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I would like to introduce an ideal 5th-order Bessel filter to my simulation. My initial thought is to use a voltage-dependent voltage source and Laplace transform. I copied the Bessel filter coefficients from Wikipedia. Bessel LPF filter coefficients.

s+1

s^2+3s+3

s^3+6s^2+15s+15

s^4+10s^3+45s^2+105s+105

s^5+15s^4+105s^3+420s^2+945s+945

I used coefficients to write down the transfer function: enter image description here How it is written in LTspice:

Laplace = 945 / (pow((s/w0),5)+15 * pow((s/w0),4)+105 * pow((s/w0),3)+420 * pow((s/w0),2)+945 * (s/w0)+945)

[Spaces between integers and “pow” are there so the “*” sign appears in the LTspice formula spaces are not present.]

945 in the numerator corresponds to a DC gain of 0 dB.

ω = 2πf(-3 dB)

f(-3dB) = 3.5 kHz

ω = 2 * π * 3500

However, after simulating the circuit I have noticed that it does not behave as it should. My passband is going far beyond 3.5 kHz. enter image description here My -3dB point is at 8.5 kHz. However, the behavior (even group delay in passband) is of the Bessel filter. Therefore, I scaled down the ω by the ratio of 3.5 kHz/8.5 kHz: ω = 2 * π * 1440. This resulted in a -3 dB point at 3.5 kHz and even group delay in the passband. enter image description here This approach however is a bodged one to fit the desired result without fixing the fundamental problem somewhere in the transfer function.

I welcome all the suggestions as to where the mistake might be.

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    \$\begingroup\$ Instead of using pow(), you can use double asterisk **. Actually there is a special case in LTspice where the carrot ^ symbol can be used for exponentiation in a Laplace expression. See the built-in help for more details. ltwiki.org/LTspiceHelp/LTspiceHelp/… \$\endgroup\$
    – Ste Kulov
    Dec 1, 2023 at 16:35
  • 1
    \$\begingroup\$ @SteKulov It's called a caret, not a carrot. \$\endgroup\$
    – Hearth
    Dec 2, 2023 at 4:00
  • \$\begingroup\$ @Hearth But that doesn’t sound as delicious! \$\endgroup\$
    – Ste Kulov
    Dec 2, 2023 at 6:41

3 Answers 3

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Save some grief: set \$\omega=1\$

Just save yourself some algebra grief and set \$\omega=1\$ when focused on a particular filter shape (like Bessel.) The filter shape can be moved around freely (at least mathematically, if not in practical terms) on the frequency axis.

So just give yourself a break. Don't fret \$\omega\$. Set it to 1.

That allows you to focus on the remaining important details without getting bogged down with extra symbols and it will be easier to think more clearly. (Super-genius types exempted.)

(NOTE: In the following, I'll use freely available SymPy and SageMath and Python.)

Butterworth at \$\omega=1\$ is \$-3.0103\:\text{dB}\$ regardless of order

It's possible that you became confused by the Butterworth polynomial family. It has a unique property that makes it special in terms of its \$-3.0103\:\text{dB}\$ cutoff.

Let's look at 2nd, 3rd, 4th, 5th, and 6th order Butterworth polynomials in order to illustrative this unique property. I'm going to ask SymPy to tell me what value of \$\omega\$ provides the \$-3.0103\:\text{dB}\$ cutoff:

# Butterworth
#
solve(Eq(abs(expand(prod(Butterworth(2))).subs({s:I*w})),sqrt(2)),w)[0].n()
1.00000000000000
solve(Eq(abs(expand(prod(Butterworth(3))).subs({s:I*w})),sqrt(2)),w)[0].n()
1.00000000000000
solve(Eq(abs(expand(prod(Butterworth(4))).subs({s:I*w})),sqrt(2)),w)[0].n()
1.00000000000000
solve(Eq(abs(expand(prod(Butterworth(5))).subs({s:I*w})),sqrt(2)),w)[0].n()
1.00000000000000
solve(Eq(abs(expand(prod(Butterworth(6))).subs({s:I*w})),sqrt(2)),w)[0].n()
1.00000000000000

As you can see, it doesn't matter what order I choose. A Butterworth polynomial always has a \$-3.0103\:\text{dB}\$ corner right at \$\omega=1\$.

It's my guess that you've been given to believe that all filter polynomials have this same feature. But the reality is that no other filter polynomial does.

