Charles Proteus Steinmetz.

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L = al -^

1 1

7X8XV 8

and substituting the numerical values,
L = ai (10-2) +^(0.125-0.001)

-^(0.0078-0) +-^(0.0001 -0)|
= a{8 +0.0207-0.0001} =8.0206a.

As seen, in this series, only the first two terms are appreciable
in value, the third term less than 0.01 per cent of the total,
and hence negligible, therefore the series converges very
rapidly, and numerical values can easily be calculated by it.

POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 63

For x\ <2 a; that is, v\ <2, the series converges less rapidly,
and becomes divergent for x\<a] or ; v\<l. Thus this series
(17) is convergent for v > 1, but near this limit of convergency
it is of no use for engineering calculation, as it does not converge
with sufficient rapidity, and it becomes suitable for engineering
calculation only when v\ 'approaches 2.

EXAMPLE 2.

48. (log 1=0, and, therefore log (1+z) is a small quantity
if x is small. / log (1+x) shall therefore be developed in such
a series of .powers of x, which permits its rapid calculation
without using logarithm tables.

It is

fdu
logu=J-;

then, substituting (l+x) for u gives,

...... (18)

From equation (4)

l+x
hence, substituted into (18),

log (1+*),= (l-z+z 2 -:r 3 + . . .)dx

= (dx - Cxdx + Cx 2 dx - Cx 3 dx + . . .

X 2 X 3 X 4

-*- 2+9~ **'"'> '

X 2

hence, if x is very small, is negligible, and, therefore, all

Zi

terms beyond the first are negligible, thus,

log(l+a;)=z;. . . ..... (20)

while, if the second term is still appreciable in value, the more
complete, but still fairly simple expression can be used,

= x- = xl- (21)

64 ENGINEERING MATHEMATICS.

If instead of the natural logarithm, as used above, the
decimal logarithm is required, the following relation may be
applied :

logio a = logi log a = 0.4343 log a, . . (22)

logio a is expressed by log a, and thus (19), (20) (21) assume
the form

- + ...; . (23)
or, approximately,

logio(l+o:)=0.4343:r; (24)

or, more accurately,

logio (l+o;)=0.4343a;(l-|). . ; . (25)

B. DIFFERENTIAL EQUATIONS.

49. The representation by an infinite series is of special
value in those cases, in which no finite expression of the func-
tion is known, as for instance, if the relation between x and y
is given by a differential equation.

Differential equations are solved by separating the variables,
that is, bringing the terms containing the one variable, ?/, on
one side of the equation, the terms with the other variable x
on the other side of the equation, and then separately integrat-
ing both sides of the equation. Very rarely, however, is it
possible to separate the variables in this manner, and where
it cannot be done, usually no systematic method of solving the
differential equation exists, but this has to be done by trying
different functions, until one is found which satisfies the
equation.

In electrical engineering, currents and voltages are dealt
with as functions of time. The current and c.m.f. giving the
power lost in resistance are related to each other by Ohm's
law. Current also produces a magnetic field, and this magnetic
field by its changes generates an e.m.f. the e.m.f. of self-
inductance. In this case, e.m.f. is related to the change of
current; that is, the differential coefficient of the current, and
thus also to the differential coefficient of e.m.f., since the e.m.f.

POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 65

is related to the current by Ohm's law. In a condenser, the
current and therefore, by Ohm's law, the e.m.f., depends upon
and is proportional to the rate of change of the e.m.f. impressed
upon the condenser; that is, it is proportional to the differential
coefficient of e.m.f.

Therefore, in circuits having resistance and inductance,
or resistance and capacity, a relation exists between currents
and e.m.fs., and their differential coefficients, and in circuits
having resistance, inductance and capacity, a double relation
of this kind exists; that is, a relation between current or e.m.f.
and their first and second differential coefficients.

