Week 9: Alternating Current Circuits
- AC Generator:If one spins a coil withNturns and cross-sectional areaAat angular velocity
ωin a uniform magnetic fieldBoriented so that it passes straight through the coil at one point
in its rotation, one generates analternating voltageaccording to:
φm = B~·N Anˆ=N BAcos(ωt) (634)V(t) = −dφm
dt =N BAωsin(ωt) (635)
We will from now on treat “arbitrary” harmonic alternating voltage sources as having the
form:
V(t) =V 0 sin(ωt) (636)
where of course we can introduce an arbitrary phase (corresponding to the choice of when we
start our clock).- 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) (637)and 240 is similarly the difference between two 120 volt lines that are completely out of phase.
Donotuse this table as an authoritative guide to electrical main supplies around the world;
there are many such authoritative guides and tables available on theinternet^84.
It is worth mentioning that (unfortunately) 60 Hz is aparticularly unfortunatechoice for
distribution frequency because it is in “resonance” with certain cardiac frequencies and hence(^84) Wikipedia: http://www.wikipedia.org/wiki/Mainselectricity. See also the many links in this article.
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 else
Table 4: Common alternating voltages and frequencies in use aroundthe world. There is a dazzling
array of plug types in use around the world as well.