The following are design formulas for a number of different parallel crossover topologies. Most are two way formulas, but some contain hints at three way designs. I don't know where I found these, but I know I did some work on this file extracting factors of pi and smaller square roots.
These formulas have a major limitation in that they assume the driver acts as a resistor, so when using these, the use of zobel and impedance peak filters is pretty much required. The resulting crossover can get expensive fast... I don't recommend these as a way to design crossovers other than for finding starting points.
Two-way Network Design Formulas
The values are in henries (L), farads (C), ohms (R) and hertz (f). To convert to standard component values:
Multiply L with 1000 to get mH.
Multiply C with 1000000 to get uF.
Symbols used:
Rh = DC Resistance of the tweeter
Rl = DC Resistance of the bass driver
f = Crossover frequency
pi = 3.14159265
First-Order Networks
First-order filter consists of two components, an inductor and a capacitor. The filter type damps 6 dB per octave.
+ -----C1-----+
| +
Tweeter
|
- ------------+
+ -----L1-----+
| +
Bass
|
- ------------+
+ --C2---L2---+
| +
HP LP Mid
|
- ------------+
Butterworth
1 Rl
C1 = --------- L1 = --------
2 pi Rh fh 2 pi fl
C2 and L2 are calculated using the woofer and tweeter crossover points, respectively. One source (Weems) recommends calculating the ratio fl/fh and adding that much to L2 and subtracting that much from C2 to compensate for bandpass gain.
Second-Order Networks
Second-order filter consists of four components, two inductors and two capacitors. The filter type damps 12 dB per octave.
+ -----C1---+------+
| | +
L1 Tweeter
| |
- ----------+------+
+ -----L2---+------+
| | +
C2 Bass
| |
- ----------+------+
+ ----C3---+---L4---+------+
| | |
HP L3 LP C4 Mid
| | | +
- ---------+--------+------+
For the midrange, capacitors are multiplied by sqrt(2), inductors divided by sqrt(2) C3 and L3 are related to the woofer cross frequency and C4,L4 to the tweeter cross frequency. The midrange will probably need polarity inversion.
Bessel
1 3 Rh
C1 = --------------- L1 = ------------
2 pi sqrt3 Rh f 2 pi sqrt3 f
1 3 Rl
C2 = --------------- L2 = ------------
2 pi sqrt3 Rl f 2 pi sqrt3 f
Butterworth
1 Rh
C1 = --------------- L1 = -----------
2 sqrt2 pi Rh f sqrt2 pi f
1 Rl
C2 = --------------- L2 = -----------
2 sqrt2 pi Rl f sqrt2 pi f
Chebychev
1 Rh
C1 = --------- L1 = ---------
2 pi Rh f 2 pi f
1 Rl
C2 = --------- L2 = ---------
2 pi Rl f 2 pi f
Linkwitz-Riley
1 Rh
C1 = --------- L1 = -------
4 pi Rh f pi f
1 Rl
C2 = --------- L2 = -------
4 pi Rl f pi f
Third-Order Networks
Third-order filter consists of six components, three inductors and three capacitors. The filter type damps 18 dB per octave.
+ -----C1---+---C2-----+
| | +
L1 Tweeter
| |
- ----------+----------+
+ -----L2---+---L3-----+
| | +
C3 Bass
| |
- ----------+----------+
+ ----C4---+---C5---L5---+---L6-----+
| | | +
HP L4 LP C6 Mid
| | |
- ---------+-------------+----------+
Midrange Scaling factors:
C4:0.667, C5:2.0, L4:0.75, L5:1.5, L6:0.5, C6:1.333
Bessel
0.07911 0.1317 Rh
C1 = -------- L1 = -----------
Rh f f
0.3953 0.3294 Rl
C2 = -------- L2 = -----------
Rh f f
0.1897 0.06592 Rl
C3 = -------- L3 = -----------
Rl f f
Butterworth
0.1061 0.1194 Rh
C1 = -------- L1 = -----------
Rh f f
0.3183 0.2387 Rl
C2 = -------- L2 = -----------
Rh f f
0.2122 0.0796 Rl
C3 = -------- L3 = -----------
Rl f f
Fourth-Order Networks
Fourth-order filter consists of eight components, four inductors and four capacitors. The filter type damps 24 dB per octave.
+ -----C1---+---C2---+------+
| | | +
L1 L2 Tweeter
| | |
- ----------+--------+------+
+ -----L3---+---L4---+------+
| | | +
C3 C4 Bass
| | |
- ----------+--------+------+
A 4th order three way could be done, but would probably cost more and perform worse than an active crossover and extra amplifiers.
Bessel
0.0702 0.0862 Rh
C1 = -------- L1 = -----------
Rh f f
0.0719 0.4983 Rh
C2 = -------- L2 = -----------
Rh f f
0.2336 0.3583 Rl
C3 = -------- L3 = -----------
Rl f f
0.0504 0.1463 Rl
C4 = -------- L4 = -----------
Rl f f
Butterworth
0.1040 0.1009 Rh
C1 = -------- L1 = -----------
Rh f f
0.1470 0.4159 Rh
C2 = -------- L2 = -----------
Rh f f
0.2509 0.2437 Rl
C3 = -------- L3 = -----------
Rl f f
0.0609 0.1723 Rl
C4 = -------- L4 = -----------
Rl f f
Gaussian
0.0767 0.1116 Rh
C1 = -------- L1 = -----------
Rh f f
0.1491 0.3251 Rh
C2 = -------- L2 = -----------
Rh f f
0.2235 0.3253 Rl
C3 = -------- L3 = -----------
Rl f f
0.0768 0.1674 Rl
C4 = -------- L4 = -----------
Rl f f
Legendre
0.1104 0.1073 Rh
C1 = -------- L1 = -----------
Rh f f
0.1246 0.2783 Rh
C2 = -------- L2 = -----------
Rh f f
0.2365 0.2294 Rl
C3 = -------- L3 = -----------
Rl f f
0.0910 0.2034 Rl
C4 = -------- L4 = -----------
Rl f f
Linear-Phase
0.0741 0.1079 Rh
C1 = -------- L1 = -----------
Rh f f
0.1524 0.3853 Rh
C2 = -------- L2 = -----------
Rh f f
0.2255 0.3285 Rl
C3 = -------- L3 = -----------
Rl f f
0.0632 0.1578 Rl
C4 = -------- L4 = -----------
Rl f f
Linkwitz-Riley
0.0844 0.1000 Rh
C1 = -------- L1 = -----------
Rh f f
0.1688 0.4501 Rh
C2 = -------- L2 = -----------
Rh f f
0.2533 0.3000 Rl
C3 = -------- L3 = -----------
Rl f f
0.0563 0.1500 Rl
C4 = -------- L4 = -----------
Rl f f