The most common differential equations of electrical engineer-
ing thus are the relations between the function and its differential
coefficient, which in its simplest form is,

!-* ........ <

or

-"*

and where the circuit has capacity as well as inductance, the
second differential coefficient also enters, and the relation in
its simplest form is,

or

and the most general form of this most common differential
equation of electrical engineering then is,

& = ...... (30)

The differential equations (26) and (27) can be integrated
by separating the variables, but not so with equations (28),
(29) and (30); the latter require' solution by trial.

So. The general method of solution may be illustrated with
the equation (26) :

66 ENGINEERING MATHEMATICS.

To determine whether this equation can be integrated by an
infinite series, choose such an infinite series, and then, by sub-
stituting it into equation (26), ascertain whether it satisfies
the equation (26) ; that is, makes the left side equal to the right
side for every value of x.

Let,

(31)

be an infinite series, of which the coefficients ao, fli, a 2 , as. . .
are still unknown, and by substituting (31) into the differential
equation (26), determine whether such values of these coefficients
can be found, which make the series (31) satisfy the equation (26).
Differentiating (31) gives,

...... (32)

The differential equation (26) transposed gives,

Substituting (31) and (32) into (33), and arranging the terms
in the order of x, gives,

(ai ao) + (2a 2 ai)x + (3a 3 a 2 )x 2

+ (4a 4 -^ 3 ).r 3 + (5a5-a 4 ):r 4 + . ,.-0. . (34)

If then the above series (31) is a solution of the differential
equation (26), the expression (34) must be an identity; that is,
must hold for every value of x.

If, however, it holds for every value of x, it does so also
for x = Q, and in this case, all the terms except the first vanish,
and (34) becomes,

a\ a =0; or, a\=o.o ...... (35)

To make (31) a solution of the differential equation (a\ a )
must therefore equal 0. This being the case, the term (ai ao)
can be dropped in (34), which then becomes,

or

POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 67

Since this equation must hold for every value of x, the second
term of the equation must be zero, since the first term, x, is
not necessarily zero. This gives,

= 0.

As this equation holds for every value of x, it holds also for
x = 0. In this case, however, all terms except the first vanish,
and,

2a 2 -ai=0; (36)

hence,

a i

and from (35),

a

Continuing the same reasoning,

3a 3 2 = 0, 4a 4 as = 0, etc.

Therefore, if an expression of successive powers of x, such as
(34), is an identity, that is, holds for every value of x, then all
the coefficients of all the powers of x must separately be zero*

Hence,

a\ a = ; or a\ = a Q ;

a\ a
2a 2 ai=0; or 0,2 = -^=-^]

. (37)

4a 4 3a 3 =0; or fl '4* !S "T* s 'Tj

etc.,

etc.

* The reader must realize the difference between an expression (34), as
equation in x, and as substitution product of a function; that is, an as
identity.

Regardless of the values of the coefficients, an expression (34) as equation
gives a number of separate values of x, the roots of the equation, which
make the left side of (34) equal zero, that is, solve the equation. If, however,
the infinite series (31) is a solution of the differential equation (26), then
the expression (34), which is the result of substituting (31) into (26), must
be correct not only for a limited number of values of x, which are the roots
of the equation, but for all values of x, f that is, no matter what value is
chosen for x, the left side of (34) must always give the same result, 0, that
is, it must not be changed by a change of x, or in other words, it must not
contain x, hence all the cpefficients of the powers of x must be zero.

68

ENGINEERING MATHEMATICS.

Therefore, if the coefficients of the series (31) are chosen
by equation (37), this series satisfies the differential equation
(18); that is,

2 X 3 X 4

(38)

is the solution of the differential equation,

51. In the same manner, the differential equation (27),

-<*> ....... < 39 >

is solved by an infinite series,
z = (ir> + aix + av

+ . . .,

(40)

and the coefficients of this series determined by substituting
(40) into (39), in the same manner as done above. This gives,

(ai aa ) + (2a 2 aai)x + (3a 3 aa 2 )x 2

+ (4a 4 -aa 3 ):r 3 + ...=0, . (41)

and, as this equation must be an identity, all its coefficients
must be zero; that is,

ai aao =
2a 2 aa\ =i

or
or

a

a d2

3

a*

or

etc.,

a a 3
a^ T = tt orr?

etc.

