4237:' etc., must remain dual impedances as the component is adjusted, so both parts of the pair must be adjusted together. With the Zobel Bode equaliser, this is a simple matter of ganging two pots together - a component configuration available off-the-shelf. Ganging together a variable capacitor and inductor is not, however, a very practical solution. These equalisers tend to be "hand built", one solution being to select the capacitors on test and fit fixed values according to the measurements and then adjust the inductors until the required match is achieved. The furthest element of the ladder from the driving point is equalising the lowest frequency of interest. This is adjusted first as it will also have an effect on higher frequencies and from there progressively higher frequencies are adjusted working along the ladder towards the driving point.
3071:
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.
3841:
3622:
3426:
3418:
789:
3590:
3305:
3080:
3088:
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.
3368:
3097:
903:
321:
1003:
2736:
2017:
1611:
1205:
4213:
3346:
4152:
3954:
3672:
3329:
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.
3647:). The X–section is a kind of bridge circuit, but usually drawn as a lattice, hence the name. Its topology makes it intrinsically balanced but it is never used to implement the constant resistance filters of the kind described here because of the increased component count. The component count increase arises out of the transformation process rather than the balance. There is however, one common application for this topology, the
4193:
4052:
4060:
3062:
4008:
3354:
4205:
to correct an attenuation increasing with frequency and for this a Cauer ladder network consisting of series resistors and shunt capacitors is required. Optionally, there may be an inductor included in series with the first capacitor which increases the equalisation at the high end due to the steeper slope produced as resonance is approached. This may be required on longer lines. The shunt resistor R
283:
3859:
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
3799:
will have a variable (and probably unknown) impedance. Likewise, two half-sections cannot be connected directly to each other as these both will have variable impedances. However, if a sufficiently large attenuator is placed between the two variable impedances, this will have the effect of masking the effect. A high value attenuator will have an input impedance
827:. None of the current flowing through the LHS (left-hand side) 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;
4020:
terminated in a variable resistor can be used to produce a variable impedance at the input terminals of the network. A useful property of this network is that the input impedance can be made to vary from a capacitive impedance through a purely resistive impedance to an inductive impedance all by adjusting the single load
4160:
noting that the transfer function of the Bode equaliser is identical to the transfer function of the general form of Zobel equaliser. Note that the dual of a constant resistance bridge T network is the identical network. The dual of a Bode network is therefore the same network except for the load resistance R
3618:(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 return leg of the circuit.
4204:
is used as the Z impedance of a Zobel constant resistance network. Just as the input impedance of a Bode network is used as the Z impedance of a Zobel network to form a Zobel Bode equaliser, so the input impedance of a Cauer network is used to make a Zobel Cauer equaliser. The equaliser is required
4019:
condition. It does not however meet the constant resistance condition, that is, the bridge is not in balance. Any impedance network, Z, can be used in a Bode network, just as with a Zobel network, but the high pass section shown for correcting high-end frequencies is the most common. A Bode network
3858:
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 represents the energy transferred from the amplifier to
3831:
no matter what the terminating impedance on the other side. In the example practical chain shown above there is a 22 dB 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
3634:
A C–section is a balanced version of an L–section. The balance is achieved in the same way as a balanced full bridged T section by placing half of the series impedance in, what was, the common conductor. C–sections, like the L–section from which they are derived, can come in
3998:
might be used. Another driver for this approach is the fact that a video signal occupies a large number of octaves, around 20 or so. If equalised with simple basic sections, a large number of filter sections would be required. Simple sections are designed, typically, to equalise a range of one or
3948:
over a wide band (usually nominally 75 Ω). On the other hand, audio cable may be nominally 600 Ω (300 Ω and 150 Ω are also standard values), but it will only actually measure this value at 800 Hz. At a lower frequencies it will be much higher and at higher frequencies will
3993:
This more predictable response of video allows a different design approach. The video equaliser is built as a single bridged T section but with a rather more complex network for Z. For short lines, or for a trimming equaliser, a Bode filter topology might be used. For longer lines a network with
3070:
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
3988:
These characteristics result in a smoother, more well behaved response for video lines with none of the nasty discontinuities typically found with audio lines. These discontinuities in the frequency response are often caused by the habit of the telecom companies of forming a connection by joining
3798:
It must be emphasised that these half sections are far from being constant resistance. They have the same transfer function as a constant resistance network, but only when correctly terminated. An equaliser will not give good results if a half-section is positioned facing the line since the line
3328:
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
2583:
3556:
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 L-section. Facing the amplifier is acceptable since the input impedance of the amplifier is normally arranged to be
3332:
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
3295:
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
3181:
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
4159:
The Bode equaliser can be converted into a constant resistance filter by using the entire Bode network as the Z branch of a Zobel network, resulting in a rather complex network of bridge T networks embedded in a larger bridge T. It can be seen that this results in the same transfer function by
4071:, is in series with the load. Since the impedance of the Bode network can be either capacitive or inductive depending on the position of the adjustment potentiometer, the response may be a boost or a cut to the band of frequencies it is acting on. The transfer function of this arrangement is:
3384:. 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. No one part of the incoming signal band will be amplified by the full
3375:
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 usable level,
3087:
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
286:
BBC engineers equalising audio landlines circa 1959. The boxes with two large black dials towards the top of the equipment racks are adjustable Zobel equalisers. They are used both for temporary outside broadcast lines and for checking the engineer's calculations prior to building permanent
3462:
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, called L–sections. These are shown above for the case of a high pass section with basic loss.
3392:
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
3659:
In respect of constant resistance filters, the term half section has a somewhat different meaning to other kinds of image filter. Generally, a half section is formed by cutting through the midpoint of the series impedance and shunt admittance of a full section of a
3341:
An adjustable attenuation high-pass filter 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.
3664:. It is literally half a section. Here, however, there is a somewhat different definition. A half section is either the series impedance (series half-section) or shunt admittance (shunt half-section) that, when connected between source and load impedances of R
3907:
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
3921:
Zobel networks can be used for the equalisation of video lines as well as audio lines. There is, however, a noticeably different approach taken with the two types of signal. The difference in the cable characteristics can be summarised a s follows;
993:
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.
252:
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.
4220:
The dual of a RC Cauer network is a LR Cauer network which is required for the Z' impedance as shown in the example. Adjustment is a bit problematic with this equaliser. In order to maintain the constant resistance, the pairs of components
2432:
3357:
The internal components of a temperature equaliser. The inductor and capacitor on the right set the frequency at which the equaliser starts to operate, the banks of selectable resistors on the left set the basic loss and hence amount of
2725:
3051:
621:
2185:
3065:
A transparent mask used to aid design of Zobel networks. The mask is laid over a plot of the line response and the component values corresponding to the closest fitting curve can be chosen. This particular mask is for high-pass
3182:
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.
2456:
2294:
3668:, will result in the same transfer function as some arbitrary constant resistance circuit. The purpose of using half sections is that the same functionality is achieved with a drastically reduced component count.
1194:
3324:
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.
1869:
2899:
3784:
which is precisely the same as a bridged T section with series element Z. The series half-section is thus a series impedance of 2Z. By corresponding reasoning, the shunt half-section is a shunt impedance of
4216:
Cauer Zobel bridge T implemented video equaliser. The impedance Z of this example consists of a three section ladder and is suitable for the equalisation of short lines (between nearby buildings for instance)
2825:
3779:
3912:
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.
509:
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
817:
can be calculated for the section. Only the RHS (right-hand side) branch needs to be considered in this calculation. The reason for this can be seen by considering that there is no current flow through
3333:
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.
1585:
1451:
388:
3695:
at the input of the constant resistance circuit. If now the constant resistance circuit is replaced by an impedance of 2Z, as in the diagram above, it can be seen by simple symmetry that the voltage V
4146:
1117:
237:. 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
1319:
2636:
2461:
892:
2953:
2093:
2003:
1946:
3894:
3172:
1511:
1375:
677:
1725:
3829:
3981:
3651:, which is also constant resistance and also invented by Zobel. This circuit differs from those described here in that the bridge circuit is not generally in the balanced condition.
