Charles Proteus Steinmetz.

Engineering mathematics; a series of lectures delivered at Union college online

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However, the first part of this series is cos u, the latter part
sin u, by (105); that is,

JU = cos u+j sin u.
Substituting u for +u gives,

s~i u = cos uj sin u.
Combining (106) and (107) gives,

(106)
(107)

and

cos

sin u-

C108)

Substituting in (106) to (108), jv for u, gives,
s ~ v = cos jv -f j sin jv, j
c + y = cos jv j sin / v. j

and,

. . (109)

84 ENGINEERING MATHEMATICS.

and

sin "iv =

. . . . (110)

By these equations, (106) to (110), exponential functions
with imaginary exponents can be transformed into trigono-
metric functions with real angles, and exponential functions
with real exponents into trignometric functions with imaginary
angles, and inversely.

Mathematically, the trigonometric functions thus do not
constitute a separate class of functions, but may be considered
as exponential functions with imaginary angles, and it can be
said broadly that the solution of the above differential equa-
tions is given by the exponential function, but that in this
function the exponent may be real, or may be imaginary, and
in the latter case, the expression is put into real form by intro-
ducing the trigonometric functions.

EXAMPLE 1.

60. A condenser (as an underground high-potential cable)
of 20 mf. capacity, and of a voltage of e = 10,000, discharges
through an inductance of 50 mh. and of negligible resistance,
What is the equation of the discharge current?

The current consumed by a condenser of capacity C and
potential difference e is proportional to the rate of change
of the potential difference, and to the capacity; hence, it is

de

C , and the current from the condenser; or, its discharge
dt

current, is

The voltage consumed by an inductance L is proportional
to the rate of change of the current in the inductance, and to the
inductance; hence, ,

POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 85

4

Differentiating (112) gives,

de d?i

di~ dt 2 '
and substituting this into (111) gives,

as the differential equation of the problem.

This equation (113) is the same as (102), for c 2== 777, thus
is solved by the expression,

. . .

and the potential difference at the condenser or at the inductance
Is, by substituting (114) into (112),

These equations (114) and (115) still contain two unknown
constants, A and B, which have to be determined by the terminal
conditions, that is, by the known conditions of current and
voltage at some particular time.

At the moment of starting the discharge; or, at the time
f = 0, the current is zero, and the voltage is that to which the
condenser is charged, that is, i = 0, and e = e .

Substituting these values in equations (114) and (115)
gives,

=*/T7J3;

= A and
hence

and, substituting for A and B the values in (114) and (115),
gives

1C t

and

...... (116)

86 ENGINEERING MATHEMATICS.

Substituting the numerical values, e = 10,000 volts, C = 20
mf. = 20 X 10- 6 farads, L = 50 mh.=0.05h. gives,

/^=0.02 and

hence,

^ = 200 sin 1000 t and e= 10,000 cos 1000 t.

61. The discharge thus is alternating. In reality, due to
the unavoidable resistance in the discharge path, the alterna-
tions gradually die out, that is, the discharge is oscillating.

The time of one complete period is given by,

10(XM =2,; or, t,-^.
Hence the frenquency,

1 1000

/= = -x = 159 cycles per second.
to *x

As the circuit in addition to the inductance necessarily
contains resistance r, besides the voltage consumed by the
inductance by equation (112), voltage is consumed by the
resistance, thus

e r = ri, . (117)

and the total voltage consumed by resistance r and inductance
L, thus is

e = ri+L- (118)

dt

Differentiating (118) gives,

and, substituting this into equation (111), gives,

, = 0, ..... (120)
at at 2

as the differential equation of the problem.

This differential equation is of the more general form, (22):'"
62. The more general differential equation (22)~ 3^

0, ...... (121)

POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 87
can, by substituting,

2/+JW ........ (122)

(Jj

which gives

dy dz
dx^dx'

be transformed into the somewhat simpler form,

= ....... (123)

It may also be solved by the method of indeterminate
coefficients, by substituting for z an infinite series of powers of
x, and determining thereby the coefficients of the series.

