ECE 102Engineering Computation

Chapter 5 SI Notation, Units, Unit Conversion

Dr. Herbert G. Mayer, PSUStatus 10/2/2015

For use at CCUT Fall 2015

Syllabus Scientific Engineering Notation Dimensions Physical Quantities Units Unit Conversion

Scientific & Engineering Notation

Scientific notation is a compact method for expressing very small or very large numbers.Format:

3

ba 10

mantissa base

exponent• The mantissa conveys the

number’s value and accuracy

• The base and exponent define the scaling factor

Scientific Engineering

exponent multiple of 1 multiple of 3

mantissa -10 < a < 10 -1000 < a < 1000

Example

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Number Scientific Engineering

0.000001234567 1.23456710-6 1.23456710-6

0.00001234567 1.23456710-5 123.456710-3

0.0001234567 1.23456710-4 12.3456710-3

0.001234567 1.23456710-3 1.23456710-3

0.01234567 1.23456710-2 0.01234567

0.1234567 1.23456710-1 0.1234567

1.234567 1.234567 1.234567

12.34567 1.23456710 12.34567

123.4567 1.234567102 123.4567

1234.567 1.234567103 1.234567103

12345.67 1.234567104 12.34567103

123456.7 1.234567105 123.4567103

1234567 1.234567106 1.234567106

Describing Physical Quantities

A physical quantity has three components: Dimension (e.g., length, time, etc.) Magnitude (quantity) Unit (reference amount)

Example: 12.5 m

A measurement determines the number of multiples of a unit that are contained within a physical quantity

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length

magnitude

unit

Dimensions

Dimensions describe physical quantities

Dimensions are independent of units

Each dimension may have a variety of units

Dimensions are divided into two areas: Fundamental (e.g., Length L or Time t) Derived (e.g., Velocity = Length / Time)

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Units

Commonly used unit systems: Metric (decimal: meter, kilogram, second) Engineering System (US: foot, pound-force, second)

Système International d′Unités (SI) is the adopted world standard, except United States

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SI Base Units Length: meter (m) Time: second (s) Mass: kilogram (kg) Electric current: ampere (A) Temperature: kelvin (K) Amount of substance: mole (mol) Luminous intensity: candela (cd)

SI Supplementary Units Plane angle: radian (rad) Solid angle: steradian (sr)

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SI Unit Prefixes

1024 yotta Y 10-1 deci d1021 zetta Z 10-2 centi c1018 exa E 10-3 milli m1015 peta P 10-6 micro 1012 tera T 10-9 nano n109 giga G 10-12 pico p106 mega M 10-15 femto f103 kilo k 10-18 atto a102 hecto h 10-21 zepto z101 deka da 10-24 yocto y

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Common Electrical Engineering Units

Resistance (ohm): MΩ kΩ Ω mΩ μΩ nΩ Inductance (henry): kH H mH μH nH pH Capacitance (farad): kF F mF μF nF pF fF aF Voltage (volt): MV kV V mV μV nV Current (ampere): MA kA A mA μA nA pA fA Power (watt): MW kW W mW μW nW pW Frequency (hertz): THz GHz MHz kHz Hz mHz Wavelength (m): km m cm mm μm nm

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M→106, k→103, 1, m→10-3, →10-6, n→10-9, p→10-12, f→10-15

Example:0.01 F = ? pF

0.009 mV versus 40.5 V. Which one is bigger?→ (0.009 mV)(103 V/mV) = 9 V. 40.5 V is bigger.

