Chapter 4: Metric Prefixes & Powers of Ten Gigafun with nanoeffort

1. Chapter 4:Metric Prefixes & Powers of TenGigafun with nanoeffort Presented by: James, VE3BUX

2. Base-10: Quick review of “tens” • We count in base 10 where there are 1s, 10s, 100s, etc .. Columns • We also count in base-24 and base-60 … we are just more familiar with base-10 for math • Reconsider the columns in terms of powers of 10 as follows:

3. Counting in Base-10 • 9 = 9 x 100 • 71 = 7 x 101+ 1 x 100 • 123 = 1 x 102+ 2 x 101+ 3 x 100 • 5860 = 5 x 103+ 8 x 102+ 6 x 101+ 4 x 100 • 721645 = 7 x 105 + 2 x 104+ 1 x 103+ 0 x 102+ 4 x 101+ 5 x 100 9 7 1 1 2 3 5 8 6 4 2 1 0 5 7 4

4. Base-10: Quick review of “tenths” • What about “decimal values” ? • 0.9 = 0 x 100 + 9 x 10-1 • 0.71 = 0 x 100 + 7 x 10-1+ 1 x 10-2 • 0.123 = 0 x 100 + 1 x 10-1+ 2 x 10-2+ 3 x 10-3 • 0.5864 = 0 x 100 + 5 x 10-1+ 8 x 10-2+ 6 x 10-3+ 4 x 10-4

5. Scientific / Engineering Notation • Is there a more effective method of expressing a large (or small) value such as: • 300 000 000m s-1 • (Speed of light) • 0.000000000000000000160217657C • The charge (in Coulombs) of an electron

6. Base and Index: A Brief Review • Any number A which is multiplied by itself “b times” can be expressed in the base-index form: Ab • A = base • b = index (or power) • Eg: 10 x 10 x 10 can be expressed as 103 • Tip: Count the zeros!

7. Base and Index: Example • Given the following constant (the speed of light in a vacuum): how can we express this in terms of base and index? Or re-written as: 3 x 108m s-1 • The 3 term preceding the base 10 is the coefficient and is generally what you will perform basic arithmetic on, saving exponent math for the base and index 300000000m s-1 300000000m s-1 x10 300000000m s-1 6 5 4 2 3 8 7 1

8. Scientific Notation & Its Uses • When dealing with large numbers, or converting between bases, it is helpful to use the base-index (scientific notation) form • Eg: λ = 300000000m s-1/ 30000000Hz λ= 3 x 108m s-1 3 x 107 s-1 λ = 108m / 107 • λ (lambda) is wavelength in m • 1Hz = 1 cycle per second .. so 1 reciprocal second (ie. s-1) … okay, but how do we solve that?

9. Exponent Math: Mult. & Div. • When you multiply or divide exponential values, (ie. λ = 108m / 107) from the previous slide we must observe some special yet simple practices: • When multiplying, simply add the indices (powers): 103 x 104 = 10(3 + 4) = 107 • When dividing, subtract the indices: 107 / 102 = 10(7-2) = 105 Take note: This can only be done when the bases are the same. Ie. 102 x 23≠ 205 λ = 108m / 107 λ = 10(8-7)m = 101m … or 10m

10. Base and Index: Small numbers • So we can express very large numbers using the Abformat, how about very small numbers? • Consider for a moment what a number such as 0.1 means • One tenth • 1/10 • 1 . 101

11. Reciprocal Values • What can we say about a value such as: 1 . 101 • What about making it: 100. 101

12. Exponent Math: Division 100. 101 • Recall that when we divide exponential values, we subtract them 100.= 100 – 101 = 10(0-1) = 10-1 101

13. Decimal Values & Scientific Notation • Since we know 0.1 can be express as 10-1, what about 0.000001 ? • Again, count the number of times you move the decimal place to the right in order to make 1.0 x 10? 0.000001 = 10-6

14. Metric Prefixes

15. Metric Prefixes: Practical Examples • 3.5MHz = ? M = mega = 106 therefore … 3.5 x 106Hz • 1.5mA = ? m = milli = 10-3 so … 1.5 x 10-3A • 3.3kV = ? k = kilo = 103 thus … 3.3 x 103V • 220μH = ? μ = micro = 10-6 … 2.2 x 10-4H … did I catch you on that one?

16. Engineering Notation • Scientific notation is nice and all, but it has its ease-of-use limitations in practice • Engineering notation works in “groups of three” such that the unit value will respect 10n where n is a multiple of 3 • Eg: 220μH from the previous slide was presented in engineering notation • Scientific would have read 2.20x10-4 from the start

17. Engineering Notation • Values are given in “base” units such as M, k, m, μ, n, p□ (where □ represents an SI unit of measure such as metres or Hz) • 500μH as opposed to 5.0 x 10-4H • 33kV as opposed to 3.3 x 104V • 0.5nF or 500pF as opposed to 5 x 10-11F • In this example, the 500pF is preferred over 0.5nF because it avoids using a decimal value

18. Mind your 0s! • Often when dealing with components, values will be listed on a schematic such that: “all values of capacitance will be given as μF” • It may be necessary to become comfortable working in “hybrid units,” eg: • 0.1μV = 100nV • 5000nF = 5μF • 1000μH = 1mH

19. Conversion between prefixes • Is there a foolproof way to convert between any two prefixes? • Absolutely! Use known ratios! • 1 MHz = ?? μHz • 1Mhz = 106Hz and 1μHz = 10-6Hz • Put another way, there are 106Hz in 1MHz and 106μHz per 1Hz • 1MHz x 106Hz x 106μHz = 1MHz1Hz = 1012μHz 106 x106μHz

20. Conversions made simple • A quicker method of base conversion is to look at the absolute “distance” between two units • Beware .. you must know something about the “direction” you are converting. • Large to small means +veexponent (index) • Small to large means –veexponent • 1G□ is 10+18n□ • (G = 109 & n= 10-9 so |9| + |-9| = 18) • Large unit to smaller, so the index is +ve • 1n□ is 10-15M□ • (n = 10-9 & M = 106 so |-9| + |6| = 15) • Small unit to larger, so the index is -ve

21. Questions?