Passive low-pass crossover

Resistive load

An ideal fourth-order passive low-pass crossover, with a resistive load, looks like this:

The response of this network is

The desired response for a fourth-order Linkwitz-Riley filter is

This is obtained from the circuit if

.

Impedance correction with a Zobel network

The Zobel network for impedance compensation consists of a capacitor CZ and a resistor RZ, and

and ,

where Ld is the inductance of the midrange driver, and Rd its DC resistance.

The HM100Z0 is quoted as having RD=6.4 ohms, and LD=0.19 mH, which gives CZ=4.64 uF. I used six 39 kohm resistors in parallel, and a 4.7uF polypropylene capacitor, and this gave a net driver impedance of 7.0 ohms from 1kHz to 10kHz.

L-Pad for driver attenuation

The HM100Z0 midrange driver is at least 3dB more sensitive than the Scan-Speak tweeter it is paired with, so with a passive crossover the midrange need to be padded down to match it. For convenience I built an L-pad which provides an input impedance which is independent of the attenuation setting.

Consider a driver with a nominal resistive impedance Rd in its passband, whose level needs to be attenuated by a factor a. If the desired net resistance seen by the crossover is Rin, then the desired values of the resistors in the L-pad are:

and

If Rin = Rd, these become

and

The measured impedance of the HM100Z0 with Zobel attached is approximately 7ohms across the passband, and the passive filter was designed with a target load of 6.75ohms. For the purpose of easy adjustment during the development process, I built an L-pad with six equal steps of 1dB, starting at -3dB. The network components were as follows:

Atten
Alpha
R1
(R1 used)
R2
(R2 used)
(Rin)
(Atten)
-3.000
0.708
1.971
2.00
15.058
15.00
6.773
-3.04dB
-4.000
0.631
2.491
2.50
10.876
10.75
6.739
-4.03dB
-5.000
0.562
2.954
3.00
8.292
8.25
6.787
-5.07dB
-6.000
0.501
3.367
3.33
6.547
6.60
6.730
-5.94dB
-7.000
0.447
3.735
3.75
5.296
5.40
6.798
-6.97dB
-8.000
0.398
4.063
4.00
4.362
4.40
6.702
-7.89dB

Once I had sorted the rest of the system, I settled on a 3dB attenuation.

The final circuit and component values

Using R=6.75 ohms, and w0=2 pi x 3.38kHz, the required reactive components are

L1 = 0.60mH ; L2 = 0.30mH

C1 = 11.2uF ; C2=2.47uF.

The final circuit looks like this:

and the values I used were:
C1 11uF ( = 10uF+1uF)
C2 2.42uF ( = 2.2uF+0.22uF)
CZ 4.7uF
L1 0.6mH
L2 0.3mH
R1 2.5 ohms = 30 ohms/12)
R2 10.9 ohms (=120 ohms/11)
RZ 6.4 ohms (=82 ohms/13)

Total filter impedance

The impedance of the crossover with load resistance RL is

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

As , ;

At ,

As , .