(42)

and the solution of differential equation (39) is,

a 2 x 2 a?x 3

-2-+-g-+- r + .... . . (43)

52. These solutions, (38) and (43), of the differential equa-
tions (26) and (39), are not single solutions, but each contains
an infinite number of solutions, as it contains an arbitrary

POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 69

constant ao; that is, a constant which may have any desired
numerical value.

This can easily be seen, since, if z is a solution of the dif-
ferential equation,

dz_ =

then, any multiple, or fraction of z, bz } also is a solution of the
differential equation;

oX^) =

since the b cancels.

Such a constant, ao, which is not determined by the coeffi-
cients of the mathematical problem, but is left arbitrary, and
requires for its determinations some further condition in
addition to the differential equation, is called an integration
constant. It usually is determined by some additional require-
ments of the physical problem, which the differential equation
represents; that is, by a so-called terminal condition, as, for
instance, by having the value of y given for some particular
value of x, usually for x = 0, or x = oc.

The differential equation,

thus, is> solved by the function,

y = a ij , ....... (45)

where,

X 2 X 3 X 4

-+++-

-

and the differential equation,

is solved by the function,

z = a G ZQ, . . ..... (48)

where,

a 2 x 2 aV a 4 * 4

+-rr - {- ....... (49)

70 ENGINEERING MATHEMATICS.

yo and ZQ thus are the simplest forms of the solutions y and z
of the differential equations (26) and (39).

53. It is interesting now to determine the value of y n . To
raise the infinite series (46), which represents y , to the nth
power, would obviously be a very complicated operation.

However,

dy n t dy

L = ny n-i-J. ......

dx dx'

du
and since from (44) ~ = ^> * .....

by substituting (51) into (50),

hence, the same equation as (47), but with y n instead of z.
Hence, if y is the solution of the differential equation,

*-*

dx y '

then z = y n is the solution of the differential equation (52),

dz

-j- = nz.

dx

However, the solution of this differential equation from (47),
(48), and (49), is

z = l+nx + + -73 +. . . ;
that is, if

X 2 X 3

then,

^2^2

- + - + ... ; . . . (53)

therefore the series y is raised to the nth power by multiply-
ing the variable x by n.

POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 71
Substituting now in equation (53) for n the value - gives

JC

^ = 1 + 1+2 + J3 + J4 + '* ; . . . (54)

that is, a constant numerical value. This numerical value
equals 2.7182828. . ., and is usually represented by the symbol e.
Therefore,

hence,

!fo = ^l+s+|+j3+j4+- (55)

and

72 ^X^

-r- + ... ; (56)

therefore, the infinite series, which integrates above differential
equation, is an exponential function with the base

= 2.7182818 ......... (57)

The solution of the differential equation,

thus is,

2/ = <V x , ....... (59)

and the solution of the differential equation,

!-<*, ....... <

is,

2/ = a c- ....... (61)

where % is an integration constant.

The exponential function thus is one of the most common
functions met in electrical engineering problems.

The above described method of solving a problem, by assum-
ing a solution in a form containing a number of unknown
coefficients, a , a\, a 2 . . ., substituting the solution in the problem
and thereby determining the coefficients, is called the method
of indeterminate coefficients. It is one of the most convenient

72 ENGINEERING MATHEMATICS.

and most frequently used methods of solving engineering
problems.

EXAMPLE 1.

54. In a 4-pole 500- volt 50-kw. direct-current shunt motor ,
the resistance of the field circuit, inclusive of field rheostat, is
250 ohms. Each field pole contains 4000 turns, and produces
at 500 volts impressed upon the field circuit, 8 megalines of
magnetic flux per pole.

