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If you didn't had the faintest idea what the mathematical parts in the last section meant, then read this! It's a quick introduction to basic math for electronics. There's nothing difficult, but without understanding it you won't get far. |
You may, if you are out of practice with mathematics, be confused by some of the conventions I will use in equations, so a quick refresher might be useful. Indeed, if you are reading this material you might well not have learnt this stuff at all, and may not have understood what I've said so far, so we'll have a very quick look at some basic operations and conventions - nothing complicated (I was consistently 31st in a mathematics class of 31). If you are 'au fait' please feel free to skip this chapter, but I would not like anyone to fail to understand electrics because of lack of familiarity with basic mathematics. It won't be complex - I am quite inept at it myself.
I'll start off, though, by telling you that 'we have a problem'. Well, two problems, because I don't know what you know, so I risk boring you or confusing you. But the problem I'm referring to is that this material is being re-set in HTML form on a computer, which is where I suspect you are probably reading it! The problem is that it is very hard to set conventional mathematical expressions on a computer. Don't worry, there's only basic mathematics here, but it's still a problem to put it on the browser page in the same way that it is written down. For example, typically a book would show mathematical operations something like this:
You might 'say' this as follows: 'Equation: x equals, minus b plus root all of b squared minus 4 times a times c, all over 2 times a'. If you don't follow that, carry on reading - that's what this section is going to explain.
Now, I've had to place this example (which includes all the
operations we're likely to use) as an image - which won't
show up if you are using a text-only browser. It's not
currently possible to do the notation in the conventional way
without either setting all the symbols out as graphics (which
doesn't work if you have no graphics turned on) or by using
some special extension (MathCAD, Matlab, whatever) which you may
not have or want, and is excessive anyway for what we're going
to do here. The next generation of browsers just might have some
fixes for this, or they might not.
This leads to quite a few problems with a text on electricity and electronics, where there's quite a few special symbols normally used. Here's a few we'll have problems with:
The solution is to type the stuff in the way it is set out for operations on a computer. You probably already know about this (and anyway that's what I've been doing so far), but just in case you don't understand it, here is a ten-second guide.
How can an equation be set out on a computer? Well, the equation above, when entered in a programming language, might be typed:
x=(-B+SQRT(B**2-4*A*C))/(2*A)
You may be more familiar with the spreadsheet, where it might be entered like this:
+(-B1+SQRT(B1*B1-4*A1*C1))/(2*A1) [NOTE 1]
Both these different ways of entering mathematics have features in common. For example, the symbol for multiplication or 'times' is the asterisk, '*' and that for division - 'divide by', the forward slash, '/'. The plus 'add' and minus 'subtract' signs are conventional: '+', '-'. We'll use all these symbols in this way to represent these operations. Raising a number to a power - that is, multiplying itself by itself a stated number of times - will be represented by the 'superscript' or 'raised' number, so 2 to the power 2 ('2' times itself twice, or 'two squared') will initially be shown like this: 22. This will lead to some early problems in browsers such as Lynx, I'm afraid, but I can't actually solve all the problems of mathematical presentation by myself! I do have a solution for this problem, though - as we will see a little later. Other symbols will be introduced as we get to them. Let's start...
An equation states that two things each are equal to the other. A very silly equation (but quite true) is 2 = 2. A slightly more complicated one is 17.5 = 87.5 / 5. However, this sort of equation, whilst true, is not very general. To produce generally-useful equations we need to go about it differently. What we often do is to use letters in equations to stand for the position of a number, and to tell us what to do with the numbers when we eventually know them. For example:
V / I = R
We don't actually try to divide the letter V by the letter I: V stands for the position in which to insert the quantity of volts, I for the position for the quantity of amps (these are explained elsewhere in the text). The letters are called variables and the whole equation is an algebraic equation, one that uses letters to stand for numbers. Yes, you are actually doing algebra. You might like to know that the word 'algebra' comes from the title of a book written in the year 825 (Western Calendar) by Abu Ja'far Mohammed ibn Musa al-Khowarizmi, called "Kitab al jabr w'al-muqabala" - moreover, his name gave rise to the term algorithm used often in computing, and his book also was responsible for familiarising the West with the number notation (0, 1, 2 ...9) we use now!
