December 1942 QST
Table of Contents
Wax nostalgic about and learn from the history of early electronics. See articles
QST, published December 1915 - present. All copyrights hereby acknowledged.
If you are just starting out in the realm of electronics or maybe just need a little freshening up of your basic math skills, this rather extensive article from a 1942 issue of QST magazine is just what you need. Author Dawkins Espy does a really nice job of laying out the basics of algebraic operations, Ohm's law, trigonometry, and logarithms. Examples are provided for each category. In this day of calculators doing all the hard work of calculating logs, antilogs, and trig functions, it does even seasoned veterans at electronics calculations a bit of good to do a quick read-through to knock off cobwebs in the gray matter. How long has it been since you have seen tables of sine, cosine, and tangent values and/or tables of logarithms? Not long enough, you say?
How's Your Math?
A Brief Review of Some Fundamental Operations
By Dawkins Espy,* W6UBT
There's no real need to he frightened away from articles which introduce such outlandish expressions as cos, sin and log and use marks apparently unrelated to ordinary punctuation. Mathematical symbols are not used to confuse the uninitiated but to give compact and precise expression to ideas which often would he pretty long-winded and confusing when expounded in "plain" English. Here's a chance to get a little initiation.
The increasing importance of the amateur to the war effort makes it highly desirable for him to understand some of the fundamentals of mathematics, since mathematical relationships will be encountered sooner or later in practically every kind of technical work. Obviously this article cannot attempt to present a complete summarization of all types of mathematics used in radio. Rather, it is the intention to treat simply a few of the more common processes.
Mathematics is merely a shorthand way of expressing relationships so that they may be manipulated more easily. It is not unusual to find an equation or relation such as
y=ax + b
where x and yare the related quantities and a and b are constants. The constants may be very simple or very complex, but the mathematician does not allow himself to become confused by the complexity of the involved quantities. What he wants is an explicit and simple relationship between the variables x and y. Soon he arrives at an ordinary equation such as that given above, and examination of the quantities - as by plotting a graph or making a table of values - enables him to get a mental picture of their relationship.
Simplicity is the essence of mathematics.
An equation is simply a statement that the two quantities on the opposite sides of the "equals mark" are identical, e.g.:
A = B
If this is true B can be replaced by its equal A and the equation becomes
A = A
which we know is true.
Four important properties of equations are: (1) Both sides of the equation can be multiplied by the same number without changing the equality.
(2). Both sides of the equation can be divided by the same number without changing the equality.
(3) The same number may be added to both sides of the equation without changing its equality.
(4) The same number may be subtracted from both sides of the equation without changing the equality.
To illustrate these principles, consider the equation
6 = 4 + 2 (check 6 = 6)
multiplying both sides by 2
12 = (4 + 2) 2
12 = 8 + 4 (check 12 = 12)
dividing both sides by 4
3 = (8 + 4) / 4 = 2 + 1 (check 3 = 3)
adding 3 to both sides
3 + 3 = 2 + 1 + 3 (check 6 = 6) subtracting 4 from both sides
3 + 3 - 4 = 2 + 1 + 3 - 4 (check 2 = 2)
Thus, the equality of the equation was maintained at every step.
Many amateurs have wondered at the various forms that Ohm's Law takes on. By using the first two rules outlined and choosing the proper quantities by which to multiply and divide, the various relations can be shown to be consistent.
Fig. 1 - Ohm's law
A familiar form of Ohm's Law is
E = IR
by dividing both sides by I we have
E / I = IR / I
or, since I divided by I is equal to 1,
E / I = R
Reverting back to the original form
E = IR
and dividing both sides by R, we get
E/R = I
Note that dividing by R is the same as multiplying by 1/R, so that either Rule (1) or Rule (2) could have been used to perform the operation. The quantity 1/R is known as the reciprocal of R.
Considering the power form of Ohm's Law, and recalling the form most familiar to the amateur, the one used to compute power input:
Replacing E by its equivalent IR we have,
P = IRI
P = I2R
and substituting for I its equivalent E/R we have,
noting that R2 = R X R
Incidentally, a good helper to assist in remembering Ohm's Law is the triangle shown in Fig. 1. By placing a finger over the desired quantity, the quantities left give the proper formula. For example, if one wishes to know the current I, covering it up leaves E/R which we've shown to be a valid form of Ohm's Law.
