How To Obtain Data To Evaluate Noise Frequency For Setting The Notch Filter

Filters in Control Systems

George Ellis , in Command Arrangement Blueprint Guide (Quaternary Edition), 2012

ix.1.1.3 Using Notch Filters for Noise and Resonance

Notch filters are also used in controllers. Where depression-laissez passer filters benumb all signals to a higher place a specified frequency, notch filters remove only a narrow ring of frequencies; as seen in the Bode plot of Figure ix.3, notch filters pass the frequency components below and above the notch frequency. The fact that notch filters pass high frequencies leads to their strongest attribute: They unremarkably cause fiddling phase lag at the proceeds crossover frequency, assuming the notch frequency is well in a higher place that. Notch filters tin can be useful on the command for a fixed-frequency noise source such equally that from line frequency (l or lx Hz) racket. Notch filters are also used to remove resonances from the system. Both notch and low-pass filters can cure resonance; notch filters practice so while creating less phase lag in the command loop.

Figure nine.3. Bode plot for a notch filter.

Although notch filters meliorate the primary shortcoming of low-pass filters (the reduction of stage margin), they are yet used less regularly. Notch filters work on only a narrow band of frequencies. To be useful, the notch filter must be tuned to the frequency of resonance or of noise generation. If the offending frequency is known, digital notch filters tin be ready to filter it with great accuracy. Still, there are many cases where the noise or resonant frequency will vary. For example, resonant frequencies ofttimes vary slightly from 1 arrangement to another. In such a case, the notch filter may accept to be manually tuned for every control arrangement. In the case of analog systems, notch filters become more complicated to configure. Because the values of analog passive components vary from i unit to another and, to a lesser extent, over time and temperature, analog notch filters oft must be "tweaked in" on every awarding. Worse all the same, time and temperature variation may force adjustment of the filter frequency subsequently the controller is placed in performance.

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Phase-locked loops and their design

Wenzhao Liu , Frede Blaabjerg , in Control of Power Electronic Converters and Systems, 2021

10.3.3 Notch filter–based PLLs

A notch filter (NF) is a band-rejection filter that significantly attenuates specific frequency signals but passes all other frequency components with negligible attenuation. This feature makes the NF attractive in order to cancel the selected desired harmonic components presented in the input signal [ix,14,20]. In fact, the NFs can be divided into adaptive or nonadaptive filters. The one-time ane is very preferred by designers because it is easier to select a narrow bandwidth for NFs to minimize the phase delay in the control loop of the PLL. Nevertheless, this reward increases computational burden of the PLL. The structure of NF-PLL is like to the standard MAF-PLL, except that the MAF is replaced with NFs equally shown in Fig. ten.eight.

Figure 10.eight. Schematic diagram of the notch filter–based phase-locked loop.

In industry, more one NF in the PLL command loop tin can be extended with cascaded topology [21] and parallel-connecting topology [22]. The main difference between these topologies is their frequency estimation method, the latter topology uses the aforementioned frequency figurer for all of the NFs. Nonetheless, in the cascaded topology, every NF is equipped with its ain frequency reckoner and in that location is a tradeoff between the filtering capability and computational brunt in both topologies. To attain a satisfactory compromise, iii NFs with notch frequencies at iiω _yard , half dozenω _{one thousand} , and 12ω _k are usually suggested past designers to obtain a robust PLL [nine].

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FilterCAD user'south manual, version ane.x

Bob Dobkin , Jim Williams , in Analog Circuit Design, 2013

Notches…the last frontier

Notch filters, specially those with high Qs and/or high attenuations, are the well-nigh difficult to implement with universal switched-capacitor filter devices. Y'all may pattern a notch filter with FilterCAD, with specifications that purport to yield a stopband attenuation of greater than 60dB, and detect that in practice an attenuation of 40dB or less is the result. This is primarily due to the sampled data nature of the universal filter blocks; signals of equal aamplitude and opposite phase do non ideally cancel when summed together equally they would do in a purely analog system. Notches of up to 60dB can exist obtained, but to practise so requires techniques not covered past this version of FilterCAD. Some of these techniques will be examined here. We will start by using FilterCAD to enter the parameters for an elliptic notch response. Nosotros'll specify a maximum passband ripple of 0.1dB, an attenuation of 60dB, a center frequency of 40kHz, a stop bandwidth of 2kHz, and a laissez passer bandwidth of 12kHz. Given these parameters, FilterCAD synthesizes the response shown in Table 23.12. This 8th order filter claims an bodily stopband attenuation of greater than 80dB, a level of functioning that would be exceedingly difficult to achieve in the real world. A working filter with an attenuation of 60dB tin be achieved, only only be deviating significantly from the advice provided past FilterCAD.

