Copyright
Charles Proteus Steinmetz.

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

. (page 3 of 17)
Online LibraryCharles Proteus SteinmetzEngineering mathematics; a series of lectures delivered at Union college → online text (page 3 of 17)
Font size
QR-code for this ebook


if c>l, and is negative, if c<l, but >0. The reverse is the
case, if a<l. Thus, the logarithm traverses all positive and
negative values for the positive values of c, and the logarithm
qfci^iegative number thus can be neither positive nor negative.

log a ( c)=log a c+loga ( 1), and the question of finding
the logarithms of negative numbers thus resolves itself into
finding the value of log a ( 1).

There are two standard systems of logarithms one with
the base = 2.71828. . .*, and the other with the base 10 is
used, the former in algebraic, the latter in numerical calcula-
tions. Logarithms of any base a can easily be reduced to any
other base.

For instance, to reduce 6 = log a c to the base 10:6 = log a c
means, in the form of involution : a b = c. Taking the logarithm
hereof gives, b logic a = logio c, hence,

, logio c logio c

o = -j - : or loga c = , .
logio a' logio a

Thus, regarding the logarithms of negative numbers, we need
to consider only logio ( 1) or log ( 1).

If jx = lo& (-1), then '*= -1,

and since, as will be seen in Chapter II,

e jx = cos x + j sin x t
it follows that,

cos x + j sin x = 1,



* Regarding e, see Chapter II, p. 71.



22 ENGINEERING MATHEMATICS.

Hence, x = x, or an odd multiple thereof, and

Ioge(-l)=j7r(2n



where n is any integer number.

Thus logarithmation also leads to the quadrature number
y, but to no further extension of the system of numbers.



Quaternions.

16. Addition and subtraction, multiplication and division,
involution and evolution and logarithmation thus represent all
the algebraic operations, and the system of numbers in which
all these operations can be carried out under all conditions
is that of the general number, a+j'6, comprising the ordmary
number a and the quadrature number jb. The number a as
well as 6 may be positive or negative, may be integer, fraction
or irrational.

Since by the introduction of the quadrature number jb,
the application of the system of numbers was extended from the
line, or more general, one-dimensional quantity, to the plane,
or the two-dimensional quantity, the question arises, whether
the system of numbers could be still further extended, into
three dimensions, so as to represent space geometry. While
in electrical engineering most problems lead only to plain
figures, vector diagrams in the plane, occasionally space figures
would be advantageous if they could be expressed algebra-
ically. Especially in mechanics this would be of importance
when dealing with forces, as vectors in space.

In the quaternion calculus methods have been devised to
deal with space problems. The quaternion calculus, however,
has not yet found an engineering application comparable with
that of the general number, or, as it is frequently called, the
complex quantity. The reason is that the quaternion is not
an algebraic quantity, and the laws of algebra do not uniformly
apply to it.

17. With the rectangular coordinate system in the plane,
Fig. 11, the x axis may represent the ordinary numbers, the y
axis the quadrature numbers, and multiplication by j=Vl
represents rotation by 90 deg. For instance, if PI is a point



THE GENERAL NUMBER. 23

= 3+2/s, the point P 2 , 90 deg. away from PI, would



be:



To extend into space, we have to add the third or z axis,
as shown in perspective in Fig. 12. Rotation in the plane xy,
by 90 deg., in the direction +x to +y, then means multiplica-
tion by j. In the same manner, rotation in the yz plane, by
90 deg., from +y to +z, would be represented by multiplica-




FIG. 11. Vectors in a Plane.

tion with h, and rotation by 90 deg. in the zx plane, from +z
to +x would be presented by k, as indicated in Fig. 12.

All three of these rotors, j, h, k, would be V-l, since each,
applied twice, reverses the direction, that is, represents multi-
plication by (1).

As seen in Fig. 12, starting from +x, and going to +y }
then to +2, and then to +x, means successive multiplication
by jj h and k, and since we come back to the starting point, the
total operation produces no change, that is, represents mul-
tiplication by ( + 1). Hence, it must be,

jhk= +1.



24



ENGINEERING MATHEMATICS.



Algebraically this is not possible, since each of the three quan-
tities is V 1, and V IxV ixV 1= V 1, and not

( + 1).