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M k 1 m n p f

M 1 103 106 109 1012 1015 1018 1021

k 10-3 1 103 106 109 1012 1015 1018

1 10-6 10-3 1 103 106 109 1012 1015

m 10-9 10-6 10-3 1 103 106 109 1012

10-12 10-9 10-6 10-3 1 103 106 109

n 10-15 10-12 10-9 10-6 10-3 1 103 106

p 10-18 10-15 10-12 10-9 10-6 10-3 1 103

f 10-21 10-18 10-15 10-12 10-9 10-6 10-3 1

From ↓To →

Multipliers for SI Prefix Conversion

→ (0.01 F)(106 pF/F) = 10000 pF

Frequency & Wavelength for EM Waves

Electromagnetic waves:

(n=10-9, M=106, G=109, T=1012, P=1015, E=1018)

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Name Frequency f Wavelength Radio 3 Hz – 300 GHz 100 Mm – 1 mm

Microwave 300 MHz – 300 GHz 1 m – 1 mm

Infrared 300 GHz – 405 THz 1 mm – 750 nm

Visible 405 THz – 790 THz 750 nm – 390 nm

Ultraviolet 790 THz – 30 PHz 400 nm – 10 nm

X-Ray 30 PHz – 30 EHz 10 nm – 0.01 nm

Gamma ray more than 30 EHz Less than 0.01 nm

f

c

Speed of light

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Unit Conversions

A conversion factor relates the same physical quantity in two different units.

A conversion factor is always equal to one (1).

The quantity’s value is multiplied by the conversion factor to change unit systems.

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BA

BA unit

unit1

unitunit NValue

NValue

1unit1

unitand1

unit

unit1unitunit1

A

B

B

ABA

N

NN

Exact conversion factors are set by definition.

Example:

Non-exact conversion factors can be derived from measured values.

An exact conversion factor becomes non-exact if it is rounded off.

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1in 12

ft 1

ft 1

in 12 in 12 ft 1

1cm 2.54

in 1

in 1

cm 2.54 cm 2.54 in 1

1m 1

cm 100

cm 100

m 1 m 1 cm 100

If a direct conversion factor does not exist, use several intermediate conversion factors.

If done correctly, the intermediate units will “cancel” out.

Example:

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ft 3.2808

ft 48.30

100

in 12

ft 1

cm 2.54

in 1

m 1

cm 100m 1:ft tom

ft 1:m ft to

ft 1

in 12

cm 100

m 1m 0.3048

in 1

cm 2.54

Conversions that involve raising values to a power can be tricky.

Example:A = πR2 Let R = 1.5 cm. Find A in m2.

Wrong →

Right →

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cm 100

m1cm5.1 2A

Conversion factor should be squared

Improper units.

Wrong exponent

242

2 m 101.7cm 100

m1cm5.1

A

mcm 101.7 2

Conversion factors should always include units

Including units allows the conversion to be checked quickly for consistency

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2

2

fth

Btu428

1

12

1

54.2

100

1

1

3600

06.1055

1

1

11350

Example:

Just outside the earth’s atmosphere, the solar heat flux is approximately 1350 W/m2. Express the heat flux in units of Btu/hft2. Note: W = J/s and 1 Btu 1055.06 J

Avoid this!

2

2

2 fth

Btu428

ft1

in12

in1

cm54.2

cm100

m1

hr1

s3600

J06.1055

Btu1

W1

sJ1

m

W1350

Example

a) 525 L (liters) = ? ft3 (1000 L = 1 m3)

b) 12 days = ? ms

c) 65.9 C = ? F

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3

3

3

3

ft5.18m3048.0

ft1

L10

m1L525

ms100368.1s1

ms10

m1

s60

h1

m60

day1

h24day12 9

3

F151F32C

F

5

9C9.65

Offset

Are there really 24 hours in a day? Actually, no!1 day 23 h 56 m 4 s

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Example

For the following dimensional equations, find the base dimensions of the parameter k:

M-mass, L-length, t-time, T-temperature

1.[M][L][t]–2 = k[M][L]–1[t]–2

2.[L]2[t]–2 = k[M]4[T]2

3. k3[T]6[M]3[L]–5 = [T] –3[t]–6[L]

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Example

Lead has one of the highest densities of all the pure metals. The density of lead is 11,340 kg/m3. What is the density of lead in units of lbm/in3?

Note: 1 kg = 2.20462 lbm

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Example

A solid cylinder of polyethylene plastic ( = 930 kg/m3) has a diameter of 12.5 mm. If the cylinder is 0.750 yards long, with is the mass and weight of the cylinder in base SI units?