Extracting Square Root
The problem of finding the square root of a number occurs frequently, and it is surprising how many of us have forgotten the method that we learned in high-school arithmetic. Though the procedure given here is somewhat different and is actually an approximation, it is quite accurate enough for most purposes and is certainly a simplification over the old method.
If n2 is the number whose square root is sought, and a2 is the nearest perfect square, as given in Table I, to n2, then b, the third quantity involved in this method, is given by the equation
By choosing the proper value for a2 and determining b from the above formula, we can find n, the desired square root, from the formula
n = a + b
All the squares given in Table I are what are known as perfect squares; that is, each is the square of some whole number such as 1, 2, 3, 4, etc. The square of 2.5 would not be a perfect square.
Suppose we should like to find the square root of 372. By referring to Table I we see that the nearest perfect square is 361, the square root of which is 19. Thus, in our method a = 19 and a2 = 361. Now, to find b we substitute in the formula given above
Thus, the square root of 372 equals a + b = 19 + 0.289 = 19.289.
Suppose the number does not appear in the table; for example, n2 = 78,921. By moving the decimal point successively to the left we can cause the number to come within the range of our table twice, first as n2 = 7892.1 (a = 88, a2 = 7744) and secondly as n2 = 789.21 (a = 28, a2 = 784). The rule is: Move the decimal point to the left two places at a time, and use the largest a2 within the range of the table that is obtainable by this method. In our example this would indicate the use of a2 = 784, a = 28. Now add ciphers to a2 until the total number of digits is the same as in the original number, n2, and add half that many ciphers to a. This gives us a2 = 78400 and a = 280; in other words, = 280.
Table I - Squares of numbers from 1 to 100
Now determine b:
Thus, n = a + b = 280 + 0.87 = 280.87.
If the number in the first example had been 392 instead of 372, then the nearest perfect square would be a2 = 400, a = 20, and
n = a + b = 20 + ( -0.2) = 20 - 0.2 = 19.8.
A common radio problem involving square root is the calculation of the turns ratio of a transformer when its impedance ratio is known; in this case
Trigonometry is a method of expressing the relations between the sides and angles of a triangle so that calculations may be simplified. While at first glance there might seem to be no obvious connection between a triangle and radio, it is a fact that trigonometric methods are extremely useful in the practical measurement of distance, height, etc., and that the oscillations of radio waves, alternating currents and the like are closely related to certain properties of triangles.
Everyone" knows" what an angle is, but it is worth our while to define angle from the trigonometric standpoint. An angle is considered to be "generated" by the rotation of a line about a fixed point on which one end of the line terminates, the size of the angle being measured with reference to a second fixed line. For example, the complete rotation of a spoke about the axle of a wheel so that it returns to its original position constitutes a 360° rotation. For convenience, a diagram can be drawn representing this rotation divided into fourths. Each of these fourths or quarters consists of 1/4 X 360° or 90° and is called a quadrant. The quadrants are designated in a counterclockwise direction as the first, second, third, and fourth, as shown in Fig. 2. Fig. 2 also shows the construction of a right-angled triangle in each of the four quadrants. When constructing such triangles, one side adjacent to the right angle is always made coincident with the horizontal axis, but the vertical, or other side adjacent to the right angle, does not in general coincide with the vertical axis. This causes one of the variable angles always to have its apex at the intersection of the horizontal and vertical axes.
