The Alternate Current Transformer Vol. 1: The Induction of Electric Currents (1896) — part 11 of 35

From 0 draw 0 C at right angles to 0 Q, and set off 0 C' equal to M.p times 0 T. 0 C is then equal to M p Ilt and 0 C represents the impressed electromotive force in the secondary circuit. On 0 C describe a semi-circle, and set off 0 B, making

an angle COB with 0 C such that tan C 0 B = 5L?, or tan

b

C 0 B = the ratio of the inductive to the ohmic resistance for the secondary circuit. Join B C. On the same scale on which 0 C represents the impressed electromotive force in the secon- dary circuit, viz., M^I^ OB will then represent the effective electromotive force in the secondary, or will represent S I2, and hence, if 0 D is taken equal to one Sth part of 0 B, 0 D will represent I2, or the secondary current. Next draw a line 0 K perpendicular to 0 B, and therefore parallel to B C, and on it set off a length, 0 K, equal to Np times 0 D or to Mp I2. 0 K represents then the back inductive electromotive force of the secondary on the primary.

The impressed electromotive force which has to be applied to the primary to produce in it the primary current 0 T and

SIMPLE PERIODIC CURRENTS.

177

to induce in the secondary the secondary current 0 D has therefore to be equal and opposite to the resultant of three

\

M

Mp',

/ 3

' 3 t

a

s

/ ^

X*

electromotive forces, or to equilibrate three electromotive forces, viz., the effective electromotive force of the primary

178 SIMPLE PEEIODIG CURRENTS.

0 Q, the electromotive force of self-induction in the primary P Q, and the back electromotive force in the primary due to, the inductive effect of the secondary on the primary, viz., 0 K.

The resultant of 0 Q and Q P is 0 P. If, then, we draw P P' from P, and make it equal and parallel to 0 K, and join O P', 0 P' will be the resultant of 0 Q, Q P, and 0 K ; and hence 0 P' will represent E, or the impressed electromotive force required to be applied to the primary to maintain the currents Ij and I^.

It is to be understood that in this diagram a unit of length stands for a volt, or unit of electromotive force ; and hence, on that assumption, E represents in volts the impressed E.M.F. — that is, the maximum of the simple periodic E.M.F. required to maintain the currents I: la, of which 0 T, 0 D represent the maximum values. The relative phases are indicated by the positions of these lines. To obtain the actual values of the E.M.F. and currents at any instant we have only to take the projections of 0 P', 0 T, and 0 D on any line drawn through 0 suitably placed, and the magnitudes of these projections will give the required quantities. We must then suppose the whole diagram to be enlarged or diminished without distortion until the length of 0 P' is numerically equal to the maximum value in volts of the impressed E.M.F. E, and then 0 T and 0 D, will represent the currents Ix and L in magnitude. We may consider the two right-angled triangles 0 Q P, 0 B C as pivoted together at 0, and revolving round 0 ; the fluctuations of the projections of 0 P', 0 T, 0 D on any line will give us the cyclic values of E, I:, and I2. We can next obtain some useful relations between these quantities from the geometry of the figure. In the triangle 0 B C,

Hence I? W f = S2 L2 + N2 f I22 ;

or ji_x/S*+^N*.

I2 ' Mp

primary current impedance of secondary secondary current" \.p

M£> might by analogy be called the mutual reactance.

To obtain the value of \ in terms of E and the inductances and resistances, we project the lines OP' and OP on the

SIMPLE PERIODIC CURRENTS. 179

-vertical line 0 K, and express the fact that 0 P' or E is in all cases the resultant of 0 K and 0 P. Let the angle P' 0 Q' be called <£. cj> is the angle by which the primary current lags behind the total impressed electromotive force. Then COB is Bit and T 0 K = C 0 B = 9,, since Q 0 C and K 0 B are both right angles.

Hence we have by resolution on 0 K E cos (<£+ 0a)

but. since L

we have by substitution

E cos (* + 0,} = {7== + ^B*+^L' co3 (6, + &)} I,,

and therefore a relation established between E and Ij which is known when <£ is known.