However, a note: That doesn't mean that other filter polynomials cannot be frequency-scaled so that they do meet this behavior. But then the polynomial becomes order-dependent and will no longer look the same as its canonical polynomial.

Canonical Bessel polynomials

Before I apply SymPy on the canonical Bessel polynomials you provided, let's first take a look at curves (partly from Desmos and partly using Paint) that plot out where the canonical Bessel functions intersect with a \$-3.0103\:\text{dB}\$ horizontal line:

enter image description here

It's plain to see above that different Bessel orders intersect at different values of \$\omega\$.

Now, we are ready to let SymPy quickly do the calculations for these canonical Bessel polynomials you provided:

# Bessel
#
solve(Eq(abs(((s**2+3*s+3)/3).subs({s:I*w})),sqrt(2)),w)[0].n()
1.36165412871613
solve(Eq(abs(((s**3+6*s**2+15*s+15)/15).subs({s:I*w})),sqrt(2)),w)[0].n()
1.75567236868121
solve(Eq(abs(((s**4+10*s**3+45*s**2+105*s+105)/105).subs({s:I*w})),sqrt(2)),w)[0].n()
2.11391767490422
solve(Eq(abs(((s**5+15*s**4+105*s**3+420*s**2+945*s+945)/945).subs({s:I*w})),sqrt(2)),w)[0].n()
2.42741070215263

(I skipped order 1 because it is exactly the same shape as the Butterworth.)

Note that none of these came up with \$\omega=1\$. And the last one says that \$\omega=2.42741070215263\$ when the transfer function is \$-3.0103\:\text{dB}\$.

That should be at \$f=0.386334412161760\$. So let's take a quick peek at what LTspice says:

enter image description here

Yeah. I'd say that's a match.

Your expectations about these canonical Bessel polynomials are the problem. That's all. (Likely because the canonical Butterworth polynomials led you into conflated thinking about all filter shapes.)

Frequency-scaling Bessel polynomials

Practical filter designers focused on solving a problem at hand and looking for pre-computed filter tables to save time will want those tables set up so that they provide \$-3.0103\:\text{dB}\$ at \$\omega=1\$ and not where the canonical form would instead put it.

So, in response, textbook authors typically provide table values for each order with filter-order-based adjustments already applied so that the expected \$-3.0103\:\text{dB}\$ cutoff does occur at \$\omega=1\$.

But such adjustments also leave you with a different polynomial, as well. It will no longer match up with the canonical form.

To make this point, let's look at one of those practitioner-tables from the 2nd edition of "Intuitive Analog Circuit Design" by Dr. Marc T. Thompson:

enter image description here

(I had to use Paint to fix errors in the table layout before posting it.)

Let's pick the 3rd order entry and take their parameters to see what we get as the polynomial:

expand((s-(-1.0509-I*1.0025))*(s-(-1.0509+I*1.0025))*(s-(-1.327)))
s**3 + 3.4288*s**2 + 4.89848566*s + 2.79916989862

Hmm. Now, that's definitely not the polynomial you listed for 3rd order!

(It's also slightly wrong. More on that, later.)

To see what the table's author did in preparing that table, let's take those \$\omega\$-factors I computed with SymPy above and use them to make filter-order-based adjustments that I think the author applied to create that table.

Here's those frequency-adjusted Bessel functions where they have been scaled (stretched on the frequency axis) so that they all wind up crossing at \$\omega=1\$:

enter image description here

These are still Bessel functions appropriate for their order. But they are no longer in their canonical form. Each order has been stretched on their frequency axis, and differently for each based upon its order, so that they wind up crossing where a practical designer would likely want.

But as I earlier showed with the 3rd order case from the table, the resulting polynomial isn't canonical anymore. It's been moved around on the frequency axis. Which explains the differences in constants.

If all that is true, then how do you know it's really a Bessel anymore?

Well, there are two constants present in the 2nd order version. It's how they relate together that tell you what kind of filter shape you have. And for higher order, that's still true. It's just that you have more constants and how they all relate to each other still matters.

It's time to investigate that.

Filter shape

Filter shape is easier to follow with 2nd order because a 2nd order filter has only two basic ideas: \$\omega_{_0}\$ and either \$Q\$ or \$\zeta\$ (each determines the other one.) \$\omega_{_0}\$ just moves the shape left or right on the frequency axis. It doesn't change the shape. So the only parameter determining shape with 2nd order is \$Q\$ or \$\zeta\$. That makes it easy.