Fig. 2 - 4 quadrants
Fig. 3 - Right triangle
Suppose we have an angle α (the Greek letters α (alpha), β (beta), and θ (theta) are most frequently used to represent the angles) in the first quadrant, as shown in Fig. 3. The longest side of a right triangle is always known as the hypotenuse, the side opposite the angle under consideration is known as the opposite side, and the third side is known as the adjacent side. This is illustrated in Fig. 3. We can now define the three most important and fundamental trigonometric functions, the sine, the cosine, and the tangent, commonly abbreviated as sin, cos, and tan.
sin α = opp/hyp
cos α = adj/hyp
tan α = opp/adj
Two Fundamental Relations
1. Dividing sin α by cos α
2. Remember the old rule from plane geometry that the square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the lengths of the other two sides? Well, here's a use for it. Applying the rule to the triangle in Fig. 3, we have
now dividing through by hyp2
but (hyp/hyp)2 is 12 which is just 1 and so
and substituting the trigonometric values for the left-hand terms, we have
sin2 α + cos2 α = 1
The exponent is written after the sin and cos instead of after the angle, as one might be tempted to write it. This is because it is the sin or cos that is squared and not the angle α, as writing it the other way would indicate.
Range of Values
Now that we have a general idea of the form of the important trigonometric relations, the range of values of the sine, cosine and tangent should be examined.
The sine is the ratio of the opposite side to the hypotenuse. Referring to Fig. 4 (b) we see that as the angle α is made large the length of the opposite side approaches the length of the hypotenuse. If the limiting case where α is made 90° is considered, the sides would coincide and the ratio would be 1 to 1, so the sine would have a value of 1 for a equal to 90°. On the other hand, referring to Fig. 4 (c), we see that as α gets smaller and is made to approach zero, the ratio of opposite to hypotenuse approaches zero. Thus, at α = 0°, sin α = 0.
Table II - Trig functions
Fig. 4 - Sine and cosine functions
Fig. 5 - Sinewave plotted
For the cosine, we find just the opposite occurring; as a approaches 90° the ratio approaches zero and as a approaches 0° the cosine approaches
1. Thus for a = 90°, cos α = 0 and for a = 0°, cos α = 1. These relations can also be seen by referring to Fig. 4.
The tangent, according to derivation 1, is equal to the ratio of sin to cos. For α = 90°, sin α = 1 and cos α = 0 and thus tan α = 1/0, which is infinitely large. At α = 0°, sin α = 0 and cos α = 1, so tan α = 0/1 = O. Thus the tangent varies from zero at 0° to infinity at 90°.
That tan α = ∞ (symbol for infinity) at a = 90° can perhaps be seen more clearly by referring to Fig. 4 (b) again. The tangent is equal to the ratio of the opposite side to the adjacent side, and thus as the angle a approaches 90°, the opposite side approaches the length of the hypotenuse and the length of the adjacent side approaches zero. Therefore the ratio keeps getting larger and larger. Consider the ratio of 100/x; if x = 50, the ratio is 2; if x = 10, the ratio is 10; if x = 1, the ratio is 100; if x = 0.5, the ratio is 200, etc. Thus as x gets smaller and smaller, approaching zero, the ratio gets progressively larger. Such a ratio is said to be infinite or equal to infinity at x = 0.
The sine, cosine and tangent relations are known as "circular functions" since their physical properties are connected with the circle. The construction given in Fig. 5 shows how a sine wave can be constructed from a circle and how the intermediate values between 0° and 90° may be obtained. Corresponding points in the construction are lettered a and a', band b', etc.
Table II gives a chart of sine, cosine and tangent values for every 2°. This should be sufficient for a number of uses. Standard textbooks and handbooks have more complete tables.
Ex. 1. The opposite side of a right triangle is 5 feet long and the hypotenuse is 10 feet. Find the value of the angle α.
Ex. 2. An observer is 30 feet away from an antenna pole and from his position there is an angle of 45° between the bottom and top of the pole. Determine the height of the pole.
Thus a rule may be formed: If one sights an angle of 45° between the ground and the top of a pole and the ground is level, the distance from the observer to the bottom of the pole is equal to the pole height.
Fig. 6 - Logarithm graph
Fig. 7 - Mantissa graph
The logarithm of a number is the power to which a second number, called the base, must be raised in order to produce the given number.
