# Loudspeaker Crossover

The goal of an audio speaker is to reproduce the sound of the input as best as possible. This may sound like a simple statement, but it is a notoriously difficult one to engineer. Furthermore, sound to peoples’ ears is subjective, which only serves to complicate this task.

Today, with currently available technology, it is not possible to reproduce a flat frequency response across the entire spectrum of sound within the human-audible regime (i.e. ~20Hz to ~20kHz) with a single driving cone. This is why a decent set of speakers usually comprise two or more components, for example, a sub-woofer and a tweeter. The sub-woofer plays low-frequency sound, while the tweeter plays the high-frequency sound, which you would probably guess just from the onomatopoeic value of their names. Here is an example of a sound pressure level vs. frequency graph for this sub-woofer:

Black Line: Sound Pressure Level in dB vs. Frequency

You can see that the sound level dies off dramatically at higher frequencies, which is why we need the tweeter. For those that are curious, the plots like the one above are usually obtained by hanging a microphone in front of the speaker cone in an anechoic chamber and sweeping through the frequencies with a frequency generator. Due to the directional nature of sound from a speaker, the microphone is hung directly in front of the cone at a distance of 1 meter.

Because sound has to be routed through two separate cones, the speaker manufacturer has to make an electronics decision about how to do this. The electronics components that are used to do this are usually referred to as the “crossover circuit”. This is really just a fancy name for a high- and low-pass filter (or band-pass filter if one is also using a mid-range speaker). Crossover circuits tend to come in two varieties: active and passive. Active crossovers usually use operational amplifiers (or op-amps) to make the filters (which require the use of external power), while passive crossovers use inductors and capacitors (and do not need to be plugged into the wall).

For the sake of simplicity, let’s suppose that we have both 8 Ohm tweeters and sub-woofers for our speaker system. Suppose we want our cross-over frequency to be 500Hz. We can make the world’s simplest high-pass filter like so for the tweeter:

High-Pass Filter

So what would the capacitance need to be? Well, simply enough, it’s just the cut-off frequency of an RC circuit. In this case, for the 500 Hz crossover, we can calculate it as so:

$f_c = \frac{1}{2 \pi R*C} = \frac{1}{2 \pi *8*C}$

Solving for $C$ with $f_c$ = 500 Hz gives 39.79$\mu F$.

For the sub-woofer’s low-pass filter, we would replace the capacitor with the inductor and the equation would be $f_c = \frac{L}{2 \pi R}$. I should mention that these simple circuits also give one a 6dB/octave rolloff. This point is explained well on Wikipedia, so I don’t think I need to repeat it here.

The stunning thing about this simple passive crossover is that manufacturers can make a set of pretty high-end speakers with just these circuits. Of course, one would have to use some reasonable capacitors and inductors — but it can be that simple.

Manufacturers have many other design considerations besides the electronics to make as well (such as the enclosure!), but I hope this helps you understand the basics of what’s inside your speaker box.