In the previous SECTIONs it was shown how vectors can be resolved in terms of horizontal and vertical components, considering them as X and Y co-ordinates. It was also shown that a vector can be expressed in two different ways with respect to the co-ordinate axes. As an example, a vector could be ex pressed in polar form, such as 5/53: 1°; or the same vector can be expressed in rectangular form with an X value of 3 and a Y value of 4. With the rectangular form it is difficult to express the quantities with convenient algebraic notation. Therefore, in this SECTION a type of algebra, in which these vectors can be expressed, is introduced.
This is called Vector Algebra. When it is used, vectors can be added, subtracted, multiplied, or divided, all in algebraic form. If required, the vector can be raised to a power, or a specific root of the vector can be determined. These operations include both polar and rectangular form, plus conversion from one form to the other when that may be required.
Most of the groundwork for Vector Algebra has been introduced in previous SECTIONs. This includes the two forms, polar and rectangular, plus the basic idea of plotting vectors on co ordinate axes. The numbers above the horizontal (X) axis are positive; those below it are negative. Those numbers to the right of the vertical (Y) axis are positive; those to the left are negative. This enables us to plot angles in any of the four quadrants. Also X and Y can be positive or negative, depending on the quadrant in which the vector lies.
In this SECTION the algebraic expressions for vectors are introduced, and then the various mathematical operations involving such numbers are explained. The main requirement for set ting up such a system is that there be some way to show which is the X and which is the Y co-ordinate in the rectangular form.
The standard notation for polar form has already been introduced, and it also forms a part of Vector Algebra. In addition, Vector Algebra draws both from conventional algebra and from right-angle trig in order to determine problem solutions.
IMAGINARY NUMBERS
Algebraic notation for vectors in rectangular form is based on a concept called imaginary numbers. These numbers are not imaginary because they don't exist, but rather because they cannot be evaluated in real numbers. Imaginary numbers occur when the square root (or any even-numbered root) of a negative number is taken. When a number is squared (multi plied by itself), the resulting product is always a positive number. As examples:
3 x 3 = 9 (-3) x (-3) = 9
The results are the same regardless of whether a positive or a negative number is squared; the product is always positive.
Conversely, to take the square root of a positive number gives either a positive or a negative root. In a given problem it may not be known which is correct, or even if there is a correct one.
For example, Vg = 3, but V - 9 - could also be -3. Usually -\/. is shown as - ± -3, indicating that either root would satisfy the equation. In practice, however, the positive root is considered as the principal root because it has more meaning. To illustrate, suppose that a problem were solved and the answer showed 15 ohms of resistance. The negative value has no meaning, so the principal root, positive 15 ohms, would be assumed to be the answer. In the solution of an equation, however, say x 2 = 25, both +5 and -5 satisfy the equality so we say that there are two roots, ± -5.
In some calculations it is necessary to determine the square root of a negative quantity, for example VLS. When determining the square root of a number, the process is to find the number which when squared produces the quantity under the radical sign. Squaring always produces a positive quantity, so there can be no value for nor for the square root of any negative number. Hence, these are called imaginary numbers. It can be considered that an imaginary quantity consists of a real number and an imaginary component multiplied together, as shown by this example:
= V -1 = V -1
As long as this is true, the only imaginary number connected with this operation is \Pi. Any other quantity can be resolved similarly, indicating that the only imaginary number is -1.
A small i, representing the word "imaginary," is used as a symbol for V -1. This was formulated by mathematicians many years ago, and it could be considered that imaginary was an unfortunate choice of adjective. The word often creates misconceptions, especially to the beginning student of mathematics.
When electricity and electronics became practical realities, the letter "i" was used for instantaneous current, so a possibility of confusion arose. For this reason, the lower-case j is used for N/T. in electronics although most math books still use the conventional i. We can therefore assume that:
V-1 = i = j
Since this text is primarily concerned with electronics, the j will be used to indicate the imaginary number in every case.
This means that for example, would be written ±3/T or - ±3j, with the latter form preferred.
Before proceeding, one minor modification should be noted.
There is a possibility that 3j could be confused with terms such as 3a, 3x, etc. in an algebraic expression. So in electronics expressions the j is listed before its numerical coefficient, and 3j is noted as j3. But a word of caution with respect to this; be careful in writing these or else j3 could resemble j 3. Also, j should not be used in any algebraic expression unless the imaginary V-1 is intended. Any other usage could introduce doubt on the part of the reader and definite difficulties in problem solution.