3135:
1675:
1779:
439:
469:
3256:
3224:
723:
3584:
3551:
3522:
3493:
3460:
3285:
1262:
953:
771:
495:
3904:). 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.
3349:
A collection of various designs of temperature compensation equaliser. Some can be adjusted with pluggable links, others require soldering. Adjustment is not very frequent.
2319:
1050:
910:
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;
197:
3524:) but the output port no longer presents a constant impedance. Both the open-circuit and the short-circuit L–sections are capable of being reversed so that
2647:
2964:
2578:{\displaystyle {\begin{aligned}\Delta \omega &={\frac {R_{0}}{L}}\\\omega _{m}&={\sqrt {{\frac {R_{0}^{2}}{4L^{2}}}+{\frac {1}{LC}}}}\end{aligned}}}
531:
4315:, Georgia Institute of Technology, School of Electrical and Computer Engineering, J. Audio Eng. Soc., Vol. 52, No. 4, April 2004. Available on-line here
3635:
both open-circuit and short circuit varieties. The same restrictions apply to C–sections regarding impedance terminations as to L–sections.
2196:
3989:
two shorter lines of differing characteristic impedance. Video lines on the other hand tend to roll off smoothly with frequency in a predictable way.
264:. In this case, only half the network is implemented as fixed components, the other half being the real and imaginary components of the loudspeaker
3194:. 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
1128:
190:
2104:
1790:
3258:
can become suddenly steeper. This situation then requires (at least) two high-pass sections to compensate each operating at a different
788:
2764:
3705:
4381:
3896:
because the centre point between the inductor and resistor is inaccessible (and, in fact, fictitious - the resistor and inductor are
328:
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;
183:
1533:
1386:
4077:
279:. This term is often used to mean a Zobel network, sometimes incorrectly when the circuit implementation is not a bridged T.
4067:
The Bode network is used in an equaliser by connecting the whole network such that the input impedance of the Bode network, Z
334:
229:, who published a much-referenced paper on image filters in 1923. The distinguishing feature of Zobel networks is that the
3177:
The most significant effect that needs to be compensated for is that at some cut-off frequency the line response starts to
2836:
3675:
A general Zobel series half section showing the equality of transfer function to an equivalent constant resistance section
4554:
2594:
833:
2910:
2045:
1957:
1899:
3862:
3140:
1469:
4549:
1061:
3644:
978:
is to make the filter section symmetrical. This has the advantage that it will then present the same impedance,
3802:
1273:
1330:
632:
3960:
3190:
Quite commonly in telecomms networks, a circuit is made up of two sections of line which do not have the same
3103:
1639:
320:
4374:
3897:
902:
4471:
3995:
108:
3615:
3397:
is 600 Ω. A good quality transformer (usually essential, but not shown on the diagram), known as a
3229:
3197:
958:
then the circuit to the right results. This is referred to as a bridged T circuit because the impedance
688:
242:
3560:
3527:
3498:
3469:
3436:
3261:
272:
correction circuits used in electrical power distribution, hence the association with
Boucherot's name.
4246:
1736:
1686:
406:
214:
1233:
916:
734:
447:
125:
78:
73:
41:
3840:
3586:
within acceptable tolerances. In other words, variable impedance must not face variable impedance.
478:
4544:
4539:
4367:
3945:
3191:
3699:
will appear halfway along the impedance 2Z. The output of this circuit can now be calculated as,
3648:
400: = 1 as is conventionally done in filter tables, then the balance condition is simply;
119:
2427:{\displaystyle \omega _{c}={\frac {1}{2LC}}\left(\pm R_{0}C+{\sqrt {R_{0}^{2}C^{2}+4LC}}\right)}
3931:
131:
91:
4316:
1027:
803:
in the bottom right of the bridge is taken to be the output load then a transfer function of
3934:
for the filters whereas audio commonly uses twisted pair which requires a balanced topology.
3409:
Besides the
Bridged T, there are a number of other possible section forms that can be used.
2720:{\displaystyle \omega _{m}^{2}=\left({\frac {\Delta \omega }{2}}\right)^{2}+\omega _{0}^{2}}
4492:
4015:
A Bode network, as with a Zobel network, is a symmetrical bridge T network which meets the
3046:{\displaystyle \omega _{m}={\sqrt {\left({\frac {1}{2R_{0}C}}\right)^{2}+{\frac {1}{LC}}}}}
294:
265:
4313:
Impedance
Compensation Networks for the Lossy Voice-Coil Inductance of Loudspeaker Drivers
3308:
Low frequency equaliser section with compensation for inductor resistance. The resistance
8:
4451:
3941:
specification which in turn results in a tighter dimensional specification for the cable.
3621:
4343:
Distortion correction in electrical circuits with constant resistance recurrent networks
3643:
It is possible to transform a bridged–T section into a
Lattice, or X–section (see
3425:
3083:
A practical high pass section incorporating basic loss used to correct high end roll-off
2904:
The transfer function and bandwidth can be found by analogy with the band-pass section.
517:
for reasons which will become clear in the discussion of bridged T circuits further on.
302:
4466:
4256:
3938:
3417:
298:
249:
136:
48:
3553:
is then presented at the output and the variable impedance is presented at the input.
4446:
4409:
4261:
4016:
3944:
The tighter specifications for video cable tends to produce a substantially constant
3901:
777:
616:{\displaystyle {\frac {1}{Z_{\text{in}}}}={\frac {1}{Z_{0}+Z'}}+{\frac {1}{Z+Z_{0}}}}
234:
222:
98:
30:
19:
3304:
2289:{\displaystyle A(\omega )={\frac {i\omega CR_{0}}{1+i\omega CR_{0}-\omega ^{2}LC}}}
306:
103:
53:
4497:
4425:
4251:
4196:
Cauer topology network to be used as the Z impedance of a Zobel network equaliser
3602:
The Zobel networks described here can be used to equalise land lines composed of
230:
218:
68:
58:
3371:
An example of a typical chain of Zobel networks being used for line equalisation
4487:
4461:
4456:
4390:
4266:
4201:
3927:
3909:
3851:
3661:
3589:
3398:
3079:
780:
in band and out of band no matter what complexity of filter section is chosen.
472:
293:
Parts of this article or section rely on the reader's knowledge of the complex
261:
1599:
4533:
4404:
4032:, is chosen to equal the nominal impedance so that in the special case when R
4021:
3832:
be split into two or more parts and used for masking more than one mismatch.
3611:
3429:
Short circuit derived Zobel L-section for a high-pass section with basic loss
63:
3421:
Open circuit derived Zobel L-section for a high-pass section with basic loss
3296:
described above. Dips can also be caused by ground temperature variations.
4513:
3603:
3367:
3096:
269:
245:. However, any impedance achievable with discrete components is possible.
3625:
A balanced Zobel high-pass short-circuit derived C-section with basic loss
3312:
represents the stray resistance of the non-ideal inductor. The resistance
4430:
3345:
3100:
High-pass Zobel network response for various basic losses. Normalised to
257:
169:
4212:
18:
For the wave filter invented by Zobel and sometimes named after him see
4151:
1189:{\displaystyle L=20\log \left({\frac {R}{R_{0}}}+1\right)\,{\text{dB}}}
1002:
164:
159:
154:
3953:
2735:
2016:
1610:
728:
The input impedance can be designed to be purely resistive by setting
3671:
3607:
2180:{\displaystyle Y'={\frac {1}{Z'}}=i\omega C'+{\frac {1}{i\omega L'}}}
1864:{\displaystyle A(\omega )={\frac {i\omega CR_{0}}{1+i\omega CR_{0}}}}
1204:
238:
226:
282:
4518:
4155:
Bode equaliser implemented as a Zobel constant resistance equaliser
3178:
4359:
4192:
962:
is seen to "bridge" across the T section. The purpose of setting
4051:
3061:
2820:{\displaystyle Y={\frac {1}{Z}}=i\omega C+{\frac {1}{i\omega L}}}
4287:
Theory and Design of
Uniform and Composite Electric Wave Filters
4059:
3774:{\displaystyle {\frac {V_{O}}{V_{in}}}={\frac {R_{0}}{Z+R_{0}}}}
4289:, Bell System Technical Journal, Vol. 2 (1923), pp. 1–46.
