Rollin’s Twin-T Notch Filter
confess to having a bit of an evil sense of humor. I once had a stereo
buddy who was obsessed with his listening room. In a sense, he was ahead of his
time, because this was long enough ago that few people paid attention to such
things and you couldn't just order room treatment panels off the internet...
because there were no panels and there was no Web/Internet, and wouldn't be for
years. He managed to work himself into a bit of a lather over room resonances
and all I heard about for months was how he thought that he had a resonance at
this frequency or that, and to be honest, it got a bit tedious. Being of a
wicked bent, I decided to test his mental stability.
In those days I worked with mainframe computers, so I took a
few minutes and wrote a program in PL/1 (I told you this was a long time ago)
that calculated the resonance modes in his room. All of them. From the lowest
mode his room would support up to some arbitrarily high limit — I forget what,
probably 5 kHz or 10 kHz.
Pages and pages of the old-fashioned green bar computer paper,
printed at 8 lines to the inch (that being as tight as you could space the
print). When I handed it to my friend, his eyes lit up — he thought I was
handing him an officially computer-calculated solution to his problem. As the
realization dawned that it was no solution; that I had actually damned him to
one of the Seven Levels of Audio Hell, his face fell. I then helpfully pointed
out that the problem was even worse than that... that the modes went from
the front wall to the back wall, from side wall to side wall, and from floor to
ceiling. In short, his room was a three dimensional waffle of pulsing, seething,
ever-changing nodes of energy.
And you know what? It cured him of his obsession. (Which is
what I'd secretly hoped all along.) Once he got all that jittery, frittery
nervousness out of the way, he buckled down and started working on the real
problems, which were actually far fewer in numbers than you might suppose. Why?
Because most rooms only have two or three resonances that are significant; all
the rest just kinda fade into insignificance.
Here's the deal: The ones you have to worry about are the ones
that persist long enough to be noticed. It is one thing to have a resonance
that's relatively low Q (see the last two articles for a little more about Q),
because the resonance doesn't last long enough to be a problem. The annoying
ones are higher Q and persist like the ringing of a bell — or perhaps a better
analogy would be a large, low frequency gong. Let us simplify matters by
ignoring anything over, say, 200Hz. Yes, there's a lot of hash up there, but
it's more appropriate to deal with that sort of thing by using room treatments.
Egad, back in the day, we had to use thick curtains and upholstery foam. Now you
can buy nifty stuff that actually looks nice and performs much more
Low frequencies present more of a challenge because, well,
because they're lower.
Uh...right. Wanna beer with that tautology?
Don't mind if I do.
The problem arises because low frequencies have longer
wavelengths. How long? Well, a 20 Hz tone has a wavelength of 56 feet. A 200 Hz
tone is somewhat more manageable, at 5.6 feet. The wavelength is important
because your ability to tame a rogue wave is defined by how much absorptive
material you can throw at it. A 20 kHz wave is less than an inch long. You can
pretty much quell any problems there by putting up a half-inch thick piece of
foam with pretty fabric stretched over it. A 56 foot wavelength laughs at such
puny efforts. It zips right through the absorptive material, bounces off the
wall behind it, and comes roaring back past you, headed for the opposite wall.
And if you have the misfortune to have a room proportioned such that a
floor-to-ceiling resonance happens at that same frequency, then you've got
double trouble. If you've got one from side-to-side at the same time, you're
going to be plumb miserable.
So how do you determine where the potential resonances are? By
plugging the dimensions of your room into a simple formula:
Resonance frequency = 1125 / (2 x Length)
So, for example, let us say your listening room is 20 feet
long, from front to back. Two times that is 40, and 1125 (the speed of sound)
divided by that is a little over 28 Hz. Now, that's the lowest frequency you
have to worry about, at least between those two walls. Anything below that is
compression, which is something you can feel, but the room isn't long enough to
properly allow the wavelength to develop. (Car stereo folks spend a lot of time
thinking about compression, given that average-sized car interiors won't allow
real bass below 100Hz or so.) Above that first resonance, you're going to see
more resonance modes at integer multiples of 28 Hz, so expect potential problems
at 56, 84, 112 Hz, etc.
So why only 'potential' problems? Why aren't all these
frequencies driving you crazy at the same time? Because some of the energy goes
right through the wall to drive your neighbors nuts. Some gets absorbed by the 2"
x 4" studs and wallboard that make up the wall itself. And so forth. You can't
just calculate and say that you will definitely, positively, absolutely have a
problem at 28Hz. Real listening rooms are not that simple. For instance, take
the vertical dimension. Here in the United States, the vast majority of ceilings
are eight feet from the floor. That means that nearly everyone will have a
potential resonance mode at 70 Hz. But those who have sloped ceilings in their
listening rooms (aka cathedral ceilings) will have no vertical problems at all.