Tabular array 23.12. f₀, Q, and f_north Values for 40kHz, 60dB Notch

STAGE	f0	Q	fn
1	35735.6793	3.3144	39616.8585
two	44773.1799	3.3144	40386.8469
3	35242.9616	17.2015	39085.8415
4	45399.1358	17.2105	40935.5393

Switched-capacitor filter devices give the all-time performance when certain operating parameters are kept within detail ranges. Those conditions which produce the best results for a particular parameter are called its "figure of merit." For instance, in the case of the LTC1064, the best specs for clock to center frequency ratio (f_CLK/f₀) accuracy are published for a clock frequency of 1MHz and a Q of 10. Equally nosotros depart from this "effigy of merit" (as we must practise to produce the 40kHz notch in our example), performance will gradually deteriorate. One of the bug that we will encounter is "Q-enhancement." That is, the Qs of the stages volition appear slightly greater than those prepare by resistors. (Note that Q-enhancement is mostly a problem in modes 3 and 3A and is not express to notches only occurs in LP, BP and HP filters besides.) This results in peaking above and beneath the notch. Q-enhancement can be compensated for by placing small capacitors (3pF to 30pF) in parallel with R4 (way two or iii). With this modification, Q-enhancement tin can be compensated for in notch filters with middle frequencies every bit loftier equally 90kHz. The values suggested here are compromise values for a broad-range clock-tunable notch. If you want to produce a fixed-frequency notch, you can utilize larger caps at college frequencies. At least in the instance of the LTC1064, Q-enhancement is unlikely to exist a problem beneath 20kHz. Adding capacitors at lower frequencies volition have the upshot of widening the notch.

As mentioned previously, the other problem in implementing notch filters is inadequate attenuation. For low frequency notches, stopband attenuation may exist increased past boosting the clock to notch frequency upward to 250:i. Attenuation may also be improved by adding external capacitors, this time in parallel with R2 (modes ane, two, and 3A). Capacitors of 10pF to 30pF in this position tin increase stopband attenuation by 5dB to 10dB. Of course, this capacitor/resistor combination constitutes a passive 1st club lowpass phase with a corner frequency at 1/(2πRC). In the case of the values indicated higher up, the corner frequency volition appear and so far out in the passband that information technology is unlikely to be significant. However, if the notch is needed at a frequency below 20kHz, the capacitor value will demand to be increased and the corner frequency of the 1st guild phase will be lowered proportionally. For a capacitor of 100pF and an R2 of 10k, the corner frequency volition be 159kHz, a value that is still unlikely to cause bug in most applications. For a capacitor of 500pF (a value that might testify necessary for a deep notch at a low center frequency) and an R2 of 20k, the corner frequency drops down to fifteen.9kHz. If maximum stopband attenuation is more important than a wide passband, such a solution may testify acceptable. Adding resistors in parallel with R2 produces one additional problem: it increases the Q that we just controlled with the capacitors across R4. Resistor values must exist adjusted to bring the Q down over again.

Table 23.13 contains the parameters for a existent notch filter which actually meets our 60dB attenuation spec using the techniques previously outlined. This is essentially the clock-tunable 8th guild notch filter described in the LTC1064 information sheet. Note the mixture of modes used. This is a solution that FilterCAD is incapable of proposing.

Table 23.13. f₀, Q, and f_{due north} Values for 40kHz, 60dB Notch

Stage	f0 (kHz)	Q	fn (kHz)	MODE
i	forty.000	ten.00	40.000	i
ii	43.920	11.00	40.000	ii
three	40.000	10.00	40.000	1
4	35.920	8.41	40.000	3

Information technology should exist apparent that the methods for notch filters described here are primarily empirical at this point, and that the account given hither is far from comprehensive. We have non even touched on optimizing these filters for noise or baloney, for instance. No unproblematic rules tin exist given for this process. Such optimization is possible, but must be addressed on a case-past-case basis. If you need to implement a high performance notch filter and the tips to a higher place prove inadequate, please call the LTC applications department for boosted aid.