As, however, the simpler forms of this equation were solved
by exponential functions, the applicability of the exponential
functions to this equation (123) may be directly tried, by the
method of indeterminate coefficients. That is, assume as solu-
tion an exponential function,

z = Ae~ bx , ....... (124)

where A and b are unknown constants. Substituting (124)
into (123), if such values of A and b can be found, which make
the substitution product an identity, (124) is a solution of
the differential equation (123).
From (124) it follows that,

7 7rt

f-=-bAs- b *- and 5^- = &M e -te, . . (125)
dx d 2 x

and substituting (124) and (125) into (123), gives,

..... (126)

As seen, this equation is satisfied for every value of x, that
is, it is an identity, if the parenthesis is zero, thus,

0, ..... (127)

and the value of 6, calculated by the quadratic equation (127),
thus makes (124) a solution of (123), and leaves A still undeter-
mined, as integration constant.

ENGINEERING MATHEMATICS.

From (127),

b=-cVc' 2 -a; ..... (128)
or, substituting,

Vc 2 -a = p, ...... (129)

into (128), the equation becomes,

b=-cp ....... (130)

Hence, two values of b exist,

&i= c + p and b 2 =cp, . . . (131)
and, therefore, the differential equation,

dz

(132)

is solved by Ae bix ; or, by Ae bzx , or, by any combination of
these two solutions. The most general solution is,

z = Aie blx +A 2 e b2X ;
that is,

6-]

a [ . . . (131)

a J

As roots of a quadratic equation, 61 and b 2 may both be
real quantities, or may be complex imaginary, and in the
latter case, the solution (131) appears in imaginary form, and
has to be reduced or modified for use, so as to eliminate the
imaginary appearance, by the relations (106) and (107).

EXAMPLE. A

63. Assume, in the example in paragraph 9, the discharge
circuit of the condenser of C = 20 mf. capacity, to contain,
besides the inductance, L = 0.05 h, the resistance, r = 125 ohms.

The general equation of the problem, (120), dividing by
C L, becomes,

POTENTIAL SERIES' AND EXPONENTIAL FUNCTION. 89
This is the equation (123), for:

T =2500:
LJ

. . . . (133)

p=\/c 2 a, then

r \ 1

and, writing

and since

. . . (134)

(135)

1

and ^7 = 2500,
and

. (136)

s = 75 and p = 750.
The equation of the current from (131) then is,

~t

(137)

This equation still contains two unknown quantities, the inte-
gration constants A\ and A 2 , which are determined by the
terminal condition: The values of current and of voltage at the
beginning of the discharge, or = 0.

This requires the determination of the equation of the
voltage at the condenser terminals . This obviously is the voltage
consumed by resistance and inductance, and is expressed by
equation (118),

'I ("*>

90 ENGINEERING MATHEMATICS.

di
hence, substituting herein the value of i and -r from equation

(137), gives

{r s.
A^-zL'

f r _ Q r

; _^ 1 r

r-s - r -* t
- 2L

(138)

and, substituting the numerical values (133) and (136) into
equations (137) and (138), gives

and,

(139)

At the moment of the beginning of the discharge, = 0,
the current is zero and the voltage is 10,000; that is,

^ = 0; i = 0] e= 10,000 . . . . . .(140)

Substituting (140) into (139) gives,

= ^1+^2, 10,000 = WOA 1 +25A 2 -
hence,

A 2 =-A l ] A! = 133.3; A 2 = -133.3.

Therefore, the current and voltage are,

;=133.3!- 500 < - 2000 M, 1

.... (142)

values of i and e, and of their two components, for,

t = 0, 0.2, 0.4, 0.6, 1, 1.2, 1.5, 2, 2.5, 3, 4, 5, 6xlO~ 3 sec.

POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 91

64. Assuming, however, that the resistance of the discharge
circuit is only r = 80 ohms (instead of 125 ohms, as assumed
above :

r 2 -^ in equation (134) then becomes 3600, and there-

fore : _

* = V-3600 - 60 V-l = 60/,
and

The equation of the current (137) thus appears in imaginary
form,

. . . (143)

The same is also true of the equation of voltage.

