RACK/FILTERS & ENCLOSURES

L\!\cdot\!C

Passive LC Crossover

Exact inductor and capacitor values for 2-way crossover networks.

MOD-12•PASSIVE FILTER SYNTHESIS & BODE ALIGNMENT BENCH

Passive LC Crossover Network & Phase Alignment Designer

Synthesize exact capacitor ( $C$ in $\mu\text{F}$ ) and air-core inductor ( $L$ in $\text{mH}$ ) component values for 1st, 2nd, and 3rd-order Linkwitz-Riley, Butterworth, and Bessel loudspeaker crossover networks.

2-Way Loudspeaker Presets

HIGH-PASS (TWEETER) VALUES

C_1,\ L_{\text{shunt}}

C₁ = 6.22 μF

Shunt L_H: 1.59 mHZ_H = 8 Ω

LOW-PASS (WOOFER) VALUES

L_1,\ C_{\text{shunt}}

L₁ = 1.59 mH

Shunt C_L: 6.22 μFZ_L = 8 Ω

SLOPE & HF POLARITY ALIGNMENT

\Delta\phi(f_c)

12dB / oct

HF Driver Polarity: 180° INVERTED (Ø)fc = 1600 Hz

FILTER TOPOLOGY, CROSSOVER FREQUENCY & IMPEDANCE

Filter Order & Alignment Topology

Crossover Frequency

f_c

100 Hz6000 Hz

HF Compression Driver / Tweeter Impedance

Z_H

4 Ω16 Ω

LF Woofer Nominal Impedance

Z_L

4 Ω16 Ω

DYNAMIC LC CIRCUIT SCHEMATIC & BODE TRANSFER FUNCTION

HOW IT WORKS: ELECTRO-ACOUSTIC PHYSICS & MATHEMATICAL DERIVATION

Passive LC Reactance, Filter Alignments & 2nd-Order Phase Inversion

Passive loudspeaker crossovers divide the full-range amplifier signal into frequency bands using reactive components: Capacitors exhibit capacitive reactance $X_C = \frac{1}{2\pi f C}$ that decreases with frequency (passing highs while blocking lows), whereas Inductors exhibit inductive reactance $X_L = 2\pi f L$ that increases with frequency (passing lows while choking highs).

Each reactive element introduces a $90^\circ$ asymptotic phase shift ( $45^\circ$ per pole at $f_c$ ). Consequently, in any **2nd-Order ( $12\text{ dB/octave}$ ) crossover**, the Low-Pass branch shifts by $-90^\circ$ at $f_c$ while the High-Pass branch shifts by $+90^\circ$ , placing the woofer and compression driver **exact $180^\circ$ out of phase** at the crossover frequency! Reversing the $+ / -$ leads on the high-frequency driver brings both transducers back into $0^\circ$ coherent summation. A **Linkwitz-Riley 2nd-order ( $Q = 0.5$ )** crossover attenuates both branches by $-6.02\text{ dB}$ ( $0.5\times$ voltage) at $f_c$ so their coherent sum yields a ruler-flat $0\text{ dB}$ acoustic response.

1st-Order Butterworth (6 dB/octave, −3 dB @ fc)EQ-01

C_1 = \frac{1}{2\pi f_c Z_H}, \qquad L_1 = \frac{Z_L}{2\pi f_c}

2nd-Order Linkwitz-Riley (12 dB/octave, −6 dB @ fc, Q = 0.5)EQ-02

C_1 = \frac{1}{4\pi f_c Z_H}, \quad L_{\text{shunt},H} = \frac{Z_H}{\pi f_c}, \qquad L_1 = \frac{Z_L}{\pi f_c}, \quad C_{\text{shunt},L} = \frac{1}{4\pi f_c Z_L}

2nd-Order Butterworth (12 dB/octave, −3 dB @ fc, Q = 0.707)EQ-03

C_1 = \frac{\sqrt{2}}{4\pi f_c Z_H}, \quad L_{\text{shunt},H} = \frac{\sqrt{2}\,Z_H}{2\pi f_c}, \qquad L_1 = \frac{\sqrt{2}\,Z_L}{2\pi f_c}, \quad C_{\text{shunt},L} = \frac{\sqrt{2}}{4\pi f_c Z_L}

Nomenclature & Physical Variables

$f_c$
Acoustic and electrical crossover transition frequency
Unit: Hz
$Z_H, Z_L$
Nominal impedance of HF driver and LF woofer at $f_c$
Unit: Ω
$C_1, C_2$
Non-polarized metallized polypropylene crossover capacitors
Unit: F (or μF)
$L_1, L_2$
Low-DCR air-core or laminated-iron crossover inductors
Unit: H (or mH)

FOH & System Tech Field Application Notes

Because a woofer’s voice-coil inductance ( $L_e$ ) causes its impedance to rise steadily above $500\text{ Hz}$ , adding a Zobel Impedance Equalization Network ( $R_z \approx Z_{\text{nom}}$ in series with $C_z = L_e / R_z^2$ ) across the woofer terminals ensures the passive low-pass filter sees a constant resistive load at $f_c$ .
Mount crossover inductors at $90^\circ$ right angles to one another on the PCB to prevent magnetic flux coupling between the high-pass and low-pass coils.