312 Week 9: Alternating Current Circuits
Resonance for this circuit is a bit unusual – it is the frequencyω=ω 0 =√LC^1 as before, but
nowf rac 1 Zislargestat resonance and the current increases away from resonance. The power
delivered to the resistance no longer depends onLorCand only depends on the frequency
as:
PR=V 02 sin^2 (ωt)
R(687)
so that the average power delivered to the circuit is:< P >=< PR>=
V 02
2 R
(688)
independent of frequency altogether. Away from resonance, one simply generates a large (but
irrelevant) current in eitherL(for low frequencies) orC(for high frequencies) that is out of
phase with the voltage and hence dissipates zero average power per cycle.9.1: Introduction: Alternating Voltage
As we have seen in the previous chapter, if one spins a coil withNturns and cross-sectional areaA
at angular velocityωin 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) (689)V(t) = −dφm
dt=N BAωsin(ωt) (690)This is, in fact, the functional form of the voltage that comes out of wall receptacles in your
house, no matter what the voltage or frequency used by your particular country of residence. It is
also the general functional form of electrical signals generated by many other means in (for example)
radio transmitters.
In this chapter, then, we will learn to treat “arbitrary” harmonic alternating voltage sources as
having the form:
V(t) =V 0 sin(ωt) (691)
where of course we can introduce an arbitrary phase (corresponding to the choice of when we start
our clock). In this expression, remember that:
ω= 2πf=^2 π
T(692)
wherefis thefrequencyof the harmonic oscillation in units ofHertz(cycles per second) andTis
the correspondingperiod.
We will also look at slightly more general voltage sources that arenearlyharmonic, in particular
amplitude modulatedharmonic sources such as:
V(t) =A(t) sin(ωt) (693)whereA(t) is aslowly varying function of time(making only small changes over many periodsTof
the harmonic part). More advanced students should note well that we will notproperlytreat this
problem by means of e.g. a Fourier Transform, as knowledge of Fourier Transforms (however useful!)
is not a requirement for this course. We willbarelyexplore some of the benefits of treating voltages
or currents given in a complex form:
V(t) =V 0 eiωt (694)
whereV 0 may be a general complex number,V 0 =|V 0 |eiδbut again, advanced students should keep in
mind the fact that this often makes things mucheasieronce one has paid the price of learning how to