As it is obvious, however, physically, that a real current
must be coexistent with a real e.m.f., it follows that this
imaginary form of the expression of current and voltage is only
apparent, and that in reality, by substituting for the exponential
functions with imaginary exponents their trigononetric expres-
sions, the imaginary terms must eliminate, and the equation
(H6)~ appear in real form.
'^ According to equations (106) and (107),

= cos 600*+ /sin 600*: 1

} ..... (144)
e - eooi* = cos GOO* -j sin 600*. j

Substituting (144) into (143) gives,

i = e~ 8Q()t \ Bi cos 600* + 2 sin 600* j, . . (145)

where BI and B 2 are combinations of the previous integration
constants A\ and A 2 thus,

Bi = Ai+A 2 , and B 2 =j(Ai-A 2 ). . . (146)

By substituting the numerical values, the condenser e.m.f.,
given by equation (138), then becomes,

e== -8oo<{ (40+30/)Ai(cos 600*+; sin 600*)

+ (40-30/)A 2 (cos 600*-/sin 600*) }
= - 800 ' { (40Bi + 30B 2 )cos 600* + (40B 2 - 30BO sin 600* } . (147)

92 ENGINEERING MATHEMATICS.

Since for t=Q, i = Q and e = 10,000 volts (140), substituting
into (145) and (147),

= i and 10,000 = 40 i+30 B 2 .
Therefore, #i = and 2 = 333 and, by (145) and (147),

. . (148)
e = 10,000- 800 < (cos 600 + 1.33 sin 600 t. J

As seen, in this case the current i is larger, and current
and e.m.f. are the product of an exponential term (gradually
decreasing value) and a trigonometric term (alternating value) ;
that is, they consist of successive alternations of gradually
decreasing amplitude. Such functions are called oscillating
functions. Practically all disturbances in electric circuits
consist of such oscillating currents and voltages.

= 2n gives, as the time of one complete period,

27T

and the frequency is

y = = 95 .3 cycles per sec.

In this particular case, as the resistance is relatively high,
the oscillations die out rather rapidly.

values of i and e, and of their exponential terms, for every 30

T T T

degrees, that is, for = 0, -^, 2-r^, 3 TT>, etc., for the first two

periods, and also to derive the equations, and calculate and plot
the numerical values, for the same capacity, C = 20 mf., and
same inductance, L = 0.05/1, but for the much lower resistance,
r = 20 ohms.

65. Tables of e +x and e~ x , for 5 decimals, and tables of
log e +x and log e~ x , for 6 decimals, are given at the end of
the book, and also a table of e~ x for 3 decimals. For most
engineering purposes the latter is sufficient; where a higher
accuracy is required, the 5 decimal table may be used, and for

POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 93

highest accuracy interpolation by the logarithmic table may be
employed. For instance,

-13.6847 = ?

From the logarithmic table,

log s-io =5.657055,
log ~ 3 =8.697117,
log - 6 =9.739423,
log -o.o8 = 9.965256,

[ interpolated,

log -o.oo4 7 = 9.998133 { between log - 004 = 9.998263,
[andlogr- 0005 =9.997829),

log -13-6847 = 4.056984 = 0.056984 - 6.
From common logarithmic tables,

-13-6847 =

CHAPTER III.

TRIGONOMETRIC SERIES.

A. TRIGONOMETRIC FUNCTIONS.

66. For the engineer, and especially the electrical engineer,
a perfect familiarity with the trigonometric functions and
trigonometric formulas is almost as essential as familiarity with
the multiplication table. To use trigonometric methods
efficiently, it is not sufficient to understand trigonometric
formulas enough to be able to look them up when required,
but they must be learned by heart, and in both directions; that
is, an expression similar to the left side of a trigonometric for-
mula must immediately recall the right side, and an expression
similar to the right side must immediately recall the left side
of the formula.

Trigonometric functions are defined on the circle, and on
the right triangle.

Let in the circle, Fig. 28, the direction to the right and
upward be considered as positive, to the left and downward as
negative, and the angle a be counted from the positive hori-
zontal OA, counterclockwise as positive, clockwise as negative.

The projector s of the angle a, divided by the radius, is
called sin a; the projection c of the angle a, divided by the

The intercept t on the vertical tangent at the origin A,
divided by the radius, is called tan a ; the intercept ct on the
horizontal tangent at J5, or 90 deg., behind A } divided by the

Thus, in Fig. 28,

s c

sma=-; cosa = -

t ct

tan = -; cot a = .