4209:
provides the basic loss of the Zobel network in the usual way.
4007:
3353:
3401:, is at the beginning of the chain where the line terminates.
4330:, US patent 2 002 216, filed 7 June 1933, issued 21 May 1935.
1598:) is falling away in the stop band at the classic 6 dB/
1580:{\displaystyle A(\omega )\approx {\frac {R_{0}}{i\omega L}}}
275:
A common circuit form of Zobel networks is in the form of a
3593:
A balanced bridged T high-pass full section with basic loss
1446:{\displaystyle A(\omega )={\frac {R_{0}}{i\omega L+R_{0}}}}
4176:' must be ganged, or otherwise kept in step such that as R
792:
Equivalent circuit of a Zobel network for calculating gain
3835:
776:
The input impedance will then be real and independent of
4141:{\displaystyle A(\omega )={\frac {R_{0}}{Z_{in}+R_{0}}}}
2588:
Note that this is different from the resonant frequency
4345:, Bell System Technical Journal, Vol. 7 (1928), p. 438.
4302:, The English Universities Press Ltd, 1961, pp117-121.
3964:
3866:
3806:
3564:
3531:
3502:
3473:
3440:
3265:
3233:
3201:
3144:
3107:
482:
451:
383:{\displaystyle {\frac {Z}{Z_{0}}}={\frac {Z_{0}}{Z'}}}
4080:
3963:
3865:
3805:
3708:
3563:
3530:
3501:
3472:
3439:
3264:
3232:
3200:
3143:
3106:
2967:
2913:
2894:{\displaystyle Z'=i\omega L'+{\frac {1}{i\omega C'}}}
2839:
2767:
2650:
2597:
2459:
2322:
2299:
The 3 dB point occurs when |1 −
2199:
2107:
2048:
1960:
1902:
1793:
1739:
1689:
1642:
1536:
1472:
1389:
1333:
1276:
1236:
1131:
1064:
1030:
919:
836:
737:
691:
635:
534:
481:
450:
409:
337:
241:. For this reason, Zobel networks are also known as
2008:
falling at 6 dB/8ve with decreasing frequency.
2631:{\displaystyle \omega _{0}={\sqrt {\frac {1}{LC}}}}
887:{\displaystyle A(\omega )={\frac {Z_{0}}{Z+Z_{0}}}}
4168:', in the dual circuit. To adjust the equaliser R
4140:
3975:
3957:Response plot of a video line showing the typical
3888:
3823:
3773:
3578:
3545:
3516:
3487:
3454:
3279:
3250:
3218:
3166:
3129:
3045:
2948:{\displaystyle \Delta \omega ={\frac {1}{CR_{0}}}}
2947:
2893:
2819:
2719:
2630:
2577:
2426:
2313:so the 3 dB cut-off frequencies are given by
2288:
2179:
2088:{\displaystyle Z=i\omega L+{\frac {1}{i\omega C}}}
2087:
1997:
1940:
1863:
1773:
1719:
1669:
1579:
1505:
1445:
1369:
1313:
1256:
1188:
1111:
1044:
947:
886:
765:
717:
671:
615:
489:
463:
433:
382:
2190:The transfer function of the section is given by
1784:The transfer function of the section is given by
1769:
1715:
1380:The transfer function of the section is given by
1252:
1041:
944:
761:
4531:
3404:
3380:. The gain of the amplifier is usually at most
3316:is a real resistor calculated to compensate for
1998:{\displaystyle A(\omega )\approx i\omega CR_{0}}
256:When used to cancel out the reactive portion of
3679:If a constant resistance circuit has an input V
1941:{\displaystyle \omega _{c}={\frac {1}{CR_{0}}}}
1893:so the 3 dB cut-off frequency is given by
1463:so the 3 dB cut-off frequency is given by
3889:{\displaystyle \scriptstyle R_{B}\;=\;\infty }
3844:Zobel network correcting loudspeaker impedance
3466:The input port still presents an impedance of
3393:insignificant overlap. A typical choice for R
3167:{\displaystyle \scriptstyle \omega _{c}\;=\;1}
1506:{\displaystyle \omega _{c}={\frac {R_{0}}{L}}}
393:If this is expressed in terms of a normalised
4375:
4200:To equalise long video lines, a network with
3937:Video requires a wider bandwidth and tighter
2641:the relationship between them being given by
191:
4040:the network behaves as a Zobel network and Z
4011:Bode network for high frequency equalisation
1122:The attenuation of the section is given by;
1112:{\displaystyle Z'=R'={\frac {R_{0}^{2}}{R}}}
260:impedance, the design is sometimes called a
248:Zobel networks were formerly widely used in
233:is fixed in the design independently of the
3495:(provided that the output is terminated in
3336:
1215: ' for a Zobel low-pass filter section
897:
4382:
4368:
4300:An Introduction to Linear Network Analysis
3881:
3877:
3824:{\displaystyle \scriptstyle \approx R_{0}}
3159:
3155:
3122:
3118:
1314:{\displaystyle Z'={\frac {1}{i\omega C'}}}
198:
184:
4055:Bode network used in an equaliser circuit
3687:must have an open-circuit voltage of E=2V
1770:
1716:
1370:{\displaystyle C'={\frac {L}{R_{0}^{2}}}}
1253:
1180:
1040:
985:, at both the input and the output port.
943:
762:
672:{\displaystyle Z'={\frac {Z_{0}^{2}}{Z}}}
4211:
4191:
4150:
4058:
4050:
4006:
3976:{\displaystyle \scriptstyle {\sqrt {f}}}
3952:
3839:
3670:
3620:
3588:
3424:
3416:
3366:
3352:
3344:
3303:
3130:{\displaystyle \scriptstyle R_{0}\;=\;1}
3095:
3078:
3060:
2734:
2015:
1670:{\displaystyle Z={\frac {1}{i\omega C}}}
1609:
1203:
1001:
281:
221:design principle. They are named after
3362:
3299:
3091:
4532:
3836:Zobel networks and loudspeaker drivers
3683:, then a generator with an impedance R
3629:
3614:nature of these lines delivers a good
4363:
4351:, William Collins Sons & Co, 1969
4063:Bode trimming equaliser response plot
3597:
3056:
2746:for a Zobel band-stop filter section
2027:for a Zobel band-pass filter section
1621:for a Zobel high-pass filter section
783:
626:Substituting the balance condition,
268:. This network is more akin to the
4389:
3916:
3251:{\displaystyle \scriptstyle f_{c2}}
3219:{\displaystyle \scriptstyle f_{c1}}
3185:
988:
718:{\displaystyle Z_{\text{in}}=Z_{0}}
13:
4187:
3882:
3579:{\displaystyle \scriptstyle R_{0}}
3546:{\displaystyle \scriptstyle R_{0}}
3517:{\displaystyle \scriptstyle R_{0}}
3488:{\displaystyle \scriptstyle R_{0}}
3455:{\displaystyle \scriptstyle Z_{B}}
3280:{\displaystyle \scriptstyle f_{c}}
2914:
2677:
2464:
901:
787:
520:
324:Zobel network as a balanced bridge
319:
14:
4566:
4002:
3290:
2437:from which the centre frequency,
2035:is a series resonant circuit and
1774:{\displaystyle L'=CR_{0}^{2}\!\,}
1720:{\displaystyle Z'=i\omega L'\!\,}
434:{\displaystyle Z={\frac {1}{Z'}}}
4184:' will decrease and vice versa.
3654:
2754:is a shunt resonant circuit and
2750:For a band-stop filter section,
2031:For a band-pass filter section,
1874:The 3 dB point occurs when
1625:For a high-pass filter section,
1456:The 3 dB point occurs when
525:The input impedance is given by
1520:is in the stop band well above
1257:{\displaystyle Z=i\omega L\!\,}
1219:For a low-pass filter section,
948:{\displaystyle Z_{B}=Z_{0}\,\!}
766:{\displaystyle Z_{0}=R_{0}\!\,}
464:{\displaystyle \scriptstyle Z'}
4320:
4305:
4292:
4279:
4164:, which must be the inverse, R
4090:
4084:
3795:Z' (or twice the admittance).
