Week 9: Alternating Current Circuits 317
to renewable resource electrical generation, conversion of e.g. sunlight or the power of the wind in
suitable locations into electrical current and its transmission acrossthousands of milesfrom those
locations to where it will be consumed.
So here’s the trick of the power grid, Tesla’s solution. The resistance of a wire is (recall)R=ρLA
(whereAis the effective cross section at a given frequency). A copper wire just under a quarter
inch thick has a resistance of roughly 1 Ohm/mile (rule of thumb). A wire a third of an inch thick
has a resistance of roughly 0.1 Ohms/mile. Wires this thick are heavy and expensive and have to
carry alot of energy. Now, suppose we have a power station a mere ten miles from your home. The
total resistance of all the wires between that power station and your home is easily order of an ohm.
Now imagine that you turn on a single 100 Watt bulb (drawing roughly 1 Ain current. The power
station must provide 101 Watts for your bulb to burn – 100 Watts used by the bulb andI^2 R≈ 1
Watt used in thesupply line.
However, you then turn on therestof your lights, your refrigerator kicks on, your AC starts up.
Your house is now drawing more like 100 Amperes (delivered in parallel to the many appliances)
and is using order of 10000 Watts.So is the supply line!Half of the energy being delivered to your
home is wasted as heat along the way. A second consequence is thatthevoltageat your house is
reduced to a fraction of the nominal voltage as you turn on more appliances and more of the voltage
drop occurs across the supply resistance!
The solution is totransmit at high voltage and low currentanduse at low voltage and high
current. If we step up the voltage by (say) 10,000 Volts (real long distancetransmission is at much
higher voltages than this) then in order to deliver the samepowerat the far end, instead of delivering
100 Amps at 100 volts one can deliver 1 Amp at 10,000 Volts! The resistive heating of the supply line
is back to 1 Watt out of 10,000 delivered. Here the square inI^2 Rbecomes yourfriend– delivering
10 kW at 100,000 V requires only 0.1 A and uses only 0.01 W heating the wire.
This is good for transmission, but bad for utilization. 100,000 volts can arc an appreciable
distance through evendryair; that’s why the insulators on high voltage transmission towers are so
long! We’d hate to get electrocuted every time we changed a light bulbas power arced out of the
socket through our bodies on the way to ground. With an entire power plant delivering the energy,
even the (mere) 16,000 volt lines that run down the streets can literally make your body explode if
you should stray within a few cm of a supply line.
In one of the few instances in my memory of a power outage at Duke,a squirrel was recently
crispy-fried when it got inside the barbed wire fences at a major step-down transformer serving part
of the campus. It strayed too near to the main power buses, whicharced over (through the squirrel)
blowing the transformer and shutting down power to the campus for a time. Imagine how exciting
life would be if every time you went to plug in an electric light into your 16,000 volt household wiring
or flick a switch on a humid day, you risked being electrocuted by a lightning bolt!
“Exciting” isn’t quite the right word for it. Consequently, there isalwaysa step-down transformer
at the very end of the line, that drops the voltage in our houses to themuchsafer but still dangerous
120 volts (relative to ground). We use currents on the order of 1-20 Amps within the house, which is
low enough that the resistive heating of the order of 30-50 meter longhouseholdsupply lines remains
low. Even “low” can waste a lot of heat! 12 gauge copper wire has a resistance of a bit less than
0.25 Ohms in 50 meters, wasting around 100 watts heating the wire allalong its length when one
draws 20 Amps of current (and reducing the line voltage available to the∼2000 watt appliance at
the end that is drawing all of that power by roughly 5%). Personally,I prefer to do primary runs
in household wiring with the even thicker 10 gauge wire (and not to usethe thinner 14 gauge wire
at allto minimize heat loss in the household wiring. As you can see, with thinner wiring you can
easily waste anywhere from 1% to 5% of your energy bill simply heatingthe space inside your walls
when you run appliances!
All of this will make sense when you work out the algebra for yourself. One of the homework