Excursion vs. Frequency

Last week, I was doing a lecture about the basics of audio and I happened to mention one of the rules of thumb that we use in loudspeaker development:

If you have a single loudspeaker driver and you want to keep the same Sound Pressure Level (or output level) as you change the frequency, then if you go down one octave, you need to increase the excursion of the driver 4 times.

One of the people attending the presentation asked “why?” which is a really good question, and as I was answering it, I realised that it could be that many people don’t know this.

Let’s take this step-by-step and keep things simple. We’ll assume for this posting that a loudspeaker driver is a circular piston that moves in and out of a sealed cabinet. It is perfectly flat, and we’ll pretend that it really acts like a piston (so there’s no rubber or foam surround that’s stretching back and forth to make us argue about changes in the diameter of the circle). Also, we’ll assume that the face of the loudspeaker cabinet is infinite to get rid of diffraction. Finally, we’ll say that the space in front of the driver is infinite and has no reflective surfaces in it, so the waveform just radiates from the front of the driver outwards forever. Simple!

Then, we’ll push and pull the loudspeaker driver in and out using electrical current from a power amplifier that is connected to a sine wave generator. So, the driver moves in and out of the “box” with a sinusoidal motion. This can be graphed like this:

Figure 1: The excursion of a loudspeaker driver playing a 1 kHz sine wave at some output level.

As you can see there, we have one cycle per millisecond, therefore 1000 cycles per second (or 1 kHz), and the driver has a peak excursion of 1 mm. It moves to a maximum of 1 mm out of the box, and 1 mm into the box.

Consider the wave at Time = 0. The driver is passing the 0 mm line, going as fast as it can moving outwards until it gets to 1 mm (at Time = 0.25 ms) by which time it has slowed down and stopped, and then starts moving back in towards the box.

So, the velocity of the driver is the slope of the line in Figure 1, as shown in Figure 2.

Figure 2: The excursion and velocity of the same loudspeaker driver playing the same signal.

As the loudspeaker driver moves in and out of the box, it’s pushing and pulling the air molecules in front of it. Since we’ve over-simplified our system, we can think of the air molecules that are getting pushed and pulled as the cylinder of air that is outlined by the face of the moving piston. In other words, its a “can” of air with the same diameter as the loudspeaker driver, and the same height as the peak-to-peak excursion of the driver (in this case, 2 mm, since it moves 1 mm inwards and 1 mm outwards).

However, sound pressure (which is how loud sounds are) is a measurement of how much the air molecules are compressed and decompressed by the movement of the driver. This is proportional to the acceleration of the driver (neither the velocity nor the excursion, directly…). Luckily, however, we can calculate the driver’s acceleration from the velocity curve. If you look at the bottom plot in Figure 2, you can see that, leading up to Time = 0, the velocity has increased to a maximum (so the acceleration was positive). At Time = 0, the velocity is starting to drop (because the excursion is on its was up to where it will stop at maximum excursion at time = 0.25 ms).

In other words, the acceleration is the slope of the velocity curve, the line in the bottom plot in Figure 2. If we plot this, it looks like Figure 3.

Figure 3: The excursion, velocity and acceleration of the same loudspeaker driver playing the same signal.

Now we have something useful. Since the bottom plot in Figure 3 shows us the acceleration of the driver, then it can be used to compare to a different frequency. For example, if we get the same driver to play a signal that has half of the frequency, and the same excursion, what happens?

Figure 4: Comparing the excursion, velocity and acceleration of the same loudspeaker driver playing two different signals with the same excursion. (The red line is the same in Figure 4 as in Figure 3.)

In Figure 4, two sine waves are shown: the black line is 1/2 of the frequency of the red line, but they both have the same excursion. If you take a look at where both lines cross the Time = 0 point, then you can see that the slopes are different: the red line is steeper than the black. This is why the peak of the red line in the velocity curve is higher, since this is the same thing. Since the maximum slope of the red line in the middle plot is higher than the maximum slope of the black line, then its acceleration must be higher, which is what we see in the bottom plot.

