ENEE 313, Fall ’08
Supplement I
Units and Unit Conversions
Zeynep Dilli, Sep. 2008
In any engineering context, we need to carefully keep track of the units involved in our calculations. Matching the units on both sides of an equation is a good way to make sure there are
no egregious errors. Also, especially in calculations involving solid state devices, we often need to
scale units to match each other before proceeding. This document presents a conversion method
and lists some units we will use in class.
1
Using Units to Remember Expressions (And Vice Versa)
An equation I have never bothered to memorize is the relationship between wavelength, frequency,
and the speed of light. I depend on the units to get the conversion right. Wavelength, being a
length, is measured in units of distance, for example, in meters. Frequency is measured in ”timesper-second” (or sec−1 , or 1/sec, or in Hertz (Hz) if we are being formal). Speed of light, c, being a
speed, is given in distance per time; let’s say meters/second (m/sec).
Let’s say I know the frequency ν and would like to get the wavelength λ. This means I have
the quantity given in 1/sec and know the constant speed of light given in m/sec. And since I am
looking for the quantity given in meters,
m
[ sec
]
c m/sec
= m ⇒ λ [m] = [
]
1
ν 1/sec
[ sec ]
This principle is obviously applicable to any calculation. Conversely, if you do remember the
expression, you can keep track of the units to remember the unit of a quantity you did not recall.
For example, take the expression for the capacitance of a parallel-plate capacitor:
C=
A
d
where is the permittivity of the dielectric material between the plates, A is the plate area, and d
is the distance between the plates. We can pull the unit for permittivity out of this expression:
C [F ] = [?]
A [m2 ]
F
⇒ [ ]
d [m]
m
where F is Farads, the unit of capacitance. If we need it in more basic units, we might remember
the definition of capacitance as linking the charge stored in a capacitor to the potential across the
capacitor:
Q [C] = CV [F · V ] ⇒ [F ] = [
where C is Coulombs, the unit of charge, and V is volts.
1
C
]
V
2
Making Sure Conversions Are Error-Free
Often, different quantities involved in a problem are provided with different units or at different
scales. Let’s take the capacitor example above for an illustration. Parallel-plate capacitors implemented on integrated circuits are very small, with edges and inter-plate distance measured in
micrometers. The permittivity of the silicon dioxide in between the plates can be given in F/cm or
F/m, however. If the engineer does not keep track of the units, the outcome will be wrong.
Assume that the parallel plates are squares, 50 microns on one side, and the spacing between
them is filled with silicon dioxide, 2 µm thick. The permittivity of silicon dioxide is given as
3.45×10−13 F/cm. What is the capacitance of this capacitor?
The trick is to include the units in your equation and also in your conversion factors. First find
the area, then the capacitance:
A = 50 µm × 50 µm = 2500 µm2
C [F ] = A
d
F 2500 [µm2 ]
]
cm
2 [µm]
F
1 [cm] 2500 [µm2 ]
[
] × 4
cm
10 [µm] 2 [µm]
= 3.45 × 10−13 [
= 3.45 × 10−13
Here, the expression 1014[cm]
is the conversion factor. By inspection, we can see that it should be
[µm]
there in order for all the units on the right hand side to cancel as required and leave only Farads,
which is the correct unit for the capacitance we are aiming for. Then, pulling all units out,
1
F cm µm2
×
1250
[
][
][
]
104
cm µm µm
F = 43.1 × 10−15 F = 43.1 f F
C = 3.45 × 10−13 ×
= 4.31 × 10−14
where at the end, F (Farads) are converted to femtoFarads (1 fF=10−15 F) with a similar process.
As an exercise, carry out that conversion explicitly.
Note that without the conversion in the units of , the calculation would wrongly have yielded
431 pF (picoFarads; 1 pF=10−12 F), which is a large capacitance, possibly the capacitance of a
capacitor you can buy to use on a breadboard or printed circuit board. (For comparison, good
oscilloscope probes have capacitances ”as low as” 1 pF.) This is too high for anything that could
(or should) be implemented in the scales of an integrated circuit.
3
Final Tricks
In doing calculations, wait until the last possible step to plug in the numbers and do the arithmetic.
While solving a problem, stay with the letter symbols for as long as you can. Make sure the units on
both sides match, then cancel whatever you can cancel, then plug in the numbers. After that, make
sure all the unit scaling/conversion factors are in place; pull the units out and do the cancellations
to make sure everything is as should be and you are getting what you should get as the result. As
the final step use the calculator on the remaining numbers.
And the following are worth a reminder:
10x × 10y = 10(x+y) ;
2
10x
= 10(x−y)
10y
4
Most Common Units
• Distance units
Unit
Meter
Symbol
m
Centimeter
cm
Millimeter
Micrometer
mm
µm
Nanometer
Angstrom
nm
Å
Common conversions
1 m = 102 cm
1 m =103 mm
1 m =106 µm
1 m =109 nm
1 m =1010 Å
1 cm = 10−2 m
1 cm =104 µm
1 mm = 10−3 m
1 µm = 10−6 m
1 µm =10−4 cm
1 nm = 10−9 m
1 Å = 10−10 m
(Note: Micrometers are often just called ”micron”s in semiconductor technology.)
• Area and volume units
Unit
Square Meter
Symbol
m2
Square Centimeter
cm2
Square Millimeter
mm2
Square Micrometer
µm2
Cubic Meter
m3
Cubic Centimeter
cm3
Cubic Micrometer
µm3
Common conversions
1 m = 102 cm × 102 cm = 104 cm2
1 m =106 mm2
1 m =1012 µm2
1 cm2 = 10−4 m2
1 cm2 = 108 µm2
1 mm2 = 10−6 m2
1 mm2 = 106 µm2
1 µm2 = 10−12 m2
1 µm2 = 10−8 cm2
1 m3 = 102 cm × 102 cm × 102 cm = 106 cm3
1 m =109 mm3
1 m =1018 µm3
1 cm3 = 10−6 m3
1 cm3 = 1012 µm3
1 µm3 = 10−18 m3
1 µm3 = 10−12 cm3
2
3
• Other units
Unit
Joule
Electron Volts
Kilograms
Seconds
Amperes
Coulombs
Volts
Farads
Ohms
Symbol
J
eV
kg
sec or s
A
C
V
F
Ω
In more basic units
kg·m2 /sec2
A.sec
kg.m2 /(A.sec3 )
C/V (Coulombs per volt)
V/A
4
Measures
Energy
Energy
Mass
Time
Current
Charge
Potential
Capacitance
Resistance
Common conversions
1 J = 6.25×1018 eV
1 eV = 1.6×10−19 J
1 kg = 1000 gr (grams)