Radio Mathematics - ARRL
Radio MathematicsThis supplement is a collection of tutorial information on av ariety of mathematics used in Amateur Radio. Many of the fundamental relationships in electronics and radio are bestexpressed in the language of math. This chapter will expand insubsequent editions as it becomes clearer what information willbe useful to readers. Material in this chapter has been collectedfrom many sources, authors, and instructors over the years— the ARRL appreciates their contributions.1 The Metric System1.1 Metric Prefixes —the Language of RadioThe units of measurement employed in radio use the metric systemof prefixes. The metric system is used because the numbers involvedcover such a wide range of values. Table 1 shows metric prefixes,symbols, and their meaning. The prefixes expand or shrink the units,multiplying them by the factor shown in the table. For example, akilo-meter (km) is one thousand meters and a milli-meter (mm) isone-thousandth of a meter.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. Forexample, M means one million and m means one-thousandth. Usingthe wrong case would make a big difference!The metric system uses a basic unit for each different typeof measurement. For example, the basic unit of length is themeter (or metre). The basic unit of volume is the liter (or litre).The unit for mass (or quantity of matter) is the gram. The newton is the metric unit of force, or weight, but we often use thegram to indicate how “heavy” something is by assuming a standardvalue of gravity.Table 1 summarizes the most-used metric prefixes. The metric system expresses larger or smaller quantities by multiplyingor dividing the basic unit by factors of 10 (10, 100, 1000, 10,000and so on). These multiples result in a standard set of prefixes,which can be used with all the basic units. Even if you comeacross some terms you are unfamiliar with, you will be able torecognize the prefixes.We can write these prefixes as powers of 10, as shown inTable 1. The power of 10 (called the exponent) shows how manytimes you must multiply (or divide) the basic unit by 10. Forexample, we can see from the table that kilo means 103. Let’s usethe meter as an example. If you multiply a meter by 10 three times,you will have a kilometer. (1 m 103 1 meter 10 10 10 1000 meters, or 1 kilometer.)If you multiply 1 meter by 10 six times, you have a megameter.(1 meter 106 1 m 10 10 10 10 10 10 1,000,000 metersor 1 megameter.)Notice that the exponent for some of the prefixes is a negativenumber. This indicates that you must divide the basic unit by 10that number of times. If you divide a meter by 10, you will havea decimeter. (1 meter 10–1 1 m 10 0.1 meter, or 1 decimeter.) When we write 10–6, it means you must divide by 10 sixRadio Mathematics 1
Table 1International System of Units (SI) — Metric MicroNanoPicoSymbolTGMkhdadcmµnpMultiplication Factor1012 1,000,000,000,000109 1,000,000,000106 1,000,000103 1000102 100101 1010-1 0.110-2 0.0110-3 0.00110-6 0.00000110-9 0.00000000110-12 0.0000000000011 M 1000 k; 1 m 1000 μ 1,000,000 n; 1 μ 1000 n 1,000,000 ptimes. (1 meter 10–6 1 m 10 10 10 10 10 10 0.000001 meter, or 1 micrometer.)We can easily write very large or very small numbers with thissystem. We can use the metric prefixes with the basic units, or wecan use powers of 10. Many of the quantities used in basic electronics are either very large or very small numbers, so we usethese prefixes quite a bit. You should be sure you are familiar atleast with the following prefixes and their associated powers of10: giga (109), mega (106), kilo (103), centi (10–2), milli (10–3),micro (10–6) and pico (10–12).Let’s try an example. For this example, we’ll use a term thatyou will run into quite often in the Handbook: hertz (abbreviatedHz, a unit that refers to the frequency of a radio wave). We havea receiver dial calibrated in kilohertz (kHz), and it shows a signalat a frequency of 28,450 kHz. Where would a dial calibrated inhertz show the signal? From Table 1 we see that kilo means times1000. The basic unit of frequency is the hertz. That means thatour signal is at 28,450 kHz 1,000 28,450,000 hertz. There are1000 hertz in 1 kilohertz, so 28,450,000 divided by 1000 gives us28,450 kHz.If we have a current of 3000 milliamperes, how many amperesis this? From Table 1 we see that milli means multiply by 0.001or divide by 1000. Dividing 3000 milliamperes by 1000 gives us3 amperes. The metric prefixes make it easy to use numbers thatare a convenient size simply by changing the units. It is certainlyeasier to work with a measurement given as 3 amperes than as3000 milliamperes!Notice that it doesn’t matter what the units are or what theyrepresent. Meters, hertz, amperes, volts, farads or watts make no2 Radio Mathematicsdifference in how we use the prefixes. Each prefix represents acertain multiplication factor, and that value never changes.1.2 Converting Between Types of UnitsConverting between different types of units requires a conversionfactor. The conversion factor is a value representing how many unitsof Type A are equivalent to a single unit of Type B, generally givenas (amount of units of Type B) per (unit of Type A).For example, the