And here on the introduction, circuit quantities and component values have extreme ranges in electronic circuits. It is not uncommon to have values such as, and here we have a very large frequency, 1,200,000,000Hz, kind of a large number.
Then here we have a number in Farads and it's 0.00000000047F in the same circuit and these numbers are quite large. For this reason, metric prefixes and engineering notation are used to simplify communications and computations. In this particular value here we will probably express this as rather, than this huge we will probably say 1.2GHz. This one would probably be 470PF. These are much more manageable values than these giant numbers.
Engineering Notation is a technique for expressing very large and very small numbers in an easy to understand and interpret form. Engineering notation is based on powers of ten, and examples of which are shown to the right, so we have a few examples here. Remember 10 to the 0 is always going to be equal to one. Then we have powers of 10 to the 1st = 10, 10 to the 2nd = 100, 10 to the 3rd is 1000 as equivalent to 10*10*10.
Going in the negative direction, if we have 10-1 remember that's 1/10 equivalent to one tenth. 10-2 is 100th, 10-3 that will be 1000th—the equivalent of 1/10*10*10.
Powers of Ten
Any number can be expressed as a series of digits multiplied by a power of ten. The procedure to express a number as a power of ten is, and this is the procedure, there are three steps here:
Move the decimal point to any desired position while counting the numbers of positional changes. If the decimal point is moved to the left, the exponent will be positive. If to the right, the exponent will be negative.
The value of the exponent will be the same as the number of positional changes made by the decimal point. I want to look at the examples in 3.1.
Let's convert the following numbers to equivalent numbers expressed in powers of 10 form. We start out, here we have five values and you don't usually do this but we could say all of these numbers you could say *100 because ten to the zero was one so any of these numbers times one is in fact in itself. You don't usually do that but we could say that.
Let's take this first number and let's move the decimal place one point so that would be 12.35 and since we moved the decimal point one place to the left, the exponent will be a positive one. If we had moved the extra of the decimal point let's say two points then it would have been 1.235*102. If we'd have moved it three places over here it would have been 0.1235*103 and all of these are representative of the same value.
Here we have a decimal point, a decimal, in this case rather than going to the left we will go to the right. Let's move this one, two, three points so with that we will say 4.2*10-3 and since we went to the right it will be a negative exponent. If we probably could go over one more and that case we would say 42*10-4 and again all of these would be equivalent. Another decimal let's say point, let's move this two places so we'll say 2.26 and this as we moved 2*10-2.
Here we have a large value 78,200,000 we could move our decimal point one, two, three, four, five, six places. How about that? 78.2*106 and since we moved six places to the left, positive 6. Then -2.77, if we moved to the right one point, we could say -27.7*10-1 and since we moved to the right 1 to the -1. We could have gone the other direction, we could have gone like this and in that case, we would have said -0.277*10 and since we went to the left, the exponent would be a positive one. These are some alternative ways to express these basic numbers as powers of ten.
Then in example 3.2 let's just do a couple of these. In this example, the number is given to us as a power of 10 and we are asked to convert it to a different power of 10. The method, one method of doing this is to simply subtract the exponents and that is how many decimal places you would move the current decimal place. Here we have 104 so we start out with the decimal exponent of four and we want to convert from two so if we subtracted that minus two we will have plus two.
To convert this we would simply need to move that exponent plus two places so that would be to the left one, two and that will be 1.44*104 is equivalent to this value here. Then the second one we have a -0.09*10-1 and we want to convert that to *10-5 so we start out with minus five and then we're going to subtract this exponent it's going to be minus, in this case, it's a minus one. Since they're minus and minus, that's going to effectively make this a plus so the result will be minus four. To convert this we would need to move four places so we'll go one, two, three, four and it's still minus so minus that is going to be -900 and we have that conversion.
Engineering notation is similar to standard powers of ten operations, with restrictions. In engineering notation, the decimal point can be moved to any convenient position as long as the resulting exponent is either zero or a multiple of three and so here you see some examples, - 6, -3, 0, 3, 6 and 9. Could be 12, 15 etc.
The procedure goes like this, write the number using powers of 10, move the decimal point right or left while increasing or decreasing the exponent. The final exponent must be a zero or divisible by three. Here are some examples. In fact, you have steps of examples, 3.3, 3.4, 3.5 and 3.6. I want to do a couple or a few of the problems from each of the sections.
The first one is to express 129*105 in engineering notation. Now since the exponent is five here this would be unacceptable in engineering notation so we could either go to three or six. Six is the closest one so let's go that way. If we are going to increase, we have to go plus one. That would mean we would have to move our decimal point one place. We could say, 12.9x106 is the same as this value but now that is with this exponent of six it is now an engineering notation. Then they want you to use a calculator to here multiply 125.8*103*0.6. I think I have a little calculator here. This particular calculator is available as a free download. It's called Calc98. Let me just clear the screen here and let's see the options here. Let's go to… I don't want options, I want… display.
Currently, we are in Engineering Mode. Let's go to Normal Mode and do this and just see what we would see is normal calculation and then we're going to change it to engineering. We are going to say 125.8 and then we want to go *103, so one way to this would just go down here to Exponent and then three and then we're going to multiply that *0.6, so it would go *0.6 equals, and so there we see the results 75,480. Now, that is just in the standard normal calculator value. Let's go to our options again and now let's change that to Engineering and we'll leave it to three decimal places. You'll see here the calculator is done it for us. The value 75.48 with the exponent three is the resultant value. Then there is more there. Please do those with your calculator.
Section 3-5. In 3-5 it says write using the nearest standard prefix and you want to refer to Table 3-2 when you do this. The first one is 120*103V. We know that 103 is 1,000. We will typically write this as 120 and we would say KV. Here we have 47*10-12 Farads. Now Farads has to do with capacitors, capacitance and we would typically write that as 47-12 would be PF and then we would say F for farads or 47 picofarads.
Then write using symbols in Table 3-2. These comes from Exercise 3-6 and let's see, so we are saying 125*10-3 so the way we would typically write that would be 125 millivolts. Now that is equivalent to 0.125V. We would typically go with electrons. We would write that like this 125mv. Then we have 5*103Hz. Remember 103 is 1,000 so that would be a K. So we would say 5k and Hz for hertz.
Then we have one down here 0.001*10-11 and remember -11th is not in scientific notations so we want to change that to -12. We want to go one more negative so we would need to move that decimal point one more to the right, so we would move it over here to 0.01 and then we could make this to the 12th. Then we could say 0.01 and that would be in a PF. Please do all of the problems that are in 3-3, 3-4, 3-5 and 3-6.
Prefixes for Unites of Measure
Metric prefixes are used to communicate the value of very large and very small quantities. Most metric prefixes are used in places of powers of 10. Metric prefixes simplify the expression of numbers in electronic systems.
Powers of Ten and Metric Expressions
Here we have powers of 10 and metric expressions and as we go through this course, you're going to become very familiar with almost all it. You probably won't use this one very much. The fento which is -15. That's not commonly used in electronics but all of the rest of these from -12 to +12 you are going to get familiar with.
Now, centi actually falls out of the engineering notation because it's the minus the minus two. Minus zero is just the number at whatever the number is *-1 so that would… you don't typically see 100 but it's understood that all values unless they have one of the prefixes is in 100 notation. But we will use all if this pico, nano, micro, milli, giga, mega, tera etc.
Okay, so in this lesson, we've introduced scientific notation with the powers of 10. Let's just backtrack here, powers for units of measure. We did some calculations and make sure you do all of these on your own. Engineering notations, we did some conversions from one power to another power and then we expressed some values on the alternatives powers of 10. We looked at what is scientific notation.