Since tan 9, =*£, and tan 02 =^,

lii O

it follows by an easy transformation that RS-»2LN

.hence E = — a . .

y {5* _[. p- ]'2 cos (<p + tt>)

To find the value of <£, suppose that whilst E and Ij remain the same, the secondary circuit is suppressed. We should then only have an impressed electromotive force, E, creating a current, I15 and from the diagram and from what has been before explained it is obvious that the effective and self-induc- tive electromotive forces in the circuit would then be repre- sented by 0 Q' and Q' P'. If we denote these by the symbols B/I! and L'^Ii we may properly call E' and L' the equivalent resistance and inductance ; that is to say, these quantities are the resistance and inductance which the primary circuit should have in order that, when there is no secondary circuit, the primary impressed electromotive force may generate in it the same current which it does when the secondary circuit is present and the primary has its natural resistance R and inductance L. We see, then, that the effect of bringing up the secondary and allowing it to be acted upon and react

180 SIMPLE PERIODIC CURRENTS.

upon the primary is to increase the effective resistance and diminish the effective inductance of the primary ; in other words, the equivalent resistance of the primary circuit is- greater and the equivalent inductance is less by reason oi the presence of the secondary circuit.

We have then to find the value of cos (<£ + ft2). Cos (<f> + ft2) = cos <£ cos ft2 - sin <£ sin ft, ;

Rf T '•"

but cos 6 = — — . sin <i =

v'R'2+p2L'2 and cos ft, = ^— sinft2- N^

hence cos (</> + ft2) = -^ B'S-L'Njf

Substituting this in the equation connecting E and I1} we arrive at

Es=I (MV + B S - jt)2LN) ^B^+y'L'"

K'S-L'N^2

Returning to Fig. 72, we see from it that P' Q' is parallel to P Q, and hence, if we draw P'V parallel to Q' Q, we have P'Q' = PQ-PY

= PQ-PP'cosP'PV = PQ-PP'sin6>2,

or

or,smce

we have

Also, again, 0 Q represents E Ij and 0 Q' on the sama- scale R' Ix, and

Y

OQ + PP'sinP'PV 0 Q + P P' cos ft, ;

hence Bf ^-

and; therefore; R' =

SIMPLE PERIODIC CURRENTS. 181

These formulae (42) and (43) give us the effective inductance and resistance of the primary circuit as affected by the secon- dary. They were first given by Clerk Maxwell in a paper in the Philosophical Transactions of the Koyal Society in 1865, •entitled " A Dynamical Theory of the Electromagnetic Field " {Phil. Trans., 1865, p. 475).*

If we form from (42) and (43) the function E' S — L' N^, we find it to be M'2^?2 + E S— />2LN; and hence, by substitution in the expression already given connecting E and Ii, we arrive finally at the result

Following thejisual nomenclature, we may call the expres- sion ^/E''2+^2L'2 the equivalent impedance of the primary circuit, and we have as the final result for the induction coil of constant inductance

Primary current ^pressed electromotive force streno-th = equivalent impedance of primary circuit

Secondary current _ -*r primary current

strength " impedance of secondary circuit*

The angle of lag of primary current behind impressed E.M.F. = <j>, where tan </>=-??--_, and the angle of lag of the secondary current behind the primary is seen to be 90°+ 02 and tan #2 = i_jP ; hence we have the values and relative phases of

h

the currents and the impressed electromotive force.

In the above equations we are to understand current strengths and electromotive forces to be the maximum values during the period. If ^ and i.2 be the actual values at any time t, reckoning time from the instant of the zero value of the electromotive force, then, from the principles previously

• Sec also Lord Rayleigh on " Forced Harmonic Oscillations of Various Periods," Phil. May., May, 1886, p. 375.

182 SIMPLE PERIODIC CURRENTS.

explained in this chapter, it is obvious that *! = !! sin (pt-$),

' P

or it = — — sin (p t - <i) ;

x/E'- + ^L'a

and I,-—^1'

or

The student will find the above expressions for the primary and secondary currents can be deduced by analytical processes from the simultaneous equations.

, . , (44) a t a t

N^L + M^A + S?2 = 0, . . , , (45) d t d t

which equations can be established for two circuits by analo- gous methods to that by which in § 18 a current equation was arrived at for one circuit, subject to a simple periodic electromotive force.* It is easily seen that if n is very great, or the alternations extremely rapid, then I:_N T2~M'

If the primary and secondary circuits consist of two equal circuits, so interwound that for these circuits L = M = N, then for very rapid alternations we see that the secondary current I2 is equal in magnitude, and exactly opposite in phase, to the primary current I1} and the magnetic fields due to these currents respectively are also equal and opposite in direction at every instant.