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Audio

Ian Hickman , in Analog Circuits Cookbook (2d Edition), 1999

Some other filter types

Unproblematic notch filters – where the gain is unity everywhere either side of the notch – can be very useful, eastward.m. for suppressing fifty  Hz or threescore   Hz hum in measurement systems. The passive TWIN TEE notch is well known, and tin exist sharpened up in an agile circuit so that the gain is abiding, say, beneath 45   Hz and higher up 55   Hz. Even so, information technology is inconvenient for tuning, due to the apply of no less than six components. An ingenious alternative (Ref. viii) provides a design with limited notch depth, but compensating advantages. A notch depth of 20   dB is easily achieved, and the filter can be fine tuned past means of a single pot. The frequency aligning is independent of attenuation and bandwidth.

Finally, a discussion on linear stage (constant group delay) filters. These are hands implemented in digital form, FIR filters existence inherently linear stage. But most analog filter types, including Butterworth, Chebychev and elliptic, are anything but linear phase. Consequently, when passing pulse waveforms, considerable ringing is experienced on the edges, particularly with high-order filters, fifty-fifty of the Butterworth variety. The linear stage Bessel design tin be used, but this gives only a very gradual transition from pass-to stopband, even for quite high orders. All the same, a fact that is non widely known is that information technology is possible to design truthful linear phase filters in analog applied science, both bandpass (Ref. 9) and lowpass (Ref. ten). These can use passive components, or – as in Ref. 10 – agile circuitry.

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Circuit drove, book Five

Richard Markell Editor , in Analog Circuit Design, 2013

Narrow-band notch filter blueprint reaches 80dB notch depth

Narrow-band notch filters are specially challenging designs. The requirement for virtually notch filters is to remove a particular tone and not bear upon any of the remaining signal bandwidth. This requires an infinitesimally narrow filter that can just be approximated by a reasonably narrow bandwidth. These types of filters, like the narrow-band bandpass discussed higher up, require precision f _O accuracy. Figure 38.171 shows the schematic of this type of filter. This filter is a 1.02kHz notch filter that is often used in telecommunications test systems.

Figure 38.171. Narrow-Band Notch Filter

1 of the challenges of designing a switched capacitor notch filter i nvolves the broad-ring nature of a notch filter. The broad-ring dissonance can be aliased down into the band of interest. Optimal high performance notch filters should utilize some class of noise-band limiting. To attain the noise-band limiting, the design in Effigy 38.171 places capacitors in parallel with the R2 resistors of each 2nd order section. This forms a pole, set at f_P = 1/(ii • π • R2 • C2), that will limit the bandwidth. This pole frequency must be low enough to take a band-limiting issue but must not exist so low as to affect the notch filter'due south response. The pole should be greater than thirty times the notch frequency and less than lxx-v times the notch frequency for the all-time results. Figure 38.172 shows the frequency response of the filter. Note that the notch depth is greater than −80dB. Without the apply of the C21 and C22, the notch depth is only about −35dB.

Figure 38.172. Measured Frequency Response of Figure 38.171's Narrow-Band Notch Filter

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Filter Realizations

In Op Amp Applications Handbook, 2005

Bainter Notch

A simple notch filter is the Bainter circuit (run across Reference 21). It is composed of simple circuit blocks with 2 feedback loops as shown in Effigy 5-59. Likewise, the component sensitivity is very low.

Figure 5-59. Bainter notch filter

This circuit has several interesting properties. The Q of the notch is not based on component matching as it is in every other implementation, but is instead only dependant on the proceeds of the amplifiers. Therefore, the notch depth volition not migrate with temperature, aging, and other environmental factors. The notch frequency may shift, but not the depth.

Amplifier open loop gain of ten⁴ will yield a Q_z of > 200. Information technology is capable of orthogonal tuning with minimal interaction. R6 tunes Q and R1 tunes ω_Z. Varying R3 sets the ratio of ω₀/ω_Z produces low pass notch (R4 > R3), notch (R4 = R3) or high laissez passer notch (R4 < R3).

The design equations of the Bainter circuit are given in Effigy 5-77.

