Theory of Sound; Braun, Phys.
Za. 3, 1901; Drude, Ann. d. Phys. 13, 1904; Jones, Phil. Mag. 1907; Cohen, Bui. Bu. Stds.
S, 1909; Pierce, Proc. Am. Ac. A. & Sc. 46, 1911; Fleming, Proc. Phys. Soc. 1913-

t Eccles, Phys. Soc. Proc. 24, 1912. Kennelly, Proc. IRE, 1915.

The solution of this fourth degree equation is laborious, and may be
avoided in the following manner. The equation may be written in the form :

/hyp. rad.X 4 ,

and if we treat the same circuit without resistance we obtain the easily
solved equation:

/hyp- rad.V

I -^- - )

\ sec. /

The roots of this last equation will differ but little in absolute value from
the absolute values of the roots of the complete equation. If (i -\-q) and
(i q) are the correction factors to be applied to the absolute values of the
resistanceless roots in order to obtain the absolute values of the complete
roots, we may find an expression for q by means of the algebraic relations
between the roots and coefficients of the above equations; and in deriving
this relation the square of q may be neglected. This expression is:

P-yt-ctS

- - - numeric

where s is the sum and / the difference of the squares of the roots of the
resistanceless equation.

In a similar manner the relation may be derived:

27 as+aqt
4t8qs

where ( -+p } and ( - p } are the decrements in the solution of the com-
\4 / \4 /

plete circuit. In this manner the frequencies and decrements of the oscil-
lations in the coupled circuit may be obtained without the necessity of
solving the fourth degree equation.

This method, tested on a typical circuit with constants:

Ci = io~ 9 farads
C 2 = io- 10 farads
RI =1000 ohms
RZ =2000 ohms
LI =0.025 henries
L 2 =0.040 henries
M =0.020 henries

gave by exact solution:

17638.9=117192683.

and by the approximate method :

- 57361. o; 664750.
-17639.0^192680.

The amplitudes of oscillation may be readily found by the use of the
threshold impedance. If we consider the discharge of the primary con-
denser, so that the primary is the main branch, the threshold impedance is:

ohms Z

Inserting into this expression the four roots MI, w 2 , w 3 , w 4 of the equation Z = o,
gives four particular values of the threshold impedance. Dividing the
initial primary condenser voltage by each of these values gives the four
complex amplitudes for the primary current expression:

t nit , n%t . . n z t . . w 4 / /

^l = Al +A 2 e +A 3 e + A t e , amperes L

This expression may be readily reduced to trigonometric form, when the
imaginary portions of the expression cancel out.

An examination of the generalized impedance of the several elements of
the circuit gives for the ratio between the primary and secondary ampli-
tudes: v

Mn . /

- numeric Z.

The four values of this ratio applied to the primary amplitudes give the
corresponding secondary amplitudes*

The results above were checked by means of oscillograms taken upon
a typical coupled .circuit. The constants chosen

^i =I -937 ohms

Rz =2.531 ohms

LI =7.52 x io~ 3 henries

Lt =7.63 x i o~ 3 henries

M =3475 x io~ 3 henries

C\ =13.51 microfarads

2 =24.62 microfarads

gave frequencies of oscillation 609.5 an d 339- 2 which were within con-
venient range for the oscillograph. The computed points checked the os-
cillograms within the errors of measurement. A solution was also made by
differential equations as a check.

APPLICATION TO THE ARTIFICIAL LINE.

There has been much difficulty encountered in the analysis of smooth line
transients. The -cable has been comparatively easily handled,* but the
analysis of the aerial line has given in general results too complicated for
engineering use. For experimental analysis for steady state phenomena
the lumped artificial line has proved invaluable,! but there has been much
doubt as to just how far such a line of a given number of sections could be
trusted for transient effects, t

The method of generalized angular velocities is applied in the thesis to
the analysis of the oscillations of the artificial line under certain typical con-
ditions. The distant-end current on a grounded artificial line, when a
steady voltage is suddenly applied at the home end, is considered for the
artificial cable, and the artificial aerial line. The IT line is used, but the
formulas apply also to the T line with small changes.

The purpose of this analysis is to determine, for specific cases, the number
of sections requisite in an artificial line, in order that it may represent its
corresponding smooth line, not only for the steady state, but also for certain
transient effects, to a sufficient degree of approximation for engineering
investigations.

The method used is simply to analyze artificial lines with various num-
bers of sections, considered simply as networks with concentrated constants.
The results of these successive solutions are grouped, and from them is
derived the solution for the general case of m sections.

The general solutions for the application of a steady electromotive force
to the grounded artificial line obtained in this way follow:

For the cable of m sections containing resistance and capacitance only:

[ m*hit m*fe* "~|

<* I+ !^=l ( RC -^Z2e RC + . . . amperes

RL 22 J

where

*# is the received current
E the applied steady voltage
R the total resistance
Cthe total capacitance.

