Zobel network

Zobel networks are a type of filter section based on the image-impedance design principle. They are named after Otto Zobel of Bell Labs, who published a much-referenced paper on image filters in 1923. The distinguishing feature of Zobel networks is that the input impedance is fixed in the design independently of the transfer function. This characteristic is achieved at the expense of a much higher component count compared to other types of filter sections. The impedance would normally be specified to be constant and purely resistive. For this reason, Zobel networks are also known as constant resistance networks. However, any impedance achievable with discrete components is possible.
Zobel networks were formerly widely used in telecommunications to flatten and widen the frequency response of copper land lines, producing a higher performance line from one originally intended for ordinary telephone use. Analogue technology has given way to digital technology and they are now little used.
When used to cancel out the reactive portion of loudspeaker impedance, the design is sometimes called a Boucherot cell. In this case, only half the network is implemented as fixed components, the other half being the real and imaginary components of the loudspeaker impedance. This network is more akin to the power factor correction circuits used in electrical power distribution, hence the association with Boucherot's name.
A common circuit form of Zobel networks is in the form of a bridged T network. This term is often used to mean a Zobel network, sometimes incorrectly when the circuit implementation is not a bridged T.

Derivation

The basis of a Zobel network is a balanced bridge circuit as shown in the circuit to the right. The condition for balance is that;
If this is expressed in terms of a normalised Z₀ = 1 as is conventionally done in filter tables, then the balance condition is simply;
Or, is simply the inverse, or dual impedance of.
The bridging impedance Z_B is across the balance points and hence has no potential across it. Consequently, it will draw no current and its value makes no difference to the function of the circuit. Its value is often chosen to be Z₀ for reasons which will become clear in the discussion of bridged T circuits further on.

Input impedance

The input impedance is given by
Substituting the balance condition,
yields
The input impedance can be designed to be purely resistive by setting
The input impedance will then be real and independent of ω in band and out of band no matter what complexity of filter section is chosen.

Transfer function

If the Z₀ in the bottom right of the bridge is taken to be the output load then a transfer function of V_o/V_in can be calculated for the section. Only the RHS branch needs to be considered in this calculation. The reason for this can be seen by considering that there is no current flow through Z_B. None of the current flowing through the LHS branch is going to flow into the load. The LHS branch, therefore, cannot possibly affect the output. It certainly affects the input impedance but not the transfer function. The transfer function can now easily be seen to be;

Bridged T implementation

The load impedance is actually the impedance of the following stage or of a transmission line and can sensibly be omitted from the circuit diagram. If we also set;
then the circuit to the right results. This is referred to as a bridged T circuit because the impedance Z is seen to "bridge" across the T section. The purpose of setting Z_B = Z₀ is to make the filter section symmetrical. This has the advantage that it will then present the same impedance, Z₀, at both the input and the output port.

Types of section

A Zobel filter section can be implemented for low-pass, high-pass, band-pass or band-stop. It is also possible to implement a flat frequency response attenuator. This last is of some importance for the practical filter sections described later.

Attenuator

For an attenuator section, Z is simply
and,
The attenuation of the section is given by;

Low pass

For a low-pass filter section, Z is an inductor and Z ' is a capacitor;
and
where
The transfer function of the section is given by
The 3 dB point occurs when ωL = R₀ so the 3 dB cut-off frequency is given by
where ω is in the stop band well above ω_c,
it can be seen from this that A is falling away in the stop band at the classic 6 dB/8ve.

High pass

For a high-pass filter section, Z is a capacitor and Z' is an inductor:
and
where
The transfer function of the section is given by
The 3 dB point occurs when ωC = so the 3 dB cut-off frequency is given by
In the stop band,
falling at 6 dB/8ve with decreasing frequency.

