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====== Appendix B ======

===== Radio Math =====

Ham radio involves a lot of technology. You don’t need to be a scientist or engineer to use and enjoy ham radio, of course. Nevertheless, math is a big part of the hobby and you’ll need to speak a little of the language. This appendix provides a little background on common bits of math you’ll encounter as you study for your license and then start using it.

:!:
A number of license exam questions involve these technical details. Your license study guides will show you how to use this information.
Online videos abound. The Khan Academy (www.khanacademy.org) has many tutorials to help you. For topics in electricity, magnetism, and electronics, Georgia State University hosts a great website called Hyperphysics (hyperphysics.phy-astr.gsu.edu/hbase). Each lesson features clear graphics and many are animated.

:!:
When you are just learning a subject or perhaps having difficulty with a key point, review two or three different tutorials. Perhaps you will understand one better than the others. Seeing the material from different points of view often makes it clearer, as well.

===== The Metric System =====

Radio of all types uses the metric system because the values and quantities cover such a wide range of values. Table B-1 shows metric prefixes, symbols, and their meaning. Each prefix is the name of a factor that multiplies quantities by amounts shown in the table. For example, a kilo-meter (km) is one thousand meters and a milli-meter (mm) is one-thousandth of a meter.

==== Table B-1	International System of Units (SI) — Metric Units ====

^ Prefix | Symbol | Multiplication Factor |
| Tera | T | 1012 = 1,000,000,000,000 |
| Giga | G | 109 = 1,000,000,000 |
| Mega | M | 106 = 1,000,000 |
| Kilo | k | 103 = 1,000 |
| Hecto | h | 102 = 100 |
| Deca | da | 101 = 10 |
| Deci | d | 10-1 = 0.1 |
| Centi |c | 10-2 = 0.01 |
| Milli |m | 10-3 = 0.001 |
| Micro | μ | 10-6 = 0.000001 |
| Nano | n | 10-9 = 0.000000001 |
| Pico | p | 10-12 = 0.000000000001 |
| 1 M = 1,000 k; 1 m = 1,000 μ = 1,000,000 n; 1 μ = 1,000 n = 1,000,000 p |||
The most common prefixes you’ll encounter in radio are pico (p), nano (n), micro (μ), milli (m), centi (c), kilo (k), mega (M), and giga (G). It is important to use the proper case for the prefix letter. For example, M means one million and m means one-thousandth.

=== <TechnicalStuff> ===

Here in the United States, ham radio uses a mix of the old Imperial Units (feet, pounds, gallons, degrees Fahrenheit, and so forth) and the International System (or SI) Units (meter, kilogram, liters, degrees Kelvin, and so forth). You will often find a mix of units used in articles and construction projects. Fasteners, wire sizes, and other hardware and materials are usually not metric (for example, 1/4-inch screws instead of 6 mm). If you have equipment manufactured outside the U.S., however, it will probably use metric hardware and material dimensions.

:!:
Converting from one type of units to another has become very simple thanks to the Google Convert facility. Just type the word convert and the two units into the Google search window and a conversion tool appears. There are specialized websites for converting units, such as www.unitconverters.net and there are free Android and iPhone apps, as well.

:!: __<Warning>__
If you are unfamiliar with a particular type of unit, it is easy to reverse the order of conversion or use the wrong conversion factor. This can lead to some crazy results! For example, applying the conversion from yards to centimeters “backward” (0.0109 cm per yard instead of 91 cm per yard) means you’ll be off by a factor of 8,348! Double-check your work and do a sanity check to be sure your answer is reasonable.

==== Scientific Notation
 ====
You’ll find numbers in ham radio that are very, very large and very, very small. At either extreme, it is difficult to write the numbers as decimal values because of all the zeros. For example, the speed of light at which radio waves travel in a vacuum is 300,000,000 m/s (meters per second). The value of a 22 pF capacitor would be written as 0.000000000022 F.

