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 The Crossover Design Cookbook Chapter 3: Speaker Motors and Crossovers by Mark Lawrence 
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Up until now we've been designing our crossovers as if the drivers were just resistors. Unfortunately, the truth is far different. We're going to have to compensate for the strange electrical behavior of the drivers so that our crossovers have a chance to work correctly. To do this, we're going to have to measure the performance of our drivers. The parameters we need are traditionally called the ThieleSmall parameters, after the two engineers who made speaker design a science.
In order to proceed, you're going to need an audio frequency sine wave generator and a digital volt meter. A dualtrace oscilloscope makes this easier, but is not required. Included below and in appendix 3 is a measurement form, which will help you make the correct measurements and calculations in the correct order.
We need to hook up the speaker to our measurement gear. If we are trying to measure the speaker's free air ThieleSmall parameters, we hang the driver in the air. If we are interested in how the speaker performs in it's box, then these tests are done with the driver mounted. For box design programs, we need free air parameters. For cross over design, we need inbox parameters. Most tweeters have sealed backs, so for tweeters free air and inbox parameters are the same.
Connect up your gear to the driver as shown below. Don't hook up through the cross over. Turn on your signal generator at 200Hz. Using your DVM, set Vr to precisely 1 volt AC. Now, connect the DVM across the driver to measure Vd.
If you have a dualtrace oscilloscope, hook both probes ground leads to the ground shown. Hook one probe between the driver and the .5Ω resister, and the second probe between the driver and the 1,000Ω resistor. Set the oscilloscope in XY mode. Your oscilloscope should be showing a diagonal ellipse, which is called a Lissajous figure. As you turn down the generator's frequency, the ellipse will become more and more narrow until at one particular frequency, the trace becomes a diagonal line. This frequency is exactly Fr. Fill it into the form, and turn off your oscilloscope. Leave your generator set at Fr.
If you don't have a dual trace oscilloscope, you may omit the .5Ω resistor. Fill in Fr from the manufacturer's sheet. Set your generator to Fr. Now, with your DVM measuring Vd, turn the frequency up and down a small amount until you get the maximum reading.
Measure the AC voltage across the driver at Fr. Each millivolt equals 1 ohm of driver resistance, so 32 millivolts means 32 ohms. Write this down on your form as Rr. If you're worried about precision, check that Vr is precisely 1V RMS before you measure Vd. Vr will vary slightly as you change the signal generator's frequency.
Now, we calculate R1 = √(Rr * Re), and fill it into the form. Next, we find the two frequencies where the speaker impedance is equal to R1. One of these frequencies will be at about two thirds Fr, and the other will be at 1.5 * Fr. Turn your signal generator down in frequency until Vd equals R1 millivolts, and fill this frequency onto the form as F1. Next, set your generator back to Fr, and turn the frequency up until Vd equals R1 millivolts, and fill this frequency onto the form as F2. Calculate Fr = √(F1 * F2). and write the result on the form in the second Fr blank. The calculated Fr and the oscilloscope Fr should match within 2Hz. If they don't, something is wrong. The manufacturer's Fr is likely to be off by two to five Hz.
Now, calculate Qf = Fr / (F2  F1). Use the oscilloscope Fr if you have it, otherwise use the calculated Fr. Finally, calculate Qtc = Qf / √(Rr / Re). If you need a Zobel, you'll also need the driver resistance at your selected cross over frequency (Rc), and the driver inductance measured at your selected cross over frequency L = √(Rc²  Re²) / 2π Fc.
Parameter  Value  

Re  DC resistance  
Fr  resonant frequency  
Rr  resistance at resonance  
R1  √(Rr * Re)  
F1  driver resistance is R1  
F2  driver resistance is R1  
Fr  √(F1 * F2)  
Qf  Fr / (F2  F1)  
Qtc  Qf / √(Rr / Re)  
Fc  cross over frequency  
Rc  driver resistance at Fc  
L  √(Rc²  Re²) / 2πFc 
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Copyright © 20022019 Mark Lawrence. All rights reserved. Reproduction is strictly prohibited.
Email me, mark@calsci.com, with suggestions, additions, broken links. Revised Thursday, 15Aug2019 09:30:53 CDT 
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