Simple Nature - Light and Matter

ag/Example 37.

A second-order stereo crossover filter example 37

A stereo crossover filter ensures that the high frequencies go to

the tweeter and the lows to the woofer. This can be accomplished

simply by putting a single capacitor in series with the tweeter and

a single inductor in series with the woofer. However, such a filter

does not cut off very sharply. Suppose we model the speakers

as resistors. (They really have inductance as well, since they

have coils in them that serve as electromagnets to move the di-

aphragm that makes the sound.) Then the power they draw is

I^2 R. Putting an inductor in series with the woofer, ag/1, gives

a total impedance that at high frequencies is dominated by the

inductor’s, so the current is proportional toω−^1 , and the power

drawn by the woofer is proportional toω−^2.

A second-order filter, like ag/2, is one that cuts off more sharply:

at high frequencies, the power goes likeω−^4. To analyze this

circuit, we first calculate the total impedance:

Z=ZL+ (ZC−^1 +ZR−^1 )−^1

All the current passes through the inductor, so if the driving volt-

age being supplied on the left isV ̃d, we have

V ̃d= ̃ILZ,

and we also have

V ̃L= ̃ILZL.

The loop rule, applied to the outer perimeter of the circuit, gives

V ̃d=V ̃L+V ̃R.

Straightforward algebra now results in

V ̃R=

V ̃d

1 +ZL/ZC+ZL/ZR

.

At high frequencies, theZL/ZCterm, which varies asω^2 , dom-

inates, soV ̃Rand ̃IRare proportional toω−^2 , and the power is

proportional toω−^4.

638 Chapter 10 Fields