Resistive load
An ideal fourthorder passive lowpass crossover, with a resistive load, looks like this:
The response of this network is
The desired response for a fourthorder LinkwitzRiley 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 C_{Z} and a resistor R_{Z}, and
and ,
where L_{d} is the inductance of the midrange driver, and R_{d} its DC resistance.
The HM100Z0 is quoted as having R_{D}=6.4 ohms, and L_{D}=0.19 mH, which gives C_{Z=}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.
LPad for driver attenuation
The HM100Z0 midrange driver is at least 3dB more sensitive than the ScanSpeak 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 Lpad which provides an input impedance which is independent of the attenuation setting.
Consider a driver with a nominal resistive impedance R_{d} in its passband, whose level needs to be attenuated by a factor a. If the desired net resistance seen by the crossover is R_{in}, then the desired values of the resistors in the Lpad are:
and
If R_{in} = R_{d,} 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 Lpad with six equal steps of 1dB, starting at 3dB. The network components were as follows:
























































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 w_{0}=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 
R_{1}  2.5 ohms = 30 ohms/12) 
R_{2}  10.9 ohms (=120 ohms/11) 
R_{Z}  6.4 ohms (=82 ohms/13) 
Total filter impedance
The impedance of the crossover with load resistance R_{L} is
If the component values are substituted in for the 4thorder LR filter,
As , ;
At ,
As , .