Resonances only occur between parallel walls. Another thing you sometimes see is
an open archway into a hall or another room. That reduces the severity of the
standing wave since the amount of wall at a given distance is reduced; the arch
is going to give you a different effective room length, hence a different
Incidentally, these two concepts, non-parallel walls and
staggered walls, are extremely effective at reducing or even eliminating
resonances, but if your listening room wasn't built that way to begin with,
you're going to have to do some serious (and expensive) reworking of your
household. Hint: A very effective cheat, at least for mid and high frequencies,
is lots of bookshelves full of lots and lots of books. The rounded spines
reflect incident sound waves in myriad directions. Where isn't important, just
so long as it isn't towards the opposite wall.
Note that this is a splendid reason to keep all those
old-fashioned real books. Kindles and Nooks won't perform this function for you.
But let us suppose that you've got a resonance problem in your
listening room and you were silly enough to give away all your Harry Potter
books when you got your Kindle. What can you do?
When faced with a Norwegian Ridgeback (that's a type of
dragon, for those unfamiliar with the Harry Potter universe), it's best not to
poke it in the eye with a sharp stick. Similarly, when your listening room has a
resonance, it is best just not to excite it in the first place. If you reduce
the amount of energy going into your listening room at the resonant frequency,
the resonance will not ring. If you're sufficiently clever, you can reduce the
energy just the right amount to produce flat response. Okay, that sounds
reasonable. How do you go about doing it?
The answer is a notch filter. A notch filter reduces — or
even knocks out entirely — a narrow frequency band from a wider bandwidth
signal. They're quite useful, really, and find application in things like taking
out 60 Hz hums and annoying high-pitched whistles in recordings. (I once had a
fellow in an unhappy relationship ask me for the values to notch out his whiny
girlfriend's voice — discretion being the greater part of valor; I made myself
scarce until the storm blew over...)
Your garden variety Twin-T notch filter is a fairly simple
circuit. For convenience, I've put the filter between two JFET buffers (V1 and
V2 allow you to adjust DC offset). Feel free to improvise on the buffers —
there is nothing sacred about them. Indeed, many folks use chip OpAmps arranged
as voltage followers. The idea is to keep the filter and the outside world at
arm's-length from one another.
There are two branches to the circuit. The upper one, R4, R6,
and C2, is a low pass filter. The lower one, C1, C3, and R5, is a high pass
filter. That's it. The formula is:
F = 1/(2*PI*R*C)
Where F is the notch frequency, R is the resistance, and C is
the capacitance in Farads. Note that C2 is twice the value of C in the formula,
and R5 is half the value of R in the formula. R4, R6, C1, and C3 are the
straight values as calculated. The values shown will give a notch at about
160Hz. If you want to make the Q of the circuit somewhat variable, use a dual
pot to vary R4 and R6 at the same time. The textbooks insist that the values of
the components be matched to a fare-thee-well, but their goal is as near
infinite attenuation as possible. For this application, the Q doesn't need to be
anywhere near infinite, and most times you'll find that a Q of 2 to 5 or so will
be quite adequate.
A variation on that circuit uses a technique called
bootstrapping to sharpen up the knees of the filter's response. The circuit is
identical, except for the addition of a third buffer that allows for some of the
output signal being fed back to the ground point in the middle of the filter.
The erstwhile ground point now follows the signal, rather than being tied to a
hard reference. Every time the signal moves, "ground" moves with it. The pot,
V4, allows you to adjust the amount of signal fed back to the filter, which
changes the Q.
If you feel that you want a variable frequency design, the
circuit can be rearranged somewhat:
In this circuit, V2 tunes the frequency of the notch, and V3
allows for tuning the Q of the circuit. The notch frequency for this circuit is
1/(2*PI*C*√3*(total resistance to the left of V2's
wiper)*(total resistance to the right of V2's wiper))
As a general starting point, the sum of R5 and V3 should be
about six times the sum of R4, V2 and R6. The values given will give the circuit
a frequency range from roughly 30 to 200 Hz.
This leaves hanging the question of how to tell where you need
your notch filters. Modern technology comes to the rescue in the guise of your
AV receiver. These days, it's become common to have a setup program in the
receiver that executes an analysis of the acoustics of your listening room. If
yours doesn't have such a circuit, ask around — chances are someone you know
has a receiver that will do the job. Although the audio electronics in AV
receivers aren't equivalent to high end stereo componentry, there's no reason
you can't use an AV receiver as a test instrument. Once it determines where you
need filters, you can remove it and put it back to use decoding the footfalls of
dinosaurs and exploding planets, where it belongs.