Every analog signal when reduced to its most basic form will be a sum of sines. Even a square wave is the sum of a series of the odd harmonics (integer multiples) of the fundamental.
It is the harmonics that give different instruments their unique sounds. Change the harmonics and you change the sound. Take the trumpet and the clarinet. Both reproduce essentially the same range of musical notes on a scale, but hardly anyone could mistake one for the other. There are other non-harmonic cues as well, but absent these, the distinction is still very clear. Sending all these frequencies to the correct drivers (woofers, mids, and tweeters) in the correct proportions ensures that the resulting sound is a faithful reproduction of the original.
Filter design is necessary for every Klipsch system, and every analog filter, whether passive or active, is made up of a few basic components: resistors, capacitors, and inductors (active filters add a transistor or op amp)
Each of these components has its own particular relationship with voltage (V) and current (I), described by the following equations:
V = I*R where R is the value of the resistor.
I = C(dV/dt) where C is the value of the capacitor.
V = L(dI/dt) where L is the value of the inductor.
Engineering textbook convention says that when you see capital “V” and “I”, they describe DC voltages and current, and when you see lower case “v” and “i” (usually in a script font), they describe AC voltages and currents. I will ignore that for the most part here to make it easier to read. (Likewise the “d” is usually a greek "δ" and “t” is sometimes printed in a script font).
Capacitors and inductors are reactive components. In a nutshell they store in energy in an electric or magnetic field, respectively. From the above equations you infer that you can not change the voltage across a capacitor instantaneously, nor can you change the current through an inductor instantaneously (in other words, no step functions), and when AC voltage signals are applied, there is a relative phase difference between the voltage and current through these components. In a capacitor, the current leads the voltage by 90 degrees, and in an inductor the current lags the voltage by 90 degrees.
The unit of measure of capacitance is the Farad, and is a rather large quantity. Most of the time we deal with microfarads (10-6 Farads) and picofarads (10-12 Farads). The unit of measure of inductance is the Henry, and mostly we deal with millihenries (10-3 Henries) and microhenries (10-6 Henries).
“Impedance,” which is frequency dependant, is the term we use in place of “resistance” when we’re dealing with AC signals and reactive components, and is denoted by the letter “Z”. “Reactance” is used to describe the impedance of capacitors and inductors and is the purely imaginary component of impedance, and is denoted by the letter “X”.
What is imaginary? In mathematics, the square roots of negative numbers are described as imaginary numbers, and the term “i” is given to be the value √-1. In engineering we use the term “j” instead (“i” was already taken, see above).
When you combine the real (resistive) components with the imaginary (reactive) components, you end up with complex numbers to describe the impedance.
Ideal capacitors and inductors do not dissipate any power, they store all the energy, and will give it back to the circuit whenever allowed to do so. Real world components always have “parasitic” components, so they are not purely reactive, and we have to take that into consideration in the design, especially any resistive parasitics, since this directly affects the magnitude of the signal going to the loudspeaker.
The reactance of capacitors and inductors is given by the following:
XC = -j/ωC or 1/jωC (where ω is angular velocity in radians per second)
XL = jωL
Impedances work just like resistances when using Ohm’s Law, but you end up doing math on the complex numbers to obtain the result.
So, a couple quick examples are in order. What is the reactance of a 1uF capacitor and a 1mH inductor at 1kHz? (Remember, ω=2πf where f is frequency.)
XC = -j/ωC = -j/(2*π*1000*0.000001) = -j/0.006283 = -j159.2 or -159.2j
XL = jωL = j*2*π*1000*0.001 = j6.283 or 6.283j, whichever you prefer.
Notice in both examples the “real” part of the overall impedance is zero, i.e. 0-j159.2 and 0+j6.283.
Now for a little fun with the previous two components, and a 1kΩ resistor in a series circuit.
For a series circuit ZTOTAL = Z1 +Z2 + Z3 + … + ZN So,
ZTOTAL = ZR + ZC +ZL = (1000 + j0) + (0-j159.2) + (0+j6.283) = 1000-j152.9
Notice that at this frequency, the reactive part is overall capacitive (we know this from the “-j” term), meaning the current through this series circuit will lead the voltage somewhat. How much? We’ll have to find the phase angle, but since 152.9 is “small” compared to 1000, we expect to see a small phase angle.
A really cool trick for finding the phase angle can be done geometrically, using what is called a “phasor diagram”. Simply put a phasor diagram is a graphical representation using vector summing of the real and imaginary parts of the result, and will give us both the magnitude of the impedance, |Z|, and the “direction” or phase, φ, of the current. On the “x-axis” we put the real part of the term, and on the “y-axis” we put the reactive (imaginary) part of the term. See http://en.wikipedia.org/wiki/Complex_plane
The phase angle of the current then is, φ = tan-1(152.9/1000) = 8.7°
And the magnitude of the impedance is found using the Pythagorean Theorem, a2 + b2 = c2, and
|Z| = √[10002 + (152.9)2] = 1012.6 Ohms.
At 100Hz the impedance is Z = (1000+j0) + (0-j1592) + (0+j0.6283) = 1000-j1591, having much more influence from the rising impedance of the capacitor at low frequency. The magnitude of the impedance has also changed, increasing to 1879 Ohms, with a corresponding reduction of current flow.
At 10kHz the impedance is Z = (1000+j0) + (0-j1.592) + (0+j628.3) = 1000+j680.7 with more influence coming from the rising impedance of the inductor at high frequency. The magnitude of the impedance at this frequency is 1209 Ohms.
The phase angle at 100Hz is 57.8°, while at 10kHz the phase angle is -34.2° and the current lags the voltage.
In a series RLC circuit there always exists an impedance minimum. This occurs when the magnitude of the inductive and capacitive reactance are equal (graphically they will cancel out on a phasor diagram and the contribution to the magnitude of the LC part of the circuit is zero Ohms). This occurs at the frequency ω = 1/√(LC), and is called the resonant frequency.
In our example, this happens at f = 1/[2*π*√(0.001*0.000001)] = 5033Hz. At this frequency the magnitude of the impedance is 1000 Ohms and the phase angle of the current is zero, φ = tan-1(0/1000) = 0°.
Knowing what you know now and applying Ohm’s Law you can pretty much describe any RLC circuit in various series and parallel combinations at any given frequency. Powerful stuff. Yes, it can get quite tedious with several circuit components, especially when you throw the driver model into mix. But that’s why we invented computers.
All for now, when I get time to get back to it we’ll look at some simple passive filters.