Band pass

For a band-pass filter section, Z is a series resonant circuit and Z' is a shunt resonant circuit;
and
The transfer function of the section is given by
The 3 dB point occurs when |1 − ω²LC| = ωCR₀ so the 3 dB cut-off frequencies are given by
from which the centre frequency, ω_m, and bandwidth, Δω, can be determined:
Note that this is different from the resonant frequency
the relationship between them being given by

Band stop

For a band-stop filter section, Z is a shunt resonant circuit and Z' is a series resonant circuit:
and
The transfer function and bandwidth can be found by analogy with the band-pass section.
And,

Practical sections

Zobel networks are rarely used for traditional frequency filtering. Other filter types are significantly more efficient for this purpose. Where Zobels come into their own is in frequency equalisation applications, particularly on transmission lines. The difficulty with transmission lines is that the impedance of the line varies in a complex way across the band and is tedious to measure. For most filter types, this variation in impedance will cause a significant difference in response to the theoretical, and is mathematically difficult to compensate for, even assuming that the impedance is known precisely. If Zobel networks are used however, it is only necessary to measure the line response into a fixed resistive load and then design an equaliser to compensate it. It is entirely unnecessary to know anything at all about the line impedance as the Zobel network will present exactly the same impedance to line as the measuring instruments. Its response will therefore be precisely as theoretically predicted. This is a tremendous advantage where high quality lines with flat frequency responses are desired.

Basic loss

For audio lines, it is invariably necessary to combine L/C filter components with resistive attenuator components in the same filter section. The reason for this is that the usual design strategy is to require the section to attenuate all frequencies down to the level of the frequency in the passband with the lowest level. Without the resistor components, the filter, at least in theory, would increase attenuation without limit. The attenuation in the stop band of the filter is referred to as the "basic loss" of the section. In other words, the flat part of the band is attenuated by the basic loss down to the level of the falling part of the band which it is desired to equalise. The following discussion of practical sections relates in particular to audio transmission lines.

6 dB/octave roll-off

The most significant effect that needs to be compensated for is that at some cut-off frequency the line response starts to roll-off like a simple low-pass filter. The effective bandwidth of the line can be increased with a section that is a high-pass filter matching this roll-off, combined with an attenuator. In the flat part of the pass-band only the attenuator part of the filter section is significant. This is set at an attenuation equal to the level of the highest frequency of interest. All frequencies up to this point will then be equalised flat to an attenuated level. Above this point, the output of the filter will again start to roll-off.

Mismatched lines

Quite commonly in telecomms networks, a circuit is made up of two sections of line which do not have the same characteristic impedance. For instance 150 Ω and 300 Ω. One effect of this is that the roll-off can start at 6 dB/octave at an initial cut-off frequency, but then at can become suddenly steeper. This situation then requires two high-pass sections to compensate each operating at a different.

Bumps and dips

Bumps and dips in the passband can be compensated for with band-stop and band-pass sections respectively. Again, an attenuator element is also required, but usually rather smaller than that required for the roll-off. These anomalies in the pass-band can be caused by mismatched line segments as described above. Dips can also be caused by ground temperature variations.

Transformer roll-off

Occasionally, a low-pass section is included to compensate for excessive line transformer roll-off at the low frequency end. However, this effect is usually very small compared to the other effects noted above.
Low frequency sections will usually have inductors of high values. Such inductors have many turns and consequently tend to have significant resistance. In order to keep the section constant resistance at the input, the dual branch of the bridge T must contain a dual of the stray resistance, that is, a resistor in parallel with the capacitor. Even with the compensation, the stray resistance still has the effect of inserting attenuation at low frequencies. This in turn has the effect of slightly reducing the amount of LF lift the section would otherwise have produced. The basic loss of the section can be increased by the same amount as the stray resistance is inserting and this will return the LF lift achieved to that designed for.
Compensation of inductor resistance is not such an issue at high frequencies were the inductors will tend to be smaller. In any case, for a high-pass section the inductor is in series with the basic loss resistor and the stray resistance can merely be subtracted from that resistor. On the other hand, the compensation technique may be required for resonant sections, especially a high Q resonator being used to lift a very narrow band. For these sections the value of inductors can also be large.