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.

Adding and subtracting gives respectively,

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)

The reader is advised to calculate and plot the numerical

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.

The reader is advised to calculate and plot the numerical

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),

added

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

radius, is called cos a.

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

radius, is called cot a.

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.

FIG. 30. Second Quadrant.

FIG. 31. Third Quadrant.

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.

In the fifth quadrant all

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

added to, or subtracted from

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.

FIG. 32. Fourth Quadrant.

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

3d and 4th quadrant

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,

I cos ada = H-sina.

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;

cosada= +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

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.

Adding and subtracting gives respectively,

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)

The reader is advised to calculate and plot the numerical

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.

The reader is advised to calculate and plot the numerical

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),

added

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

radius, is called cos a.

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

radius, is called cot a.

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.

FIG. 30. Second Quadrant.

FIG. 31. Third Quadrant.

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.

In the fifth quadrant all

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

added to, or subtracted from

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.

FIG. 32. Fourth Quadrant.

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

3d and 4th quadrant

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,

I cos ada = H-sina.

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;

cosada= +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

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