This is a very inconvenient format for calculation and makes it easy to goof up.
Instead, the values are written in a special way called scientific notation. Numbers in scientific notation consist of a value multiplied by 10 raised to an integer power, like this:
@@pmD.DD @@ts 10EE
where D.DD is a decimal value between 1 and 10, such as 3.14 or 7.07. EE is an exponent of 10, generally between 0 and 99. Here are a few ways of writing the same number (567 kHz) in scientific notation:
567 kHz = 5.67 @@ts 105 Hz
	=	5.67 @@ts 102 kHz
	=	5.67 @@ts 10-1 MHz
	=	5.67 @@ts 10-4 GHz

Decibels (dB)
Decibels were introduced in Chapter 18 and are also described in the QST magazine article “Untangling the Decibel Dilemma” that is available as a downloadable PDF at www.arrl.org/files/file/Education/Untangling_the_Decibel_Dilemma.pdf.
Decibels are a convenient way to represent and work with ratios of power or voltage over a very wide range. The basic formulas are:
dB = 10 log (power ratio) = 20 log (voltage ratio)
If you want to find the gain of an amplifier or circuit, the ratio should be output divided by input. For example, if an amplifier’s power output is 200 watts and the input power is 40 watts, the gain in decibels is:
10 log (200 / 40) = 10 log (5) = 6.9 dB
If a filter’s input voltage is 10 volts and the output is 3 volts, the attenuation (the opposite of gain) is:
20 log (3 / 10) = 20 log (0.3) = -10.5 dB
<Tip>
Most calculators, including those included as an app with your phone or tablet, use the same process to calculate log values. Start by calculating the ratio: Enter the output and divide it by the input. Find and press the LOG key. Then multiply by 10 (if it’s a power ratio) or 20 (for voltage ratios).
You can get an approximate value of dB by knowing common ratios, such as those in Table B-2.
Table B-2	Common dB Values For Ratios of Power and Voltage
P2/P1	dB	V2/V1	dB
0.1	-10	0.1	-20
0.25	-6	0.25	-12
0.5	-3	0.5	-6
1	0	0.707	-3
2	3	1	0
4	6	1.414	3
10	10	2	6
		4	12
		10	20
<Remember>
Positive values of dB represent gain or amplification. Negative values of dB represent loss or attenuation.
<Tip>
To find the dB equivalent of two ratios multiplied together, add their dB equivalents. For example, to find the power ratio equivalent of 40 @@ts 4 = 160 in dB, convert them to dB (16 dB and 6 dB) and add them together: 16 + 6 = 22 dB.
<TechnicalStuff>
It is often useful to specify a power or voltage level with respect to some common value, such as 1 milliwatt or 1 microvolt. When a certain reference value is used, a letter is added to “dB” to indicate what the reference value is. Here are several common references:
	*	dBm: The power level in decibels compared to a milliwatt
	*	dBd: Decibels of gain with respect to a dipole antenna in its preferred direction
	*	dBμV: Voltage level in decibels compared to a microvolt (V)
	*	dBμW: Power level in decibels compared to a microwatt (W)
Gain and loss in dB can be added or subtracted to these power or voltage levels directly to get the output power or voltage. For example, if you had an amplifier with a gain of 20 dB and applied an input signal of 6 dBm, you would have an output power of 6 dBm + 20 dB or 16 dBm, which is 40 W.
You can also convert between dB and %:
dB = 10 log (percentage of power / 100) = 20 log (percentage of voltage / 100)
For example, what is the decibel equivalent of a 30% power ratio?
dB = 10 log (30 / 100) = 10 log (0.3) = -5.2 dB
Similarly, you can convert dB to %:
Percentage of power = 100% @@ts  log-1 (dB/10)
Percentage of voltage = 100% @@ts  log-1 (dB/20)
What is the percent equivalent of a 2 dB voltage gain?
Percentage = 100% @@ts  log-1 (2 / 20) = 100% @@ts  log-1 (0.1) = 100% @@ts  1.26 = 126%
<Tip>
To calculate an inverse log (written log-1), see the log tutorial web page www.chem.tamu.edu/class/fyp/mathrev/mr-log.html. The web page also includes other useful tips on log calculations.
Miscellaneous Tutorials