Let's test this idea with the 2nd order Bessel. I'll use the canonical values and then also the re-positioned values that I used in the above plot.

Let's give it a whirl:

cbessel = 3/(s**2+3*s+3)                    # canonical 2nd order Bessel

k0 = 2/(-1 + sqrt(5))
k1 = sqrt(6)/sqrt(-1 + sqrt(5))
sbessel = k0/(s**2+k1*s+k0)                 # scaled 2nd order     
sbessel.n()
1.61803398874989/(s**2 + 2.20320266118432*s + 1.61803398874989)

nsimplify(tf2(cbessel)[zeta])
sqrt(3)/2                                   # canonical 2nd order shape

nsimplify(tf2(sbessel)[zeta])
sqrt(3)/2                                   # scaled 2nd order shape

You can see that the shape didn't change. It's the exact same shape. So it is still a Bessel. But the constants are certainly different!! That's because the scaled 2nd order Bessel was frequency-stretched in just the right way to get the cross-over where, as a practical designer, you'd likely want it placed.

I hope that's enough to help you at least accept the idea here; that polynomials may need to be scaled/stretched on the frequency axis to help designers achieve particular goals. And more, that textbook authors will often reflect such desires when laying out design tables.

But at least one thing should now be evident: mathematical families of polynomials may have canonical forms more commonly used by mathematicians and frequency-stretched versions more commonly used by electronics designers and they may look differently -- especially for any family other than Butterworth (which is special in this way.)

How to develop a frequency-stretched Bessel polynomial

So, this beggars another question:

How did I come up with \$\frac2{\sqrt{5}\,-\,1}\$ and \$\sqrt{\frac6{\sqrt{5}\,-\,1}}\$ for a frequency-stretched 2nd order Bessel?

Well, I am not reading this from any textbook (nor have I ever.) In fact, I don't even have an idea what textbook to look in. It's just how I'd handle it. Maybe it works for you to see it in action:

$$\begin{align*} H\left(s\right)&=\frac3{s^2+3\,s+3} \\\\&=\frac{\omega_{_0}^{\,2}}{s^2+2\zeta\,\omega_{_0}\,s+\omega_{_0}^{\,2}}, \text{ where }\omega_{_0}=\sqrt{3} \text{ and } \zeta=\frac{\sqrt{3}}2 \end{align*}$$

Note that the shape matches earlier computation using SymPy.

For the frequency-scaling, we must first find out how much to scale the frequency. I already did that far above from here. But let's do it again, but this time exactly:

nsimplify(solve(Eq(abs(((s**2+3*s+3)/3).subs({s:I*w})),sqrt(2)),w)[0])
sqrt(-3/2 + 3*sqrt(5)/2)

Let's scale all frequencies by that factor, calling it \$K_f=\sqrt{\frac32\left[\vphantom{\frac32}\sqrt{5}-1\right]}\$ so that \$\hat{s}=K_f\cdot s\$.

$$\begin{align*} H\left(\hat{s}\right)&=\frac{\omega_{_0}^{\,2}}{\hat{s}^2+2\zeta\,\omega_{_0}\,\hat{s}+\omega_{_0}^{\,2}} \\\\ &=\frac{\omega_{_0}^{\,2}}{\left(K_f\,s\right)^2+2\zeta\,\omega_{_0}\,\left(K_f\,s\right)+\omega_{_0}^{\,2}} \\\\ &=\frac{\omega_{_0}^{\,2}}{K_f^{\,2}\,s^2+2\zeta\,\omega_{_0}\,K_f\,s+\omega_{_0}^{\,2}} \\\\ &=\frac{ \frac1{K_f^{\,2}} }{ \frac1{K_f^{\,2}}}\cdot\frac{\omega_{_0}^{\,2}}{K_f^{\,2}\,s^2+2\zeta\,\omega_{_0}\,K_f\,s+\omega_{_0}^{\,2}} \\\\ &=\frac{\left(\frac{\omega_{_0}}{K_f}\right)^2}{s^2+2\zeta\left(\frac{\omega_{_0}}{K_f}\right)s+\left(\frac{\omega_{_0}}{K_f}\right)^2} \\\\ &=\frac{\frac2{\sqrt{5}\,-\,1}}{s^2+\sqrt{\frac6{\sqrt{5}\,-\,1}}\,s+\frac2{\sqrt{5}\,-\,1}} \end{align*}$$

That last denominator factors as \$s_\pm=\frac14\sqrt{2\vphantom{\sqrt{5}+1}}\sqrt{\sqrt{5}+1}\left[\vphantom{\sqrt{\sqrt{5}+1}}-\sqrt{3}\pm j\right]\$. There's only a factor of \$\sqrt{3}\$ difference between the two constants: \$s_\pm=-1.10160133059216\pm j\, 0.636009824757034\$.

