Transcript 4: - ve3bux.com

Chapter 4:
Metric Prefixes & Powers of Ten
Gigafun with nanoeffort
Presented by: James, VE3BUX
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:
Column
100’000s
10’000s
1000s
100s
10s
1s
Exponent 105
104
103
102
101
100
# of 0s
4
3
2
1
0
5
2
Counting in Base-10
Column
100’000s
10’000s
1000s
100s
10s
1s
Exponent
105
104
103
102
101
100
9
9
.
7
1
.
1
2
3
.
5
8
6
4
.
1
0
4
5
.
71
123
5864
721045
–
–
–
–
7
2
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
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Base-10: Quick review of “tenths”
• What about “decimal values” ?
Column
1s
10ths
100ths
1000ths
10’000ths
Exponent
100
10-1
10-2
10-3
10-4
0.9
0
.
9
0.71
0
.
7
1
0.123
0
.
1
2
3
0.5864
0
.
5
8
6
–
–
–
–
4
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
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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
5
Base and Index: A Brief Review
• Any number A which is multiplied by itself “b times”
can be expressed in the base-index form:
b
A
• A = base
• b = index (or power)
• Eg: 10 x 10 x 10 can be expressed as 103
– Tip: Count the zeros!
6
Base and Index: Example
• Given the following constant (the speed of light in a
vacuum): 300000000m s-1 how can we express this in
terms of base and index?
x10
300000000m s-1
8 76 5 4 3 2 1
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
7
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 … okay, but how do we solve that?
– λ (lambda) is wavelength in m
– 1Hz = 1 cycle per second .. so 1 reciprocal second (ie. s-1)
8
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):
8m /4 107 (3 + 4)
3
λ = 10
10 x 10 = 10
= 107
λ = 10(8-7)m = 101m … or 10m
• 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
9
Base and Index: Small numbers
• So we can express very large numbers using
the Ab format, how about very small numbers?
• Consider for a moment what a number such
as 0.1 means
– One tenth
– 1/10
– 1 .
101
10
Reciprocal Values
• What can we say about a value such as:
1 .
101
• What about making it:
100 .
101
11
Exponent Math: Division
100 .
101
• Recall that when we divide exponential values,
we subtract them
100 .= 100 – 101 = 10(0-1) = 10-1
101
12
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
13
Metric Prefixes
Prefix
Symbol
Scientific
Notation
Decimal
Common
Word
tera
T
1012
giga
G
109
1000000000 billion
mega
M
106
1000000 million
kilo
k
103
1000000000000 trillion
1000 thousand
100
1 one
milli
m
10-3
0.001
thousandth
micro
μ
10-6
0.000001
millionth
nano
n
10-9
0.000000001
billionth
pico
p
10-12
0.000000000001
trillionth
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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?
15
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
16
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
17
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
18
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 = 106 x 106μHz = 1012μHz
1MHz
1Hz
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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 +ve exponent (index)
– Small to large means –ve exponent
– 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
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• Questions?
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