2758:is a series resonant circuit:
2209:
2203:
1970:
1964:
1803:
1797:
1590:it can be seen from this that
1546:
1540:
1399:
1393:
906:Zobel bridged T implementation
846:
840:
490:{\displaystyle \scriptstyle Z}
1:
4272:
3930:which requires an unbalanced
3412:
3405:Other section implementations
3074:
2039:is a shunt resonant circuit;
997:
315:
74:Optimum "L" (Legendre) filter
4472:Transformer ratio arm bridge
4419:Resistance–Capacitance
3645:Bartlett's bisection theorem
3638:
2730:
2011:
1605:
243:constant resistance networks
7:
4240:
4028:. The bridging resistor, R
3949:be lower and more reactive.
3616:common mode rejection ratio
1199:
1017:For an attenuator section,
10:
4571:
4555:Electronic filter topology
4247:Electronic filter topology
471:is simply the inverse, or
4506:
4480:
4439:
4418:
4397:
4349:Redifon Radio Diary, 1970
309:representation of signals
126:Bridged T delay equaliser
42:Network synthesis filters
3946:characteristic impedance
3337:Temperature compensation
3192:characteristic impedance
1602:(or 20 dB/decade).
1227: ' is a capacitor;
898:Bridged T implementation
305:and on knowledge of the
4550:Image impedance filters
3649:lattice phase equaliser
1045:{\displaystyle Z=R\,\!}
500:The bridging impedance
92:Image impedance filters
59:Elliptic (Cauer) filter
4217:
4197:
4156:
4142:
4064:
4056:
4012:
3984:
3977:
3890:
3845:
3825:
3775:
3676:
3626:
3594:
3580:
3547:
3518:
3489:
3456:
3430:
3422:
3372:
3359:
3350:
3321:
3281:
3252:
3220:
3174:
3168:
3131:
3084:
3067:
3047:
2949:
2895:
2821:
2747:
2721:
2632:
2579:
2428:
2290:
2181:
2089:
2028:
1999:
1942:
1865:
1775:
1721:
1671:
1622:
1581:
1507:
1447:
1371:
1315:
1258:
1216:
1190:
1113:
1046:
1014:
1013:for a Zobel attenuator
949:
907:
888:
793:
767:
719:
673:
617:
491:
465:
435:
384:
325:
288:
132:Composite image filter
4215:
4195:
4154:
4143:
4062:
4054:
4010:
3996:Cauer filter topology
3978:
3956:
3891:
3843:
3826:
3776:
3691:in order to produce V
3674:
3624:
3592:
3581:
3548:
3519:
3490:
3457:
3428:
3420:
3370:
3356:
3348:
3307:
3282:
3253:
3221:
3169:
3132:
3099:
3082:
3064:
3048:
2950:
2896:
2822:
2738:
2722:
2633:
2580:
2450:, can be determined:
2429:
2291:
2182:
2090:
2019:
2000:
1943:
1866:
1776:
1722:
1672:
1613:
1582:
1508:
1448:
1372:
1316:
1259:
1207:
1191:
1114:
1047:
1005:
950:
905:
889:
791:
768:
720:
674:
618:
492:
466:
436:
385:
323:
285:
217:section based on the
109:General image filters
79:Linkwitz–Riley filter
4078:
3961:
3926:Video commonly uses
3863:
3803:
3706:
3561:
3528:
3499:
3470:
3437:
3433:As mentioned above,
3363:Typical filter chain
3300:Transformer roll-off
3262:
3230:
3198:
3141:
3104:
3092:6 dB/octave roll-off
2965:
2911:
2837:
2765:
2648:
2595:
2457:
2320:
2197:
2105:
2046:
1958:
1900:
1791:
1737:
1687:
1640:
1534:
1470:
1387:
1331:
1274:
1234:
1129:
1062:
1028:
917:
834:
735:
689:
633:
532:
479:
448:
407:
335:
4452:Carey Foster bridge
4311:Leach, W. M., Jr.,
3900:quantities as in a
3630:Balanced C-sections
2716:
2665:
2533:
2394:
1768:
1629:is a capacitor and
1364:
1223:is an inductor and
1103:
663:
116:(constant R) filter
4467:Murray loop bridge
4326:Bode, Hendrik W.,
4257:Constant k filters
4218:
4198:
4157:
4138:
4065:
4057:
4044:is also equal to R
4013:
3985:
3973:
3972:
3939:differential phase
3886:
3885:
3846:
3821:
3820:
3771:
3677:
3627:
3598:Balanced bridged T
3595:
3576:
3575:
3543:
3542:
3514:
3513:
3485:
3484:
3452:
3451:
3431:
3423:
3373:
3360:
3351:
3322:
3277:
3276:
3248:
3247:
3216:
3215:
3175:
3164:
3163:
3127:
3126:
3085:
3068:
3057:Practical sections
3043:
2945:
2891:
2817:
2748:
2717:
2702:
2651:
2628:
2575:
2573:
2519:
2446:, and bandwidth, Δ
2424:
2380:
2286:
2177:
2085:
2029:
1995:
1951:In the stop band,
1938:
1861:
1771:
1754:
1717:
1667:
1623:
1577:
1503:
1443:
1367:
1350:
1311:
1254:
1217:
1186:
1109:
1089:
1042:
1015:
945:
908:
884:
794:
763:
715:
669:
649:
613:
487:
486:
461:
460:
431:
380:
326:
297:representation of
289:
250:telecommunications
49:Butterworth filter
33:electronic filters
4527:
4526:
4493:Anderson's bridge
4447:Bridged T circuit
4410:Wheatstone bridge
4262:m-derived filters
4136:
3970:
3902:transmission line
3769:
3734:
3041:
3039:
3011:
2943:
2889:
2815:
2782:
2687:
2626:
2625:
2569:
2567:
2549:
2492:
2417:
2352:
2284:
2175:
2132:
2083:
1936:
1859:
1665:
1575:
1501:
1441:
1365:
1309:
1184:
1167:
1107:
882:
784:Transfer function
699:
667:
611:
583:
550:
547:
429:
378:
353:
277:bridged T network
235:transfer function
208:
207:
99:Constant k filter
20:m-derived filters
4562:
4384:
4377:
4370:
4361:
4360:
4331:
4324:
4318:
4309:
4303:
4296:
4290:
4283:
4147:
4145:
4144:
4139:
4137:
4135:
4134:
4133:
4121:
4120:
4107:
4106:
4097:
3982:
3980:
3979:
3974:
3971:
3966:
3917:Video equalisers
3895:
3893:
3892:
3887:
3876:
3875:
3830:
3828:
3827:
3822:
3819:
3818:
3794:
3793:
3789:
3780:
3778:
3777:
3772:
3770:
3768:
3767:
3766:
3750:
3749:
3740:
3735:
3733:
3732:
3720:
3719:
3710:
3612:balanced circuit
3585:
3583:
3582:
3577:
3574:
3573:
3552:
3550:
3549:
3544:
3541:
3540:
3523:
3521:
3520:
3515:
3512:
3511:
3494:
3492:
3491:
3486:
3483:
3482:
3461:
3459:
3458:
3453:
3450:
3449:
3391:
3387:
3383:
3379:
3286:
3284:
3283:
3278:
3275:
3274:
3257:
3255:
3254:
3249:
3246:
3245:
3225:
3223:
3222:
3217:
3214:
3213:
3186:Mismatched lines
3173:
3171:
3170:
3165:
3154:
3153:
3136:
3134:
3133:
3128:
3117:
3116:
3052:
3050:
3049:
3044:
3042:
3040:
3038:
3027:
3022:
3021:
3016:
3012:
3010:
3006:
3005:
2989:
2982:
2977:
2976:
2954:
2952:
2951:
2946:
2944:
2942:
2941:
2940:
2924:
2900:
2898:
2897:
2892:
2890:
2888:
2887:
2869:
2864:
2847:
2826:
2824:
2823:
2818:
2816:
2814:
2800:
2783:
2775:
2726:
2724:
2723:
2718:
2715:
2710:
2698:
2697:
2692:
2688:
2683:
2675:
2664:
2659:
2637:
2635:
2634:
2629:
2627:
2624:
2613:
2612:
2607:
2606:
2584:
2582:
2581:
2576:
2574:
2570:
2568:
2566:
2555:
2550:
2548:
2547:
2546:
2532:
2527:
2518:
2516:
2507:
2506:
2493:
2488:
2487:
2478:
2433:
2431:
2430:
2425:
2423:
2419:
2418:
2404:
2403:
2393:
2388:
2379:
2371:
2370:
2353:
2351:
2337:
2332:
2331:
2295:
2293:
2292:
2287:
2285:
2283:
2276:
2275:
2263:
2262:
2237:
2236:
2235:
2216:
2186:
2184:
2183:
2178:
2176:
2174:
2173:
2155:
2150:
2133:
2131:
2120:
2115:
2094:
2092:
2091:
2086:
2084:
2082:
2068:
2004:
2002:
2001:
1996:
1994:
1993:
1947:
1945:
1944:
1939:
1937:
1935:
1934:
1933:
1917:
1912:
1911:
1892:
1891:
1882:
1870:
1868:
1867:
1862:
1860:
1858:
1857:
1856:
1831:
1830:
1829:
1810:
1780:
1778:
1777:
1772:
1767:
1762:
1747:
1726:
1724:
1723:
1718:
1714:
1697:
1676:
1674:
1673:
1668:
1666:
1664:
1650:
1633:is an inductor:
1586:
1584:
1583:
1578:
1576:
1574:
1563:
1562:
1553:
1512:
1510:
1509:
1504:
1502:
1497:
1496:
1487:
1482:
1481:
1458:ωL = R
1452:
1450:
1449:
1444:
1442:
1440:
1439:
1438:
1416:
1415:
1406:
1376:
1374:
1373:
1368:
1366:
1363:
1358:
1346:
1341:
1320:
1318:
1317:
1312:
1310:
1308:
1307:
1289:
1284:
1263:
1261:
1260:
1255:
1195:
1193:
1192:
1187:
1185:
1182:
1179:
1175:
1168:
1166:
1165:
1153:
1118:
1116:
1115:
1110:
1108:
1102:
1097:
1088:
1083:
1072:
1051:
1049:
1048:
1043:
989:Types of section
954:
952:
951:
946:
942:
941:
929:
928:
893:
891:
890:
885:
883:
881:
880:
879:
863:
862:
853:
772:
770:
769:
764:
760:
759:
747:
746:
724:
722:
721:
716:
714:
713:
701:
700:
697:
678:
676:
675:
670:
668:
662:
657:
648:
643:
622:
620:
619:
614:
612:
610:
609:
608:
589:
584:
582:
581:
570:
569:
556:
551:
549:
548:
545:
536:
496:
494:
493:
488:
470:
468:
467:
462:
459:
440:
438:
437:
432:
430:
428:
417:
389:
387:
386:
381:
379:
377:
369:
368:
359:
354:
352:
351:
339:
307:frequency domain
200:
193:
186:
104:m-derived filter
54:Chebyshev filter
27:
26:
4570:
4569:
4565:
4564:
4563:
4561:
4560:
4559:
4545:Bridge circuits
4540:Analog circuits
4530:
4529:
4528:
4523:
4502:
4476:
4435:
4426:Schering bridge
4414:
4393:
4391:Bridge circuits
4388:
4356:
4335:
4334:
4325:
4321:
4310:
4306:
4297:
4293:
4284:
4280:
4275:
4252:Image impedance
4243:
4236:
4232:
4228:
4224:
4208:
4190:
4188:Cauer equaliser
4183:
4179:
4175:
4171:
4167:
4163:
4129:
4125:
4113:
4109:
4108:
4102:
4098:
4096:
4079:
4076:
4075:
4070:
4047:
4043:
4039:
4035:
4031:
4027:
4005:
3965:
3962:
3959:
3958:
3919:
3871:
3867:
3864:
3861:
3860:
3838:
3814:
3810:
3804:
3801:
3800:
3791:
3787:
3786:
3762:
3758:
3751:
3745:
3741:
3739:
3725:
3721:
3715:
3711:
3709:
3707:
3704:
3703:
3698:
3694:
3690:
3686:
3682:
3667:
3657:
3641:
3632:
3600:
3569:
3565:
3562:
3559:
3558:
3536:
3532:
3529:
3526:
3525:
3507:
3503:
3500:
3497:
3496:
3478:
3474:
3471:
3468:
3467:
3445:
3441:
3438:
3435:
3434:
3415:
3407:
3396:
3389:
3385:
3381:
3377:
3365:
3339:
3302:
3293:
3270:
3266:
3263:
3260:
3259:
3238:
3234:
3231:
3228:
3227:
3206:
3202:
3199:
3196:
3195:
3188:
3149:
3145:
3142:
3139:
3138:
3112:
3108:
3105:
3102:
3101:
3094:
3077:
3059:
3031:
3026:
3017:
3001:
2997:
2993:
2988:
2984:
2983:
2981:
2972:
2968:
2966:
2963:
2962:
2936:
2932:
2928:
2923:
2912:
2909:
2908:
2880:
2873:
2868:
2857:
2840:
2838:
2835:
2834:
2804:
2799:
2774:
2766:
2763:
2762:
2733:
2711:
2706:
2693:
2676:
2674:
2670:
2669:
2660:
2655:
2649:
2646:
2645:
2617:
2611:
2602:
2598:
2596:
2593:
2592:
2572:
2571:
2559:
2554:
2542:
2538:
2534:
2528:
2523:
2517:
2515:
2508:
2502:
2498:
2495:
2494:
2483:
2479:
2477:
2470:
2460:
2458:
2455:
2454:
2445:
2399:
2395:
2389:
2384:
2378:
2366:
2362:
2358:
2354:
2341:
2336:
2327:
2323:
2321:
2318:
2317:
2312:
2271:
2267:
2258:
2254:
2238:
2231:
2227:
2217:
2215:
2198:
2195:
2194:
2166:
2159:
2154:
2143:
2124:
2119:
2108:
2106:
2103:
2102:
2072:
2067:
2047:
2044:
2043:
2014:
1989:
1985:
1959:
1956:
1955:
1929:
1925:
1921:
1916:
1907:
1903:
1901:
1898:
1897:
1890:
1884:
1880:
1879:
1852:
1848:
1832:
1825:
1821:
1811:
1809:
1792:
1789:
1788:
1763:
1758:
1740:
1738:
1735:
1734:
1707:
1690:
1688:
1685:
1684:
1654:
1649:
1641:
1638:
1637:
1608:
1564:
1558:
1554:
1552:
1535:
1532:
1531:
1526:
1492:
1488:
1486:
1477:
1473:
1471:
1468:
1467:
1462:
1434:
1430:
1417:
1411:
1407:
1405:
1388:
1385:
1384:
1359:
1354:
1345:
1334:
1332:
1329:
1328:
1300:
1293:
1288:
1277:
1275:
1272:
1271:
1235:
1232:
1231:
1202:
1181:
1161:
1157:
1152:
1151:
1147:
1130:
1127:
1126:
1098:
1093:
1087:
1076:
1065:
1063:
1060:
1059:
1029:
1026:
1025:
1000:
991:
984:
977:
970:
937:
933:
924:
920:
918:
915:
914:
900:
875:
871:
864:
858:
854:
852:
835:
832:
831:
826:
816:
809:
802:
786:
755:
751:
742:
738:
736:
733:
732:
709:
705:
696:
692:
690:
687:
686:
658:
653:
647:
636:
634:
631:
630:
604:
600:
593:
588:
574:
565:
561:
560:
555:
544:
540:
535:
533:
530:
529:
523:
521:Input impedance
516:
508:
480:
477:
476:
452:
449:
446:
445:
421:
416:
408:
405:
404:
399:
370:
364:
360:
358:
347:
343:
338:
336:
333:
332:
318:
231:input impedance
219:image-impedance
204:
175:
174:
150:
142:
141:
137:mm'-type filter
94:
84:
83:
69:Gaussian filter