What is the equation of the field current, and how much
time after closing the field switch is required for the field cur-
rent to reach 90 per cent of its final value?

Let r be the resistance of the field circuit, L the inductance
of the field circuit, and i the field current, then the voltage
consumed in resistance is,

In general, in an electric circuit, the current produces a
magnetic field; that is, lines of magnetic flux surrounding the
conductor of the current ; or, it is usually expressed, interlinked
with the current. This magnetic field changes with a change of
the current, and usually is proportional thereto. A change
of the magnetic field surrounding a conductor, however, gen-
erates an e.m.f. in the conductor, and this e.m.f. is proportional
to the rate of change of the. magnetic field; hence, is pro-
portional to the rate of change of the current, or to

~-r , with a proportionality factor L, which is called the induct-
ance of the circuit. This counter-generated e.m.f. is in oppo-
sition to the current, L ^-, and thus consumes an e.m.f.,

di

+ L-r, which is called the e.m.f. consumed by self-inductance,
at

or inductance e.m.f.

Therefore, by the inductance, L, of the field circuit, a voltage
is consumed which is proportional to the rate of change of the
field current, thus,

di

POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 73

Since the supply voltage, and thus the total voltage consumed
in the field circuit, is e = 500 volts,

*=" +/ 4 (62)

or, rearranged,

di eri
di == ~L~'

Substituting herein,

u = e-ri' } (63)

hence,

du di

gives,

dt r di'

This is the same differential equation as (39), with a= -,

L

and therefore is integrated by the function,

therefore, resubstituting from (63),

eri = OQ L ,
and

f-*?^' ...... (65)

This solution (65), still contains the unknown quantity o;
or, the integration constant, and this is determined by know-
ing the current i for some particular value of the time t.

Before closing the field switch and thereby impressing the
voltage on the field, the field current obviously is zero. In the
moment of closing the field switch, the current thus is still
zero; that is,

7 ; = for * = 0. , (66)

74 ENGINEERING MATHEMATICS.

Substituting these values in (65) gives,

0=7*7; or <*o=+e,
hence,

<~(l~r*') (67)

is the final solution of the differential equation (62); that is,
it is the value of the field current, i, as function of the time, t,
after closing the field switch.

After infinite time, Z = oo , the current i assumes the final
value io, which is given by substituting =oo into equation
(67), thus,

io = -=~ = 2 amperes; .... (68)

hence, by substituting (68) into (67), this equation can also be
written,

= 2(1- TL 1 ), (69)

where io = 2 is the final value assumed by the field current.
The time t\, after which the field current i has reached 90
per cent of its final value i , is given by substituting i = 0.9?'
into (69), thus,

and

r"=o.i.

Taking the logarithm of both sides,

- j*i log t - l;

and

fr-r-|- (70)

rlog

POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 75

55. The inductance L is calculated from the data given
in the problem. Inductance is measured by the number of
interlinkages of the electric circuit, with the magnetic flux
produced by one absolute unit of current in the circuit; that
is, it equals the product of magnetic flux and number of turns
divided by the absolute current.

A current of i =2 amperes represents 0.2 absolute units,
since the absolute unit of current is 10 amperes. The number
of field turns per pole is 4000; hence, the total number of turns
71 = 4X4000 = 16,000. The magnetic flux at full excitation,
or i' =0.2 absolute units of current, is given as \$=8xl0 6 lines
of magnetic force. The inductance of the field thus is:

the practical unit of inductance, or the henry (h) being 10 9
absolute units.

Substituting L = 640 r = 250 and e = 500, into equation (67)
and (70) gives

and

Therefore it takes about 6 sec. before the motor field has
reached 90 per cent of its final value.

values of i from equation (71), for
* = 0, 0.1, 0.2, 0.4, 0.6, 0.8, 1.0, 1.5, 2.0, 3, 4, 5, 6, 8, 10 sec.

