Passive high-pass crossover

Resistive load

An ideal fourth-order high-pass filter with a resistive load, looks like this:

The response of such a passive network is

The corresponding target high-pass response is

This is obtained if

,

, .

Zobel network

If the driver has DC resistance Rd and inductance Ld, the component values for the Zobel network are then

and

If the crossover frequency is much higher than the driver resonance, the driver will behave approximately as an inductor, and the Zobel network will equalise the driver impedance to a pure resistance Rd. Unfortunately, with a tweeter the resonance is unlikely to be more than an octave or two lower than the crossover frequency, and even with the Scan-Speak D2905/9300 the resonance is 600HZ, a factor of five lower than the crossover frequency 3.3kHz. I found that with a simple Zobel network the measured response of the high-pass leg of the filter was substantially incorrect, and this was due to the impedance of the driver plus Zobel increasing sharply as the frequency dropped below about 2.5kHz.

Peter Dahl's impedance correction fix

Peter Dahl suggested the following circuit as a substitute for the Zobel network, which he has used himself with a D2905/9300:

where R1 = 7.5 ohms (I used two 15 ohms in parallel), C3 = 66uF (I used a 60uF reversolytic in parallel with a 6u0 polypropylene) , C4 = 3.3uF and L3 = 0.8mH. The LC circuit (L3 and C3) has a resonance frequency at

at which the impedance of the compensating circuit has a minimum value of R1, which presumably compensates for the impedance maximum corresponding the main mechanical resonance at 600Hz of the D2905/9300. I have my doubts about putting a poorly-damped LC resonant circuit in the crossovers, but fr is well outside the tweeter passband and is quite well isolated from the midrange driver.

I constructed this, and found it gave a much better-behaved impedance characteristic, with an impedance of 5.0 ± 0.3 ohms between 1kHz and 10kHz. I think the impedance could be linearised still further, but I wonder how worthwhile this would be, given the non-ideal responses of the drivers themselves.

The final circuit and component values

Using R=5 ohms and w0=3.38kHz, I get the following desired component values:

L1 = 0.148mH, L2 = 0.666mH C1 = 5.0uF, C2 = 10.0uF

The final circuit looks like this:

The values I used were:
C1 5uF
C2 10uF
C3 68uF
C4 0.33uF
L1 0.15mH
L2 0.667mH (=1.0mH//2.0mH)
L3 0.8mH
R1 7.5ohms (=15 ohms/2)

Network Impedance

The impedance of the high-pass crossover, with load resistance RH is

If the component values are substituted in for the 4th-order L-R filter,

As , and as , .