Appendix B
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.
There are additional resources available to help you dig deeper. The ARRL publishes a downloadable “Radio Mathematics” supplement as a PDF document you can save for reference: www.arrl.org/files/file/ARRL%20Handbook%20Supplemental%20Files/2018%20Edition/Radio%20Supplement.pdf
.
Online videos abound, as well. 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.
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.
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:
±D.DD × 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:
Decibels are introduced in Chapter 12. They 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 30 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
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 |
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 %:
What is the percent equivalent of a 2 dB voltage gain?
Percentage = 100% x log-1 (2 / 20) = 100% x log-1 (0.1) = 100% x 1.26 = 126%
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.
www.intmath.com/Numbers/3_Order-of-operations.php
www.intmath.com/Numbers/4_Powers-roots-radicals.php
www.intmath.com/numbers/6-scientific-notation.php
www.intmath.com/Numbers/7_Ratio-proportion.php
www.equationsheet.com/sheets/Equations-4.html
en.wikipedia.org/wiki/Metric_system
en.wikipedia.org/wiki/Conversion_of_units
https://brownmath.com/bsci/convert.htm
en.wikipedia.org/wiki/Conversion_of_units
www.intmath.com/factoring-fractions/5-equivalent-fractions.php
www.intmath.com/factoring-fractions/6_multiplication-division-fractions.php
www.intmath.com/factoring-fractions/7_addition-subtraction-fractions.php
www.intmath.com/factoring-fractions/8_equations-involving-fractions.php
www.intmath.com/Functions-and-graphs/Functions-graphs-intro.php
Polar Coordinates — www.intmath.com/plane-analytic-geometry/intro.php
www.intmath.com/Exponents-radicals/Exponent-radical.php
www.intmath.com/exponential-radicals/exponen-radical.php
www.intmath.com/Basic-algebra/Basic-algebra-intro.php
www.intmath.com/trigonometric-functions/trig-functions-intro.php
www.intmath.com/trigonometric-graphs/trigo-graph-intro.php
www.intmath.com/Complex-numbers/imaginary-numbers-intro.php
www.intmath.com/Complex-numbers/4_Polar-form.php
e = 2.71828; π = 3.14159; 2π = 6.28318; π/2 = 1.5708
Height of the object = Your eye level + (tan (Angle to top of object)) × Distance to base of object)