This is a good moment to re-visit the table I pulled from a textbook where the author supplied \$-1.1030\pm j\, 0.6368\$. These constants also differ from each other by about a factor of \$\sqrt{3}\$. And that's important.

But while they are similar in possessing the same \$\sqrt{3}\$ ratio, the constant magnitudes are not the same. And that does have a (slight) impact on frequency scaling.

Let's perform a simple test to see which of the two is right.

(tf2(((c1**2+c2**2)/((s-c1+I*c2)*(s-c1-I*c2))).subs({c1:-1.103,c2:0.6368}))[zeta]).n()
0.866031301401309                  # zeta from textbook table

(tf2(((c1**2+c2**2)/((s-c1+I*c2)*(s-c1-I*c2))).subs({c1:-1.10160133059216,c2:0.636009824757034}))[zeta]).n()
0.866025403784438                  # zeta from my process

(sqrt(3)/2).n()
0.866025403784439                  # true zeta for 2nd order Bessel

Both are close and especially also share the same relative ratio.

The key here is how one might frequency-scale a filter so that its \$-3.0103\:\text{dB}\$ cutoff occurs when \$\omega=1\$.

As a final note in this section, let's see if there is a simpler way to frequency-scale. I'll use the 3rd order Bessel for that. Look back up to find the scaling factor:

# Compute new factors (s^3 factor is always 1):
(1/1.75567236868121)**1 * 6           # s^2 factor is scaled once
3.41749412192832
(1/1.75567236868121)**2 * 15          # s factor is scaled twice
4.86636086392275
(1/1.75567236868121)**3 * 15          $ units factor is scaled thrice
2.77179327460633

# Plug these in and work out the roots:
roots(s**3 + 3.41749412192832*s**2 + 4.86636086392275*s + 2.77179327460633,s)
{-1.32267579991045: 1,
 -1.04740916100894 - 0.999264436280635*I: 1,
 -1.04740916100894 + 0.999264436280635*I: 1}

That's it. We get about \$-1.3227\$ and \$-1.04741\pm j\,0.9993\$.

By comparison, the textbooks's table values are \$-1.327\$ and \$-1.0509\pm j\,1.0025\$. These also don't match in magnitude.

It's time to discuss errors.

Note about errors

The above errors in the textbook table are minor. And while getting errors in the 3rd decimal place seems a bit odd to me there's probably no real harm here.

A point I'd like to make is that textbooks can make errors that have greater impact. Sometimes, the errors are just wrong. So it helps to be able to find your own way, if and when needed.

(I have tracked these constants at least going back to a 1997/1998 edition of "Filter Design" by Steve Winder [Newnes Press 1998, also EDN 1997, I think.] But it may go well before then, too.)

Although these errors are minor, let's use LTspice to make a quick comparison. (It's useful to know how to do this in LTspice, so that errors may be quickly identified.)

I'll test the textbook version against the one I developed above:

enter image description here

As you can see, it's not a lot of difference.

Still? Never rest easy. Trust, but verify.