§30. The Flow of Simple Periodic Currents into a Con- denser. — The electrical capacity of a conductor of any kind is measured by the quantity of electricity required to charge it to unit potential. Two conducting surfaces so arranged as to have constant capacity are generally called a condenser. The most simple and familiar form of this appliance is the Ley den jar, in which two surfaces of tin foil are separated by a

• See Mascart and Joubert's " Electricity," Vol. L, p. 521.

SIMPLE PERIODIC CURRENTS. 183

sheet of glass. Condensers for practical purposes are generally formed of sheets of tinfoil, separated by some dielectric, such as paraffined paper, mica, gutta-percha, tissue, or ebonite. If a quantity of electricity, Q, measured in microcoulombs, is given to one of the plates or sets of plates of a condenser, and if the other set are kept at zero potential, the charged plates will be raised to a certain potential, say V volts, above that of the earth. The capacity of the condenser C in microfarads is then such that by definition

CV = Q.

If the potential difference of the condenser terminals at any instant is v, and if it is changing, and if q is the charge or quantity of electricity in the condenser at that same instant, then

C = ; dt dt

but ?-? is the time rate at which the charge is changing or

dt

is the value of the current i at that instant flowing into or out of the condenser. Hence

-t f^v .

and C — =z.

dt

Suppose the condenser is being charged through a circuit whose resistance is E, then i is the current which is flowing through this resistance at the instant considered ; and the fall of potential down the resistance is R i volts. If a constant potential, V, is being applied to the outer extremity of the resistance, so that the total difference of potential between one plate of the condenser and the outer end of the resistance attached to the other plate is V volts, we have at any instant the equation

v + Ei = V,

or v+CR(-=V,

dt

or CR'-+t?=V ...... (4G)

a t

184 SIMPLE PERIODIC CURRENTS.

This is the differential equation for the potential v of the condenser at any instant, t, when being charged through a resistance, E, by a constant potential, V. The equation (46) is easily integrated as follows : — Multiply all through

by en~c , where e is the base of the Napierian logarithms, viz., 2-71828. Then we have, as a result, the equation

or .

dt

Both sides of this last equation can then be integrated. Hence, integrating with respect to time, we obtain the equation

t_ t v e™= V<?RO + a constant,

The constant C is determined by the condition that when the time * is zero then v is also zero. Hence C= - V, and, therefore,

' .... (47)

The student should compare equations (46) and (47) for the value of the instantaneous potential of a condenser charged by a constant voltage with the equations (26), page 127, and (27), page 132, for the instantaneous value of a current in an inductive circuit acted on by a constant pressure, and it will be seen that they are similarly constructed. The quantity

CE is called the time-constant of the condenser just as _

is called the time-constant of the inductive coil. The greater the condenser time-constant the larger will be the time taken by the condenser potential to arrive at a given fraction of the steady impressed voltage applied to charge it. Practically, the condenser is fully charged in a time equal to eight or ten times its true constant. The reader must note that if the capacity C of the condenser is measured in microfarads, then the resistance E must be measured in megohms to obtain the correct measure of the time-constant C E in seconds.

SIMPLE PERIODIC CURRENTS. 186

Suppose, for example, that a condenser of one microfarad is being charged through a resistance of one megohm by an impressed voltage of 100 volts. It is required to find the potential difference at the terminals of the condenser at the end of the 1st, 2nd, 3rd, 4th, nth second. Let vlt r2l r3, vn be these potential differences. Then t\ = 100 (1 - e~l) = 63 volts nearly,

where « = 2-71828,

and r2 = 100 (1 - e~~) = 86 volts nearly,

Since e~w is an exceedingly small number, in ten seconds the condenser potential is practically equal to 100 volts. We see, therefore, that in charging condensers through resistances sufficient time must always be allowed for the charge, depend- ing on the value of the quantity C E, and the charge is not practically complete unless contact with the source of im- pressed voltage endures at least for a time equal to 5 C E, or, better still, 10 C E.