Figure five-77. Bainter notch

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Linear and Nonlinear Control of Switching Power Converters

José Fernando Silva , Sónia F. Pinto , in Power Electronics Handbook (Fourth Edition), 2018

Proportional-integral derivative (PID), plus loftier-frequency poles

The PID notch filter type (35.52) scheme is used in converters with ii lightly damped complex poles, to increase the response speed, while ensuring zero steady-state fault. In about switching power converters, the two complex zeros are selected to have a damping gene greater than the converter complex poles and slightly smaller oscillating frequency. The high-frequency pole is placed to achieve the needed phase margin [9]. The design is correct if the complex pole loci, heading to the complex zeros in the system root locus, never enter the right half plane:

(35.52) $\begin{array}{l}{C}_{\mathit{PIDnf}}\left(\mathrm{due\; south}\right)=T_{cp}\frac{{\mathrm{due\; south}}^2+\mathrm{ii}{\xi }_{cp}{{\omega }_0}_{cp}s+{{\omega }_0^2}_{cp}}{s\left(1+\mathrm{southward}/{\omega }_{p1}\right)}\\ =\frac{T_{cp}s}{1+\mathrm{due\; south}/{\omega }_{p1}}+\frac{\mathrm{two}{T}_{cp}{\xi }_{cp}{\omega }_{0cp}}{1+s/{\omega }_{p1}}+\frac{T_{cp}{{\omega }_0^{\mathrm{two}}}_{cp}}{\mathrm{south}\left(1+\mathrm{southward}/{\omega }_{p1}\right)}\\ =\frac{T_{cp}s}{\mathrm{ane}+s/{\omega }_{p\mathrm{ane}}}+\frac{T_{cp}{\omega }_{0pc}^2\left(1+\mathrm{ii}\mathrm{southward}{\xi }_{cp}/{\omega }_{0cp}\right)}{\mathrm{due\; south}\left(\mathrm{i}+\mathrm{southward}/{\omega }_{p\mathrm{i}}\right)}\end{array}$

For systems with a high-frequency cypher placed at to the lowest degree one decade above the two lightly damped complex poles, the compensator (35.53), with ω _z1  ≈ ω _z2<   ω_p , can be used. Usually, the two real zeros present frequencies slightly lower than the frequency of the converter circuitous poles. The 2 high-frequency poles are placed to obtain the desired phase margin [9]. The obtained overall functioning will often be inferior to that of the PID type notch filter:

(35.53) $C_{PID}\left(s\right)=W_{cp}\frac{\left(1+s/{\omega }_{z1}\right)\left(\mathrm{ane}+s/{\omega }_{z2}\right)}{\mathrm{south}{\left(1+s/{\omega }_p\right)}^{\mathrm{two}}}$

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Space Impulse Response Filter Design

Li Tan , Jean Jiang , in Digital Signal Processing (Second Edition), 2013

eight.7.2 Second-Order Bandstop (Notch) Filter Design

For this type of filter, the pole placement is the aforementioned as the bandpass filter (Figure 8.33). The zeros are placed on the unit circle with the same angles with respect to poles. This will better passband performance. The magnitude and the angle of the complex conjugate poles determine the 3 dB bandwidth and eye frequency, respectively.

Figure viii.33. Pole-zip placement for a second-order notch filter.

Design formulas for bandstop filters are given in the following equations:

(8.45) $r\approx \mathrm{i}-\left(B{W}_{3dB}/f_s\right)\times \pi ,\text{good}\text{for}0.9\le r<\mathrm{ane}$

(viii.46) $\theta =\left(\frac{f_0}{f_{\mathrm{south}}}\right)\times 360\mathrm{°}$

(eight.47) $H(z)=\frac{G(z-e^{j\theta })(z+e^{-j\theta })}{(z-r{e}^{j\theta })(z-r{\mathrm{due\; east}}^{-j\theta })}=\frac{\mathrm{Chiliad}(z^2-\mathrm{two}z\text{cos}\theta +\mathrm{one})}{(z^{\mathrm{two}}-2rz\text{cos}\theta +r^2)}$

The scale cistron to arrange the bandstop filter then it has a unit passband gain is given by

(8.48) $K=\frac{\left(1-\mathrm{two}r\text{cos}\theta +r^2\right)}{\left(\mathrm{two}-\mathrm{ii}\text{cos}\theta \right)}$

EXAMPLE 8.xviii

A second-order notch filter is required to satisfy the following specifications:

•

Sampling rate = 8,000 Hz

•

3 dB bandwidth: $B\mathrm{Due\; west}=100$ Hz

•

Narrow passband centered at $f_0=\mathrm{ane},500$ Hz

Find the transfer function using the pole-zero placement approach.