Sounds complicated? Well, let's see. The definition speaks of a number, any number; let's choose 100 for argument's sake. It also mentions a base; that, too, may be any number1 but for simplicity 10 is usually used. All right, we've got to raise the base to a power to get the number, but since the quantities are interrelated, the fixing of any two of the three quantities involved automatically fixes the third. If you square a number, that's called "raising the number to the second power." Raising a number to any power means multiplying the number by itself that many times. Well, 102 equals 100, so if 10 is the base and 100 is the number, then 2 is the logarithm. Or the log (abbreviation for logarithm) to the base 10 of 100 is 2. Written mathematically
log10 100 = 2
This is read "log to the base 10 of 100 equals 2." We may also write
log10 10 = 1
because 10 raised to the 1st power gives 10, and
log10 1.0 = 0
Since any number raised to the zero power is 1, 10 raised to the zero power is 1.
We can form a table of relations. Since:
Now it becomes obvious that, if we desire to determine the log of just any number, we will have to have some intermediate values. If, for example, we want the log of 423, we know that it will lie between 2 and 3 since 2 is the log of 100 and 3 is the log of 1000. We can determine the log of 423 by plotting on semilogarithmic paper some of the values given in Table B above, and drawing a smooth curve through the resulting points. Such a curve is shown in Fig. 6. Referring to this curve, we find that the log of 423 is 2.63. Upon further examination of Fig. 6 we find that the log of 42.3 is 1.63 and the log of 4.23 is 0.63. In other words the fraction, or part to the right of the decimal point - the "mantissa," as it is called - is the same if the number is made up of a fixed group of digits regardless of the position of the decimal point. The portion of the logarithm to the left of the decimal point is known as the "characteristic" and is the only part of the log that varies as one shifts the decimal point about.
Fig. 7 shows a portion of Fig. 6 enlarged so that it may be used to determine the mantissa of any number. A simple rule may be formed to find the characteristic of any number: The characteristic is one less than the number of digits to the left of the decimal point for a number greater than 1, and is a negative number equal to one more than the number of ciphers between the decimal point the actual beginning of the number for a number less than 1. This rule can be verified by referring back to Tables A and B above.
If greater accuracy than can be obtained from Fig. 7 is desired, the log tables found in many texts and reference books may be used. The tables in every case consist of a list of mantissa values.
Table III - Use of Logarithms
Many problems require the examination of the antilog. This is the inverse operation to finding the log. In other words, you are given a log and you are to find a number to correspond. The antilog of 3.26 is 1820, for example.
Now that we understand the manipulation of logarithms let's turn to some of their practical uses. Logarithms may be used to multiply, divide, raise to powers, and extract roots. To multiply two numbers their logs are added and the antilog of the sum taken. To divide one number by another, the log of the denominator is subtracted from the log of the numerator and the antilog taken. To raise a number to a power, the log of the number is multiplied by the power and the antilog taken. To extract a root of a number, the log of the number is divided by the root and the antilog taken. In the case of raising to a power or extracting a root, the multiplying or dividing may also be done with logs. This will involve taking the log of a log or "log log" of the number as it is called. Table III gives a summation of these operations.
Ex. 1. To multiply 3.42 by 60.8
Antilog (answer) = 208
Ex. 2. To divide 3.42 by 60.8
Antilog = 0.0562
Ex.3. To multiply 0.0342 by 60.8
Antilog = 2.08
Ex. 4. To raise 60.8 to the 3.42 power or (60.8)3.42
3.42 X log 60.8 = 3.42 X 1.784
If we perform the multiplication by logs also, then we shall have to take the antilog twice.
Antilog = 6.10 (Result of 3.42 X 1.784)
Antilog = 4,070,000 (Result of (60.8)3.42)
Ex. 5. To extract the 3.42 root of 60.8 or
Antilog = 0.521
Antilog = 3.32
* In this case it is necessary to add and subtract 10 to the characteristic as shown. This is a standard procedure for this type problem.
** Here the characteristic is actually negative and is most conveniently expressed in the manner shown. Using our rule one plus the number of ciphers in this case is two which makes the characteristic - 2 or 8 - 10 .where the - 10 is written to the right as shown.
Ex. 6. To solve the following problem
In this example the simple multiplications and divisions were done by longhand:
Combining the numerator and denominator -
Antilog = 1100
* Columbia University, Division of War Research, New London, Conn.
1 1 cannot be used as a base because as it is raised to various powers the resultant is always 1.
Posted April 5, 2016