Simple Nature - Light and Matter

ae/Power in a capacitor:

the rate at which energy is being

stored in (+) or removed from (-)

the electric field.

af/We wish to maximize the

power delivered to the load,Zo,

by adjusting its impedance.

and based on figure ae, you can easily convince yourself that over
the course of one full cycle, the power spends two quarter-cycles
being negative and two being positive. In other words, the average
power is zero!
Why is this? It makes sense if you think in terms of energy.
A resistor converts electrical energy to heat, never the other way
around. A capacitor, however, merely stores electrical energy in an
electric field and then gives it back. For a capacitor,

Pav= 0 [average power in a capacitor]

Notice that although the average power is zero, the power at any

given instant isnottypically zero, as shown in figure ae. The capac-

itordoestransfer energy: it’s just that after borrowing some energy,

it always pays it back in the next quarter-cycle.

An inductor

The analysis for an inductor is similar to that for a capacitor: the

power averaged over one cycle is zero. Again, we’re merely storing

energy temporarily in a field (this time a magnetic field) and getting

it back later.

10.5.9 Impedance matching

Figure af shows a commonly encountered situation: we wish to

maximize the average power,Pav, delivered to the load for a fixed

value ofVo, the amplitude of the oscillating driving voltage. We

assume that the impedance of the transmission line,ZT is a fixed

value, over which we have no control, but we are able to design the

load,Zo, with any impedance we like. For now, we’ll also assume

that both impedances are resistive. For example,ZT could be the

resistance of a long extension cord, andZocould be a lamp at the

end of it. The result generalizes immediately, however, to any kind

of impedance. For example, the load could be a stereo speaker’s

magnet coil, which is displays both inductance and resistance. (For

a purely inductive or capacitive load,Pavequals zero, so the problem

isn’t very interesting!)

Since we’re assuming both the load and the transmission line are

resistive, their impedances add in series, and the amplitude of the

current is given by

Io=

Vo

Zo+ZT

,

Section 10.5 LRC circuits 635