Supposing, in the next place, that the impressed voltage acting on the condenser is periodic in character or alternating, and that the condenser, instead of being charged through a resistance, has its terminals shunted by a resistance. Let •the capacity of the condenser be, as before, C microfarads, and let the resistance of the shunting conductor be _ ohms ; that

is, let K be the shunt conductance. Let the impressed •voltage be periodic, and let its frequency be n, and write p for 2-«, as usual. Then, as above, the current flowing into the condenser at the instant when its terminal potential

•difference is v is C f— . Also the current flowing through the

a t

shunt is Kt-. We have, therefore, as the equation for the total current flowing into the shunted condenser, the -expression

•C+Kt-t,

a t

where i is the total current flowing into condenser and through the shunt at the instant when the terminal potential difference of the condenser is r. Let time be measured from

186 SIMPLE PERIODIC CURRENTS.

the instant when the total current denoted by i is zero, then,. if i is a simple periodic current, it may be expressed in terms- of its maximum value by the equation i = Isinpt.

Hence C— + Kv = lsmpt ..... (49)

at

On comparing these equations, (48) and (49), for the- instantaneous value of the condenser potential difference when a periodic current is flowing through it with the equations (29), page 133, and (31), page 135, it will be seen that they are structurally the same, and we can at once write down the solution of equation (49) by imitating the solution of (29), writing C instead of L, K instead of R,> instead of t, and I instead of E. Hence the solution of

C— K dt

sin (p t — 6} + a constant, . (50)

where 0 is an angle such that tan 6 = _^.

K

The quantity /K- + G2^>2 has not yet received an acknow- ledged name, but it is analogous to the impedance in the case of the current flow in inductive circuits. It has been suggested that this quantity should be called the admittance of the condenser.

From equation (50) we see that the terminal voltage of the condenser lags behind the charging current in phase ; or otherwise that the current is in advance of the potential difference. If the shunt wire is removed this is equivalent to making K equal to zero, and under these circumstances w& have

v = _ sin (p t - 90), Gp

or v = ---cospt ....... (51)

This shows that in the case of a condenser having a di- electric with no true conducting power, the charging current is exactly 90deg. in advance in phase of the terminal potential difference of the condenser.

SIMPLE PERIODIC CURRENTS. 187

It is clear, therefore, that as regards producing lag or lead of current capacity acts in the opposite direction to inductance^ and may be considered to be equivalent to a negative inductance.

It is also evident that if a shunted condenser is placed in series with an inductive circuit a proper relation may be formed between the inductance, resistance, capacity, and conductance, such that the combination of inductive resistance and shunted condenser acts to an impressed simple periodic electromotive force just as if it were totally non-inductive. This annulment of inductance has important applications in practice in tele- graphy, and it is therefore desirable to define these conditions carefully:

§ 31. A Shunted Condenser in Series with an Inductive Resistance. — Let a condenser of capacity C have its terminals closed by a circuit whose conductance is K, and hence its resistance K-1. Let this shunted condenser be placed in series with an inductive resistance whose inductance is L and resistance E. At any instant let vt be the potential difference of the terminals of the condenser and v» those of the inductive resistance when a periodic current having at this same instant a value i is flowing through the system. Let v be the value of the over-all potential difference. Then suppose that v is a simple periodic potential difference, or that

r = V sin^ t.

Then the. current i will differ in phase from v, and may be represented by the expression

Capital letters represent maximum values as usual. We can then form the fundamental equation for the shunted condenser in series with an inductive circuit as follows : — We have for the shunted condenser terminal potential i\ the equation

C^ + Kr^-i, (52)

and for the inductive resistance the equation

L'-+Rf = *a, (53)

and also t'i + 1-4 = v, (54)

188 SIMPLE PERIODIC CUBRENT8.

since the over-all potential v is the sum of the separate potential steps vi and vt. Eliminate Vi and va from these equations, and we get

• KV, . (55) also, since v = V sin p t,

we have Cd—= Cp Vcospt,

at

and K v = K V siup t.

