Zobel network

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Linear analog electronic filters
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For the wave filter invented by Zobel and sometimes named after him see m-derived filters

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 them in 1923.^[1] 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 they 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 quality line from one originally intended for ordinary telephone use. However, as analogue technology has given way to digital 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. This term is often used to mean a Zobel network, sometimes incorrectly when the circuit implementation is, in fact, something other than a bridged T.

Parts of this article or section rely on the reader's knowledge of the complex impedance representation of capacitors and inductors and on knowledge of the frequency domain representation of signals.

[edit] 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;

$\frac{Z}{Z_0}=\frac{Z_0}{Z'}$

If this is expressed in terms of a normalised $Z 0 = 1$ as is conventionally done in filter tables, then the balance condition is simply;

$Z=\frac{1}{Z'}$

In other words, $Z'$ is simply the inverse, or dual impedance of $Z$ .

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. However, its value is often chosen to be $Z 0$ for reasons which will become clear in the discussion of bridged T circuits.

[edit] Input impedance

The input impedance is given by;

$\frac{1}{Z_{in}}=\frac{1}{Z_0+Z'}+\frac{1}{Z+Z_0}$

Substituting the balance condition;

$Z'=\frac{Z_0^2}{Z}$

Yields;

$Z_{in}=Z_0\,\!$

The input impedance can be designed to be purely resistive by setting;

$Z_0=R_0\,\!$

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.

[edit] Transfer function

If the $Z 0$ in the bottom right of the bridge is taken to be the output load then a transfer function of $V i n / V o$ 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 $R B$ . 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 (and hence the input terminal voltage) but not the transfer function. The transfer function can now easily be seen to be;

$A(\omega)=\frac{Z_0}{Z+Z_0}$

[edit] 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;

$Z_B=Z_0\,\!$

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 $Z B = Z 0$ is to make the filter section symmetrical. This has the advantage that it will then present the same impedance, $Z 0$ , at both the input and the output port.

[edit] 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.

[edit] Attenuator

Z and Z' for a Zobel attenuator

For an attenuator section, $Z$ is simply;

$Z=R\,\!$ and,

$Z'=R'=\frac{R_0^2}{R}$

The attenuation of the section is given by;

$L=20\,Log(\frac{R}{R_0}+1) \quad dB$

[edit] Low pass

Z and Z' for a Zobel low-pass filter section

For a low-pass filter section, $Z$ is an inductor and $Z'$ is a capacitor;

$Z=i \omega L\,\!$ and,

$Z'=\frac{1}{i \omega C'}$ where $\quad$ $C'=\frac{L}{R_0^2}$

The transfer function of the section is given by;

$A(\omega)=\frac{R_0}{i \omega L+R_0}$

The 3dB point occurs when $ω L = R 0$ so the 3dB cut-off frequency is given by,

$\omega_c = \frac{R_0}{L}$

Where $ω$ is in the stop band well above $ω c$ ,

$A(\omega) \approx \frac{R_0}{i \omega L}$

it can be seen from this that $A (ω)$ is falling away in the stop band at the classic 6dB/8ve (or 20dB/decade).

[edit] High pass

Z and Z' for a Zobel high-pass filter section

For a high-pass filter section, $Z$ is a capacitor and $Z'$ is an inductor;

$Z=\frac{1}{i \omega C}$ and,

$Z'=i \omega L'\,\!$ where $\quad$ $L'=CR_0^2\,\!$

The transfer function of the section is given by;

$A(\omega)=\frac{i \omega C R_0}{1 + i \omega C R_0}$

The 3dB point occurs when $\omega C = \frac{1}{R_0}$ so the 3dB cut-off frequency is given by,

$\omega_c = \frac{1}{C R_0}$

In the stop band,

$A(\omega) \approx i \omega C R_0\,\!$

falling at 6dB/8ve with decreasing frequency.