As I said, an equation is an expression meaning that the quantities on one side are equal to those on the other. For this reason, we can manipulate the equation in various ways, as long as we do the same thing to the quantities on both sides of the equation. An example:
I = V / R
The equation is unchanged if both sides are changed in the same way, like in this example:
10 I = 10 (V / R)
Two conventions immediately appear. The example means:
10 * I = 10 * (V / R)
but it is unnecessary to write the '*' sign. It is implied whenever two quantities are directly written together in algebra. (Another way of looking at this is that the other possible signs are '+','-', or '/', meaning plus, minus, or divided by, and if these are absent then it has to be the remaining one,'*'). So we can normally leave out the times sign (sometimes a dot is used instead, '.' but this is easily overlooked or mistaken for spilt coffee :).
The brackets specify that everything inside the bracket is to be treated as a single quantity. There are rules for the order in which operations are done, and these are known as rules of precedence. The order is:
(for example, +1, +10, -71, -1045)
and the equation won't work out if these are not done in the right order (because the equations are designed on the basis of this order). Unary plus and minus means the sign of the number, +1 or -71, and the symbol is written right up against the number. When we intend addition or subtraction, a space should be written between the sign and the number, 1(space)+(space)1. Now, +1 + -1 of course is 0: and -1 * -1 is +1: but +1 * -1 is -1. In these particular equations you could read the '*' sign as 'of'. So the last two equations read, 'minus one of minus one is plus one, but plus one of minus one is minus one'
So, continuing with the above example, we multiply the quantities inside the brackets by the quantities outside before we can do anything else. Expanding the equation a bit,
10 I = 10 (V / R) = 10V / R
or alternatively:
10 I = 10 (V / R) = V / (R/10)
(if we multiplied both V and R by ten, the value would not change since 10 / 10 = 1. V must be multiplied by ten, OR R divided by ten).
I'm sure you can see that we could divide everything by ten to get back to where we started.
If you want to move a quantity from one side of the equation to the other, that can be done using the same principle. Let's get V on the other side, using the starting equation: I=V/R
This process is called transposition. Many simple formulae are of this form: a = b * c and they can be transposed to:
b = a / c and c = a / b
For example, we may remember the rather infamous equation E = mc2 (E equals M C squared) which can be written E = m * c2 - so we can also write
m = E / c2
- which tells us that matter is equivalent to energy divided by the fundamental constant c (the speed of light) multiplied by itself - or
c2 = E / m
- which says that the fundamental constant c (the speed of light) multiplied by itself expresses the constant ratio of Energy to matter, thus itself a fundamental constant. That's all quite interesting, philosophically, but unlikely to be of immediate practical relevance unless your yacht is an ex-navy nuclear powered submarine (which is, I suspect - on the whole - improbable).
I also mentioned simple proportion. Proportion takes the form
a:b = c:d
which can also be written:
a/c = b/d
and here there are four variables, a, b, c, d instead of three. To change this sort of formula around, we can cross multiply like this:
a*d = b*c
(check it with some numbers if you don't see it - for example a = 20, b = 2, c = 10, d = 1). Usually we know three of the values and want the other one. To transpose the equation we use the cross-multiplied form, and then multiply or divide both sides by the same value until it is in the right form. Say we wanted a:
a*d = b*c
Divide both sides by d:
(a * d) / d = (b * c) / d
d/d is of course 1, so:
a * 1 = (b * c) / d or a = (b*c) / d
This sort of re-arrangement applies to any four-part equation of this type.
You will have noticed that we have already run into some rather large and rather small quantities and ratios when dealing with resistance. This very wide range of numbers is commonplace in electrics and electronics (in physics and engineering at large, in fact) so there are various conventions for dealing with these big numbers. In engineering in general, quantities larger than 100 or less than 0.01 are usually given in standard form, as a power of ten. A power is just a number multiplied by itself a specified number of times. Two to the power two or two squared (22) means 2 times 2 = 4. Ten to the power five (105) means 10 * 10 * 10 * 10 * 10 or 100,000. 3 times 10 to the power 5 (3*105) is 3*100,000 or 300,000. We can also use negative powers of a number, for example 10 -2 means 1 / 10 * 10 or 1 / 100th or 0.001. Standard form makes use of powers of ten to simplify calculations. A number is modified so that it consists of two parts:
The opposite (or perhaps 'converse') to a power is a root. A root is the number that, when multiplied by itself the specified number of times, gives us the number that it is the root of. To clarify, the square root of 4 is 2, 2 * 2 = 4. But there is another square root of 4, -2, since the rule is that -2 * -2 = 4. The square root of 9 is 3 (or -3). The cube root of 8 is 2 (NOT -2 ...-2*-2*-2 = -8!), 2 * 2 * 2 = 8 and so on. The root is represented by the symbol:√ (as explained above) with a superscript number in front of it to indicate which root it is, thus: 3 √ 9 = 3 (which you'd read as 'the third root of 9 is 3') When it is a square root the superscript is usually left out. Now there is a fair to middling chance that you are not seeing the root symbol at all, so I will avoid using it in the rest of the text. We'll use the word root instead. As an alternative, a convenient way to write a root is to write it as a superscript fraction: 91/3 = 3, but this is just as problematic as the other options, on a computer screen.
