Zobel Network Design
The voice-coil impedance of a loudspeaker driver is not purely resistive. This can have a major perturbation on the performance of crossover networks. At high frequencies, the voice-coil impedance becomes inductive. For odd-order crossover networks, this inductance can be utilized as part of the crossover network. Because the inductance is lossy, some experimentation may be necessary. For even-order networks, the inductance can be canceled by using a simple RC matching network as discussed below.
The impedance rise near the resonance frequency of closed-box midrange and tweeter drivers can have a major perturbation on the performance of the high-pass crossover networks. The effect is to cause a peak to appear in the pressure output of the driver at or near its resonance frequency. It can be very difficult to pull down this peak without causing a depression in the frequency response over a much wider band. To minimize the problem, the lower crossover frequency for the midrange and the tweeter should be greater than the fundamental resonance frequency of the drivers. The matching network described below can be used to cancel the impedance rise, but the element values may not be practical. A matching network, sometimes called a Zobel network, between the crossover network and the voice-coil terminals of a driver can be used to cause the effective load on the crossover network to be resistive.

Fig. 1 shows the network connected to the voice-coil equivalent circuit for a closedbox driver.

The high-frequency part of the network consists of R1, C1, R2, and C2. This network can be designed to correct for the lossy voice-coil inductance in an equal ripple sense between two specified frequencies in the band where the impedance is dominated by Ze (ω). At the fundamental resonance frequency of the driver, L1 and C3 resonate and put R3 in parallel with the voice coil. This cancels the rise in impedance at the fundamental resonance frequency fC.
Let the lossy voice-coil inductance have the impedance Ze (ω) = Le (jω)n, where Le and n
are defined in the paper “Loudspeaker Voice-Coil Inductance Losses: Circuit Models, Parameter
Estimation, and Effect on Frequency Response.” Let the network consisting of R1, C1, R2, and C2
correct for the lossy voice-coil inductance over the frequency band from f1 to f2. The frequency
f1 might be chosen to be the frequency above the fundamental resonance frequency fC where the voice-coil impedance exhibits a minimum before the high-frequency rise caused by the voice-coil inductance. The frequency f2 might be chosen to be 20 kHz. In order for the input impedance to the network plus the driver to be approximately equal to RE at all frequencies, the matching 1 network elements are given by

where RE is the voice-coil resistance, fC is the closed-box resonance frequency, QEC is the electrical quality factor, and QMC is the mechanical quality factor. The above equations are derived under the assumption that C1 and C2 are open circuits in the low-frequency range where R3, C3, and L1 are active and that L1 is an open circuit in the high-frequency range where R1, C1, R2, and C2 are active. For a lossless inductor, n has the value n = 1. In this case, C1 = Le/R2 E, and both R2 and C2 are open circuits.
Example 1 A closed-box midrange driver has the parameters RE = 7.5 Ω, Le = 0.00689, n = 0.7, fC = 250 Hz, QEC = 1.1, and QMC = 4. For the high-frequency network, assume f1 = 733 Hz and f2 = 20 kHz. Calculate the element values for the matching network which will make the driver impedance look like a 7.5 Ω resistor to the crossover network. Solution: From Eqs. (1) - (3), we have R1 = 7.5 Ω, C1 = 4.44 μF, R2 = 15.8 Ω, C2 = 3.52 μF, R3 = 9.56 Ω, L1 = 5.25 mH, and C3 = 77.2 μF. The figure below shows the calculated voice-coil impedance over the frequency range from 20 Hz to 20 kHz without the Zobel network (red curve) and with the Zobel network (blue curve). The frequency f1 corresponds to the frequency at which the red curve exhibits a minimum in its midband region.

Figure 2: Calculated voice-coil impedance with and without the Zobel network.