conversion factors between pounds (Type A) andkilograms (Type B) are 0.455 kg/lb and 2.2 lb/kg.To decide which to use in converting units of Type A to Type B,select the factor with the Type B units in the numerator of the conversion factor. Multiply the amount of Type A units by the conversionfactor.For example, to convert 3 lb (Type A) to kg (Type B):1. Select the conversion factor with Type B units (kg) in the numerator, 0.455 kg/lb2. Multiply the amount of Type A units by the conversion factor:3 lb 0.455 lb/kg 1.37 kgSimilarly, to convert 3 kg (Type A) to lb (Type B):1. Select the conversion factor with Type B units (lb) in the numerator, 2.2 lb/kg2. Multiply 3 kg 2.2 lb/kg 6.6 lbNote that these conversion factors are reciprocals. You can changeone conversion factor into the other by calculating 1 divided by theconversion factor. Exchange the numerator and denominator at thesame time:1/ (0.455 kg/lb) 2.2 lb/kg and 1/ (2.2 lb/kg) 0.455 kg/lbYou can also convert Type B units into Type A by dividing theamount of Type B units by the conversion factor from Type A to TypeB:10 kg / 0.455 kg/lb 22 lbNote that not all conversions are simple multiplications or divisions. Additional offsets or factors may be required. For example, toconvert degrees Fahrenheit to degrees Celsius, the formula is:Deg C 5/9 (Deg F – 32)The Handbook’s Component Data and References chapterincludes a table of conversion factors for US Customary Units andbetween US Customary Units and Metric Units. Online calculatorsabound. Google (www.google.com) provides a unit converter accessible by entering “convert [abbreviation or name for Type A units]to [abbreviation or name for Type B units]” into an Internet searchwindow.
2 Numbers and Notation2.1 Accuracy, Resolution, and PrecisionThe terms accuracy, precision, and resolution are often confusedand used interchangeably, when they have very different meanings.When dealing with measurements and test instruments, it’s importantto keep them straight. Accuracy is the ability of an instrument to make a measurementthat reflects the actual value of the parameter being measured. Aninstrument’s accuracy is usually specified in percent or decibels (seebelow) referenced to some known standard. Precision refers to the smallest division of measurement that aninstrument can make repeatedly. For example, a metric ruler dividedinto mm is more precise than one divided into cm. Resolution is the ability of an instrument to distinguish betweentwo different quantities. If the smallest difference a meter can distinguish between two currents is 0.1 mA, that is the meter’s resolution.It is important to note that the three qualities are not necessarilymutually guaranteed. That is, a precise meter may not be accurate,or the resolution of an accurate meter may not be very high, or theprecision of a meter may be greater than its resolution. It is importantto understand the difference between the three.2.2 Accuracy and Significant FiguresThe calculations you will find throughout the Handbook followthe rules for accuracy of calculations. Accuracy is represented by thenumber of significant digits in a number. That is, the number ofdigits that carry numeric value information beyond order of magnitude. For example, the numbers 0.123, 1.23, 12.3, 123, and 1,230 allhave three significant digits.The result of a calculation can only be as accurate as its least accurate measurement or known value. This is important because it is rarefor measurements to be more accurate than a few percent. This limitsthe number of useful significant digits to two or three. Here’s anotherexample; what is the current through a 12-Ω resistor if4.6 V is applied? Ohm’s Law says I in amperes 4.6 / 12 0.3833333.but because our most accurate numeric information only has twosignificant digits (12 and 4.6), strictly applying the significant digitscalculation rule limits our answer to 0.38. One extra digit is oftenincluded, 0.383 in this case, to act as a guide in rounding off the answer.Quite often this occurs because a calculator is used, and the resultof a calculation fills the numeric window. Just because the calculatorshows 9 digits after the decimal point does not mean that is a morecorrect or even useful answer.2.3 Scientific and Engineering NotationModern electronics often uses numbers that are either quite largeor very close to zero. At either extreme, it is difficult to write thenumbers because of all the zeros. For example, the speed of light atwhich radio waves travel in a vacuum is 300,000,000 m/s. Similarly,a 22 pF capacitor would be written as 0.000000000022 F. These arevalues written in decimal form where all of the significant digits arepresent, including the place-holding zeros. This is a very inconvenientformat for calculation.Instead, most large and small values in electronics are written ina special type of exponential notation called scientific notation.Numbers written this way consist of a value multiplied by 10 raisedto an integer power. A number written in normalized scientific notation looks like this: D.DD 10 EEwhere D.DD is a decimal value between 1 and 10, such as 3.14 or7.07. EE is an exponent of 10, generally between 0 and 