Speaker Crossover Calculator with Real Driver Impedance

The normalized low-pass denominator is $D(s)=1+a_1s+a_2s^2+a_3s^3+a_4s^4$, with $s=jf/f_c$. Butterworth coefficients follow its stable all-pole polynomial. LR2 uses $(1+s)^2$ and LR4 uses $(1+\sqrt{2}s+s^2)^2$, as described by Linkwitz Lab, sections 2–3. Bessel uses reverse Bessel polynomials $1+s$, $1+s+s^2/3$, $1+s+2s^2/5+s^3/15$ and $1+s+3s^2/7+2s^3/21+s^4/105$, scaled so $|D(j)|^2=2$. The scale factors are 1, 1.361654128716, 1.755672368681 and 2.113917674904. This is the magnitude convention documented by SciPy.

LC synthesis, derived here. Start with a series element and alternate series/shunt. For order 2, $g_1=a_1$, $g_2=a_2/a_1$. For order 3, $g_3=a_3/a_2$, $g_1=a_1-g_3$, $g_2=a_2/g_1$. For order 4, $g_4=a_4/a_3$, $b=a_2-g_4a_1$, $g_3=a_3/b$, $g_1=a_1-g_3$, $g_2=b/g_1$. Low-pass series inductors are $gR_e/\omega_c$ and shunt capacitors $g/(R_e\omega_c)$. High-pass series capacitors are $1/(gR_e\omega_c)$ and shunt inductors $R_e/(g\omega_c)$. Ideal source, single resistive termination; no interstage buffers.

Driver and circuit solution. Set $r=f/f_s$. The small-signal impedance is

Audio Precision AN3, attached page 19, relates the resonance impedance to Qms and Qes. A Zobel has $R_z=R_e$, $C_z=L_e/R_e^2$; it is disabled at zero Le. For pad voltage ratio $k=10^{-A/20}$, $R_s=R_e(1-k)$ and $R_p=R_ek/(1-k)$. At zero attenuation both pad parts are omitted. These networks follow Elliott Sound Products, sections 3.1 and 6; the implemented ideal RC cancellation is derived from the series RL model.

The ladder is solved backwards from its complex load with unit output voltage: add shunt current, then series voltage, preserving the loading of all components. The pad's voltage division is included. The two branch input admittances add. Each inductor includes the specified branch DCR.

Acoustic approximation. For $x=jf/f_s$, the piston factor per terminal volt is $(R_e/Z_d)x^2/(x^2+x/Q_{ms}+1)$, derived from the force/current and acceleration equations. Each driver's midband sensitivity is normalized. The flat-voltage option sets this factor to one. Tweeter sensitivity, polarity and $\exp(-j2\pi f\Delta d/343)$ are applied before coherent summation. Phase is wrapped to ±180°; a jump at that boundary is not a discontinuity in pressure. Displayed magnitudes have a −180 dB floor. The 181-point logarithmic sweep includes fc exactly; minima and maximum sine losses are sampled estimates.

Resistor heating and regression checks. Zobel power is evaluated independently at each frequency before taking its maximum. Separate current-based checks compute $I_z=|V_d|/\sqrt{R_z^2+[1/(2\pi fC_z)]^2}$ and compare $I_z^2R_z$ with the conductance-based loss across the whole sweep, with both Zobels connected. The crossover-detail view interpolates the same acoustic results from $0.5f_c$ to $2f_c$, limited to 20 kHz; it includes the design frequency exactly.

• Linear, small-signal, free-air driver equivalent circuit, with frequency-independent Re and Le. Defaults are illustrative, not catalogue data.
• Ideal zero-impedance amplifier; ideal capacitors and resistors. Every filter inductor in a branch uses that branch's entered DCR. No mutual inductance, saturation, capacitor ESR or wiring resistance.
• The design termination is Re, including for an optional matched L-pad. Compensation does not change the synthesis resistance. Finite resonance and residual reactance therefore shift the actual alignment.
• Equal normalized woofer voltage sensitivity; tweeter sensitivity is specified at the same voltage and distance. One path difference accounts for delay only. Positive offset delays the tweeter.
• Cauer synthesis, backward circuit solution and normalized piston equations are derived here from Kirchhoff's laws and the linear moving-coil equations, rather than copied from a worked design.

Two-way parallel passive crossovers, orders 1–4; LR only 2 or 4. Sweep: 20 Hz–20 kHz. The suggested second-order reversed wiring is a nominal electrical rule, not an optimization of the real acoustic phase.

A T/S impedance model does not determine actual acoustic frequency response. The piston approximation omits breakup, horn loading, radiation impedance variation, diffraction, enclosure resonances and off-axis behavior. Use measured mounted-driver impedance and acoustic data for final validation. In particular, free-air parameters do not model the low-frequency load of a sealed or vented enclosure.

Zobel correction cancels an ideal RL rise, not the motional resonance or a real driver's semi-inductance. Coil heating changes Re and Qes; large-signal compression and distortion are excluded. Resistor losses refer to an RMS sine voltage at each frequency, not broadband music or driver survival. Sampled minima can miss a narrow impedance dip. Verify amplifier compatibility and component temperature, current and voltage ratings on the built network. Always test with the intended drivers connected.