Solution:

We first calculate the required magnitude of the poles

$r\approx \mathrm{one}-\left(100/\mathrm{viii,000}\right)\times \pi =0.9607$

which is a good approximation. We use the center frequency to obtain the angle of the pole location:

$\theta =\left(\frac{\mathrm{ane,500}}{\mathrm{eight,000}}\right)\times 360\mathrm{°}=67.5\mathrm{°}$

The unit of measurement-gain scale factor is calculated equally

$\mathrm{One\; thousand}=\frac{\left(\mathrm{ane}-2\times 0.9607\text{cos}67.5\mathrm{°}+{0.9607}^2\right)}{\left(2-\mathrm{two}\text{cos}67.5\mathrm{°}\right)}=0.9620$

Finally, we obtain the transfer function:

$H(z)=\frac{0.9620(z^2-\mathrm{two}z\text{cos}67.5\mathrm{°}+1)}{(z^2-\mathrm{ii}\times 0.9607z\text{cos}\mathrm{67.five}\mathrm{°}+{0.9607}^{\mathrm{ii}})}=\frac{0.9620-0.7363z^{-1}+0.9620z^{-2}}{1-0.7353z^{-1}+0.9229}$

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Agile Filter Blueprint Techniques

Bruce Carter , in Op Amps for Everyone (Fourth Edition), 2013

6.iv.4 High-Speed Notch Filters

A very like bandwidth restriction affects notch filters. Instead of eroding the amplitude of the acme, as it does bandpass filters, the bandwidth restriction erodes the depth of the notch ( Figure 6.16).

Figure 6.16. Notch Filter Response

Using the exact same op amp as in the bandpass section above, and the notch filter topology of Department six.three.six, notch filters were constructed at x   MHz, ane   MHz, and 100   kHz. No center frequency tuning was attempted. The 10   MHz results were and so terrible they are not included here. Even the 1   MHz results evidence dramatically how bandwidth was affecting the notch. At a Q of i, a thirty   dB notch is possible at 1   MHz. However, higher Q values merely erode the depth of the notch further. This erosion is even axiomatic at 100   kHz. At this bespeak, information technology is clear that even a 1   GHz op amp can only be used to construct notch filters at 1   MHz and a Q of 1. If a Q of ten is desired, a i   GHz op amp can only be used to construct a notch filter of 100   kHz. This amazing degree of limitation was totally unexpected, to say the to the lowest degree.

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Loftier-Speed Filters

Bruce Carter , Ron Mancini , in Op Amps for Anybody (Fifth Edition), 2018

18.5 High-Speed Notch Filters

A very like bandwidth restriction affects notch filters. Instead of eroding the amplitude of the pinnacle and/or shifting it in frequency as it does ring-pass filters, the bandwidth brake erodes the depth of the notch. A notch filter was constructed using the techniques of Section 17.3.5. A 1   GHz bandwidth op amp was used, for a filter center frequency of 1   MHz. No tuning of center frequency was attempted. Various values of Q were attempted, with the following results (Fig. xviii.3).

Figure 18.3. Notch filter response for 1   MHz middle frequency.

Patently, the center frequency is largely unaffected, it is the Q that takes a hit. At a Q of one, a 30   dB notch is possible at 1   MHz. However, college Q'south only erode the depth of the notch. The circuit as synthetic is completely unsuitable for a Q greater than 1.

The excursion was retuned for a center frequency of 100   kHz, and the results beneath were measured (Fig. xviii.iv).

Figure eighteen.4. Notch filter response for 100   kHz center frequency.

Notch depth erosion is even evident at 100   kHz. At this point, information technology is clear that even a 1   GHz op amp tin only exist used to construct notch filters at one   MHz and a Q of 1 and 100   kHz with a Q of 10 or so. This amazing degree of limitation was totally unexpected, to say the to the lowest degree!

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How To Obtain Data To Evaluate Noise Frequency For Setting The Notch Filter,

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