+z




FIG. 12. Vectors in Space, /A&= +1.



If we now proceed again from x, in positive rotation, but
first turn in the xz plane, we reach by multiplication with k
the negative z axis, z, as seen in Fig. 13. Further multiplica-




FIG. 13. Vectors in Space, khj= -1.

tion by h brings us to +y, and multiplication by j to -x, and
in this case the result of the three successive rotations by



THE GENERAL NUMBER. 25

90 deg., in the same direction as in Fig. 12, but in a different
order, is a reverse; that is, represents (-1). Therefore,



and hence,

jhk = khj.

Thus, in vector analysis of space, we see that the fundamental
law of algebra,



does not apply, and the order of the factors of a product is
not immaterial, but by changing the order of the factors of the
product jhk, its sign was reversed. Thus common factors
cannot be canceled as in algebra; for instance, if in the correct
expression, jhk = khj, we should cancel by /, h and k, as could be
done in algebra, we would get + 1 = 1, which is obviously wrong.
For this reason all the mechanisms devised for vector analysis
in space have proven more difficult in their application, and
have not yet been used to any great extent in engineering
practice.

B. ALGEBRA OF THE GENERAL NUMBER, OR COMPLEX

QUANTITY.

Rectangular and Polar Coordinates.

18. The general number, or complex quantity, a+jb, is
the most general expression to which the laws of algebra apply.
It therefore can be handled in the same manner and under
the same rules as the ordinary number of elementary arithmetic.
The only feature which must be kept in mind is that j 2 = 1, and
where in multiplication or other operations j 2 occurs, it is re-
placed by its value, -1. Thus, for instance,

(a + jb) (c + jd) = ac + jad + jbc + fbd
= ac+jad+jbc-bd
= (ac-bd)+j(ad + bc).

Herefrom it follows that all the higher powers of j can be
eliminated, thus:



f =+/, . . . etc.



ENGINEERING MATHEMATICS.

In distinction from the general number or complex quantity,
the^orotinary numbers, - + a and -a, are occasionally called
I or real numbers. The general number thus consists
the.jcombination of a scalar or real number and a quadrature
number, or imaginary number.

Since a quadrature number cannot be equal to an ordinary
number it follows that, if two general numbers are equal,
their real components or ordinary numbers, as well as their
quadrature numbers or imaginary components must be equal,
thus, J^^

then,

a = c and b = d.

Every equation with general numbers thus can be resolved
into two equations, one containing only the ordinary numbers,
the other only the quadrature numbers. For instance, if

x+fy-5-Bj,

then,

x = 5 and y= 3.

19. The best way of getting a conception of the general
number, and the algebraic operations with it, is to consider
the general number as representing a point in the plane. Thus
the general number a+/& = 6+2.5/ may be considered as
representing a point P, in Fig. 14, which has the horizontal
distance from the y axis, OA = BP = a = 6, and the vertical
distance from the x axis, OB = AP = b = 2.5.

The total distance of the point P from the coordinate center
then is

OP =




and the angle, which this distance OP makes with the x axis,
is given by

} AP 2.5
~~OA~ 6

- 0.417.

a



THE GENERAL NUMBER.



27



Instead of representing the general number by the two
components, a and &, in the form a+jb, it can also be repre-
sented by the two quantities:

The distance of the point P from the center 0,



and the angle between this distance and the x axis,

-.

a




Fis. 14. Rectangular and Polar Coordinates.

Then referring to Fig. 14,

a = c cos and b = c sin 0,

and the general number a+jb thus can also be written in the
form,

c(cos 0+/sin 0).

The form a+jb expresses the general number by its
rectangular components a and b, and corresponds to the rect-
angular coordinates of analytic geometry; a is the x coordinate,
6 the y coordinate.

The form c(cos0+; sin 6) expresses the general number by
what may be called its polar components, the radius c and the



28 ENGINEERING MATHEMATICS.

angle 6, and corresponds to the polar coordinates of analytic
geometry, c is frequently called the radius vector or scalar,
6 the phase angle of the general number.

While usually the rectangular form a+jb is more con-
venient, sometimes the polar form c(cos 6 +j sin 0) is preferable,
and transformation from one form to the other^therefore fre-
quently applied.

