Impedance Compensation Circuits

by Isaac MCN
f4ier@hotmail.com


0.  Introduction

DIY loudspeaker beginners usually start off with off-the-shelf crossovers.  These are designed using equations from textbooks, which assume a resistive load.  Loudspeakers are far from resistive!  To the unsuspecting, untrained, listener the results of using textbook filters may sound OK, but it will be shown that such crossovers fail to do what they're supposed to do.


1.  Textbook Crossovers -- Some Basics

Ohm's law states that voltage equals the product of current and resistance.  Given the following circuit,

<insert VR1R2 series circuit here>

The current through the circuit is simply 2V / 2Ohms = 1A and the voltage across R1 is 1V = the voltage across R2 as well -- the voltage was divided in half into the 2 resistors.  If R1 where to be bypassed [0 resistance], then the full 2V will appear across R2.  If, on the other hand, R2 were to be infinitely large [R2 is effectively disconnected], then no voltage would be measured across R2.  These overservations hold for any frequency of the applied voltage because resistors are not reactive.

If R1 were to be substituted with a fixed value inductor, which has a rising impedance with frequency, then the voltage division between both loads would varry with frequency.


1.  Off-The-Shelf Crossovers Or Textbook Crossovers

Circuit analysis books introduce filter design using examples with purely resistive loads.  


1.  The Electrodynamic Loudspeaker

An electrodynamic loudspeaker has a vibrating cone -- held in place by a spider and surround assembly -- powered by a permanent magnet-voice-coil motor.  The springiness of the suspension and mass of the cone form a 2nd-order resonant system.  A typical electical impedance measurement looks like the following graph.



where one can see a peak centered around the resonance frequency of the device under test and a rising impedance due to the voice-coil inductance.  There is always a peak because a loudspeaker tends to vibrate more at its resonance frequency and this induces more opposing current through the voice-coil.


IMPEDANCE COMPENSATION CIRCUITS

by Isaac MCN
f4ier@hotmail.com


0.  Introduction

Often an available amplifier is more comfortable with flat-impedance loads.  In reality a loudspeaker has a complex impedance.  To compensate for this, circuits can be designed and implemented.  Probably the most common impedance compensation circuit is the so-called Zobel network, which has a few derivatives the simplest being the capacitor-resistor network.  The full Zobel network would also compensate for the resonance impedance peak of the driver at the resonant frequency, Fs.

1.  Woofer

One of the most famous drivers in the DIY world is the Vifa P17WJ-00-08.  Many builders use it because it has a smooth frequency response and resonable bandwidth.  Because many constructors use it, the said driver was chosen to be modeled in this website.  The following is a picture of the the driver unit along with its Thiele-Small Parameters.

  Thiele-Small Parameters
Revc  

5.8 ohms

   DC resistance of voice coil
Levc   0.55 mH   voice coil inductance
Bl   6.5 T.m   force factor
Qts   0.35  

total Q
Qes   0.45   electrical Q
Qms   1.55   mechanical Q
Fs   37 Hz   resonant frequency
Mmd   0.014 kg  

mass of cone + voice coil + etc.
Rms   2.08   resistance of suspension
Cms   1.34 mm/N   compliance of suspension
Sd   0.0136 sq.m   effective cone area
Vas   0.0347 cu.m   equivalent acoustic volume
Xmax   0.004 m   linear travel of voice coil
FR   37 - 5000kHz   frequency response
Vd   0.0005 cu.m   driver unit volume displacement

The driver can be modelled with the following electroacoustic circuit.

Some of the circuit values above are already obvious.  Veg represents the amplifier and is assumed to have no output resistance.  The remaining values were calculated from,

Cmes = Mmd/(Bl*Bl) = electrical analog of driver mechanical cone mass
Lces = Cmd*Bl*Bl = electrical analog of driver mechanical suspension compliance
Res = Bl*Bl/Rmd = electrical analog of driver mechanical suspension resistance
Cmef = 8*po*Ad*Ad*Ad/(3*Bl*Bl) = electrical analog of air load on the driver unit's cone

where:

po    =    1.18 kg/cu.m    =    air density

and Ad is the effective radius of the driver unit's cone.  The following is a screenshot of the driver unit's calculated impedance curve.


Some Thiele-Small Parameters can be calculated from the impedance curve above.  Using equations from [1], we get,

Fs = 35.604 Hz
Fl = 19.121 Hz
Fh = 65.615 Hz
Ro = Rmax/Revc = 4.51 ohms
Rx = Revc*sqrt(Ro) = 12.32
Qms = Fs*sqrt(Ro)/(Fh-Fl) = 1.63
Qes = Qms/(Ro-1) = 0.463
Qts = Qms/Ro = 0.361

The calculated parameters are close enough and one would get near-real-world parameters with real-world impedance curves.

It is obvious that the amplifier would be more happy if the effective impedance accross the speaker terminals were flat or more resistive. It is possible to flatten the raw impedance curve of a driver unit by using impedance compensation circuits. With impedance compensation circuits, the driver unit equivalent circuit model looks like

The RC circuit formed by Ric and Cic helps to flatten the rising impedance due to the voice coil and on the other hand the LCR circuit formed by Lic1, Cic1 and Ric1 flattens the impedance peak due to Eddy currents, back emf and so on (at and around the driver unit's resonant frequency). The circuit values can be calculated from,

Ric = Revc (for a flat impedance curve) or 1.26Revc (which is a bit better for the amplifier)
Cic = Levc/(Revc*Revc)
Lic1 = Revc*Qes/(2*pi*Fs)
Cic1 = 1/(2*pi*Fs*Revc*Qes)
Ric1 = Revc + (Revc*Qes)/Qms

The voice coil is basically an RL circuit, therefore the impedance accross the voice coil is directly proportional with frequency. An RC circuit exibits a decreasing impedance with frequency. Placing such a circuit in parallel with the voice coil would help to counteract the rising impedance. The following picture shows the effect of the RC circuit.

The RC circuit has successfully compensated for the rising impedance of the voic coil. The lowest impedance is 5.45 ohms and at 10kHz, the impedance is now 5.8 ohms instead of 35 ohms.

What remains to be adjusted is the peak impedance at resonance. Ignoring the effects of the other circuit components, Lces and Cmes are essentially open-circuit at resonance, which explains the peak impedance of 26.2 ohms. A series LCR circuit is basically equal to R at the resonant frequency of L and C. Designing such a circuit to resonate at Fs would basically result in the following impedance (at resonance),

Z = (Ric1*(Revc + Res))/(Ric1 + Revc + Res) = 5.82 ohms

which is almost equal to the DC resistance of the voice coil. This is neglecting the (small) effects of Levc, Cmef, Ric and Cic. The following picture shows the effect of Lic1, Cic1 and Ric1 circuit in place.

The combined effects of the above-mentioned RC and LCR circuits results in the following impedance curve.

The green curve shows a closer look at the red curve. As you can see, the voice coil impedance is now resonably flat. The peak impedance is equal to 5.89 ohms and the trough corresponds to an impedance of 5.3 ohms, which when subtracted from the former yields an impedance difference of 0.59 ohms.

More complex impedance compensation circuits can give a more flat impedance curve, but final building cost would be higher with little to negligible returns.