Modelling Filter Networks

As you probably already have guessed I am a big fan of serial network for use in loudspeaker filters. They are "component inexpensive" and offer many desirable opportunities as those found in parallel filters. The main downside to the serially coupled loudspeakers is that they are virtually impossible to design using an anaytical approach and thus an numerical approach is required. By creating a few simple mathematical equations, you can simulate the responses of different networks and I have realized a few of those using Excel spreadsheets.

Screen dump from a 1st order serial filter design spreadsheet.

The way of using the spreadsheets is uniquely simple: First, you type in the characteristics of your loudspeaker unit (DC resistance and voice coil inductance) and then you try out some component values for the filtering. The program depicts the resulting impedance seen by your amplifier and the electrical magnitude response of highpass, lowpass and the summed response.
Bear in mind that the acoustical parameters of your loudspeakers are not included and therefore, you should carefully consider the upper rolloff of the bass and the lower rolloff of the tweeter. I prefer to cutoff the loudspeakers atleast one octave before the loudspeakers rolloff due to the distortion parameters of the loudspeaker.

1st order w. Zobel

On your right, a 1st order serial network is depicted. This network has a Zobel network included for the bass driver. I found that loudspeakers are much easier to integrate in this network than the standardized quasi-stationary filter because more design utilities are given regarding the attenuation/filtering and especially the tweaking of the overall system impedance. I like this network and have used it for some 2 way designs.
Download the Excel spreadsheet.

1st order serial network with zobel included for the bass.

Zobel network

The Zobel network is often difficult to design accurately, simply because the loudspeaker impedance is'nt linearly dependant on the frequency

The Zobel network is placed directly across the loudspeaker before any filtering networks.
The square root of the voice coil impedance (√(j2pifLe) + Re) is approximately proportional to the frequency and therefore you can't simply insert a capacitor which poses an inverse impedance as that of the inductor. Rather, experience tells us that a good approximation is that a capacitance of approximately Le/(Re)2 is suitable. The vesistor value should then be approximately Re1.25. Refer to for a thorough description of how to design a Zobel. Download the Excel spreadsheet.

2nd order serial network

The second order serial network naturally provides a stronger rolloff, but it is quite inconvenient in many situations bacause no impedance smoothing circuits are applied. However, in systems where the bass units have low inductance and the DC resistance of the tweeter is quite high it can be used. In practice this combination of loudspeakers is rare. Notice, that a perfect summation is not given, meaning that you have to be carefull in your choice of component values. This deviation from unity gain should ideally be benefitted from, obtaining the required attenuation of the tweeter.
Download the Excel spreadsheet.

Standardized second order 2 way serial network.