Now to an example: To express 47000 and 0.0015 in standard form.
47000 = 4.7 * 10000 = 4.7 * 104
0.0015 = 1.5 *1/1000 = 1.5 * 10-3
The simple way to do this is to count the number of places the decimal point has been moved, like this:
47000 is really 47000'.'
Move the decimal point to the left until the number is between 1 and ten:
[move 1] 4700.0
[move 2] 470.00
[move 3] 47.000
[move 4] 4.7000
The power of ten in the multiplier is therefore 4:
4.7*104
With small numbers do the same thing, but move the point to the right:
0.0015
[move 1] 00.015
[move 2] 000.15
[move 3] 0001.5
The number is now between 1 and ten, so is expressed as 1.5*10-3. After doing this once or twice to get the idea it can be done by inspection, rather than writing it out. The first part is called the coefficient and the second part is called the exponent or index. If you have a scientific calculator, the exponent can be directly entered using the 'E', 'EE' or 'EXP' key. There are simple rules for adding, subtracting, multiplying, dividing, raising to powers and reciprocals for numbers in exponent form:
For a number x to exponent m or n - xm, xn:
Multiplication - add: xm * xn = x +n
example: 102 * 103 = 105
Division - subtract: xm / xn = xm-n
example: 105 / 102 = 103
Raising to a power - multiply: (xm)n = xm*n
example: (103)2 = 106
Taking a root - multiply: (xm)1/n =x(m/n)
example: (106)1/3 = 10(6*1/3) = 102 = 100
This is the same as raising to a power.
Reciprocals: 1/(xm) = x-m
example: 1 / (103) = 10-3
equally: 1 / (10-6) = 10-6
Addition and subtraction can only be done if both numbers have the same exponent. If they don't they must be rewritten so that they do. For example, if you have the numbers
3.3*104 and 2*103
they must be changed to 33*103 and 2*103. Then:
Addition: xm + ym = x + ym example: 33*103 + 2*103 = 353 = 3.5*104 or 35000.
Subtraction: xm - ym = x - ym example: 3.0*103 - 1.6*103 = 1.4*103
Having got you used to this form of notation, I am now going to change it! Not much, though. I use a way of writing exponents or indices that I find much easier and quicker to use when writing them by hand, or in typing. It also clearly distinguishes between powers of a number (like V2) and powers of ten (102). The way I write them is this: 2 * 102 is written as 2E2 and 1.717 * 10-4 is written as 1.717E-4. This means you don't have to write superscript numbers all the time, which is harder to do when writing than when they are printed, and you don't have to keep writing '*10'. Moreover, it works just fine on machines that don't have a font that allows subscripts and superscripts, or with a text-to-speech browser. E means 'exponent', and I always use this (it is the way it is written with computer programming, when you don't have superscripts and subscripts available). The calculations will be done using this notation from now on. It also fits in very well with scientific calculators, if you should happen to buy one. The whole point of using exponents is that it is much easier to do arithmetic with large numbers in standard form. An example:
What is 300,000 times 0.00015? Express in standard form:
3.0E5*1.5E-4
Now just do the multiplication of the numbers:
3*1.5 = 4.5
and add the multipliers:
(+5) + (-4) = 1
The result is 4.5E1, or 45.