The Interactive Mathematics website (www.intmath.com) offers a free, online system of tutorials. The system begins with basic number concepts and progresses all the way through introductory calculus. The lessons referenced here are those of most use to a student of radio electronics.
Basic numbers and formulas
	*	Order of operations: www.intmath.com/Numbers/3_Order-of-operations.php
	*	Powers, roots, and radicals: www.intmath.com/Numbers/4_Powers-roots-radicals.php
	*	Scientific notation: www.intmath.com/numbers/6-scientific-notation.php
	*	Ratios and proportions: www.intmath.com/Numbers/7_Ratio-proportion.php
	*	Geometric formulas: www.equationsheet.com/sheets/Equations-4.html
Metric system and conversion of units
	*	Metric system overview: en.wikipedia.org/wiki/Metric_system
	*	Metric English: en.wikipedia.org/wiki/Conversion_of_units
	*	Conversion factors: https://brownmath.com/bsci/convert.htm
	*	Tables of conversion factors: en.wikipedia.org/wiki/Conversion_of_units
Fractions
	*	Equivalent fractions: www.intmath.com/factoring-fractions/5-equivalent-fractions.php
	*	Multiplication and division: www.intmath.com/factoring-fractions/6_multiplication-division-fractions.php
	*	Adding and subtracting: www.intmath.com/factoring-fractions/7_addition-subtraction-fractions.php
	*	Equations involving fractions: www.intmath.com/factoring-fractions/8_equations-involving-fractions.php
Graphs
	*	Basic graphs: www.intmath.com/Functions-and-graphs/Functions-graphs-intro.php Polar Coordinates — www.intmath.com/plane-analytic-geometry/intro.php
	*	Exponents and radicals: www.intmath.com/Exponents-radicals/Exponent-radical.php
	*	Exponential and logarithmic functions: www.intmath.com/exponential-radicals/exponen-radical.php
Algebra and trigonometry
	*	Basic algebra: www.intmath.com/Basic-algebra/Basic-algebra-intro.php
	*	Basic trig functions: www.intmath.com/trigonometric-functions/trig-functions-intro.php
	*	Graphs of trig functions: www.intmath.com/trigonometric-graphs/trigo-graph-intro.php
Complex numbers
	*	Complex numbers: www.intmath.com/Complex-numbers/imaginary-numbers-intro.php
	*	Polar-rectangular conversion: www.intmath.com/Complex-numbers/4_Polar-form.php
Handy Items
Values of e and pi
e = 2.71828; @@lcpi = 3.14159; 2@@lcpi = 6.28318; @@lcpi/2 = 1.5708
Frequency-wavelength conversion
Frequency (in Hz) = 3 @@ts 108 / wavelength (in m)
Wavelength (in m) = 3 x 108 / frequency (in Hz)
Frequency (in MHz) = 300 / wavelength (in m)
Wavelength (in m) = 300 / frequency (in MHz)
Half-wavelength in free space (in feet) = 492 / frequency (in MHz)
Quarter-wavelength in free space (in feet) = 246 / frequency (in MHz)
Length conversion
Multiply meters by 3.28 to get feet
Multiply meters by 39.4 to get inches
Multiply meters by 1.09 to get yards
Multiply yards by 0.914 to get meters
Multiply feet by 0.305 to get meters
Multiply inches by 2.54 to get centimeters
Multiply feet by 30.5 to get centimeters
Multiply centimeters by 0.0328 to get feet
Multiply centimeters by 0.394 to get inches
Trigonometry and angles
1 revolution = 360@@dg; 2 rev = 720@@dg; 3 rev = 1080@@dg
The Greek letter @@lctheta (theta) is the most often used symbol to represent an angle. The letter ϕ (phi) is also used.
sin (@@lctheta) = -sin (–@@lctheta) = cos (@@lctheta – 90@@dg)
cos (@@lctheta) = cos (–@@lctheta) = sin (@@lctheta + 90@@dg)
sin (90@@dg) = cos (0) = 1
sin (45@@dg) = cos (45@@dg) = 0.707
sin (30@@dg) = cos (60@@dg) = 0.5
sin (0) = cos (90@@dg) = 1
tan(@@lctheta) = sin (@@lctheta) / cos (@@lctheta)
tan (45@@dg) = 1; tan (0@@dg) = 0; tan (90@@dg) = infinity
<Tip>
To calculate the height of some object (such as a tree or tower) without climbing it, stand far enough away from it so that the angle from eye level to the object’s top is about 45 degrees. Measure the distance between you and the object’s base.
Height of the object = your eye level + (tan (angle to top of object)) @@ts distance to base of object)

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