A binomial is an expression that contains two terms, such as x + y, 2a - b, 3x - 4y, etc. Other binomials can consist of a real number (one that can be evaluated) and an imaginary component. As long as the imaginary term cannot be evaluated, it cannot be combined with the real number. Examples of such binomials are 3 + j5, 2 - j3, -5 - j4, -2 + j, etc. These are called complex numbers, and each consists of a real and an imaginary term. The numerical coefficient of the j term is always a real number, but an imaginary factor multiplied by a real one results in another imaginary term. We must consider also that either of these terms can have zero as its numerical coefficient. If the real-number coefficient is zero, the binomial is imaginary. If the j coefficient is zero, the complex quantity is a real number. Complex-number terms can also have literal coefficients, such as a + jb, x - jy, etc.
However, these occur more often in general expressions rather than those pertaining to specific problems.
The following assumptions must be made with regard to imaginary numbers:
j = j 2 = = -1 j 3 = j 2 • j = - 1 • j = -j j4 j2 j2 = 1) ( -1) = +1
Notice that j 2 equals -1.
This could be considered as a departure from the usual rules of algebra. This meaning, how ever, should be better understood after studying the graphic analysis of imaginary numbers, which is presented in the next section of this SECTION. Of the assumptions just made, j and j 2 occur much more often in practical problems than do the others. In every case, j 2 should be evaluated as -1 before proceeding in the problem being solved and j should be substituted for Graphic Analysis Two real numbers are plotted on the X axis in Fig. 4-1 and are labeled +A and -A. Counterclockwise rotation is assumed in designating the angles. A and -A are opposite each other and can be considered to be 180° apart. If A is multiplied by -1, the resultant product is -A. Then if -A is multiplied by -1, the resultant product is A. This means that each time a vector is multiplied by -1 the vector has been rotated 180°. Then if (-1) (A) is a rotation of 180° and j 2 equals -1, it can be assumed that a rotation of j turns the vector by 90°. This is also shown in Fig. 4-1, where A has been multiplied by j to give a product of jA. Another multiplication by j gives j 2A or -A, placing the vector at 180°. Then -A times j gives -jA which represents an angular position of 270°. And -jA times j gives -j2A or +A, after substituting -1 for the j 2 factor. A is then 360°, which is the same vector position as 0°. Note that real numbers occur on the X axis and that imaginary numbers occur on the Y axis, providing a method for plotting complex numbers on co-ordinate axes.
A complex number such as 3 + j5 can be plotted as in Fig. 4-2. The diagonal drawn from the origin is the hypotenuse of a right triangle, and the rectangular components are the shorter sides of the same triangle. Such a binomial could also be plotted in any of the other quadrants except that at least one of the rectangular components would be negative. The hypotenuse (vector length) is assumed to be positive regardless of the quadrant in which it appears.
These ideas are used in electrical and electronics calculations by plotting resistance on the real-number axis and reactance on the imaginary axis. By using the Pythagorean relationship, impedance is found to be the diagonal (hypotenuse). Such a plot can be made because of the 90° phase difference between resistance and reactance. This does not mean that reactance is imaginary, but the idea provides a convenient method of notation, as will be seen in the next SECTION.
Fig. 4-1. Multiplying A by j to give jA.
Fig. 4-2. Plotting a complex number.
Working With Imaginaries
Conventional math methods hold for imaginaries as long as j 2 is assumed to equal -1, and that substitution is made each time j 2 appears. The usual rule in most problems is to change V-1 to j each time it appears before attempting to perform any operations on the problem. This is especially evident in the example multiplication problems shown next.
1. V-3 • V-2 = iNfa • jV- 2 -= j 2V. e= -VG
Notice that (-3) multiplied by (-2) within the radicals would have given a product of +6, which is incorrect. The con version to the j form is also evident in the following examples of addition, subtraction, and division.
2. Here the j is treated as any other base number, and the numerical coefficients are added. But this may be illustrated better in the next example problem in which the radicands are perfect squares.
3. Subtraction is similar to addition, it merely involves algebraic addition of the numerical coefficients and uses the sign of the larger term.