44:
32:
12:
11:
5:
4568:
4558:
4557:
4552:
4547:
4542:
4525:
4524:
4522:
4521:
4516:
4510:
4508:
4504:
4503:
4501:
4500:
4495:
4490:
4488:Maxwell bridge
4484:
4482:
4478:
4477:
4475:
4474:
4469:
4464:
4462:Lattice filter
4459:
4457:Fontana bridge
4454:
4449:
4443:
4441:
4437:
4436:
4434:
4433:
4428:
4422:
4420:
4416:
4415:
4413:
4412:
4407:
4401:
4399:
4395:
4394:
4387:
4386:
4379:
4372:
4364:
4355:
4354:
4353:
4352:
4346:
4341:Zobel, O. J.,
4336:
4333:
4332:
4319:
4304:
4291:
4285:Zobel, O. J.,
4277:
4276:
4274:
4271:
4270:
4269:
4267:Boucherot cell
4264:
4259:
4254:
4249:
4242:
4239:
4234:
4230:
4226:
4222:
4206:
4202:Cauer topology
4189:
4186:
4181:
4177:
4173:
4169:
4165:
4161:
4149:
4148:
4132:
4128:
4124:
4119:
4116:
4112:
4105:
4101:
4095:
4092:
4089:
4086:
4083:
4068:
4045:
4041:
4037:
4033:
4029:
4025:
4004:
4003:Bode equaliser
4001:
3991:
3990:
3969:
3951:
3950:
3942:
3935:
3928:co-axial cable
3918:
3915:
3884:
3880:
3874:
3870:
3856:
3855:
3852:Boucherot cell
3837:
3834:
3817:
3813:
3809:
3782:
3781:
3765:
3761:
3757:
3754:
3748:
3744:
3738:
3731:
3728:
3724:
3718:
3714:
3696:
3692:
3688:
3684:
3680:
3665:
3662:ladder network
3656:
3653:
3640:
3637:
3631:
3628:
3599:
3596:
3572:
3568:
3539:
3535:
3510:
3506:
3481:
3477:
3448:
3444:
3414:
3411:
3406:
3403:
3399:repeating coil
3394:
3364:
3361:
3338:
3335:
3301:
3298:
3292:
3291:Bumps and dips
3289:
3273:
3269:
3244:
3241:
3237:
3226:, but then at
3212:
3209:
3205:
3187:
3184:
3162:
3158:
3152:
3148:
3125:
3121:
3115:
3111:
3093:
3090:
3076:
3073:
3058:
3055:
3054:
3053:
3037:
3034:
3030:
3025:
3020:
3015:
3009:
3004:
3000:
2996:
2992:
2987:
2980:
2975:
2971:
2956:
2955:
2939:
2935:
2931:
2927:
2922:
2919:
2916:
2902:
2901:
2886:
2883:
2879:
2876:
2872:
2867:
2863:
2860:
2856:
2853:
2850:
2846:
2843:
2828:
2827:
2813:
2810:
2807:
2803:
2798:
2795:
2792:
2789:
2786:
2781:
2778:
2773:
2770:
2732:
2729:
2728:
2727:
2714:
2709:
2705:
2701:
2696:
2691:
2686:
2682:
2679:
2673:
2668:
2663:
2658:
2654:
2639:
2638:
2623:
2620:
2616:
2610:
2605:
2601:
2586:
2585:
2565:
2562:
2558:
2553:
2545:
2541:
2537:
2531:
2526:
2522:
2514:
2511:
2509:
2505:
2501:
2497:
2496:
2491:
2486:
2482:
2476:
2473:
2471:
2469:
2466:
2463:
2462:
2441:
2435:
2434:
2422:
2416:
2413:
2410:
2407:
2402:
2398:
2392:
2387:
2383:
2377:
2374:
2369:
2365:
2361:
2357:
2350:
2347:
2344:
2340:
2335:
2330:
2326:
2310:
2306:| =
2297:
2296:
2282:
2279:
2274:
2270:
2266:
2261:
2257:
2253:
2250:
2247:
2244:
2241:
2234:
2230:
2226:
2223:
2220:
2214:
2211:
2208:
2205:
2202:
2188:
2187:
2172:
2169:
2165:
2162:
2158:
2153:
2149:
2146:
2142:
2139:
2136:
2130:
2127:
2123:
2118:
2114:
2111:
2096:
2095:
2081:
2078:
2075:
2071:
2066:
2063:
2060:
2057:
2054:
2051:
2013:
2010:
2006:
2005:
1992:
1988:
1984:
1981:
1978:
1975:
1972:
1969:
1966:
1963:
1949:
1948:
1932:
1928:
1924:
1920:
1915:
1910:
1906:
1888:
1872:
1871:
1855:
1851:
1847:
1844:
1841:
1838:
1835:
1828:
1824:
1820:
1817:
1814:
1808:
1805:
1802:
1799:
1796:
1782:
1781:
1766:
1761:
1757:
1753:
1750:
1746:
1743:
1728:
1727:
1713:
1710:
1706:
1703:
1700:
1696:
1693:
1678:
1677:
1663:
1660:
1657:
1653:
1648:
1645:
1607:
1604:
1588:
1587:
1573:
1570:
1567:
1561:
1557:
1551:
1548:
1545:
1542:
1539:
1524:
1514:
1513:
1500:
1495:
1491:
1485:
1480:
1476:
1460:
1454:
1453:
1437:
1433:
1429:
1426:
1423:
1420:
1414:
1410:
1404:
1401:
1398:
1395:
1392:
1378:
1377:
1362:
1357:
1353:
1349:
1344:
1340:
1337:
1322:
1321:
1306:
1303:
1299:
1296:
1292:
1287:
1283:
1280:
1265:
1264:
1251:
1248:
1245:
1242:
1239:
1201:
1198:
1197:
1196:
1178:
1174:
1171:
1164:
1160:
1156:
1150:
1146:
1143:
1140:
1137:
1134:
1120:
1119:
1106:
1101:
1096:
1092:
1086:
1082:
1079:
1075:
1071:
1068:
1053:
1052:
1039:
1036:
1033:
999:
996:
990:
987:
982:
975:
966:
956:
955:
940:
936:
932:
927:
923:
899:
896:
895:
894:
878:
874:
870:
867:
861:
857:
851:
848:
845:
842:
839:
822:
814:
807:
800:
785:
782:
774:
773:
758:
754:
750:
745:
741:
726:
725:
712:
708:
704:
695:
680:
679:
666:
661:
656:
652:
646:
642:
639:
624:
623:
607:
603:
599:
596:
592:
587:
580:
577:
573:
568:
564:
559:
554:
543:
539:
522:
519:
514:
504:
485:
473:dual impedance
458:
455:
442:
441:
427:
424:
420:
415:
412:
397:
391:
390:
376:
373:
367:
363:
357:
350:
346:
342:
317:
314:
313:
312:
262:Boucherot cell
213:are a type of
211:Zobel networks
206:
205:
203:
202:
195:
188:
180:
177:
176:
173:
172:
167:
162:
157:
151:
149:Simple filters
148:
147:
144:
143:
140:
139:
134:
129:
123:
120:Lattice filter
117:
111:
106:
101:
95:
90:
89:
86:
85:
82:
81:
76:
71:
66:
61:
56:
51:
45:
40:
39:
36:
35:
25:
24:
9:
6:
4:
3:
2:
4567:
4556:
4553:
4551:
4548:
4546:
4543:
4541:
4538:
4537:
4535:
4520:
4517:
4515:
4512:
4511:
4509:
4505:
4499:
4496:
4494:
4491:
4489:
4486:
4485:
4483:
4479:
4473:
4470:
4468:
4465:
4463:
4460:
4458:
4455:
4453:
4450:
4448:
4445:
4444:
4442:
4438:
4432:
4429:
4427:
4424:
4423:
4421:
4417:
4411:
4408:
4406:
4405:Kelvin bridge
4403:
4402:
4400:
4396:
4392:
4385:
4380:
4378:
4373:
4371:
4366:
4365:
4362:
4358:
4350:
4347:
4344:
4340:
4339:
4338:
4337:
4329:
4323:
4317:
4314:
4308:
4301:
4295:
4288:
4282:
4278:
4268:
4265:
4263:
4260:
4258:
4255:
4253:
4250:
4248:
4245:
4244:
4238:
4214:
4210:
4203:
4194:
4185:
4153:
4130:
4126:
4122:
4117:
4114:
4110:
4103:
4099:
4093:
4087:
4081:
4074:
4073:
4072:
4061:
4053:
4049:
4023:
4022:potentiometer
4018:
4009:
4000:
3999:two octaves.
