W9_parallel_resonance.eps

304 Week 9: Alternating Current Circuits

unusually likely to defibrillate the human heart. As little as 10 mA of 60 HzAC across the

heart can kill a person. It requires roughly five times as much DC (50mA) to be equivalently

dangerous!

• The reason for using such low frequencies is that AC does not flow uniformly through a
  conductor – it is lies within an exponential distance of theouter surfaceof a conductor, a length
  called theskin depth. At 60 Hz this length is roughly 8.5 mm in copper; copper conductors “an
  inch in diameter” or more have relatively little current transmitted along their axis, where at
  10 kHz (an arguably safer frequency) it is 0.66 mm in copper. Thicknesses comparable to the
  skin depthincrease the resistanceof a wire by effectively decreasing its cross-sectional area. 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.


• It is no exaggeration to state that this 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 toinformationin 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).


• There are two critical aspects of so-called alternating current (AC) that we will study in this
  course. The first is transformers and the electrical grid that delivers power to points distant
  from the generators with minimal loss. The second is the basis for signal processing electronics:
  theLRCband-pass circuit (or tank circuit) that can be used with rectifiersto build a simple
  amplitude-modulation (AM) radio. This circuit and its variants is ubiquitous in non-digital
  (and most digital) information processing devices.


• The Transformer:The transformer is basically a pair of flux-coupled coils, one (theprimary)
  withNpturns connected to thesourceof alternating voltage, the other (thesecondary) with
  Nsturns connected to theloadthat 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 all of the turns in both coils. The flux is usually
  coupled by wrapping the coils around e.g. a torus of soft iron that traps flux, laminated to
  preventeddy currents(called the transformercore).


• If we letφmbe the flux trapped in the core that passes through a single turn, then:

Vs = Nsdφm

dt

(638)

Vp = Np

dφm

dt

(639)

or (taking the ratios of these two equations, in order)

Vs

Vp

=

Ns

Np

(640)

Note that we omit Lenz’s law in this expression because we can wrap either coil either way

around the core so that the voltages on primary or secondary sidecan be “in phase” or “exactly

out of phase” as we wish.

• A transformer can thusstep voltage upto higher levels orstep it downto lower ones, depending
  on whetherNp< Nsor vice versa.