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    \$\begingroup\$ Thank you for your in-depth analysis of my problem. I was aware of different filter properties based on the Q factor and that there is one response that has the following property −3B at ω=1. I trusted my memory that it is Bessel. I trusted but not verified 😊 I suppose one cannot have all the benefits in one analogue filter (even group delay, −3B at ω=1 and steep roll-off without any ripple). Where would be fun in that? \$\endgroup\$
    – Wintermute
    Dec 4, 2023 at 17:52
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    \$\begingroup\$ Now I see that confusing the Butterworth filter property (-3db at ω=1) with the Bessel filter was not the only part that I got wrong. For my designs, I use AD circuit wizard which requires the following requirements: gain, passband and stopband. By manipulating the stopband, I can select the order of the filter. Therefore, I never had to worry about transfer functions and coefficients. Still, it is good to exercise the mind and gain a deeper understanding of the math behind the tool. \$\endgroup\$
    – Wintermute
    Dec 4, 2023 at 17:52
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    \$\begingroup\$ Stretching the frequency makes sense now. I will keep in mind to look for the coefficient in the right format when using the transfer function in LTSpice next time. Once again thank you for the in-depth explanation! \$\endgroup\$
    – Wintermute
    Dec 4, 2023 at 17:52
  • \$\begingroup\$ @Wintermute I'm glad the effort helped out. It also helped me, as well. I knew where I wanted to head. But I really didn't know how better to organize those latent thoughts. And I'd never before tried to do the frequency-scaling before. Nothing I read discussed a method. They'd say it was necessary. And just drop it, leaving me flat about a method. Still, I did vaguely know what was needed and had a basic idea how. Just not how better to frame it to help make it easier to gather. So I benefited from taking the effort and I appreciate the chance your question offered me in that regard. Thanks! \$\endgroup\$ Dec 4, 2023 at 20:37
  • \$\begingroup\$ @Wintermute There's probably some textbook out there that goes through all this -- perhaps in a different style. I have only read a few books that exist on the topic and just never encountered the right textbook. If someone knows of a good textbook taking on a procedure with the same goals and written for self-educating readers and not someone in a classroom setting with student groups, teachers, and aides to fill in the gaps, I'd love to know about it. Having to re-invent the wheel isn't always a good thing and I'd prefer to point to a professionally developed text. \$\endgroup\$ Dec 4, 2023 at 20:44
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I suppose that the mentioned frequency (f=3.5 kHz) is the desired 3dB-frequency (your requirement), right?

In this case, it is not true that the corresponding w value (2Pi*f) would appear in the denominator of the 5th-order transfer function. Instead, it would be necessary to find the magnitude of the function and solve it for the 3-dB point - a very involved calculation.

Therefore: The classical strategy is to use a chain of 2nd-order stages and (if necessary) a first-order section. For such a configuration you can find all necessary information (pole data) in normalizes filter tables which can be applied for a certain passband edge (3dB frequency).

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  • \$\begingroup\$ Thank you for your reply. Yes, 3.5KHz is my -3dB frequency. As a design tool, I use the AD filter wizard tool which breaks the complex filters into 2nd and 1st order parts to convert them to op-amp circuits. I will try this approach and chain three voltage-dependent voltage sources. \$\endgroup\$
    – Wintermute
    Dec 4, 2023 at 17:16
  • \$\begingroup\$ Yes - that is the most used design approach. However, in some cases another strategy is applied: Desind a passive 5th-order RLC filter (ladder topology) with parts values (normalized) as given in suitable tables. Then - as a next step - inductors are realized with opamps (gyrator principle, GIC). \$\endgroup\$
    – LvW
    Dec 5, 2023 at 8:26
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The Bessel coefficients from Wikipedia yield a filter with unit group delay, not unit cutoff frequency. It is thus not surprising that you don't get the cutoff frequency you expect.

What is your application? Bessel filters are commonly employed when smoothing signals in the time domain. If frequency domain filter requirements are driving the design, a different form of filter response may be more appropriate.

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  • \$\begingroup\$ Thank you for your reply. Yes, I confused Bessel with Butterworth response. The system is used for accurate time interval measurement in spectrometry. My understanding is that all the frequencies of interest have to have the same delay introduced by the filter. Bessel response is part of the requirements. \$\endgroup\$
    – Wintermute
    Dec 4, 2023 at 17:08
  • \$\begingroup\$ @Wintermute Makes sense. Fifth order Bessel is what I used for the shaping filters in NICER heasarc.gsfc.nasa.gov/docs/nicer/nicer_about.html \$\endgroup\$
    – John Doty
    Dec 4, 2023 at 17:35
  • \$\begingroup\$ Very interesting application! Was the filter IC, op-amp-based or digital? \$\endgroup\$
    – Wintermute
    Dec 4, 2023 at 17:55
  • \$\begingroup\$ @Wintermute Op-amp. MFB topology for the complex poles. If it was digital, I'd use a trapezoidal filter, not Bessel, see amptek.com/-/media/ametekamptek/documents/resources/…. But digital took too much power to be practical for NICER. \$\endgroup\$
    – John Doty
    Dec 4, 2023 at 18:08

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