3997:
3987:
3986:
3967:
3955:
3947:
3943:
3940:
3936:
3933:
3929:
3925:
3924:
3923:
3914:
3911:
3905:
3903:
3899:
3878:
3872:
3868:
3854:
3853:
3848:
3847:
3842:
3833:
3815:
3811:
3807:
3796:
3763:
3759:
3755:
3752:
3746:
3742:
3736:
3729:
3726:
3722:
3716:
3712:
3702:
3701:
3700:
3673:
3669:
3663:
3655:Half sections
3652:
3650:
3646:
3636:
3623:
3619:
3617:
3613:
3610:cables. The
3609:
3605:
3591:
3587:
3570:
3566:
3554:
3537:
3533:
3508:
3504:
3479:
3475:
3464:
3446:
3442:
3427:
3419:
3410:
3402:
3400:
3369:
3358:equalisation.
3355:
3347:
3343:
3334:
3330:
3326:
3319:
3315:
3311:
3306:
3297:
3288:
3271:
3267:
3242:
3239:
3235:
3210:
3207:
3203:
3193:
3183:
3180:
3160:
3156:
3150:
3146:
3123:
3119:
3113:
3109:
3098:
3089:
3081:
3072:
3063:
3035:
3032:
3028:
3023:
3018:
3013:
3007:
3002:
2998:
2994:
2990:
2985:
2978:
2973:
2969:
2961:
2960:
2959:
2937:
2933:
2929:
2925:
2920:
2917:
2907:
2906:
2905:
2884:
2881:
2877:
2874:
2870:
2865:
2861:
2858:
2854:
2851:
2848:
2844:
2841:
2833:
2832:
2831:
2811:
2808:
2805:
2801:
2796:
2793:
2790:
2787:
2784:
2779:
2776:
2771:
2768:
2761:
2760:
2759:
2757:
2753:
2745:
2741:
2737:
2712:
2707:
2703:
2699:
2694:
2689:
2684:
2680:
2671:
2666:
2661:
2656:
2652:
2644:
2643:
2642:
2621:
2618:
2614:
2608:
2603:
2599:
2591:
2590:
2589:
2563:
2560:
2556:
2551:
2543:
2539:
2535:
2529:
2524:
2520:
2512:
2510:
2503:
2499:
2489:
2484:
2480:
2474:
2472:
2467:
2453:
2452:
2451:
2449:
2444:
2440:
2420:
2414:
2411:
2408:
2405:
2400:
2396:
2390:
2385:
2381:
2375:
2372:
2367:
2363:
2359:
2355:
2348:
2345:
2342:
2338:
2333:
2328:
2324:
2316:
2315:
2314:
2309:
2305:
2302:
2280:
2277:
2272:
2268:
2264:
2259:
2255:
2251:
2248:
2245:
2242:
2239:
2232:
2228:
2224:
2221:
2218:
2212:
2206:
2200:
2193:
2192:
2191:
2170:
2167:
2163:
2160:
2156:
2151:
2147:
2144:
2140:
2137:
2134:
2128:
2125:
2121:
2116:
2112:
2109:
2101:
2100:
2099:
2079:
2076:
2073:
2069:
2064:
2061:
2058:
2055:
2052:
2049:
2042:
2041:
2040:
2038:
2034:
2026:
2022:
2018:
2009:
1990:
1986:
1982:
1979:
1976:
1973:
1967:
1961:
1954:
1953:
1952:
1930:
1926:
1922:
1918:
1913:
1908:
1904:
1896:
1895:
1894:
1887:
1878: =
1877:
1853:
1849:
1845:
1842:
1839:
1836:
1833:
1826:
1822:
1818:
1815:
1812:
1806:
1800:
1794:
1787:
1786:
1785:
1764:
1759:
1755:
1751:
1748:
1744:
1741:
1733:
1732:
1731:
1711:
1708:
1704:
1701:
1698:
1694:
1691:
1683:
1682:
1681:
1661:
1658:
1655:
1651:
1646:
1643:
1636:
1635:
1634:
1632:
1628:
1620:
1616:
1612:
1603:
1601:
1597:
1593:
1571:
1568:
1565:
1559:
1555:
1549:
1543:
1537:
1530:
1529:
1528:
1523:
1519:
1498:
1493:
1489:
1483:
1478:
1474:
1466:
1465:
1464:
1459:
1435:
1431:
1427:
1424:
1421:
1418:
1412:
1408:
1402:
1396:
1390:
1383:
1382:
1381:
1360:
1355:
1351:
1347:
1342:
1338:
1335:
1327:
1326:
1325:
1304:
1301:
1297:
1294:
1290:
1285:
1281:
1278:
1270:
1269:
1268:
1249:
1246:
1243:
1240:
1237:
1230:
1229:
1228:
1226:
1222:
1214:
1210:
1206:
1176:
1172:
1169:
1162:
1158:
1154:
1148:
1144:
1141:
1138:
1135:
1132:
1125:
1124:
1123:
1104:
1099:
1094:
1090:
1084:
1080:
1077:
1073:
1069:
1066:
1058:
1057:
1056:
1037:
1034:
1031:
1024:
1023:
1022:
1020:
1012:
1008:
1004:
995:
986:
981:
974:
971: =
969:
965:
961:
938:
934:
930:
925:
921:
913:
912:
911:
904:
876:
872:
868:
865:
859:
855:
849:
843:
837:
830:
829:
828:
825:
821:
813:
806:
799:
790:
781:
779:
756:
752:
748:
743:
739:
731:
730:
729:
710:
706:
702:
693:
685:
684:
683:
664:
659:
654:
650:
644:
640:
637:
629:
628:
627:
605:
601:
597:
594:
590:
585:
578:
575:
571:
566:
562:
557:
552:
541:
537:
528:
527:
526:
518:
513:
507:
503:
498:
483:
474:
456:
453:
425:
422:
418:
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410:
403:
402:
401:
396:
374:
371:
365:
361:
355:
348:
344:
340:
331:
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322:
310:
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300:
296:
291:
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284:
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138:
135:
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127:
124:
121:
118:
115:
114:Zobel network
112:
110:
107:
105:
102:
100:
97:
96:
93:
88:
87:
80:
77:
75:
72:
70:
67:
65:
64:Bessel filter
62:
60:
57:
55:
52:
50:
47:
46:
43:
38:
37:
34:
31:Linear analog
29:
28:
23:
21:
16:
15:
4514:Diode bridge
4498:Hay's bridge
4357:
4348:
4342:
4327:
4322:
4312:
4307:
4299:
4298:Farago, PS,
4294:
4286:
4281:
4219:
4199:
4158:
4066:
4014:
3992:
3920:
3906:
3857:
3849:
3797:
3783:
3678:
3658:
3642:
3633:
3604:twisted pair
3601:
3555:
3465:
3432:
3408:
3374:
3340:
3331:
3327:
3323:
3317:
3313:
3309:
3294:
3189:
3176:
3086:
3069:
2957:
2903:
2829:
2755:
2751:
2749:
2743:
2739:
2640:
2587:
2447:
2442:
2438:
2436:
2307:
2303:
2300:
2298:
2189:
2097:
2036:
2032:
2030:
2024:
2020:
2007:
1950:
1885:
1875:
1873:
1783:
1729:
1679:
1630:
1626:
1624:
1618:
1614:
1595:
1591:
1589:
1521:
1517:
1515:
1457:
1455:
1379:
1323:
1266:
1224:
1220:
1218:
1212:
1208:
1121:
1054:
1018:
1016:
1010:
1006:
992:
979:
972:
967:
963:
959:
957:
909:
823:
819:
811:
804:
797:
795:
775:
727:
681:
625:
524:
511:
505:
501:
499:
443:
394:
392:
327:
292:
276:
274:
270:power factor
255:
247:
210:
209:
113:
17:
4431:Wien bridge
4328:Wave Filter
4180:increases R
4036:is set to R
3910:Boucherot's
3898:distributed
258:loudspeaker
4534:Categories
4481:Inductance
4398:Resistance
4273:References
4017:constant k
3413:L-sections
3376:typically
3075:Basic loss
1021:is simply
998:Attenuator
316:Derivation
299:capacitors
223:Otto Zobel
170:RLC filter
128:(all-pass)
122:(all-pass)
4088:ω
3883:∞
3850:See also
3808:≈
3639:X-section
3608:star quad
3147:ω
3066:sections.