This calculation is best made in the form of a table, thus;

and,

logs =0.4343;
hence,

0.39 log =0.1694*;
and

= -0.1694*.

76

ENGINEERING MATHEMATICS.

The values of - 39 ' can also be taken directly from the
tables of the exponential function, at the end of the book.

t

0.1694J

-0.169*4

c -0.39

c

i_ -t).39i

i =
/ _ 3Q(x

= N 0.1694<

i

2(1- OW J

0.0
0.1
0.2
0.4
0.6
0.8

etc

0.0170
0.0339
0.0678
0.1016
0.1355

0.9830-1
0.9661-1
0.9322-1
0.8984-1
0.8645-1

1

0.962
0.925
0.855
0.791
0.732

0.038
0.075
0.145
0.209
0.268

0.076
0.150
0.290
0.418
0.536

EXAMPLE 2.

56. A condenser of 20 mf. capacity, is charged to a potential
of e = 10,000 volts, and then discharges through a resistance
of 2 megohms. What is the equation of the discharge current,

and after how long a time has
the voltage at the condenser
dropped to 0.1 its initial value?
A condenser acts as a reser-
voir of electric energy, similar
to a tank as water reservoir.
If in a water tank, Fig. 27, A
is the sectional area of the tank,
e, the height of water, or water
pressure, and water flows out
of the tank, then the height e
decreases by the flow of water;
that is the tank empties, and
the current of water, i, is proportional to the change of the

de
water level or height of water, , and to the area A of the

Cut

tank; that is, it is,

FIG. 27. Water Reservoir.

de

-rr.
dt

(72)

The minus sign stands on the right-hand side, as for positive
t; that is, out-flow, the height of the water decreases; that is,
de is negative.

POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 77

In an electric reservoir, the electric pressure or voltage e
corresponds to the water pressure or height of the water, and
to the storage capacity or sectional area A of the water tank
corresponds the electric storage capacity of the condenser,
called capacity (7. The current, or, flow out of an electric
condenser, thus is,

The capacity of condenser is,

C = 20mf = 20xlO- 6 farads.
The resistance of the discharge path is,

r = 2Xl0 6 ohms;
hence, the current taken by the resistance, r, is

and thus

de, e

^5 = 7 ;

and

de_ J_
dt~~Cr e '

Therefore, from (60) (61),

and for t = Q, e = e = 10,000 volts; hence

10,000-OQ, (74)

and

= 10,000s- - 025 < volts;
0.1 of the initial value:

Is reached at:

Cr

78 ENGINEERING MATHEMATICS.

values of e, from equation (74), for

/ = 0; 2; 4; 6; 8; 10; 15; 20; 30; 40; 60; 80; 100; 150; 200 sec.

57. Wherever in an electric circuit, in addition to resistance,
inductance and capacity both occur, the relations between
currents and voltages lead to an equation containing the second
differential coefficient, as discussed above.

The simplest form of such equation is:

To integrate this by the method of indeterminate coefficients,
we assume as solution of the equation (76) the infinite series,

(77)

in which the coefficients o, a>i, 2, 3, 4- are indeterminate.
Differentiating -(67) twice, gives

T ^ = 2a 2 + 2X3a3X + 3X4a 4 x 2 + 4X5a5X 3 + ..., . (78)
ax 2

and substituting (77) and (78) into (76) gives the identity,

or, arranged in order of x,

(79)

Since this equation (79) is an identity, the coefficients of
all powers of x must individually equal zero. This gives for
the determination of these hitherto indeterminate coefficients
the equations,

2x3a 3 -aai=0;
3x4a 4 -aa2=0;

4 X 5a t -> eras = 0, etc.

POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 79

Therefore

aa a aa\

a a 2 aar, _a\a 2

a a 3 aa 5 _a\a?

^6~ ; 7 = 6X7 = ~J7~ ;

a a 4

""

9
etc., etc.

