Week 9: Alternating Current Circuits 305
- Here’s the trick of the power grid. 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 (drawingroughly 1 A in
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 that the
voltageat your house is reduced to a fraction of the nominal voltage as youturn 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 R
becomes 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 bulb as 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. Remember the crispy-fried
squirrel story! - 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 islow 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 resistanceof a bit less than 0.25
Ohms in 50 meters, wasting around 100 watts heating the wire all along 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 thick 10 gauge wire (and not to use the thinner 14
gauge wireat allto minimize heat loss in the household wiring. As you can see, though, you
can easily waste anywhere from 1% to 5% of your energy bill simply heating the space inside
your walls! - Non-driven LC circuit:In the figure above, the capacitorCon the left is initially charged
up to chargeQ 0. At timet= 0 the switch is closed and current begins to flow. If we apply
Kirchhoff’s voltage/loop rule to the circuit, we get:
Q
C−L
dI
dt= 0 (641)
where
I=−dQ
dt