Addition and Subtraction.

20. If a\ +jbi = 6+2.5 j is represented by the point PI;
this point is reached by going the horizontal distance ai = 6
and the vertical distance &i =2.5. If a 2 +jb 2 = 3+4j is repre-
sented by th$ point P 2 , this point is reached by going the
horizontal distance a 2 = 3 and the vertical distance 62 = 4.

The sum of the two general numbers (ai +/&i) + (a 2 +jb 2 ) =
(6+2.5?) + (3+4/), then is given by point PO, which is reached
by going a horizontal distance equal to the sum of the hor-
izontal distances of PI and P 2 : a = ai + a 2 = 6+3 = 9, and a
vertical distance equal to the sum of the vertical distances of
PI and P 2 : &o == fri+&2 = 2.5 + 4 = 6.5, hence, is given by the
general number

do +fi = (ai + a 2 ) +/(&i + b 2 y
-9+6.5?.

Geometrically, point PO is derived from points PI and P 2?
by the diagonal OP of the parallelogram OPiPoP 2; constructed
with OPi and OP 2 as sides, as seen in Fig. 15.

Herefrom it follows that addition of general numbers
represents geometrical combination by the parallelogram law.

Inversely, if PO represents the number



and PI represents the number



the difference of these numbers will be represented by a point
P 2 , which is reached by going the difference of the horizontal



THE GENERAL NUMBER.



29



distances and of the vertical distances of the points PO and
PI. P 2 thus is represented by

2 = o 01 =9 6 = 3,
and



.



Therefore, the difference of the two general numbers (a +/&o)
and (ai + /&i) is given by the general number:



a 2 +fi 2 = (do di) +j(bo



as seen in Fig. 15.




FIG. 15. Addition and Subtraction of Vectors.

This difference a 2 +jb 2 is represented by one side OP 2 of
the parallelogram OPiP P 2 , which has OPi as the other side,
and OP as the diagonal.

Subtraction of general numbers thus geometrically represents
the resolution of a vector OP into two components OPi and
OP 2 , by the parallelogram law.

Herein lies the main advantage of the use of the general
number in engineering calculation : If the vectors are represented
by general numbers (complex quantities), combination and
resolution of vectors by the parallelogram law is carried out by



30 ENGINEERING MATHEMATICS.

simple addition or subtraction of their general numerical values,
that is, by the simplest operation of algebra.

21. General numbers are usually denoted by capitals, and
their rectangular components, the ordinary number and the
quadrature number, by small letters, thus:



the distance of the "point which represents the general number A
from the coordinate center is called the absolute value, radius
or scalar of the general number or complex quantity. It is
the vector a in the polar representation of the general number:

A = a(cos 6 +j sin 0),



and is given by a = ai 2 + a<?.

The absolute value, or scalar, of the general number is usually
also denoted by small letters, but sometimes by capitals, and
in the latter case it is distinguished front the general number by
using a different type for the latter, or underlining or dotting
it, thus:

A = a\ + jaz ; or A=a\ J f- jo,* ;



or = ai+a,2' or =



a = ai 2 + a 2 2 ; or
and a\ +ja,2 = a (cos 6 + j sin 0) ;

or ai 4- ja,2 = A (cos + / sin 6).

22. The absolute value, or scalar, of a general number is
always an absolute number, or positive, that is, the sign of the
rectangular component is represented in the angle 0. Thus
referring to Fig. 16,

A =a\
gives, a




and A =5 (cos 37 deg. + / sin 37 deg) .



THE GENERAL NUMBER.



31



The expression
gives



A = a\ -f ja,2 = 43;



tan 0=- =
6 = -37 deg. ; or



- 0.75;

= 180 -37 = 143 deg.




FIG. 16. Representation of General Numbers.

Which of the two values of 6 is the correct one is seen from
the condition a\ = a cos 6. As ai is positive, +4, it follows
that cos must be positive; cos (37 deg.) is positive, cos 143
deg. is negative : hence the former value is correct :

A = 5(cos( -37 deg.) +/ sin( -37 deg.)f
= 5(cos 37 deg. / sin 37 deg.).