2970:ω
2918:ω
2915:Δ
2878:ω
2855:ω
2809:ω
2791:ω
2731:Band stop
2704:ω
2681:ω
2678:Δ
2653:ω
2600:ω
2500:ω
2468:ω
2465:Δ
2360:±
2325:ω
2269:ω
2265:−
2249:ω
2222:ω
2207:ω
2164:ω
2141:ω
2077:ω
2059:ω
2012:Band pass
1980:ω
1974:≈
1968:ω
1905:ω
1843:ω
1816:ω
1801:ω
1705:ω
1659:ω
1606:High pass
1569:ω
1550:≈
1544:ω
1475:ω
1422:ω
1397:ω
1298:ω
1247:ω
1145:
844:ω
303:inductors
295:impedance
266:impedance
239:resistive
227:Bell Labs
165:LC filter
160:RL filter
155:RC filter
4519:H-bridge
4241:See also
3983:response
3932:topology
3179:roll-off
2885:′
2862:′
2845:′
2744:Z'
2171:′
2148:′
2129:′
2113:′
2025:Z'
1745:′
1712:′
1695:′
1339:′
1305:′
1282:′
1200:Low pass
1081:′
1070:′
1011:Z'
641:′
579:′
457:′
426:′
375:′
4440:General
3790:⁄
3388:. The
1883:⁄
796:If the
682:yields
1730:where
1516:where
1324:where
215:filter
4507:Other
4172:and R
3390:45 dB
3386:45 dB
3382:45 dB
3378:0 dBu
2958:And,
1055:and,
287:units
4229:', C
3137:and
2830:and
2742:and
2098:and
2023:and
1680:and
1617:and
1267:and
1211:and
1009:and
444:Or,
301:and
4024:, R
3606:or
2756:Z'
2308:ωCR
2037:Z'
1631:Z'
1619:Z'
1600:8ve
1142:log
475:of
225:of
4536::
4233:/L
4225:/L
4069:in
4048:.
4042:in
3697:in
3693:in
3689:in
3681:in
3314:r'
3287:.
2304:LC
1876:ωC
1527:,
1183:dB
1139:20
815:in
698:in
546:in
497:.
4383:e
4376:t
4369:v
4235:2
4231:2
4227:1
4223:1
4221:C
4207:1
4182:L
4178:L
4174:L
4170:L
4166:L
4162:L
4131:0
4127:R
4123:+
4118:n
4115:i
4111:Z
4104:0
4100:R
4094:=
4091:)
4085:(
4082:A
4046:0
4038:0
4034:L
4030:0
4026:L
3968:f
3879:=
3873:B
3869:R
3816:0
3812:R
3792:2
3788:1
3764:0
3760:R
3756:+
3753:Z
3747:0
3743:R
3737:=
3730:n
3727:i
3723:V
3717:O
3713:V
3685:0
3666:0
3571:0
3567:R
3538:0
3534:R
3509:0
3505:R
3480:0
3476:R
3447:B
3443:Z
3395:0
3320:.
3318:r
3310:r
3272:c
3268:f
3243:2
3240:c
3236:f
3211:1
3208:c
3204:f
3161:1
3157:=
3151:c
3124:1
3120:=
3114:0
3110:R
3036:C
3033:L
3029:1
3024:+
3019:2
3014:)
3008:C
3003:0
2999:R
2995:2
2991:1
2986:(
2979:=
2974:m
2938:0
2934:R
2930:C
2926:1
2921:=
2882:C
2875:i
2871:1
2866:+
2859:L
2852:i
2849:=
2842:Z
2812:L
2806:i
2802:1
2797:+
2794:C
2788:i
2785:=
2780:Z
2777:1
2772:=
2769:Y
2752:Z
2740:Z
2713:2
2708:0
2700:+
2695:2
2690:)
2685:2
2672:(
2667:=
2662:2
2657:m
2622:C
2619:L
2615:1
2609:=
2604:0
2564:C
2561:L
2557:1
2552:+
2544:2
2540:L
2536:4
2530:2
2525:0
2521:R
2513:=
2504:m
2490:L
2485:0
2481:R
2475:=
2448:ω
2443:m
2439:ω
2421:)
2415:C
2412:L
2409:4
2406:+
2401:2
2397:C
2391:2
2386:0
2382:R
2376:+
2373:C
2368:0
2364:R
2356:(
2349:C
2346:L
2343:2
2339:1
2334:=
2329:c
2311:0
2301:ω
2281:C
2278:L
2273:2
2260:0
2256:R
2252:C
2246:i
2243:+
2240:1
2233:0
2229:R
2225:C
2219:i
2213:=
2210:)
2204:(
2201:A
2168:L
2161:i
2157:1
2152:+
2145:C
2138:i
2135:=
2126:Z
2122:1
2117:=
2110:Y
2080:C
2074:i
2070:1
2065:+
2062:L
2056:i
2053:=
2050:Z
2033:Z
2021:Z
1991:0
1987:R
1983:C
1977:i
1971:)
1965:(
1962:A
1931:0
1927:R
1923:C
1919:1
1914:=
1909:c
1889:0
1886:R
1881:1
1854:0
1850:R
1846:C
1840:i
1837:+
1834:1
1827:0
1823:R
1819:C
1813:i
1807:=
1804:)
1798:(
1795:A
1765:2
1760:0
1756:R
1752:C
1749:=
1742:L
1709:L
1702:i
1699:=
1692:Z
1662:C
1656:i
1652:1
1647:=
1644:Z
1627:Z
1615:Z
1596:ω
1594:(
1592:A
1572:L
1566:i
1560:0
1556:R
1547:)
1541:(
1538:A
1525:c
1522:ω
1518:ω
1499:L
1494:0
1490:R
1484:=
1479:c
1461:0
1436:0
1432:R
1428:+
1425:L
1419:i
1413:0
1409:R
1403:=
1400:)
1394:(
1391:A
1361:2
1356:0
1352:R
1348:L
1343:=
1336:C
1302:C
1295:i
1291:1
1286:=
1279:Z
1250:L
1244:i
1241:=
1238:Z
1225:Z
1221:Z
1213:Z
1209:Z
1177:)
1173:1
1170:+
1163:0
1159:R
1155:R
1149:(
1136:=
1133:L
1105:R
1100:2
1095:0
1091:R
1085:=
1078:R
1074:=
1067:Z
1038:R
1035:=
1032:Z
1019:Z
1007:Z
983:0
980:Z
976:0
973:Z
968:B
964:Z
960:Z
939:0
935:Z
931:=
926:B
922:Z
877:0
873:Z
869:+
866:Z
860:0
856:Z
850:=
847:)
841:(
838:A
824:B
820:Z
812:V
810:/
808:o
805:V
801:0
798:Z
778:ω
757:0
753:R
749:=
744:0
740:Z
711:0
707:Z
703:=
694:Z
665:Z
660:2
655:0
651:Z
645:=
638:Z
606:0
602:Z
598:+
595:Z
591:1
586:+
576:Z
572:+
567:0
563:Z
558:1
553:=
542:Z
538:1
515:0
512:Z
506:B
502:Z
484:Z
454:Z
423:Z
419:1
414:=
411:Z
398:0
395:Z
372:Z
366:0
362:Z
356:=
349:0
345:Z
341:Z
311:.
199:e
192:t
185:v
22:.
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