Publisher Summary

This chapter reviews the principles of basic applied mathematics. There are many excellent books on physics, math and applied math, suitable for all levels, and it is not the intention to reproduce large tracts of math and physics here. There are certain fundamentals that are necessary to adequately understand the process of selecting, designing, procuring, installing, and testing a cable system. Some of the basic elements, such as decibels, occur over and over again in the particular subject of cable engineering. The chapter explores topics such as those and gives the reader a sufficient foundation of knowledge to make best use of the remaining chapters in the book, “Cable engineering for local area networks.”

2.1 Working with indices

To handle very large or very small numbers, we use a convention of representing these numbers in the following notation, for example:

9=100000000010−3=0.001

The small number in the superscript, or the index, if it is positive, represents how many zeroes there are, or more precisely, how many factors of ten are involved. So 109 means one thousand million, or a billion.

If the index is negative, then the number is less than one, and the index number reveals how many places after the decimal point there should be, or how many factors of divisions of ten there are. So 10−3 means one-thousandth. An expression of 6.3 × 106 means 6300000.

If two numbers in this notation are multiplied together then simply add the index numbers together. For example:

4×107=101110−3×107=104(2×104)×(8×106)=1.6×1011

To divide two numbers, subtract one index from the other. For example:

9/106=1031012/10−3=10158×108/2×104=4×104

Simple addition or subtraction of two numbers in this form can only take place if the indices are the same. For example:

8.5×106)−(3.1×106)=5.4×106

2.2 Prefixes to denote size

There are accepted prefixes we can use to denote the size of a number more simply than always writing it out or pronouncing it in its entirety, e.g. kilometres means one thousand metres; the kilo part representing one thousand. Table 2.1 gives the full list. For example:

pF is one picofarad, or 1×10−12 farads.

2.3 Logarithms

Logarithms of numbers also make the large and the small easier to manipulate. A logarithm of a number is that number that you have to raise another number to the power of to get back to the first number. For example:

This is working to base ten, but logarithms can be expressed in any base. To be precise we should write log10, but it is always assumed that if no base number is specified then we are talking about calculations made to the base ten. These are sometimes referred to as common logarithms. Sometimes base 2 is used in communications theory because digital transmission only has two states, that is ‘ones’ and ‘zeroes’. The log2 of 16 is 4, because we have to raise 2 to the power of 4 (24) to get 16.

Antilogarithms are simply working the other way round. For example:

102=100antilog103=1000antilog10−3=0.001

2.4 Decibels (dB)

Decibels are ten times the logarithm of a ratio. They are used in all branches of engineering, and can be used to represent differences in electrical power, light or even sound. Using decibels makes calculations much easier to comprehend and even do in your head. For example, if the attenuation of one piece of cable is 4dB, and you add onto it another length of cable with 3dB of attenuation then the resulting attenuation of the whole channel is 7 dB. Decibels are a convenient shorthand which show how energy is absorbed or produced regardless of what levels of energies are actually involved:

(attenuation if it is negative)=10 log10(P1/P2), [2.1]

where P1 is one power measurement and P2 is another power measurement that we wish to compare with the first.

For the remainder of this book we will adopt the convention that all logarithms are to the base 10 unless otherwise denoted. For example:

if power level 1 (output) equals 1mW and power 2 (input) equals 0.002mW then

=10 log1/.002=10 log500=10×2.7=27dB

for attenuation we have

 log 0.002(10 log P2/P1)=−27dB

Note the absolute value does not change, only the sign. Many writers leave out the negative sign altogether if it is clear they are talking about attenuation, so as to avoid the uncertainty of a double negative.

Sometimes it is presumed that the comparison is being made to 1mW of power, so P1 or P2 in the equation will always be one. To denote this the resulting answer has the units dBm.

Measuring the power in a device or a cable is not nearly as easy as measuring the voltage across it relative to ground or any other potential. We can take account of this by knowing that:

=volts×amps, or P=V×I [2.2]

But from Ohms law we also know that V = IR and hence / = V/R and therefore power also equals V × V/R or V2/R.

R is the resistance of the cable and is presumably the same from one reading to another, so we can cancel it out in the following equation:

=10 log(V12/RV22R)=10 log(V1/V2)=20 log(V1/V2) [2.3]

So we can obtain decibel readings by simple voltage measurements and incorporating a factor of 2 in the standard decibel equation.

It should be remembered that power only equals voltage multiplied by the current for the special case of direct current. For alternating current the correct formula is:

 (W)=voltage×current×the cosine of the phase difference between the two.

The cosine of the phase difference is known as the power factor. If the current and voltage are in phase then the angle is zero and the cosine of zero is one, so in that special case power does indeed equal voltage times current. Any reactive load, i.e. capacitance or inductance, will cause the current to lead or lag the voltage in phase.

2.5 Sine waves and phase

A sine wave or sinusoidal wave is the most natural representation of how many things in nature change state. A sine wave shows how the amplitude of a variable changes with time. The variable could be audible sound for example. A single pure note is a...