Two such general numbers as (4+3/) and (4 3/), or,
in general,

(a+jb) and (ajb),

are called conjugate numbers. Their product is an ordinary
and not a general number, thus : (a+jb) (a jb) = a 2 +b 2 .



32 ENGINEERING MATHEMATICS.

The expression

A = a 1 +ja 2 = -4+3;
gives



tan0=-| = -0.75;

6 = -37 deg. or - 180 -37 = 143 cleg. ;

but since a\ = a cos 6 is negative, 4, cos must be negative,
hence, = 143 deg. is the correct value, and

4=5(cos 143 deg. +;'sin 143 deg.)
= 5( -cos 37 deg. +/ sin 37 deg.)

The expression

A = ai+ja 2 = -4 3;

gives

a=

6> = 37deg.; or = 180+37 = 217 deg.;

but since a\=a cos 6 is negative, 4, cos 6 must be negative,
hence = 217 deg. is the correct value, and,

4 = 5 (cos 217 deg. +;' sin 217 deg.)
= 5( - cos 37 deg. -;' sin 37 deg.)

The four general numbers, +4+3;, +43;, 4+3;, and
4 3;, have the same absolute value, 5, and in their repre-
sentations as points in a plane have symmetrical locations in
the four quadrants, as shown in Fig. 16.

As the general number A = a\+ja2 finds its main use in
representing vectors in the plane, it very frequently is called
a vector quantity, and the algebra of the general number is
spoken of as vector analysis.

Since the general numbers A = a\+ja 2 can be made to
represent the points of a plane, they also may be called plane
numbers, while the positive and negative numbers, + a and a.



THE GENERAL NUMBER.



33



may be called the linear numbers, as they represent the points
of a line.



Example : Steam Path in a Turbine.



23. As an example of a simple operation with general num-
bers one may calculate the steam path in a two-wheel stage
of an impulse steam turbine.




FIG. 17. Path of Steam in a Two-wheel Stage of an Impulse Turbine.

Let Fig. 17 represent diagrammatically a tangential section
through the bucket rings of the turbine wheels. Wi and W 2
are the two revolving wheels, moving in the direction indicated
by the arrows, with the velocity s = 400 feet per sec. / are
the stationary intermediate buckets, which turn the exhaust
steam from the first bucket wheel Wi, back into the direction
required to impinge on the second bucket wheel W%. The
steam jet issues from the expansion nozzle at the speed s = 2200



34



ENGINEERING MATHEMATICS.



feet per sec., and under the angle = 20 deg., against the first
bucket wheel W\.

The exhaust angles of the three successive rows of buckets,
Wij I, and W 2 , are respectively 24 deg., 30 deg. and 45 cleg.
These angles are calculated from the section of the bucket
exit required to pass the steam at its momentary velocity,
and from the height of the passage required to give no steam
eddies, in a manner which is of no interest here.

As friction coefficient in the bucket passages may be assumed
^ = 0.12; that is, the exit velocity is 1 k f =O.SS of the entrance
velocity of the steam in the buckets.




FKJ. 18. Vector Diagram of Velocities of Steam in Turbine.

Choosing then as x-axis the direction of the tangential
velocity of the turbine wheels, as 2/-axis the axial direction,
the velocity of the steam supply from the expansion nozzle is
represented in Fig. 18 by a vector OS of length s = 2200 feet
per sec., making an angle 6 = 20 deg. with the .r-axis; hence,
can be expressed by the general number or vector quantity:

So = So (cos 0o +j sin )

= 2200 (cos 20 deg. +/ sin 20 deg.)
= 2070 + 750/ft. per sec.

The velocity of the turbine wheel Wi is s = 400 feet per second,
and represented in Fig. 18 by the vector OS, in horizontal
direction.



THE GENERAL NUMBER. 35

The relative velocity with which the steam enters the bucket
passage of the first turbine wheel Wi thus is :



= (2070 +750J) -400
= 1670 +7^6/ ft. per sec.

This vector is shown as OSi in Fig. 18.
The angle A , under which the steam enters the bucket
passage thus is given by

^50
tan 0i = = 0.450, as d l = 24.3 deg.



This angle thus has to be given to the front edge of the
buckets of the turbine wheel Wi.