Since the sound pressure level is proportional to the acceleration of the driver, then we can see in the top and bottom plots in Figure 4 that, if we halve the frequency (go down one octave) but maintain the same excursion, then the acceleration drops to 25% of the previous amount, and therefore, so does the sound pressure level (20*log10(0.25) = -12 dB, which is another way to express the drop in level…)

This raises the question: “how much do we have to increase the excursion to maintain the acceleration (and therefore the sound pressure level)?” The answer is in the “25%” in the previous paragraph. Since maintaining the same excursion and multiplying the frequency by 0.5 resulted in multiplying the acceleration by 0.25, we’ll have to increase the excursion by 4 to maintain the same acceleration.

Figure 5: Comparing the excursion, velocity and acceleration of the same loudspeaker driver playing two different signals at two different excursions. Notice that some of the vertical scales in the plots have changed. (The red line is the same in Figure 5 as in Figures 4 and 3.)

Looking at Figure 5: The black line is 1/2 the frequency of the red line. Their accelerations (the bottom plots) have the same peak values (which means that they produce the same sound pressure level). This, means that the slopes of their velocities are the same at their maxima, which, in turn, means that the peak velocity of the black line (the lower frequency) is higher. Since the peak velocity of the black line is higher (by a factor of 2) then the slope of the excursion plot is also twice as steep, which means that the peak of the excursion of the black line is 4x that of the red line. All of that is explained again in Figure 6.

Figure 6. A repeat of Figure 5 with some explanations that (hopefully) help.

Therefore, assuming that we’re using the same loudspeaker driver, we have to increase the excursion by a factor of 4 when we drop the frequency by a factor of 2, in order to maintain a constant sound pressure level.

However, we can play a little trick… what we’re really doing here is increasing the volume of our “cylinder” of air by a factor of 4. Since we don’t change the size of the driver, we have to move it 4 times farther.

However, the volume of a cylinder is

π r2 * height

and we’re just playing with the “height” in that equation. A different way would be to use a different driver with a bigger surface area to play the lower frequency. For example, if we multiply the radius of the driver by 2, and we don’t change the excursion (the “height” of the cylinder) then the total volume increases by a factor of 4 (because the radius is squared in the equation, and 2*2 = 4).

Another way to think of this: if our loudspeaker driver was a square instead of a circle, we could either move it in and out 4 times farther OR we would make the width and the length of the square each twice as big to get the a cube with the same volume. That “r2” in the equation above is basically just the “width * length” of a circle…

This is why woofers are bigger than tweeters. In a hypothetical world, a tweeter can play the same low frequencies as a woofer – but it would have to move REALLY far in and out to do it.

Tracking force and pickup compliance

It should not come as a surprise that, when we talk about how a vinyl record works, we can start by looking at the movement of the needle in the groove. If we simplify that connection a little (by reducing the audio signal to one channel, but we’ll come back to that point later), then we can think of this as a needle, sitting on a surface. The needle is at the end of an arm that we call the “cantilever” (because it is fixed on one end and it can move up and down on the other end where the needle is attached) and that cantilever is attached somehow to the tonearm using a springy material of some kind (like rubber, for example).

Figure 1

The simple diagram above shows that arrangement. Of course, I’ve left out a bunch of things, and nothing is to scale, but those details are not important right now.

I’ll make the “spring” in this diagram out of flexible rubber that has some “springiness” or “compliance”. The more compliant the spring, the easier it is to flex. So a stiff spring in not very compliant. (This concept is very important to understand as we go on.)

The audio signal is “encoded” into the surface of the vinyl using bumps and dips that cause the needle to move up and down. I’ve shown this in the simple diagram below.

Figure 2

Notice in that diagram that the needle is in contact with the surface of the vinyl, but the part of the system that connects back to the tonearm has not lifted. This is because the connection between the cantilever and the tonearm assembly is compliant enough to let the cantilever move upwards (or downwards) without moving the rest of the system.

Think of this like driving over a very small bump in the road in your car. The compliance of the tires and the shock absorbers will result in the tire riding over the bump, but the car doesn’t jump as a result.

Remember that the bump in the surface of the vinyl is only passing by, so the needle isn’t raised for long. As a result part of the reason the tonearm doesn’t move upwards (and your car doesn’t jump) is partly because it’s heavy. Its mass results in an inertia that “wants” to stop it from moving up and down. (The other factor that’s involved here is an adjustment in the tonearm called the “tracking force” which is a measurement of how much the tonearm is pushing downwards on the needle.)