Now in daily life we don't (except when being in the happy? state of owning a yacht) much need to deal with large numbers, but in electrics and electronics we are regularly running into things like "how much current is going though this 1,800,000 ohm resistor?" or "How many millionths of volts and billionths of amps are arriving at the radio from the aerial?". The values for electronic components are given in modified exponent form. It is disguised, mind you, but that's what it comes down to. You will undoubtedly (?!) know or at least have heard of kilometres ('kilometers' if you speak US) or kilograms. There are a standard set of multipliers and dividers in powers of three in the metric system, which are:
| Prefix | Abbreviation | Power of ten notation | E notation | Multiplier |
|---|---|---|---|---|
|
Giga |
G |
109 |
E9 |
1 000 000 000 |
|
Mega |
M |
106 |
E6 |
1 000 000 |
|
kilo |
k |
103 |
E3 |
1 000 |
|
milli |
m |
10-3 |
E-3 |
1/1 000 |
|
micro |
('mu') symbol |
10-6 |
E-6 |
1/1 000 000 |
|
nano |
n |
10-9 |
E-9 |
1/1 000 000 000 |
|
pico |
p |
10-12 |
E-12 |
1/1 000 000 000 000 |
Different electronic components tend to use different ranges of these. Instead of writing the resistor value as above, 1,800,000 ohms, it would in practice be written 1.8 M ohm. This is equivalent to 1,800 k ohm. The same multipliers and dividers are always used. For current, a current of 0.0000015 amps would be written as 1.5 mu-A (you wouldn't write mu, you'd write the mu symbol as in the image at the beginning of the chapter). This is the same as 1500 nA. And so on, and so on. The whole thing is very like the standard notation I introduced above, but with a tiny modification. Instead of the strict standard form - where the expression always consists of a number between 1 and 10 and an exponent, we adjust the exponent to be the nearest standard power of three - kilo, mega, micro, pico, whatever - and then the number is adjusted accordingly. In this way it is easy to compare values within the same range, and everything maps nicely onto standard component values and measuring scales. There's very little difference - if you can read one form, you can read the other and convert easily. Let's try out an example.
How much current is going through the 1.8 M ohm resistor when it has 1.23 volts across it?
I = V / R = 1.23 / 1.8E6 = 1.23 / 1.80 * 1 / E 6 = 0.683 * 1 /E6
Now 1 / E6 is the same as E-6 so
I = 0.683 * E -6 = 0.683 micro-Amps (mu-A) - this is the same as 683 nano-Amps (nA).
Note also that you can write the answer down directly without bothering with the millions or million-ths. In (my modified) standard form we'd have expressed it as 6.83E-7, but this does not map onto the standard power-of-three values. At the cost of allowing the number to exceed the range limit of 1 to 10, we can have 0.683 E-6 in power-of-three form [NOTE 2]
In commercial applications, numbers of more than a thousand are expressed with the 000's divided up into groups of three for easy reading, like this - $ 1,713,002.00 - but this grouping with commas is not used in engineering. This is in part because different countries use different symbols, in Spain (for example, and most of Europe) it would be written 1.713.002,00 $ so there could be confusion over the decimal point. It is better just to leave a space - 1 000 000 ohms. There is one other important point on conventions. The decimal point '.' is very easily lost in printing, so in the UK, ever since the introduction of a standard called BS1852, it has been the practice to write the multiplier instead of the decimal point. Our 1,800,000 ohm resistor would now become 1M8. The ohm is left out, as long as it is clearly understood that it is a resistor we are discussing. Exactly the same practice is used for other component values. Current and voltages are expressed with the indicator, 100mV or 2.5nA. I suppose you could write 2nA5 but it isn't done. 2A2 (2.2 amps) or 13V8 (13.8 volts) is commonplace and sensible, however. Losing the decimal point leaves us with a slight problem when expressing the value of a 1 ohm resistor. This is got round by using the letter 'R' to stand for a multiplier of one, so we write 1R0, adding the final '0' for clarity. A resistor smaller than one ohm would be written 0R1, for 0.1 ohms. I suspect printers were happy not having to set ohm. Don't get mixed up with the practice of numbering components. R1 means resistor number 1, not a 0.1 ohm resistor, and for this reason the leading zero in 0R1 is important. Another way round this is to use subscripts, R1, but this practice seems to have been dropped.
Despite appearances, electrics is a rather approximate sort of thing. Real components, whilst they will have an actual value, will not usually have the value marked on the case. They will be to a certain tolerance, usually 5% or 2%, sometimes 1% or even 0.1% Some components will have a much wider tolerance, ± 100% in some cases, so there is no point being pedantic over excessive accuracy. Three significant figures is usually more than enough for most purposes. By 'three significant figures' I mean 1.23, or 123,000, or 1M25 - there's three actual numbers expressing the value, the rest is the multiplier. If we measure a voltage with a voltmeter (soon to be described) we will be lucky if the accuracy is much better than 5% with a cheap meter, so a measurement of 13.8 volts will be 13.8(± 5%), or between 13.11 and 14.49 (the ± symbol is read as 'plus or minus', which is what it also means). Writing this as 13V8 is already optimistic. If something is known to two significant figures, 1.5 say, any calculations using this will also only be accurate to two figures, and adding any more on is spurious. 1.5 * 0.49 = 0.735, but it is better to say 0.73. It is always possible to know the accuracy of a number by the number of significant figures given; the accuracy is half of the least significant figure, so 0.5 means 0.5 ± 0.05, whilst 3.1416 means 3.1416 ± 0.00005.