The absolute value of the relative velocity of steam jet
and turbine wheel JFi, at the entrance into the bucket passage,
is

i = \ / 1670 2 + 750 2 = 1830 ft. per sec.

In traversing the bucket passages the steam velocity de-
creases by friction etc., from the entrance value Si to the
exit value

and since the exit angle of the bucket passage has been chosen
as 02 = 24 deg., the relative velocity with which the steam
leaves the first bucket wheel Wi is represented by a vector
0$2 in Fig. 18, of length 2 = 1610, under angle 24 deg. The
steam leaves the first wheel in backward direction, as seen in
Fig. 17, and 24 deg. thus is the angle between the steam jet
and the negative x-axis; hence, 02 = 18024 = 156 deg. is the
vector angle. The relative steam velocity at the exit from
wheel Wi can thus be represented by the vector quantity

$2 = 82(008 02 +j sin 2 )

= 1610 (cos 156 deg. +/ sin 156 deg.)
- 1470 + 655 /.

Since the velocity of the turbine wheel Wi is s = 400, the
velocity of the steam in space, after leaving the first turbine



36 ENGINEERING MATHEMATICS.

wheel, that is, the velocity with which the steam enters the
intermediate /, is



= 470 +655?) +400
-1070 + 655/,

and is represented by vector 08$ in Fig. 18.
The direction of this steam jet is given by



tan 3 = - = -0.613,

as

6 3 = -31.6 deg.; or, 180-31.6 = 148.4 deg.

The latter value is correct, as cos 63 is negative, and sin 63 is
positive.

The steam jet thus enters the intermediate under the angle
of 148.4 deg.; that is, the angle 180-148.4 = 31.6 deg. in opposite
direction. The buckets of the intermediate / thus must be
curved in reverse direction to those of the wheel W i} and must
be given the angle 31.6 deg. at their front edge.

The absolute value of the entrance velocity into the inter-
mediate / is



ft. per sec.

In passing through the bucket passages, this velocity de-
creases by friction, to the value :

S 4 = s 3 (l -k f ) = 1255 X0.88 = 1105 ft. per sec.,

and since the exit edge of the intermediate is given the angle:
#4 = 30 deg., the exit velocity of the steam from the intermediate
is represented by the vector OS^ in Fig. 18, of length s 4 = 1105,
and angle 4 = 30 deg., hence,

4 = 1105 (cos 30 deg. +j sin 30 deg.)
= 955+550/ ft. per sec.

This is the velocity with which the steam jet impinges
on the second turbine wheel W 2 , and as this wheel revolves



THE GENERAL NUMBER. 37

with velocity s = 400, the relative velocity, that is, the velocity
with which the steam enters the bucket passages of wheel W 2 , is,



= (955 + 550?) -400
= 555 + 550/ ft. per sec.;

and is represented by vector OS 5 in Fig. 18.
The direction of this steam jet is given by

550
tan 5 = 7^- = 0.990, as 5 = 44.8 deg.



Therefore, the entrance edge of the buckets of the second
wheel W 2 must be shaped under angle #5 = 44.8 deg.
The absolute value of the entrance velocity is



ft. per sec.

In traversing the bucket passages, the velocity drops from
the entrance value S 5 , to the exit valve,

s 6 = s 5 (l-k f ) =780X0.88 = 690 ft. per sec.

Since the exit angles of the buckets of wheel W 2 has been
chosen as 45 deg., and the exit is in backward direction, 6 =
180-45=135 deg., the steam jet velocity at the exit of the
bucket passages of the last wheel is given by the general number

$6 =SG(COS #6 +/ sin 0&)

= 690 (cos 135 deg. +/ sin 135 deg.)
= -487+487j ft. per sec.,



and represented by vector 08$ in Fig. 18.

Since s = 400 is the wheel velocity, the velocity of the
steam after leaving the last wheel W 2 , that is, the "lost"
or " rejected " velocity, is



= (487 +487?) +400
-87+487/ft. per sec.,

and is represented by vector 087 in Fig. 18.



38 ENGINEERING MATHEMATICS.

The direction of the exhaust steam is given by,

487
tan 7 = - 37- = -5.6, as ^ 7 = 180 -80 = 100 deg.,

and the absolute velocity is,



s 7 = \/87 2 +487 2 = 495 ft. per sec.
Multiplication of General Numbers.



