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The Crossover Design Cookbook
Chapter 3: Speaker Motors and Crossovers
by Mark Lawrence

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Now, we are ready to discuss our next improvement in our Watt V crossover. As we designed it many long pages ago, we see that we used the standard second-order crossover circuits, and we assumed that the speaker drivers were simple resistors. If you compare the following pictures, you'll see that we just removed the resistors, and replaced them with drivers.

Of course, actual speaker drivers have a motor - a coil of wire surrounded by a magnetic structure. We know now that there's just no way this large coil can have no inductance. In fact, things are even more complicated than that: the mechanical properties of the speakers cone and suspension also get into the act, and the resulting impedance is, well, complicated.

If you pick some woofer and measure it's impedance, you'll find it looks a lot like this:

### Impedance of an actual, uncompensated woofer

The impedance starts off at the DC resistance of the voice coil. For an 8 ohm speaker, this is normally about 6 ohms. At the woofer's resonant frequency, the impedance jumps up to a peak. This peak is normally anywhere from 30 to 100 ohms. At about three times the speaker's resonant frequency (about 120Hz for a typical 30Hz woofer), the impedance drops back down to roughly the speaker's DC resistance. This impedance will typically be about 7 ohms in an 8 ohm speaker. Finally, as the frequency increases, the speaker's impedance starts on a continual rise, typically reaching about 30 - 50 ohms by 20kHz. All these numbers are very approximate. To build an effective compensation network you have to measure your particular woofers.

So, we see that this speaker makes a rather poor 8 ohm resistor. How does this effect our crossover? If the impedance is significantly greater than 8 ohms, the speaker is drawing less current from the amplifier than we might otherwise expect. While this is good for the amplifier, it means the inductors in the crossover don't do much. Inductors "choke off" current, so if the current draw is low, the inductors don't do much "choking". In fact, at frequencies above a few hundred hertz, the woofer looks mostly like an inductor itself, so a simple first order crossover will act like a voltage divider, and not like a crossover at all.

The speaker impedance can also cause problems for the power amp. When the impedance plot is sloping upwards, as the woofer above is from about 20hz to about 30hz, the amplifier thinks it's driving an inductor. This is not a good thing: inductors make their maximum current demands when the signal crosses through ground (0 volts). This makes the amplifier dissipate a lot of heat and tends to cause amplifier crossover distortion, the distortion made by a transistor amplifier when the signal passes over from the negative portion of the amplifier to the positive portion. Also all inductors, including your woofer coil, do something called "inductive kickback." If you suddenly want the woofer to change velocity it's like wanting the water in a 4" fire hose to suddenly change velocity: this is not going to be easy. The woofer motor coil will protest by sending a large voltage spike back along the speaker wires to the amplifier. The amplifier must absorb and dissipate this quickly. If your power amplifier has a very low output impedance it can probably handle this, but it's going to get hot. If your amplifier has a large output impedance, like over 1/2 ohm, the amplifier will lose control of the woofer for a short time and your bass will get mushy. When we're designing speakers we would like the speaker impedance to be as flat as is realistic so as to help the amplifier do its job.

This same effect causes problems for the power company. If you have a large building with a large motor to handle air circulation, the power company will require you to put a large capacitor next to the motor. If chosen correctly, this capacitor will resonate out the motor inductance and the result will be very close to a pure resistive load at 60hz. Later we're going to see how to resonate the woofer motor impedance into something close to a resistor, for which the amplifier will be eternally grateful.

Similarly, when the impedance plot is trending downwards, as the woofer is from about 30hz to about 100hz, the amplifier thinks it's driving a capacitor. This is also not a good thing: capacitors make their maximum current demands when the signal crosses through ground (0 volts). Again, this causes major amounts of heat to dissipate in the heat sinks and it tends to cause crossover distortion in the power amplifier.

Capacitors are made more effective as the current demands go down, so our crossover is in trouble. Because of the motor inductance raising the woofer impedance, the capacitor in our second order crossover is not operating at it's assigned frequency - it is passing signals above the crossover frequency due to the low current demand.

These effects are so significant that it's impractical to design a reasonable crossover for a real life woofer using only the first or second order circuits. Even though our Watt V crossover was designed to operate at 2,000Hz, the woofer will probably still be active up to 3,000 or perhaps even 4,000Hz, and the tweeter will be active down below 1,000Hz. We need to do something about this problem.

There's a standard fix for these problems. In the case of our Watt V, the Scan Speak woofer resonance is at about 70Hz, and the crossover is located at about 2,000Hz, so we can safely ignore the speaker resonance. However, the voice coil inductance is a dominating factor at the 2,000Hz crossover frequency. We need to turn the speaker into a resistor at these frequencies. We do this with what's called a Zobel circuit. A Zobel circuit is just the first R-C circuit show above, placed in parallel with the speaker.

## Calculating Zobel impedance compensation networks

The impedance in the curve above is raising at high frequencies due to the motor coil inductance - as the frequency raises, the motor inductor draws less and less current to do its job. We can't change the coil properties, but we can waste some power so that overall the entire system is drawing a constant amount of current.

To do this, we put a resister in parallel with the driver. This resister is calculated to waste just the right amount of power. Then, we block off this resistor as low frequencies with a capacitor. This capacitor is calculated to start letting current through the resistor at just the right frequency to keep the impedance flat.

The Zobel is designed like this: You measure the woofer's resistance at DC, which is conventionally called Re. Then, you measure the speaker's resistance at the crossover frequency, which I'll call Rc. The voice coil inductance is L = √( Rc² - Re²) / (2πFc). The capacitor in the Zobel circuit has the value L / Re², and the resistor in the Zobel circuit has the same value as Re. The resulting impedance graph for the woofer still has the large hump at resonance, but we've decided to ignore that as insignificant for our crossover. The impedance above resonance is now much flatter. There will still be a slight curve, but now the curve will only vary about 1/2 ohm. We will neglect this slight variance for now, and discuss it later.

### Impedance including Zobel compensator

The Scan Speak woofer has a DC resistance of 5.5 ohms, and a voice coil inductance of .1mH, so we need a .0001 / 5.52 = 3.3µF capacitor and a 5.5 ohm resistor for the Zobel network. Now, our Watt V woofer crossover looks like this:

### Woofer with an attached Zobel impedance compensation circuit

The actual crossover hasn't changed - we've just added the Zobel R-C compensator so that the woofer properly loads the crossover and allows the crossover to function correctly.

Are we done with this Watt V crossover? Well, no. Will it work? It will work better than our first attempt, which we now realize was going to crossover at some strange combination of frequencies - not at all what we had intended. We've fixed the woofer problem, but we still have a couple tweeter problems to deal with.