4. j6 - j4 = j2
5. j3 - j8 = -j5 Example problem 6 involves division of imaginary terms in which the i's are divided out and do not appear in the quotient.
6.
\/-2 j N/2
As with other algebraic expressions involving radicals it is not considered proper form to leave the radical in the denominator, and j is considered to be a radical because it is equal to \/-1. The radical is eliminated from the denominator by rationalization, the process of multiplying the numerator and the denominator by the quantity that eliminates the radical from the denominator. An example of this is problem 7.
CONVERSION OF FORMS
It is frequently necessary to convert a vector from polar to rectangular form, or vice versa, when solving a particular problem. We have already seen that the polar and rectangular forms of a vector express the same thing, except in different ways. In the example previously given, 3 + j4 is the same as 5/53.1°. Each form has its uses, also its limitations, for algebraic purposes. For example, addition or subtraction of polar form vectors cannot be performed, except by graphic means.
Vector algebra is desirable in problem solutions because it pro vides greater accuracy than the graphical methods, although graphical methods provide a visual image of the problem being solved.
Fig. 4-3. Illustrating conversion of forms.
For an illustration of conversion of forms, Fig. 4-3 is used.
This shows a first-quadrant vector in which the X value is 3 and the Y value is 5. Expressed in rectangular form it is 3 + j5.
Probably the easiest way to convert to polar form is to use the Pythagorean formula to determine Z and the arc tangent to determine the angle θ. The polar form, expressed generally, would be Z/p.
Z VX2 172 = V9 ± 25 = N/34 = 5.83 Y 5 θ = arc tan x- = arc tan u = arc tan 1.66 = 58.9°
Therefore, 3 + j5 is the same as 5.83/58.9°. If the rectangular form had been 3 - j5, a fourth-quadrant angle would be described, and the polar form would be 5.83/-58.9°. The length of Z is the same, only the angle is changed.
The usual method of converting from polar to rectangular form was mentioned in the previous SECTION with regard to resolving a vector into its rectangular components. Here, the idea is expanded to include the j notation.
cos 9 = so X = Z cos 0 Z'
• Y sin 0 = - Z' so Y = Z sin 0 Y is plotted on the imaginary axis so it would be noted as jZ sin θ. The full conversion then is: Z/_9_ = Z cos 0 + jZ sin 0 or, = Z (cos 0 + j sin 0)
This is sometimes referred to as the trigonometric form, but it is merely a general statement of the rectangular form including the trig functions.
As an example, consider the vector 6/32°. 6/32° = Z cos 0 + jZ sin 0
= 6 cos 32° + j6 sin 32°
-= 6(.8480) + j6 (.5299)
= 5.088 + j3.1794
Fig. 4-4. Vector 6/148° is the same as vector 6/32 °, except it is in a different
quadrant.
Depending on the accuracy required, this answer may be rounded off to 5.09 + j3.18. If the polar expression had been 6/-32°, the rectangular equivalent would have been 5.09 - j3.18. When the angle in any problem is greater than 90°, it means that the vector lies in other than the first quadrant.
The calculations are the same, but the proper signs must be applied to the rectangular terms. As an example, consider 6/148° shown in Fig. 4-4. In this quadrant the cosine is negative and the sine positive, therefore, 6/148°= 6 cos 148° + j6 sin 148°
= -6 cos 32° + j6 sin 32°
= -6(.8480) + j6 (.5299)
= -5.09 + j3.18
This is the same vector as solved in the previous problem, but it is in a different quadrant. Similarly, when converting from rectangular to polar form, the quadrants must be considered. The signs of the X and Y components indicate the quadrant. Vector length Z and the angle are calculated similarly for all quadrants. Bear in mind, however, that the angle determined from the trig table is always 90° or less. This is what we previously called the working angle and is not always the polar-form angle. For example, in the problem just solved, except working it in reverse, the rectangular form is -5.09 + j3.18. Using the arc tangent, an angle of 32° is read from the table. The polar-form angle (for second quadrant) is 180° minus the working angle. Therefore, in this problem the correct angle would be 180° minus 32°, or 148°. Careful attention to the signs will prevent errors in both conversions.
In most phase problems involving AC the angle is less than 90°, but in other vector problems angles greater than 90° occur often; therefore they should be considered in Vector Algebra.
