Week 9: Alternating Current Circuits 313
Volts Hz Purpose Continent
120 60 lighting, small appliances, N. and S. America
electronics
208 or 240 60 heating, cooling, large N. and S. America
appliances, 3 phase motors
230 50 all household use Everywhere elseTable 5: Common alternating voltages and frequencies in use aroundthe world. There is a dazzling
array of plug types in use around the world as well.
use algebra over the field of complex numbers plus a few things such as Cauchy’s theorem and Fourier
Transforms. Some ideas, such as the importance of havingenoughbandwidth to encode an amplitude
modulated (or otherwise encoded) signal on top of a given carrier frequency while nevertheless
remaining well resolved from nearby carriers carrying information on other channels are very difficult
toprovewithout using this more advance math, so students will have to content themselves with a
few of this book’s rare it-is-so-because-I-say-so without proper derivation or justification.
One very important thing all students should learn from this chapter is just how alternativing
voltages and high-voltage transmission lines, together, are nothing less than thebasis for modern
civilization– a country’s productive capacity and the comfort of its citizens isdirectly linkedto its
ability to generate electrical energy and distribute it widely in a cost-effective way.
Nothing convinces one more of this than the not-terribly-infrequent instances ofpower outages
when hurricanes, ice storms, earthquakes, or solar storms interrupt the power grid for days or even
weeks of time. During the downtime one immediately loses all refrigeration (so stored food spoils),
heating and cooling (so one has to survive at the ambient temperature as best one can), the ability
to turn light on and off with the touch of a finger (so one can stay up later and get up earlier than
the sun), the ability to drive safely (no traffic lights), the ability to bank or shop indoors in shopping
malls (no air conditioning, lights, electronic cash registers, check card readers), the ability to listen
to music, compute, browse the internet (once local battery stores are exhausted). Over a single
week life devolves to what it was like over a century ago before the advent of universally accessible,
inexpensive electricity.
Life over a century ago, without electricity,sucked!Electrical Distribution True Facts
The most common models for household electrical distribution are represented in the following table
(note well thatω= 2πfwherefis the frequency of the source in Hertz): 209 is the potential
difference between any two phases of a three-phase “Wye” main supply in the US where the pole
voltages are 120 relative to ground:
V = 120 sin(ωt) + 120 sin(ωt± 2 π/3)
= 240 sin(π/3) sin(ωt±π/3)
= 208 sin(ωt±π/3) (695)and 240 is similarly the difference between two 120 volt lines that are completely out of phase. Do
notuse this table as an authoritative guide to electrical main supplies around the world; there are
many such authoritative guides and tables available on the internet^88.
It is worth mentioning that (unfortunately) 60 Hz is aparticularly unfortunatechoice for distri-
bution frequency because it is in “resonance” with certain cardiac frequencies and hence unusually
(^88) Wikipedia: http://www.wikipedia.org/wiki/Mainselectricity. See also the many links in this article.