Hence we have

C ^ + K v = JR2 + Cy V sin (p t-0). . (56)

This last follows at once from the lemma on page 161. Then, since i = I sm (pt-<f>),

we obtain — = I p cos (p t - <£),

•and ^=-Ij^sm(pt-^),

and by substitution of these last values in equation (55), we arrive at the equation

(K K + 1 - C Lp>) I sin (p t - «£) + (C R^ + K "Lp) I cos (pt-<$

0. . . (57)

dt

= v/K* + CV2V sin (^ « - 0). . . (58)

Since it is shown on page 161 that any function of the form A sin 6 + B cos 0 can always be expressed in the form

-r>

. ^/A'2 + B2 sin (0 + a) , where tan a = -r-,

A

it follows at once that the maximum value of the current I ia given by the equation

= _

(K R + 1 - C L /»") + (C B/> + K L p)'2' '

and it is not difficult to show that this last equation (59) may be written in the form

P_

"

SIMPLE PERIODIC CUEEENTS. 189-

Let E1 stand for and L1 for

If the condenser and shunt are removed it is clear that the maximum value of the current I will be given by

Hence the effect of adding the shunted condenser is as if the resistance of the inductive circuit were increased bv an

T7- •

amount equal to , and the inductance of the

K2 + C2£>2

inductive circuit diminished by an amount equal to C

Hence, if the quantity _ - — _ — - is equal to L, which is

the inductance of the circuit, then the total effective induct- ance of the circuit will vanish, and the inductive resistance and shunted condenser together will act as if there were no resultant inductance at all. This condition is fulfilled. when we have

which equation gives us the required magnitude of the capacity to annul the inductance. Uniting - instead of K, so that r is- the resistance of the shunt, we have L

(60)

C(l-CLj>2)

as the final equation, which tells us what must be the resistance of the shunt to be put across the terminals of a condenser of capacity C in order that the combination may just annul the inductance of a resistance in series with it whose resistance is E and inductance L. If L is measured in. henries, C must be in farads, and E and r be given in ohms.

It will be seen that the annulment is only exact for one- particular frequency, and that change of frequency means a change in the value of r requisite to neutralise the induct-

190

SIMPLE PERIODIC CURRENTS,

ance. In telegraphic work it is usual to employ a shunted condenser to annul more or less completely the self-induction of relay magnets.

§ 32. Representation of Periodic Currents by Polar Diagrams. Two methods of graphically representing simple periodic currents or electromotive forces, viz., clock diagrams and wave diagrams, have been hitherto used. Each of these methods has some peculiar advantages of its own. There is, however, a third method which has sometimes especial utility, and this is by a polar diagram.

Let a straight line 0 P (Fig. 73) revolve round one of its .extremities 0, and let the angle of inclination 6 which the

FIG. 73.

revolving line makes with another fixed straight line OA passing through this centre of revolution be called the angle of displacement of the revolving line. Let any point P be taken on the revolving line, and let the distance of this point from the centre be denoted by r. The length r is called the radius vector. Let r increase or diminish as 0 P revolves, but so that the length 0 P = r varies according to any law connecting it with the angle of displacement A 0 P. The extremity P of the line 0 P will then trace out a curve called a polar diagram.

Suppose that 0 P runs through a cycle of values beginning with a zero value when 0 = 0, and, after reaching a maximum

SIMPLE PERIODIC CURRENTS. 191

value, becoming zero again after the line 0 P has completed one half revolution, or when 6 = 7r, the polar diagram will be a closed curve passing through the origin. Since the radius 0 P runs through a cycle of values from zero to a maximum, and returns to zero again during a revolution of the radius through two right angles, or an angle TT, the radius r may suitably represent the value of any periodic quantity which completes a cycle of values in the time represented by one complete half revolution of the radius vector. Suppose, then, that 0 P varies as sin 0, where 6 is the displacement phase angle — that is, let

r=Rsin0.

It is then clear that in this case the polar curve is a circle. For if we draw the line 0 B through 0 perpendicular to 0 A, and draw PB at right angles to OP, since AOP = 0 and OP = r, and since by supposition r=R sin 0, it follows that -the angle P B 0 is always equal to 0, and that, therefore, the length 0 B is a constant length equal to E. In other words, the locus of the point P is a circle passing through 0, P, and B. Hence the circle is the curve which represents in a polar diagram a simple periodic quantity, and if the length 0 P is taken to represent the magnitude of a simple periodic current

• or electromotive force at any instant corresponding to a phase angle A 0 P, the circle passing through 0 will be the polar

• curve representing this simple periodic current or electro- motive force.