There is one other form of vector expression, the exponential form, which is used to a great degree in pure mathematics. However, it is not generally used in practical calculations and, consequently, is not considered in this text.
VECTOR ALGEBRA
Complex numbers can be handled in the same manner as we do other types of binomials as long as the special significance of j is kept in mind and as long as the proper substitutions (for example j 2 = -1) are made. These numbers can be added, subtracted, multiplied, divided, or worked with in terms of powers and roots. These are all the complex-number operations that are likely to be encountered in vector and phase analysis.
Addition and Subtraction
If complex numbers appear in rectangular form, they can be added or subtracted by combining the real parts and then the imaginary parts, taking the various plus and minus signs into account. Addition and subtraction are similar, but for the most part what is known as algebraic addition is used. This involves addition, but it also considers the signs. For example, add 2 + j7 and 3 - j3: 2 + j7 3 - j3 5 + j4
Actually, the j3 was subtracted from j7, an example of what is meant by algebraic addition. If, however, 3 - j3 were subtracted from 2 + j7, it would be necessary to change the signs of the number being subtracted. Thus:
2 + j7
-3 + j3
-1 + j10
The difference is -1 + j10.
This type of problem does not occur as often as the previous one because a problem is usually stated in mathematical terms rather than in the form of a sentence. In the previous SECTION one of the subtraction methods presented was that of changing the signs of the number being subtracted. In rectangular form the changing of signs represents a change of 180° in the direction of the vector.
Several more examples of complex-number addition follow:
4 - j3 -5 - j3 4 + j6 3 + j2 2 + j5 3 + j -3 - j 3 6 + j2 -2 - j2 1 + j6 3 – j
Polar-form vectors cannot be added or subtracted algebraically unless the lines representing the vectors are parallel. In all other cases the quantities must be converted to rectangular form, then added algebraically. If a polar resultant is needed, the rectangular sum must be converted into polar form.
Multiplication
Multiplication of rectangular-form quantities can be per formed simply as the multiplication of two binomials. Polar form quantities can also be multiplied, as shown in the previous SECTION, and this is probably the easier of the two methods.
However, all the numbers being multiplied must be in the same form, either rectangular or polar. In this SECTION multiplication in rectangular form is considered first. For example:
(3 - j2) (2 + j4)
Each term of the second factor must be multiplied by each term of the first, and the order of these operations in not important. Here is one form:
6 - j4 + j12 - j 28 -= 6 + j8 - (-1) 8 = 6 + j8 + 8 = 14 + j8
Notice that two terms result, one real and one imaginary as long as j 2 is replaced by -1. Here is another example:
(2 + j4) (1 + j) = 2 + j4 + j2 + j 24 = 2 + j6 - 4 = -2 + j6
When multiplying the sum and difference of the same two terms, the j term is eliminated and a real-number product occurs. For example:
(2 + j) (2 - j) = 4 + j2 - j2 - j 2 = 4 - (-1) = 4 + 1 = 5
This type of combination is used when dividing by a complex number, as will be shown later. Such numbers are said to be conjugates of each other; they are alike, except for the signs separating them. Thus, 2 + j is the conjugate of 2 - j, and vice versa. Here are several other examples of conjugate numbers:
-2 + j and -2 - j j4 and -j4 3 and 3
The product of conjugate numbers is always a real number; likewise, the sum of conjugate numbers.
Polar-form multiplication involves multiplying the magnitudes together and algebraically adding the angles, as previously explained. The magnitudes are always positive values, but the angles can be either positive or negative, and the products are in polar form. Here are four examples:
1. (2/40°) (3/35°) = 6/75°
2. (4-30°) (2/48°) = 8/18°
3. (1.5/10°) (2/-26°) = 3 / -16°
4. (3/-25°) (4J-120°) = 12J-145°
More than two vectors can be multiplied as evidenced by these next two examples:
5. (3/40°) (2/25°) (4/60°) = 24/125°
6. (2/35°) (5/-50°) (6 /15°) = 60L0°
Division
Division of polar-form quantities is similar to multiplication, except, of course, that they are inverse operations. The magnitudes are divided arithmetically to obtain the magnitude of the quotient. Then the angle of the divisor is subtracted from the angle of the vector being divided. This involves changing the sign of the divisor angle and adding the angles algebraically. The quotient is also in polar form, as seen in these examples:
15/45° 2 . ;_02, 1 100. 3/15. = 5/30° 6/-20° - 3 / 9/-50° 12/-32° 3 38.
