314 Week 9: Alternating Current Circuits
likely to defibrillate the human heart. As little as 10 mA of 60 Hz AC across the heart can kill a
person. It requires roughly five times as much DC (50 mA) to be equivalently dangerous!
As you can see, most power is distributed at only 50 or 60 Hz. This leads us to several important
questions. Why distribute alternating voltage at all? Why use the particular frequencies that we
use to alternate with, instead of (say) much higher frequencies ormuch lower ones (all the way down
to DC voltage).
The reason we use alternating voltage is because it makes it easy to increase or decrease the
voltage usingtransformers. In a moment we’ll cover transformers and the reasons for using them in
detail, but in a nutshell for now, we need to transmit the energy from the power station to where it
is used at as high avoltageas possible. Transformers work “better” at higher frequencies than at
lower frequencies, as they use induction; we need at least aminimalfrequency in the tens of Hz to
permit them to work at all well, but they’d work fine at 100’s or 1000’s of Hz too.
However, we cannot use these higher frequencies – in spite of the fact that they’d be much
safer biologically because alternatingcurrent(AC) does not flowuniformlythrough a (cylindrical)
conductor – most of the current flows near theouter surfaceof a conductor, and the current density
drops ofexponentiallyas one procedes further in with an exponential decay lengthδscalled theskin
depth. At 60 Hz this length is roughly 8.5 mm in copper; copper conductors “an inch in diameter”
have at least some current density throughout their cross-section. At 10 kHz (an arguably safer
frequency) it is 0.66 mm in copper, and an inch-thick cable carriesnosignificant current overmost
of its cross-section.
If a wire is much thicker than the skin depth, its resistance issignificantly increasedbecause the
effective cross-section in the
R=ρL
A(696)
expression isn’t e.g. A≈πR^2 , it is roughlyA≈ 2 πRδsforδs≪R(a much smaller number). 50
or 60 Hz are thuscompromisesbetween the need to use AC to transmit energy long distances and
the need to minimize the resistance of the transmission wires along the way by making effective use
of their entire cross-sectional areas, for cable cross-section diameter assumed to be an inch or less.
Cables thicker than this are sometimes fabricated so that they arehollow, since there is little current
carried by the central core anyway.
It is no exaggeration to state that alternating voltage generatedusing Faraday’s Law and trans-
mitted at high alternating voltages before being stepped down and used at lower voltages is the
fundamental basis for modern civilization. Power distributed over long distances using step-up and
step-down transformers has created the highest global standard of living in human history. Some
2/3 of the world’s population uses nearly ubiquitous electricity to light, heat and cool their homes,
to refrigerate and cook their food, to fuel devices that provide increasingly universal access toin-
formationin many of its sensory forms – musical, textual, visual, to provide transportation, to fuel
industry and commerce and agriculture. If the electrical grid for any reason ceased to function we
would regress to a medieval existence in a matter of weeks (as I have personally experienced as both
hurricanes and ice storms have caused weeklong power outages in North Carolina on more than one
occasion).
Let us understand the transformer and the role that it plays in thetransmission of power.The Transformer
The transformer is basically a pair of flux-coupled coils, one (theprimary) withNpturns connected
to thesourceof alternating voltage, the other (thesecondary) withNsturns connected to theload
that actually consumes the energy delivered from the source. All of the flux that passes through any
turn in the primary or secondary coils passes (with as little loss as it is possible to arrange) through