Impedance Compensation Networks for the Lossy Voice-Coil Inductance of Loudspeaker Drivers*

A two-terminal network that is connected in series or in parallel with a circuit to cause its terminal impedance to be transformed into a desired impedance is commonly called a Zobel [1] network. In loudspeaker design a Zobel network consisting of a series resistor and capacitor connected in parallel with the voice coil of a driver has been described to compensate for the impedance rise at high frequencies caused by the voice-coil inductance [2]. If the inductance is lossless, the network can be designed so that the effective high-frequency impedance is resistive. By maintaining a resistive load on the crossover network, its performance is improved. However, the voice-coil inductance of the typical loudspeaker driver is not lossless. In this case a Zobel network consisting of one resistor and one capacitor can be used to obtain a resistive input impedance at only one frequency in the high-frequency range where the voice-coil inductance dominates. In this engineering report two Zobel networks are described, one consisting of two resistors and two capacitors and the other consisting of three resistors and three capacitors. Each can be designed to compensate for the lossy voice-coil inductance of a driver. It is shown that the networks can be designed to approximate the desired impedance in an “equal-ripple” sense. Although the approximation can be improved with the use of more elements, it is shown by example that the simpler four element network can give excellent results with a typical driver. The effects of this network on the responses of second-order and thirdorder low-pass crossover networks for a specific driver are presented. At low frequencies the voice-coil impedance is dominated by its motional impedance. For infinite baffle systems the low-frequency impedance exhibits a peak at the fundamental resonance frequency of the driver. In [3] a modification of the circuit proposed in [2] is described which provides an additional compensation for this impedance peak. The circuit is also applicable to closed-box systems. Although the present report concerns impedance compensation at the high-frequencies where the voice-coil inductance dominates, the low-frequency compensation circuit proposed in [3] is reviewed. In addition, a modification of this circuit for vented-box systems is given. It is assumed that the loudspeaker driver is operated in its small-signal range. Otherwise the voice-coil inductance becomes a time-varying nonlinear function, its value varying with diaphragm displacement. This would preclude a linear circuit analysis and make it impossible to derive the compensation networks. The impedance approximation technique presented here has been used in the design of filters that convert white noise into pink noise. These circuits exhibit a gain slope of −3 dB per octave over the audio band. Example circuit diagrams of such filters can be found in [4], [5], and [6], but no design equations are given. In [5] the network is described as one in which “the zeros of one stage partially cancel the poles of the next stage.” In [7] a similar network is described to realize an operational-amplifier circuit which exhibits a gain slope of +4.6 dB per octave over the audio band. The authors stated that the network component values were selected with the aid of a software optimization routine to match the desired slope. An analytical solution is given here for the design of such networks. The general impedance compensation theorem described by Zobel can be succinctly summarized as follows. Given an impedance Z1 R0 + Z0, let an impedance Z 1 *Manuscript received 2003 January 24; revised 2003 November 11. ENGINEERING REPORTS