99. The following are a few ways of writing the same number in scientificnotation several ways:567 kHz 5.67 105 Hz 5.67 102 kHz 5.67 10-1 MHz 5.67 10-4 GHzBecause electronic units of measurements generally follow metricprefixes such as μF or MHz, it is most convenient to give values inthese units while still using the general form of scientific notation.This is called engineering notation. For engineering notation, theexponent must be a multiple of 3, such as –6, –3, 3, 6, or 12. Thesecorrespond to the various metric prefixes listed earlier. This meansthe number is written: DDD.DD 10 EE (units)Because standard units are used, engineering notation is easier towork with. For example, the most convenient units can be usedthroughout a calculation:2.47 V 2.47 100 V 2.47 103 mV 2.47 106 µVSimilarly, the value of a 151 kΩ resistor could be written severalways:151 kW 151 103 W 151 100 kW 151 10–3 MWYour calculator may have the ability to perform calculations in exponential, scientific, and engineering notation. It is worth figuring outhow to use these formats if you plan on doing any electronic valuecalculations.2.4 Rate of ChangeThe symbol represents “change in” a following variable, so that I represents “change in current” and t “change in time”. A rate ofchange per unit of time is often expressed in this manner. When theamount of time over which the change is measured becomes verysmall, the letter d replaces in both the numerator and denominatorto indicate infinitesimal changes. This notation is often used in functions that describe the behavior of electric circuits.Radio Mathematics 3
3 DecibelsDecibels (abbreviated dB) are just a way of expressing ratios, usually power ratios. If you are looking at the gain of an amplifier stage,the pattern of an antenna, or the loss of a transmission line you aregenerally interested in the ratio of output power to input power. Inantenna work, you are often concerned with the ratio of the powercoming from the front of a beam antenna to that coming from theback. These are some of the places where you will find the resultsexpressed in dB.Decibels are a logarithmic function. An important feature of logarithms is that you can perform multiplication by adding logarithmicquantities instead of multiplying them. Similarly, you can dividenumbers by subtracting in the same manner. This becomes a benefitif you are dealing with multiple stages of amplification and attenuation — as you often are doing in radio systems. In a radio receiverinstead of having to multiply and divide the gains or losses of eachstage to keep track of the signal processing — often with signallevels with many zeros to the right of the decimal point — you canjust add up all the dB and determine the total gain in the system.The deci in decibels refers to a factor of 1/10, as in deciliters for1 10 of a liter, while the bel relates to the idea of a logarithmic ratio,originally used to define sound power. The bel was named afterAlexander Graham Bell, the inventor of the telephone.3.1 Calculating Decibels from RatiosTo convert a power ratio into decibels:1. Find the base 10 logarithm of the power ratio.2. Multiply by 10.dB 10 log10 (power ratio)The same decibels can be used to represent voltage or currentratios, rather than power ratios. Since power goes up with the squareof the voltage or current, assuming the same impedance, the voltageor current ratios must be squared as well. With logarithms, ratios aresquared merely by multiplying by two. Thus everything works thesame way as for power calculations, except we multiply (or divide)by 20 instead of 10:dB 20 log10 (voltage or current ratio)If you are comparing a measured power or voltage (PM or VM) tosome reference power (PREF or VREF) the formulas are: P V dB 10 log10 M 20 log10 M P REF VREF Positive values of dB mean the ratio is greater than 1 and negativevalues of dB indicate a ratio of less than 1. Ratios greater than 1 canbe referred to as gain, while ratios less than 1 can be called a loss orattenuation. Note that loss and attenuation are often given as positivevalues of dB (for example, “a loss of 10 dB” or “a 6 dB attenuator”)with the understanding that the ratio is less than one and the calculatedvalue of change in dB will be negative.For example, if an amplifier turns a 5-watt signal into a 25-wattsignal, that’s a gain of: 25 dB 10 log10 10 log10 (5) 10 (0.7) 7 dB 5 On the other hand, if by adjusting a receiver’s volume control theaudio output signal voltage is reduced from 2 volts to 0.1 volt, that’sa change of:4 Radio MathematicsDecibels In Your HeadIncreasing power by a factor of 2 is a 3-dB increase and afactor of 4 increase is a 6-dB increase. When you increasepower by 10 times, you have an increase of 10 dB. You canalso use these same values for decreasing power. Cut powerin half for a 3-dB loss of power. Reduce power to 1 4 the original amount for a 6-dB loss. Reducing power to 1 10 of the original amount is a 10-dB loss. Power-loss values are written asnegative values: –3, –6 or –10 dB. The following tables showthese common decibel values and ratios.You can also derive all the dB equivalents of integer ratiosby adding or subtracting dB values. For example, to calculatea power ratio of 10/4 (2.5) in dB, subtract the dB equivalentsfor 10 and 4: 10 – 6 4 dB. Similarly for a ratio of 10/2 (5), 10– 3 7 dB. The ratio of 5/4 (1.25) is 7 – 6 1 dB and soforth. The same trick can be used with the voltage ratios. 