Substituting these values in (77),
f ax 2 a 2 x* a 3 x 6

. . .. (80)

In this case, two coefficients ao and a\ thus remain inde-
terminate, as was to be expected, as a differential equation
of second order must have two integration constants in its
most general form of solution.

Substituting into this equation,

that is,

b = Va, ........ (81)

and

b 3 x 3 b 5 x 5 b 7 x 7
- + - + - + . .... (83)

80 ENGINEERING MATHEMATICS.

In this case, instead of the integration constants ao and a\,
the two new integration constants A and B can be introduced
by the equations

and aib = A B]
hence,

d]b

A = ^~ - and B= 2 ,

and, substituting these into equation (83), gives,
) 2 x 2 6 3 x3
II "" li

The first series, however, from (56), for n = b is e +6z , and
the second series from (56), for n= b is e~ bx .
Therefore, the infinite series (83) is,

y = Ae +bx +Be~ bx ; (85)

that is, it is the sum of two exponential functions, the one with
a positive, the other with a negative exponent.
Hence, the differential equation,

d 2 y

-r-2 = ay, (76)

is integrated by the function,

bx , ...... (86)

where,

....... (87)

However, if a is a negative quantity, & Va is imaginary,
and can be represented by

6 = /c, ........ (88)

where

c 2 =-d ........ (89)

In this case, equation (86) assumes the form,

' cx ; ..... (90)

POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 81

that is, if in the differential equation (76) a is a positive quantity,
= -ffr 2 , this differential equation is integrated by the sum of
the two exponential functions (86) ; if, however, a is a negative
quantity, = c 2 , the solution (86) appears in the form of exponen-
tial functions with imaginary exponents (90).

58. In the latter case, a form of the solution of differential
equation (76) can be derived which does not contain the
imaginary appearance, by turning back to equation (80), and
substituting therein a= c 2 , which gives,

(91)

y=

c 2 x 2 _

,~V " 1 I A

II 15.

or, writing 1 A =ao and B=a\c,

s*2y*2 ^4^*4 C^X^

_

I? I*

-+.... (92)

The solution then is given by the sum of two infinite series,
thus,

u(cx) = l

and

as

9

z II 12

e 3 r 3 c 5 x 5

-^-+-r^ - +

15. E

. (93)

Au(cx)+Bv(cx) (94)

In the ^-series, a change of the sign of x does not change
the value of 'u,

u(-cx)=u(+cx) (95)

Such a function is called an even function.

ENGINEERING MATHEMATICS.

In the v-series, a change of the sign of x reverses the sign
of v, as seen from (93):

v(-cx) = -v(+cx) ...... (96)

Such a function is called an odd function.
It can be shown that

u(cx)=coscx and v(cx)=smcx' } . . . (97)
hence,

y = A cos ex +B sin ex, ..... (98)

where A and B are the integration constants, which are to be
determined by the terminal conditions of the physical problem.
Therefore, the solution of the differential equation

(99)

has two different forms, an exponential and a trigonometric.
If 4t is positive,

it is:

y = As + bx +Be- bx t ..... (101)

If a negative,

it is:

y = A cos ex + B sin ex (103)

In the latter case, the solution (101) would appear as ex-
ponential function with imaginary exponents;

y = Ae + i cx +Be-i cx (104)

As (104) obviously must be the same function as (103), it
follows that exponential functions with imaginary exponents
must be expressible by trigonometric functions.

POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 83

59. The exponential functions and the trigonometric func-
tions, according to the preceding discussion, are expressed by
the infinite series,

X 2 X 3 X 4 X 5

T + J3; + [4 + J5"

x 4

[T

|4

X 5 X 7

.

COS X=l 77+[T Ic +

2 4 6

. . . (105)

Therefore, substituting ju for x,

u

u

u .u

l~ J 'l + It +/ |6~j~ / lL" K "

Online LibraryCharles Proteus SteinmetzEngineering mathematics; a series of lectures delivered at Union college → online text (page 5 of 17)