[edit] Band pass

Z and Z' for a Zobel band-pass filter section

For a band-pass filter section, $Z$ is a series resonant circuit and $Z'$ is a shunt resonant circuit;

$Z=i \omega L + \frac{1}{i \omega C}$ and,

$Y'=\frac{1}{Z'}= i \omega C' + \frac{1}{i \omega L'}$

The transfer function of the section is given by;

$A(\omega)=\frac{i \omega C R_0}{1 + i \omega C R_0 - \omega^2 L C}$

The 3dB point occurs when $| 1 - ω 2 L C | = ω C R 0$ so the 3dB cut-off frequencies are given by,

$\omega_c = \frac{\pm R_0 C + \sqrt{R_0^2C^2+4 L C}}{2 L C}$

From which the centre frequency, $ω m$ , and bandwidth, $Δω$ , can be determined;

$\Delta \omega = \frac{R_0}{L}$

$\omega_m = \sqrt{\frac{R_0^2}{4 L^2}+\frac{1}{LC}}$ note that this is different from the resonant frequency $\omega_0 = \sqrt{\frac{1}{L C}}$ , the relationship between them being given by;

$\omega_m^2 = \left(\frac{\Delta \omega}{2}\right)^2 +\omega_0^2$

[edit] Band stop

Z and Z' for a Zobel band-stop filter section

For a band-stop filter section, $Z$ is a shunt resonant circuit and $Z'$ is a series resonant circuit;

$Y=\frac{1}{Z}= i \omega C + \frac{1}{i \omega L}$ and,

$Z'=i \omega L' + \frac{1}{i \omega C'}$

The transfer function and bandwidth can be found by analogy with the band-pass section.

$\Delta\omega = \frac{1}{C R_0}$

And,

$\omega_m = \sqrt{\left(\frac{1}{2 R_0 C}\right)^2+\frac{1}{LC}}$

[edit] 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.

[edit] Basic loss

A practical high pass section incorporating basic loss used to correct high end roll-off

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 (that is, the limiting maximum attenuation) 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.

[edit] 6dB/octave roll-off

High-pass Zobel network response for various basic losses. Normalised to

R 0 = 1

and

ω c = 1

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.

[edit] 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 150Ω and 300Ω. One effect of this is that the roll-off can start at 6dB/octave at an initial cut-off frequency $f c 1$ , but then at $f c 2$ can become suddenly steeper. This situation then requires (at least) two high-pass sections to compensate each operating at a different $f c$ .

[edit] 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.

[edit] 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.

[edit] Temperature compensation

An adjustable attenuation high-pass section can be used to compensate for changes in ground temperature. Ground temperature is very slow varying in comparison to surface temperature. Adjustments are usually only required 2-4 times per year for audio applications.

[edit] Typical filter chain

An example of a typical chain of Zobel networks being used for line equalisation

A typical complete filter will consist of a number of Zobel sections for roll-off, frequency dips and temperature followed by a flat attenuator section to bring the level down to a standard attenuation. This is followed by a fixed gain amplifier to bring the signal back up to a useable level, typically 0dBu. The gain of the amplifier is usually no more than 45dB maximum. Any more and the amplification of line noise will tend to cancel out the quality benefits of improved bandwidth. This limit on amplification essentially limits how much the bandwidth can be increased by these techniques. It should also be noted that no one part of the incoming signal band will be amplified by the full 45dB. The 45dB is made up of the line loss in the flat part of its spectrum plus the basic loss of each section. In general, each section will be minimum loss at a different frequency band, hence the amplification in that band will be limited to the basic loss of just that one filter section, assuming insignificant overlap. A typical choice for R₀ is 600 Ω.

[edit] Other section implementations

Besides the Bridged T, there are a number of other possible section forms that can be used.

[edit] Half-sections

Open circuit derived Zobel half section for a high-pass section with basic loss

Short circuit derived Zobel half section for a high-pass section with basic loss

As mentioned above, $Z B$ can be set to any desired impedance without affecting the input impedance. In particular, setting it as either an open circuit or a short circuit results in a simplified section circuit. These are shown above for the case of a high pass section with basic loss.

The input port still presents an impedance of $R 0$ (provided that the output is terminated in $R 0$ ) but the output port no longer presents a constant impedance. Both half sections are capable of being reversed so that $R 0$ is then presented at the output and the variable impedance is presented at the input.