If the current or electromotive force is periodic, but not simply periodic, then the polar curve representing it will be a unicursal curve passing through the origin 0, but will not be a circle. The polar diagram of an alternating current or electromotive force enables us very easily to determine the square root of the mean-square value of the periodic quantity represented by it. It is more difficult to do this with the ordinary wave-curve diagram.

Suppose, for example, that in Fig. 74 we have a wave diagram drawn representing a half wave of an alternating electromotive force which is not a simple sine curve. The ordinates represented by the dotted lines are proportional to the instantaneous values of the periodic quantity taken every 20deg. If we wish to find the Vrnean2 value of the

192

SIMPLE PERIODIC CURRENTS.

ordinates of this curve there is no other way of doing this- than by drawing a number of numerous equidistant vertical ordinates, measuring their lengths, squaring these magnitudes,, and taking the square root of the mean of these squares. This- is a troublesome arithmetical process, and in proportion as the wave curve is more irregular or complex, so much the more numerous must be the ordinates to obtain a correct result for the mean-square value. It is, however, a much easier process if the periodic quantity is represented by a polar curve. Let Fig. 75 represent the same periodic function as in Fig. 74,. drawn in a polar form. The dotted radii are proportional to the instantaneous values of the periodic quantity, and are

FIG. 74.

placed at angular intervals of lOdeg. Let any radius 0 P (see Figs. 75 and 76) be denoted by r, and let the corresponding phase angle P 0 A be- denoted by the letter 6. If we consider the radius to move forward by a small angle d 6 and to increase in length by a small amount dr, then it is obvious that the small increment of area d A swept out by the radius r is equal to | r2 d 6. To obtain the whole area, and included by the polar curve 0 P Q E, we have to integrate this quantity ^r*d6 from 0 to TT, or to obtain the integral

SIMPLE PERIODIC CURRENTS.

193

On the base line drawn through the polar centre 0 let a semicircle A C B (see Fig. 75) be described equal in area to the area included by the polar curve OPQ. Let OA = R be the radius of this semicircle. Then

R =

The expression on the right hand side of this last equation is obviously the square root of the mean of the squares of the instantaneous values of the periodic quantity r. Hence, we can obtain at once the /s/mean^ value of a periodic quantity current or electromotive force as follows. On any straight

FIG. 75.

line describe a semicircle and draw radii of this semicircle at angular intervals equal to those phase intervals for which the instantaneous values of the current or electromotive force are observed. Set off on these radii lengths equal to the respec- tive values of these instantaneous quantities and join the extremities of all these lines so set off by a curve. This curve is the polar diagram, and it is a closed curve. Take the area included by the polar diagram with an Amsler's or other plani- meter, and from a table of areas of circles find the circle whose area is double that of the polar curve. Then the radius of this circle is the square root of the mean of the squares of the periodic quantity. If, for instance, we have a periodic current

o

194 SIMPLE PERIODIC CURRENTS.

curve plotted down in polar form, this operation will give us the dynamometer value of the current, or the value which would be read on a dynamometer.

The polar curve, therefore, lends itself very easily to the determination of the mean -square value of a periodic current or electromotive force, of which the instantaneous values are known throughout a semi-period, whether those values are ia'ven at equidistant intervals of time or not. The reader w^, therefore, note that if we plot down a periodic quantity ii rectangular co-ordinates (Fig. 74) or wave form, and find the rectangle A P P' B of equivalent area to the half wave, the altitude AP of this rectangle gives us the true mean ordinate of the periodic curve ; but if we plot down the periodic quantity to a polar diayram (Fig. 75) and find the

FIG. 76.

semicircle A C B of equivalent area, the radius 0 A of this aemicircle gives us the square root of the mean-square value of the periodic quantity represented by the radii of the polar curve.