Consequently, when that bump comes along, the needle rides on top of it, and the force that is pushing it downwards comes mostly from the “spring” at the other end of the cantilever, as shown below.

Figure 3

If the spring had no compliance (in other words, if it weren’t a spring, and the cantilever were just connected directly to the tonearm) and if the cantilever and needle were strong enough to take the force, then the entire tonearm assembly would jump up and down instead, as shown below. (Imagine riding in a horse-drawn buggy with wooden wheels with steel rims, and no springs on the axles. You’d feel every single rock on the road…)

Figure 4

The tonearm is resting on two points: one is the tip of the needle and the other is at the other end at the pivot point where it also rotates horizontally as you play the album. If we were really dumb turntable designers, then half of the mass of the tonearm would be resting on the needle (and the other half would be resting on the pivot). This would be bad, since your records would wear out very fast. So, a tonearm has some kind of adjustment on it that reduces the amount of weight on the needle. The simplest way to do this is to put a counterweight on the opposite side of the pivot so it’s more like a see-saw at the playground. As you move the counterweight away from the pickup, the downwards force at the needle gets smaller. In fact, you can probably adjust the counterweight so far that the needle-end of the tonearm is lighter, and it is stuck up in the air…

We adjust the amount of downwards force at the needle (called the “tracking force”) to result in a value that is in balance with the compliance of the connection to the cantilever. If the tracking force is too high (or the compliance is too high for the tracking force) then the tonearm will sink like I’ve shown below.

Figure 5

There are lots of things wrong with this. The first is that the needle isn’t at the correct angle to the surface of the vinyl, so it’s not going to move correctly. The second is that the cantilever is at the wrong angle, so it’s not going to move upwards with the same behaviour as it moves downwards, which results in an asymmetrical distortion of the signal. But possibly the most obvious problem is that there’s just too much downwards pressure on the vinyl, so your records will wear out faster.

So, there is a balance between the tracking force and the compliance. That balance ensures that you always have contact between the tip of the needle and the surface of the vinyl as the bumps and dips go by.

Digging into the details

One of the things I do regularly is to measure the magnitude response of a turntable from the surface of the vinyl to the electrical output of the RIAA preamplifier. In order to do this, I play two tracks on a special test record (Brüel & Kjær QR 2010) which has the following audio signals:

  • Track 1
  • 2 seconds of 1 kHz sinusoidal wave, L&R channels (3.16 cm/sec lateral velocity)
  • 20 Hz to 45 kHz sinusoidal tone, log sweep, 5 sec per decade, Left channel
  • Track 2
  • 2 seconds of 1 kHz sinusoidal wave, L&R channels (3.16 cm/sec lateral velocity)
  • 20 Hz to 45 kHz sinusoidal tone, log sweep, 5 sec per decade, Right channel

Sometimes (but very rarely), I notice that the needle will skip (or jump) at the transition between the 1 kHz tone and the start of the sine sweep. If this happens, for track 1, the needle will skip forwards into the sweep.

When this happened the first time I thought “Ah hah! The tracking force isn’t high enough, so the needle is being thrown out of the groove. I just need to adjust it.” But after checking the tracking force with my meter (a very small, very precise and accurate scale), I found out that this was not the problem.

Of course, I could make the problem go away by increasing the tracking force, but then it was too high, and my records (and the needle tip) will wear down faster. This would be covering up the symptom, but not correcting the actual problem.

So, what is the problem? It’s that the compliance of the pickup is too low due to an error in the manufacturing process or the fact that it’s just old and the rubber has stiffened over time. In other words it looks more like the system shown in Figure 4, above.

Let’s take a system where the pickup compliance is too low (so the spring is too stiff), so the tonearm can be tossed up off the vinyl surface. We then combine that with the knowledge of how the needle sits in the groove on the vinyl and which channel is on which side of that groove (which I’ve shown below in Figure 6).

Figure 6

Now we can see that, if there’s a bump in the Left channel, it will push the needle on a 45º angle upwards, and if the tracking force and compliance aren’t working together as they should, then the entire tonearm can be pushed hard enough to cause the needle to lift off the surface of the vinyl, heading in towards the centre of the record (towards the left in Figure 6).

What does the signal actually look like?

