Introduction
A Zobel network (sometimes called a Boucherot cell), is in its simplest form, a series resistor-capacitor (R-C) network that is connected in parallel with an audio speaker in order to minimize the effects of the voice coil's inductance L(e).
These networks have been used for decades to make the speaker load appear resistive to the amplifier output over the operating frequency range, increasing the stability factor of the audio amplifier. Additionally, in applications where an amplifier drives multiple speakers, and a crossover network is used, the Zobel network eases the design of the crossover network by keeping the load impedance resistive.
With the advent of higher power (>100W) integrated circuit (IC) audio amplifiers, the selection of component values for this network, if not done correctly, can result in catastrophic damage to an IC and is the subject of this article.
Background
An audio loudspeaker has a complex impedance, while an audio amplifier is more comfortable driving a purely resistive load. To compensate for this, a Zobel network is placed in parallel with the speaker.
Some designers use a standard "rule of thumb" method, where the value of the resistor is between 2.7 Ω and 10 Ω, depending on the speaker DC resistance, and the capacitor is invariably 100nF.
This works for most discrete amplifiers, but selecting component values without understanding the parameters of the speaker can lead to serious consequences for IC power amplifiers.
Analysis
Figure 1 is a simple electro-acoustic circuit model of a speaker driver. The circuit values for the driver model can be obtained from parameters given in the manufacturers data sheet, or derived using a network analyzer.
Figure 1: Speaker Driver Model.
Definitions:
Revc is the Voice Coil DC resistance
Levc is the Voice Coil Inductance
Lces is the electrical analog of driver mechanical suspension compliance
Cmes is the electrical analog of driver mechanical cone mass
Res is the electrical analog of driver mechanical suspension resistance
In order to demonstrate how the values of the Zobel network should be calculated, a speaker used in a popular audio system was selected and the following Thiele-Small parameters were obtained from the manufacturer's data sheet:
Bl = 6.5 Tm (Force Factor)
Cms = 53.3 µm/N (Mechanical Compliance)
Mms = 0.0104 kg (Mechanical Compliance)
Rms = 1.56 (Mechanical Losses)
The values of the components in figure 1 may be measured or calculated as follows:
Revc = 8 Ω (Measured with an ohmmeter)
Levc = 135 µH (Measured with an inductance meter, normally where the inductance is significant)
Lces = Cms*(Bl)2 = 2.25 mH
Cmes = Mms / (Bl)2 = 246 µF
Res = (Bl)2 / Rms = 27 Ω
Constants:
j = √-1
ω(f) = 2πf
The complex impedance of the speaker can be expressed by:
(1)
This results in the following graph of Voice Coil Driver impedance vs. frequency shown in figure 2:
Figure 2: VCD Z vs. Frequency (calculated).
At resonance, Lces and Cmes are essentially open-circuit and, therefore, the peak impedance at resonance Zres is the series combination of Revc and Res, or Zres = 8 Ω + 27 Ω = 35 Ω.