§33. Initial Conditions on starting Current Flow in a Cir- cuit having Resistance and Inductance. — It has been shown in the foregoing sections that if an impressed electromotive force of simple periodic kind acts upon a circuit having in- ductance, the resulting current is a simple periodic current, but lags behind the impressed electromotive force in phase. These, however, are the conditions when the resulting current has become steady. At the instant of closing the circuit there are peculiar conditions of augmentation of the current which are called initial conditions, and which have very important

SIMPLE PERIODIC CUHEENTS. 195

consequences in practice. It will be advisable, therefore, to •examine a little more closely how these are produced, and what results may be expected at the instant of starting or stopping the current in such a circuit. To do this we will, in the first place, examine more carefully the solution of the fundamental equations for current creation in an inductive circuit. It has been shown that the differential equation for current at any instant in the circuit of constant inductance L and resistance R under an impressed simple periodic electro- motive force v — V sin p t is

. . (61)

at

To solve this equation completely, we differentiate it twice, and eliminate thereby the term V sin p t, thus obtaining the equation

L+B+^L+^Et'=0- • (62)

A differential equation of this type is called a linear differ- ential equation of the third order. It is shown in treatises on differential equations that its solution depends on the solution of a cubic equation called the auxiliary equation. The auxiliary equation in this case is

Lw3 + Bm2+^2Lm+j92E = 0. . . . (63) • This cubic equation can be split up into two factors and be written

and hence the roots of the cubic equation (63) are DI= + v — 1 p

For a linear equation of the type of (62), the roots of the auxiliary cubic being a and ± J -1(3, the solution is known io be of the form

i = A ea * + B sin /3 £ + B' cos £ t. The solution of the differential equation (62) is, then, given by

Rt

. . (64) o2

196 SIMPLE PERIODIC CURRENTS.

By the Trigonometrical Lemma on page 161 we can write, instead of the second and third terms on the right hand side of (64), the single term v/B2 + B'2 sin (p t-<f>), where >/B2 + B'2 is obviously the maximum value of the current — call it I — when the initial state is over, and <f> is the angle by which the current, when steady, lags behind the electromotive force in phase. Hence (64) may be written,

t = Ae L +Isin(pt-<f>). . . . (65)

To find the constant quantity A, we note that at the instant when the circuit is closed the current has necessarily a value zero. Let this closing of the current happen at a time t' reckoned from the instant when the electromotive force is zero. Then at the instant of closing the circuit we have

Q = ke~Lt + Isin(pt'-<t>),

R

or A= -

Hence, substituting this value of A in equation (65), we obtain

-4)4 ^ ', . . (66) and this is the complete solution of the differential equation (61) for the current in the circuit.

We note that the expression for the current at any instant in the inductive circuit is made up of two terms ; the first term, I sin (p t— </>), is a simple sine function, and represents by itself a simple periodic current having a maximum value I.

-« t) The second term, I sin (pt'-$)e~*- , is an exponential

function having a maximum value I sin (pt' - </>), when t' = tr and this term represents, therefore, a logarithmic curve begin- ning with the value I sin (p t' - <j>) and dying away gradually to zero.

Hence the resulting current curve consists of these two curves superimposed upon one another, a periodic curve and a logarithmic or diminishing curve.

In Fig. 77 are shown two such curves ; curve 1 being the sine curve, curve 2 the logarithmic curve, and the resultant curve 3, represented by the dotted line, which is obtained by adding together the ordinates of the sine curve and the logarithmic curve. It will be seen that the effect of the

SIMPLE PERIODIC CURRENTS. 197

superposition is to make the resultant curve lopsided with respect to the curve axis for a certain period, but beyond that time it is sensibly symmetrically situated with respect to the time axis. Hence, during this initial period, the maximum values of the current in opposite directions are not the same. At the instant when t = t' the value of the current is zero. It is obvious that, when pt' = 9Q + (f), sin (pt'-$) = , and that then the logarithmic curve begins with its greatest value ; but that, when pt' — <}>, then the logarithmic curve has no existence at all. When pt' = QQ + <f> — that is if the circuit

is closed at an instant t = — -L? reckoning from the instant

P

when the impressed electromotive force is zero— the disturbance •of the uniformity of the periodic curve of current is the

FIG. 77.

greatest possible. At the same time the maximum value of the current in the negative direction can never be greater than 2 1 where I is the maximum value of the steadily periodic current. For the maximum value of the logarithmic curve at the instant of closing the circuit is — I, and at that instant i = 0 ; hence the value at which the periodic component of the current must begin will be + 1. At the time when the periodic part has reached a maximum of - 1, which happens after half a period, the ordinate of the logarithmic curve will have fallen to something less than I by an amount depending on the rate

T>

of decrease, which in turn depends upon the value of =- or the reciprocal of the time-constant of the circuit.

198 SIMPLE PERIODIC CURRENTS.