4.5/15° = 2/-65° 4/-70° /
Dividing in rectangular form involves only rationalizing the denominator to eliminate the j term from it, and then simplifying the quotient as much as possible. If the denominator includes only the j term, multiply the numerator and the denominator by the denominator to determine the quotient, as in these examples:
When dividing by only a real number, the quantity can usually be left as it is. Or else it can be broken into two fractions provided that it is reduced to lowest terms, as in these examples:
3 +j 3 +j 3 j 3. - - Or - + - 4 4 4 4
4. 4 + j3 4 + j3 or 2 + j3 2 = 2 7
Dividing by a complex number involves multiplying the numerator and the denominator by the conjugate of the denominator, as in this example:
4 + j3 _ 4 + j3 . 1 + j _ 4 + j3 + j4 + j 23 5. 1 - j 1 - j 1 + j 1 - j 2
-1 + j7 + j 23 1 + j7 1 j7 = or 2 - + - 2 2 2
Powers and Roots
Determining powers and roots of polar vectors involves methods similar to those used in multiplication and division.
Raising a number to a power involves a series of multiplications. As an example, 2 4 is the same as 2 • 2 • 2 • 2, using 2 as a factor 4 times.
To raise a polar vector to a power, raise the magnitude of the vector to the indicated power. Then multiply the angle by the number representing that power. Here are three examples:
1. (4/21 0) 2 = 16/42°
2. (3/-150) 2 = 9/-30°
3. (2/25°) 3 = 8/75°
Raising rectangular-form vectors to a power can be done as a series of multiplications. The process is usually easier in polar form, especially when raising the vector to the third or higher power. For example, to raise a vector in the rectangular form to the fourth power involves three separate multiplications. Although it is not used to any great extent, DeMoivre's Theorem can be used. Stated in general form:
[Z (cos 0 + j sin 0)]" Zn (cos n 0 + j sin n 0) where, n is any real number.
This theorem will not be expanded on here because of its lack of general use.
Roots of vectors could be taken by the same formula, but in these cases n would be fractional. For example, in determining a square root, n has a value of 1 / 2 . But roots are almost always performed in polar form by determining the indicated root of the magnitude, then dividing the angle by the index of the root.
These are examples:
1. N/9/±12° = 3M.°
2. V16/_ 720° = 4/-10°
3. N'3 /8/39° = 2 /13°
It was previously indicated that square-root problems give two results. For ordinary numbers these are plus and minus values, with the positive value being considered as the principal root. This is also true when determining the square root of a polar-form vector. Only one example is considered here be cause of the limited usefulness of the idea in practical vector problems. In the first example problem it was shown that: N/9/42° = 3/21° If 360° is added to the angle, the vector direction remains unchanged, but the new notation is: N/9/402° = 3/201° The problem could also be written:
N/9/-318° = 3/-159°
This is the same result as obtained previously because: 3/-159° = 3/201° So in addition to the principal root each square root of a polar-form vector has another root having the same magnitude but with the angle changed by 180°. Powers and roots of vectors, however, are not used to nearly the extent as the other mathematical processes in practical electronics problems involving vector relationships.
1. V-5 • V-2 =
2. V-4 • V-3 =
3. 2V-5 + V-5 =
4. V-25 - V-9 =
5. j7 - j3
6. j8 + =
7. j15 - j18 = V-8 8.
9. V-3 -
10. V-8
V2 = Change to polar form: 11. 5 + j6 12. 3 - j7
QUIZ
Perform the indicated operations: 15. (3 + j2) + (4 - j3)
16. (5 - j4) - (3 + 17. 4 + j2 + j3 18. (3 + j) (2 + j2)
19. (2 - j3) (3 + j5)
20. (3 + j) (3 - j)
21. (3/36°) (2/18°) 22. (4/160°) (3/-20°)
Change to rectangular form: 13. 6/38° 14. 4/160° 23. 18/72° 9/30' 24. 15/-30° 3/-48° 25. 2 + j 3 j2 26. 2 + 1 + j 27. 3 - j3 2 j3 28. (3/15°)' 29. (2/20°)' 30. V25/82°