* Kelvin, Proc. Roy. Soc. 1855;

Poincar6, EC. Elect. 40, 1904;

Malcolm, Electrician 1911; 12.
fPupin, Trans. AIEE 1890, 1900; Trans. Am. Math. Soc. 1900;

Kennelly, Proc. Am. Acad. Arts & Sci., 44, 1908 ;
. Huxley. Thesis M. I. T., 1914.
J Cunningham and Davis, Proc. AIEE 1911, 1912;

Ricker. Thesis M. I. T. 1915.

The values of h are found as roots of the auxiliary equation:

2 !

numeric

where there are terms if m is even, and terms if m is odd.

A curve for obtaining these roots is presented for convenient use in prac-
tical cases. ,

For the aerial line containing resistance, capacitance and inductance,
the corresponding equation is:

<-![-

hm2 ^

2L 2L

J m'fe ( R\*

V ~7C " (71 ^ + ' ' J

LC \* ^/ amperes

As the line is subdivided an oscillatory term appears in this equation for
each section introduced.

This aerial line formula was checked by means of oscillograms taken
upon a typical artificial aerial power transmission line at Pierce Hall,
Harvard University.* Twelve sections were used, representing a line of
the following constants:

#000 A.W.G. aluminum stranded conductors

Overstrand diameter 0.47 inches

Interaxial distance 90.5 inches

Length 596.4 miles.

Total inductance 1.035 henries

Total capacitance 8.ioXio~ 6 farads

Total resistance 300. 1 ohms

The elements of this artificial line were grouped in such a manner that it
was arranged as a IT line of various numbers of sections. The arrival curve
of current computed and plotted to the scale of the oscillograms showed a
check to a reasonable degree of accuracy.

* Kennelly and Tabossi, Elec. World 1912.

In order to determine the relation between the artificial and smooth
cables, the cable formula was plotted for the numerical case:

=200 ohms

C= io~ 6 farads

R=2O volts

for various numbers of sections. It was found that with these constants,
the arrival curve on a six section artificial cable coincided to a sufficient
degree of accuracy, for the purposes of engineering, with the smooth line
arrival curve as plotted from Kelvin's formula:

-C-

2 e

n-n
'RC

amperes

The limiting value of the artificial aerial line formula as the number of
sections is indefinitely increased was also considered. From the fact that
the artificial cable formula approaches Kelvin's smooth line formula in the
limit, were derived the limits of the various values of m z h^ and h as m= o .
A certain approximation was also made because of the fact that in lines

encountered in practice

/ R
ice, I -

\2 Li

L
may be neglected in comparison with - .

X/C

Applying these facts to the artificial aerial line formula gave the following
expression for the received current on a grounded smooth aerial line when a
steady voltage is suddenly applied at the home end :

*-*-'

amperes

where F (/) is the discontinuous function represented in fig. 3.

HLC-+

fig. 3. The

Fft) in tfte atriaL tine formula.

The arrival curve plotted from this formula for the smooth line on which
the oscillograms were taken is shown in fig. 4.

-4-" ft /W

Time

Fig. 4. Arrival Current, Smooth Aerlat Line.

A comparison of this smooth line curve with the artificial line arrival
curves showed that the artificial line of four sections, or less, did not well
approximate the smooth line, for the transient due to the sudden application
of a steady voltage. The artificial line of twelve sections approximated
the smooth line fairly well; but a still greater number of sections would be
necessary, in order to enable the artificial line to be used for experimental
investigation with this type of transient.

SUGGESTION FOR A CONTINUATION OF THE WORK.

The method of generalized angular velocities, applied to the oscillations
of networks with concentrated constants, has proved to be valuable for
engineering purposes. It is believed that the same method may be profit-
ably applied to networks containing branches with distributed constants.
A starting point for such work would be found in Heaviside's application
of the resistance operator to the smooth line.

SUMMARY.

In addition to the theorem which determines the free angular velocities
of oscillation of a network, there is a theorem which will determine the
amplitudes. This theorem involves a "threshold impedance" which may
be formed for any circuit with concentrated constants, and which enables
the amplitudes of oscillation to be found from the initial potential of the
unbalanced energy.

An application of this method to the coupled circuit gives an easily ap-
plied and convenient complete solution for the primary condenser dis-
charge.

Applied to the artificial line, it enables the lumped line of a given number
of sections to be compared with the represented smooth line for certain
transient effects.

The writer wishes to express his thanks to Prof. D. C. Jackson, Dr. A. E.
Kennelly, and other members of the Department of Electrical Engineering
who have assisted him in the preparation of the thesis.

R-6-i 6-400.

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