24. If A = ai+ja,2 and B = bi+jb 2 , are two general, or
plane numbers, their product is given by multiplication, thus :



+fa 2 b 2 ,
and since j' 2 = -1,

AB = (ai&i -a 2 b 2 ) +j(aib 2 +$2^1),

and the product can also be represented in the plane, by a point,

C = ci+jc 2 ,
where,

d =a\b\ a 2 b 2)
and



For instance, A=2+j multiplied by .B=*l+1.5/ gives
ci=2Xl-lXl.5 = 0.5,



hence,

C =

as shown in Fig. 19.

25. The geometrical relation between the factors A and B
and the product C is better shown by using the polar expression ;
hence, substituting,



which gives
a

tan a


u,i=u, uus a
tt 2 = tt Sin OL


\ and
and


u\ ==u uu p i

b 2 = b sin/?/'




= Vai 2 +d2 2
_a 2


&2


b 2 2



THE GENERAL NUMBER.

the quantities may be written thus :

A =a(cos a+j sin a);



39



and then,

C = AB = ab(cos a+j sin a) (cos /9+ j sin /?)
= ab { (cos a cos /9 sin a sin /?) -f /(cos a sin /? +sin a cos /?)}
= ab {cos (a +/?)+/ sin (a




FIG. 19. Multiplication of Vectors.

that is, two general numbers are multiplied by multiplying their
absolute values or vectors, a and 6, and adding their phase angles
a and /?.

Thus, to multiply the vector quantity, A = ai+ja 2 = a (cos
a + / sin<$; by B = hi 4- jb 2 = b (cos /? + / sin /?) the vector OA in Fig.
19, which represents the general number A, is increased by the
factor 6 = V& 1 2 + & 2 2 , and rotated by the angle /?, which is given

b 2

by tan P = T->
bi

Thus, a complex multiplier 5 turns the direction of the
multiplicand A } by the phase angle of the multiplier J5, and
increases the absolute value or vector of A, by the absolute
value of B as factor.



40 ENGINEERING MATHEMATICS.

The multiplier B is occasionally called an operator, as it
carries out the operation of rotating the direction and changing
the length of the multiplicand.

26. In multiplication, division and . other algebraic opera-
tions with the representations of physical quantities (as alter-
nating currents, voltages, impedances, etc.) by mathematical
symbols, whether ordinary numbers or general numbers, it
is necessary to consider whether the result of the algebraic
operation, for instance, the product of two factors, has a
physical meaning, and if it has a physical meaning, whether
this meaning is such that the product can be represented in
the same diagram as the factors.

For instance, 3X4 = 12; but 3 horses X 4 horses does not
give 12 horses, nor 12 horses 2 , but is physically meaningless.
However, 3 ft. X4 ft. = 12 sq.ft. TKus, if the numbers represent

$ i loot i i i i i i o i i i
O A B C

FIG. 20.

horses, multiplication has no physical meaning. If they repre-
sent feet, the product of multiplication has a physical meaning,
but a meaning which differs from that of the factors. Thus,
if on the lirxe in Fig. 20, OA = 3 feet, OB = 4 feet, the product,
12 square feet, while it has a physical meaning, cannot be
represented any more by a point on the same line; it is not
the point 0(7 = 12, because, if we expressed the distances OA
and OB in inches, 36 and 48 inches respectively, the product
would be 36X48 = 1728 sq.in., while the distance UC would be
144 inches.

27. In all mathematical operations with physical quantities
it therefore is necessary to consider at every step of the mathe-
matical operation, whether it still has a physical meaning,
and, if graphical representation is resorted to, whether the
nature of the physical meaning is such as to allow graphical
representation in the same diagram, or not.

An instance of this general limitation of the application of
mathematics to physical quantities occurs in the representation
of alternating current phenomena by general numbers, or
complex quantities.



THE GENERAL NUMBER.



41



An alternating current can be represented by a vector 01


1 3 5 6 7 8 9 10 11 12 13 14 15 16 17

Online LibraryCharles Proteus SteinmetzEngineering mathematics; a series of lectures delivered at Union college → online text (page 3 of 17)