Everyone is familiar with the practice of presenting information
in graphic form, and we will use it as well. There are a couple of
points to note. In ordinary commercial practice, 'graphs'
consist of a set of points, yearly sales or whatever, that are
either presented as a bar-chart or have the points joined up with
lines to guide the eye from one years figures to the next, Figure 1 
. This
sort of thing is not a real graph. A graph has to
represent continuous data points, not one a year
or one a month, and the values at any point in between the ends of
the graph can therefore be taken off by measurement: look at
Figure 2 for an example.
Non-linear scales also come into electrics a
lot. Many important physical quantities do not have a
linear relationship in our perception,
for example volume of a sound, pitch of a sound, the brightness of
a light. When we are dealing with hugely variable quantities, it is
difficult to express them using a linear scale. For example, try
and show the data points 2, 12, 31, 69 000, 1 457
000 on a linear scale and you will immediately grasp the
difficultyFigure 3 . For this reason logarithmic scales are
widely used with this sort of problem. A logarithmic scale is one
where the growth of the scale follows a power-law relationship
rather than a linear one, power-law meaning exponent-style. In
other words the points at one cm, two cm, three cm, and so on would
follow a relationship like 10, 100, 1 000, 10 000 and so on Figure 4 .
Logarithmic resistance scales are common used in electronic
equipment, for example all volume controls are of this type to
allow for the way our ears work.
Wherever possible, use powers of ten to simplify the mathematics. You can take the powers out as I have shown in the calculation of current through the 1M8 resistor, and put them at the end as a separate multiplication or division. If you have to calculate something alarming like: root 0.0009 then this can be simplified by realizing that it is equal to root 9 * root E -4 = 3E-2 = 0.03. This works the same for all roots. Another shocker when the calculator is on the blink is: 14.012 + 6.232
This is of the form a2 + b2 and can be factored without working out the squares like this: a2 + b2 = (a + b)*(a-b). This only works for squares, though.
Components are not available in every possible value. There are various standard sets of values, the most common being called E12. This divides up a decade (10 : 100) or (1 000 : 10 000) into 12 different values like this:
10 12 15 18 22 27 33 39 47 56 68 82
which are available in multiples or sub-multiples of ten. For resistors, this means that you can get a 1000R (1k) or 47k or 1R2 or 3M9 resistor, but not the ones in between. There are other scales, E24 with 24 values, E96 with 96, if in-between values are needed, with E6 for coarser variations. Some components also will need a wattage rating, for resistors the common ones are 0.1W, 0.25W, 0.33W, 0.5W, 1W and so on. Components also have a voltage rating, that being the maximum volts they can handle without breaking down, temperature ratings, humidity ratings and so on and so on, all of these being important sometimes but usually irrelevant, because the limits are not often approached.
People have come up with many impressive justifications for the strange relationship between these values, 33 : 39 for example. Why not 30 : 35 ? I am assured that the true story behind it is that a war-time shipload of resistors urgently needed for production was sunk, and someone had to replace them quickly. If you just make a whole lot of resistors with random values, using this scale they can be selected on test to have the values above with a tolerance of ± 10%. That is, a 33k resistor +10% is 36k3, and a 39k -10% is 35k1 - these values overlap, so all the resistors made can be used. Since it is usually the ratios of two components that are important, this system works fine. All components will have a tolerance, nowadays 5% is usual, and 2%, 1% and better are increasingly commonly available.
I hope you followed all that without alarm. There is more complex mathematics needed for certain topics [ [NOTE 3] , but this will carry you through everything we will look at here. If you actually get enthusiastic about keeping your gear working, or even designing the odd fix for problems aboard, you will find it useful. It is almost inevitable that you either want more, or less, explanation than I've given. If you are really stuck or notice a stupid error (I was after all 31st in a class of 31 - well actually, 32nd once, I've never been quite sure how) then do let me know...
[Note 1] all these examples are taken from the MathCAD documentation, by the way. They have had the same problem - the equations are rendered as images in the help file!
[Note 2] just as a matter of interest, if your yacht happened to cost $ 95,000 you could express this as $95E3 which sounds just a little less painful...
[Note 3] If you actually get into ac electrical theory, you'll realize that I just made a very feeble joke. Sorry.
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