94

TRIGONOMETRIC SERIES.

95

In the right triangle, Fig. 29, with the angles a and ft
opposite respectively to the cathetes a and b, and with the
hypotenuse c, the trigonometric functions are :

a

-; cos

c

r; cot a=

. . (2)

By the right triangle, only functions of angles up to 90 deg.,
or -, can be defined, while by the circle the trigonometric

functions of any angle are given. Both representations thus
must be so familiar to the engineer that he can see the trigo-

FIG. 28. Circular Trigonometric
Functions.

FIG. 29. Triangular Trigono-
metric Functions.

nometric functions and their variations with a change of the
angle, and in most cases their numerical values, from the
mental picture of the diagram.

67. Signs of Functions. In the first quadrant, Fig. 28, all
trigonometric functions are positive.

In the second quadrant, Fig. 30, the sin a is still positive,
as s is in the upward direction, but cos a is negative, since c
is toward the left, and tan a and cot a also are negative, as t
is downward, and ct toward the left.

In the third quadrant, Fig. 31, sin a and cos are both

ENGINEERING MATHEMATICS.

negative: s being downward, c toward the left; but tan a and
cot a are again positive, as seen from t and ct in Fig. 31.

In the fourth quadrant, Fig. 32, sin a is negative, as s is
downward, but cos a is again positive, as c is toward the right;

tan a and cot a are both
negative, as seen from t and
ct in Fig. 32.

the trigonometric functions
again have the same values
as in the first quadrant, Fig.
28, that is, 360 deg., or 2;r,
or a multiple thereof, can be
the angle a, without changing
the trigonometric functions,
but these functions repeat
after every 360 deg., or 2?r;
or 360 deg. as their period.

that is, have

SIGNS OF FUNCTIONS

Function.

Positive.

Negative.

sin a.
cos a
tan a.
cot a

1st and 2d
1st and 4th
1st and 3d
1st and 3d

2d and 3d
2d and 4th
2d and 4th

(3)

TRIGONOMETRIC SERIES. 97

68. Relations between sin a and cos a. Between sin a and
cos a the relation,

sin 2 a+cos 2 a = l, (4)

exists; hence,

sin a = Vl cos 2 a ; ] , . v

cos a =\/l sin 2 a. J

Equation (4) is one of those which is frequently used in
both directions. For instance, 1 may be substituted for the
sum of the squares of sin a and cos a, while in other cases
sin 2 a +cos 2 a may be substituted for 1. For instance,

1 sin 2 a + cos 2 a /sina\ 2
7T- =- -J +I=tan 2 a + l.

cos 2 a cos 2 a: xcosa:/

Relations between Sines and Tangents.

sin a *

(5)

tan a =

cos a
cot a=-^

hence

cot a = 7

tan "'.[ ...... (5a)

COt a

As tan a and co^ a are far less convenient for trigonometric
calculations than sin a and cos a, and therefore are less fre-
quently applied in trigonometric calculations, it is not neces-
sary to memorize the trigonometric formulas pertaining to
tan a and cot , but where these functions occur, sin a. and
and cos a. are substituted for them by equations (5), and the
calculations carried out with the latter functions, and tan a
or cot a resubstituted in the final result, if the latter contains

sin a

- , or its reciprocal.
cos a

In electrical engineering tan a or cot a frequently appears
as the starting-point of calculation of the phase of alternating
currents. For instance, if a is the phase angle of a vector

98 ENGINEERING MATHEMATICS.

quantity, tan a is given as the ratio of the vertical component
over the horizontal component, or of the reactive component
over the power component.
In this case, if

tan a = j- ,

a *

sm a = , and coso:= .

or, if

c

cot ct = -7 .
a

sin a = ' , and cos a= , . . (5c)

Vc 2 +d 2

The secant functions, and versed sine functions are so
little used in engineering, that they are of interest only as
curiosities. They are defined by the following equations:

1

sec a =

cos a

1

cosec a = - ,

sin a

sin vers a = I sin a,
cos vers a = 1 cos a.