To retain the benefit of Zobel networks constant impedance, the variable impedance port must not face the line impedance. Nor should it face the variable impedance port of another half section. Facing the amplifier is acceptable since the input impedance of the amplifier is normally arranged to be $R 0$ . In other words, variable impedance must not face variable impedance. However, if a sufficiently large attenuator is placed between the two facing variable impedances, this will have the effect of masking the variable impedance. A high value attenuator will have an input impedance $\approx R_0$ no matter what the terminating impedance on the other side. In the example practical chain shown above there is a 22dB attenuator required in the chain. This does not need to be at the end of the chain, it can be placed anywhere desired and used to mask two mismatched impedances. It can also be split into two or more parts and used for masking more than one mismatch.

A balanced bridged T high-pass full section with basic loss

[edit] Balanced bridged T

The Zobel networks described here can be used to equalise land lines composed of twisted pair or star quad cables. The balanced circuit nature of these lines delivers a good common mode rejection ratio (CMRR). To maintain the CMRR, circuits connected to the line should maintain the balance. For this reason, balanced versions of Zobel networks are sometimes required. This is achieved by halving the impedance of the series components and then putting identical components in the "earthy" leg of the circuit.

A balanced Zobel high-pass short-circuit derived half section with basic loss

[edit] Balanced half-section

A balanced half section is achieved in the same way as a balanced full bridged T section.

[edit] Zobel networks and loudspeaker drivers

Zobel network correcting loudspeaker impedance

See also Boucherot cell

Zobel networks can be used to make the impedance a loudspeaker presents to its amplifier output appear as a steady resistance. This is beneficial to the amplifier performance. The impedance of a loudspeaker is partly resistive. The resistance representing the energy transferred from the amplifier to the sound output plus some heating losses in the loudspeaker. However, the speaker also possesses inductance due to the windings of its coil. The impedance of the loudspeaker is thus typically modelled as a series resistor and inductor. A parallel circuit of a series resistor and capacitor of the correct values will form a Zobel bridge. It is obligatory to choose $R_B = \infin$ because the centre point between the inductor and resistor is inaccessible (and, in fact, fictitious - the resistor and inductor are distributed quantities as in a transmission line). The loudspeaker may be modelled more accurately by a more complex equivalent circuit. The compensating Zobel network will also become more complex to the same degree.^[2]

Note that the circuit will work just as well if the capacitor and resistor are interchanged. In this case the circuit is no longer a Zobel balanced bridge but clearly the impedance has not changed. The same circuit could have been arrived at by designing from Boucherot's minimising reactive power point of view. From this design approach there is no difference in the order of the capacitor and the resistor and Boucherot cell might be considered a more accurate description.

[edit] See also

[edit] References

^ Zobel, O. J.,Theory and Design of Uniform and Composite Electric Wave Filters, Bell Systems Technical Journal, Vol. 2 (1923), pp. 1-46.
^ Leach, W. M., Jr., Impedance Compensation Networks for the Lossy Voice-Coil Inductance of Loudspeaker Drivers, Georgia Institute of Technology, School of Electrical and Computer Engineering, J. Audio Eng. Soc., Vol. 52, No. 4, April 2004. Available on-line here[1]

Zobel, O. J., Distortion correction in electrical networks with constant resistance recurrent networks, Bell Systems Technical Journal, Vol. 7 (1928), p. 438.
Redifon Radio Diary, 1970, William Collins Sons & Co, 1969

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Zobel network

From Wikipedia, the free encyclopedia

Contents

[edit] Derivation

[edit] Input impedance

[edit] Transfer function

[edit] Bridged T implementation

[edit] Types of section

[edit] Attenuator

[edit] Low pass

[edit] High pass

[edit] Band pass

[edit] Band stop

[edit] Practical sections

[edit] Basic loss

[edit] 6dB/octave roll-off

[edit] Mismatched lines

[edit] Bumps and dips

[edit] Transformer roll-off

[edit] Temperature compensation

[edit] Typical filter chain

[edit] Other section implementations

[edit] Half-sections

[edit] Balanced bridged T

[edit] Balanced half-section

[edit] Zobel networks and loudspeaker drivers

[edit] See also

[edit] References

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