0.1 dB 20 log10 20 log10 (0.05) 20 ( 1.3) 26 dB 2 A very useful value to remember is that any time you double thepower (or cut it in half), there is a 3 dB change. A two-times increase(or decrease) in power results in a gain (or loss) of: 2 dB 10 log10 10 log10 (2) 10 (0.3) 3 dB 1 See the sidebar “Decibels In Your Head” for a guide to easy valuesof dB you can remember and apply quickly without having to use acalculator.3.2 Calculating Ratios and Percentagesfrom DecibelsTo convert decibels to a power ratio, we do the opposite:1. Divide by 10 (or 20 if converting to a voltage or current ratio)2. Find the base 10 antilog of the result.Note that the base 10 antilog of a number is just 10 raised to thepower of the number. This is also something you probably don’t doin your head, so let’s see how you can easily perform the computations.Understanding a few characteristics of logs will help avoid problems interpreting results. Note that a gain of 0 dB, means that thereis no change to the signal — not that the signal has vanished! Theother important fact is that a power ratio of less than one (a loss ratherthan a gain) results in a negative number in decibels. dB power ratio log 1 and 10 dB voltage ratio log 1 20 Note that the antilog is the same as the inverse log (written aslog10–1 or just log–1). Most calculators use the inverse log notation.On scientific calculators the inverse log key may be labeled LOG–1,ALOG, or 10X, which means “raise 10 to the power of this value.”Some calculators require a two-button sequence such as INV thenLOG.Example 1: A power ratio of 9 dB log–1 (9 / 10) log–1 (0.9) 8Example 2: A voltage ratio of 32 dB log–1 (32 / 20) log–1 (1.6) 40
Since percentage is already a ratio, you can work directly in percentages when converting to dB: Percentage Power dB 10 log 100% Percentage Voltage dB 20 log 100% To convert back to percentages, just multiply the calculated ratioby 100%: dB Percentage Power 100% log 1 10 Table 2Common dB Values For Ratios ofPower and 1220 dB Percentage Voltage 100% log 1 20 Here’s a practical application for converting dB to percentages andvice versa. Suppose you are using an antenna feed line with a signalloss of 1 dB. You can calculate the amount of transmitter power that’sactually reaching your antenna and how much is lost in the feed line. 1 Percentage Power 100% log 1 100% log 1( 0.1) 79.4% 10 So 79.4% of your power is reaching the antenna and 20.6% is lost inthe feed line.Example 3: A power ratio of 20% 10 log (20% / 100%) 10log (0.2) –7 dBExample 4: A voltage ratio of 150% 20 log (150% / 100%) 20 log (1.5) 3.52 dBExample 5: –3 dB represents a percentage power 100% log-1(–3 / 10) 50%Example 6: 4 dB represents a percentage voltage 100% log-1(4 / 20) 158%3.3 Using the WindowsScientific CalculatorIf you have a suitable scientific calculator, it should easily do yourcalculations. Not all have an ANTILOG button, but if not, they willlikely have a button that says X Y, which can be used as above. Ifyou don’t have a handheldcalculator, you can use theCalculator accessory inthe Microsoft WindowsFigure 1 — Screenshot of the Windows10 scientific calculatorready to find the powerloss of 2 dB.operating system For Windows 7 and earlier versions, click START,then ALL PROGRAMS, then ACCESSORIES. For Windows 10, clickSTART, then look in the menu for CALCULATOR.On first opening the calculator, you may find a four-functionStandard calculator. To change to a Scientific calculator in the Windows7 and earlier versions, click on VIEW, then SCIENT
Radio Mathematics 1 Radio Mathematics This supplement is a collection of tutorial information on a variety of mathematics used in Amateur Radio. Many of the fundamental relationships in electronics and radio are best expressed in the language of math. This chapter will expand in subsequent editions as it becomes clearer what information will
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used in the ARRL Laboratory. While this is not available as a regular ARRL publication, the ARRL Technical Department Secretary can supply a copy at a cost of 20.00 for ARRL Members, 25.00 for non-Members, postpaid. Most of the tests used in ARRL product testing are derived from rec
While this is not available as a regular ARRL publication, the ARRL Technical Department Secretary can supply a copy at a cost of 20.00 for ARRL Members, 25.00 for non-Members, postpaid. Most of the tests used in ARRL product testing are derived from recognized standards and test methods. Other te
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This section contains a list of skills that the students will be working on while reading and completing the tasks. Targeted vocabulary words have been identified. There are links to videos to provide students with the necessary background knowledge. There is a Student Choice Board in which students will select to complete 4 out of the 9 activities. Student answer sheets are provided for .