69. Negative Angles. From the circle diagram of the
trigonometric functions follows, as shown in Fig. 33, that when
changing from a positive angle, that is, counterclockwise
rotation, to a negative angle, that is, clockwise rotation, s, t,
and ct reverse their direction, but c remains the same; that is,

(6)

sin ( a) = sin a, ]
cos ( )= +cos ctj
tan ( a) = tan a,
cot ( a) cot a, J

cos a thus is an " even function/' while the three others are
" odd functions."

TRIGONOMETRIC SERIES.

99

Supplementary Angles. From the circle diagram of the
trigonometric functions follows, as shown in Fig. 34, that by
changing from an angle to its supplementary angle, s remains
in the same direction, but c, t, and ct reverse their direction,
and all four quantities retain the same numerical values, thus,

sin (TT a)= +sin a, -\

COS (TT a) = cos a,
tan (TT a) = tan a,
cot (n a) cot a.

FIG. 33. Functions of Negative
Angles.

FIG. 34. Functions of Supplementary
Angles.

Complementary Angles. Changing from an angle a to its

complementary angle 90 a, or - a, as seen from Fig. 35,

&

the signs remain the same, but s and c, and also t and ct exchange
their numerical values, thus,

COS

tan
cot

x- a I = cos a-,
/

2 aj=sin a,

fi-r a J MC pt a,

(H

tan a.

(8)

100

ENGINEERING MATHEMATICS.

70. Angle (an). Adding, or subtracting n to an angle a,
gives the same numerical values of the trigonometric functions

FIG. 35. Functions of Complemen- FIG. 36. Functions of Angles Plus
tary Angles. or Minus TT.

as a, as seen in Fig. 36, but the direction of s and c is reversed,
while t and ct remain in the same direction, thus,
sin (ax) = sin a, -

COS (a7r) = cos a,
tan (a n) = +tan a,
cot (a TT) = +cot a. -

FiG. 37. Functions of Angles + 1-. FIG. 38. Functions of Angles Minus |-.

Angle(o:^). Adding ^-, or 90 deg. to an angle <*, inter-
changes the functions, s and c, and t and ct } and also reverses

TRIGONOMETRIC SERIES.

101

the direction of the cosine, tangent, and cotangent, but leaves
the sine in the same direction, since the sine is positive in the
second quadrant, as seen in Fig. 37.

Subtracting , or 90 deg. from angle a, interchanges the

functions, s and c, and t and ct, and also reverses the direction,
except that of the cosine, which remains in the same direction;
that is, of the same sign, as the cosine is positive in the first
and fourth quadrant, as seen in Fig. 38. Therefore,

jr ) = +COS ,

COS

tan la+-\ == cot a,

cot (a 4-^- )== tan a,
sin ( a ^ } = cos a,

(10)

COS a - =

tan (a - }-=-

(ii)

cot I a I = tan a.

Numerical Values. From the circle diagram, Fig. 28, etc.,
follows the numerical values :

sin =

cos = 1

tan =

cot = oo

sin 30 = \

cos 30 = *\/3

tan 45 =1

cot 45 = 1

sin 45 = i\/2

cos 45==i\/2

tan 90 = co

cot 90 =

sin 60 = \ V 3

cos 60 = \

tan 135= -1

cot 135=- 1

sin 90 =1

cos 90 =

etc.

etc.

sin 120 = \/3

cos 120= -\

etc.

etc.

(12)

102

ENGINEERING MATHEMATICS.

(13)

7i. Relations between Two Angles. The following relations
are developed in text-books of trigonometry :

sin (a +/?) =sin a cos /? + cos a sin /?, 1
sin ( /?)= sin a: cos /? cos a sin /?,
cos (a +/5) =cos a cos /? sin a sin /?,
cos ( /?) = cos a cos /?+ sin a sin /?, J

Herefrom follows, by combining these equations (13) in
pairs :

cos a cos/? = J{cos (+/?) +cos (a /?) j,

sin a: sin /? = J { cos (a /?) cos (a
sin a cos/? = J{sin (a+/?)+sin (a (
cos a sin/5 = J{sin (+/?) sin (a ^9)}.

By substituting a\ for (a+j9), and /?i for (a t 8) in these
equations (14), gives the equations,

(14)

sin i +sin /?i = 2 sin =-^
sin a i sin ,#i = 2 sin ^
cos i+cos/?i= 2cos 7p

cos

cos

cos OLI cos

.

2 sin

cos

sm

(15)

These three sets of equations are the most important trigo-
nometric formulas. Their memorizing can be facilitated by
noting that cosine functions lead to products of equal func-
tions, sine functions to products of unequal functions, and
inversely, products of equal functions resolve into cosine,
products of unequal functions into sine functions. Also cosine
functions show a reversal of the sign, thus: the cosine of a
sum is given by a difference of products, the cosine of a differ-
ence by a sum, for the reason that with increasing angle
the cosine function decreases, and the cosine of a sum of angles
thus would be less than the cosine of the single angle.

TRIGONOMETRIC SERIES. 103

Double Angles. From (13) follows, by substituting a for /?:

sn a = sn a cos a,
cos 2a = cos 2 a sin 2

= 2 cos 2 a- 1,
= 1-2 sin 2 a.

. . . . (16)

Herefrom follow

1 cos 2 a
sin 2 a =

1+ COS 2 ft

and cos 2 = ^ f . (loa)

72. Differentiation.
d

-7- (sin x)= -I- cos x,

fi 7*

d

dx

(cos z) = sin x.

(17)

The sign of the latter differential is negative, as with an
increase of angle a, the cos a decreases.

Integration.

j sin ada = cos a,

flerefrom follow the definite integrals :

Jc + 2^ ,

sin (a+a)da=Q;

^*c+2jr

I cos (a+a)da = Q;

Jc

JC + 1C
sin (a + a)da = 2 cos (c + a) ;

(18)

(18a)

JC + 1C
(

cos(a+a)da=-2sin (c+a);

. . (186)

104

ENGINEERING MATHEMATICS.

**

L
f

I

r

sin acta = ;

cos

(18c)

= +1;

(18d)

73. Binomial. One of the most frequent trigonometric
operations in electrical engineering is the transformation of the
binomial, a cos a + b sin a, into a single trigonometric function,
by the substitution, a = c cos p and & = c sin p; hence,

where

a cos a + 6 sin a=ccos (a p),

and

or, by the transformation, a = c sin q and & = c cos

a cos a + 6 sin a = csin (a+q), .
where

and

p\ . . .

(19)

b

a' ' * ' '

(20)

= c cos q }

2),

(21)

a

(22)

74. Polyphase Relations.

n /

y cos ( a:

+0

,

(23

where m and n are integer numbers.

Proof. The points on the circle which defines the trigo-
nometric function, by Fig. 28, of the angles (a + a -j,

TRIGONOMETRIC SERIES.

105

are corners of a regular polygon, inscribed in the circle and
therefore having the center of the circle as center of gravity.
For instance, for n = 7, ra = 2, they are shown as PI, P%, . . . P?,
in Fig. 39. The cosines of these angles are the projections on
the vertical, the sines, the projections on the horizontal diameter,
and as the sum of the projections of the corners of any polygon,

FIG. 39. Polyphase Relations.

FIG. 40. Triangle.

on any line going through its center of gravity, is zero, both
sums of equation (23) are zero.

cos

cos

n

= cos a ~

2min n

n

(24)

These equations are proven by substituting for the products
the single functions by equations (14), and substituting them
in equations (23).

75. Triangle. If in a triangle a, /?, and ? are the angles,
opposite respectively to the sides a, 6, c, Fig. 40, then,

sin a -T- sin /? -r- sin r = a + b + c,

i i i i i i i i i i i i

(25)

(26)

(27)

106 ENGINEERING MATHEMATICS.

cos i-Tr

or

ab sin r
Area= -g-

c 2 sin sin /?
sin 7-

B. TRIGONOMETRIC SERIES.

76. Engineering phenomena usually are either constant,
transient, or periodic. Constant, for instance, is the terminal

Online LibraryCharles Proteus SteinmetzEngineering mathematics; a series of lectures delivered at Union college → online text (page 6 of 17)