Let’s go back and look at a recording of that transition between the 1 kHz tone and the start of the 20 Hz sweep, using a pickup that is behaving properly.

Figure 7

The figure above is a screenshot from Audacity that shows the “raw” signal that I recorded at the input of my sound card which is connected to the output of the RIAA preamplifier. I’ve zoomed in to the moment when the track transitions from the 1 kHz tone to the 20 Hz tone at the start of the sweep.

Let’s now use this to go backwards and try to figure out what the surface of the vinyl looks like. I’ll start by re-creating a “perfect” version of that signal in Matlab by joining a 1 kHz cosine wave to a 20 Hz cosine wave.

Figure 8

You might notice that I’ve changed the value a little. I’m simulating one channel of a tone that has a level of at 5 cm/sec, RMS lateral velocity for two channels, instead of the 3.16 cm/sec from the B&K record. But this doesn’t really matter too much – I’ve just done it to make the numbers look nice and be a little easier to talk about.

I’m simulating a system that has a total gain set so that a modulation velocity of 3.54 cm/sec in one channel will produce 354 mV RMS (500 mV peak) at the output of the RIAA at 1 kHz.

Since the lateral velocity of a two-channel tone is 5 cm/sec, then the velocity of one channel will be 1/sqrt(2) of that value because the groove wall is 45º away from the lateral axis and cos(45º) = 1/sqrt(2).

If we take the signal in Figure 8 and filter it with a RIAA pre-emphasis filter (sometime called an “anti-RIAA” or an “inverse RIAA”) and drop the level by 40 dB (a typical gain for a RIAA preamp), then the signal looks like the plot in Figure 9.

Figure 9

As you can see there, the signal much lower in level overall (because of the -40 dB gain) and the 20 Hz tone is much lower in level than the 1 kHz tone (because of the pre-emphasis filter).

The output of the pickup is a current that is proportional to the velocity of the needle. So, we can move farther backwards in the chain and plot the velocity of the needle over time, shown in Figure 10. As you can see, the shape of this plot looks identical to the one in Figure 9. This is because I’m assuming that the current output of the pickup is in phase with the voltage at the input of the RIAA. (This is a safe assumption for the two frequencies we’re looking at here. If you want to pick a fight with me about this, drop by and do it in person. But you’re buying the beer…)

Figure 10 (I made a mistake in the Y-axis label – it should say cm/sec. I’ll come back and fix that later)

Now comes a jump… the velocity of the needle can be calculated by finding the derivative of the displacement over time, which means that the displacement can be found by integrating the velocity.

If you don’t like calculus, then you can think of it this way: In the old days, if you drove from Struer to Copenhagen, you had to take a ferry to get from the island of Fyn to the island of Zealand. Every once in a while, there would be a policeperson, walking around the parking lot as people waited to board the ferry, handing out speeding tickets to some of the people there. What happened was that the licence plates were recorded with time stamps as they crossed the bridge to Fyn from Jutland – which is about 75 km away from the parking lot. If you arrive at the ferry too early, you must have been speeding, and you get rewarded with an earlier ferry, and an extra charge…

In other words, you can calculate your speed (velocity) by your change (difference) of distance (displacement) over time.

You can also do this backwards: if you know how fast you’re going, you can calculate your displacement over time (you’ll be 100 km away in an hour if you’re driving 100 km/h the whole time, for example). If your velocity changes over time (say you drive a different speed every hour for 10 hours), then you can still calculate your displacement by dividing time into slices (in this case, 1 hour per “slice”) and adding up the individual displacements for the velocity you had during each slice of time. If you divide time into infinitely short slices, then you are integrating instead of adding, but the process is essentially the same.

Back to the story: if we take the signal in Figure 10 and integrate it (and scale it – which isn’t really important for this discussion), we get the curve in Figure 11.

Figure 11

This gives us a good idea of the actual shape of the left wall of the groove in the vinyl for that particular signal.

So, as you can see there, if the connection between the cantilever and the pickup doesn’t have a high enough compliance, it’s no wonder that the needle gets thrown out of the record groove. That’s a heck of a bump to deal with! To be honest, it’s also a little amazing to me that the needle that’s behaving (like the one that produced the output shown in Figure 7) can actually put up with that kind of abuse.

(Special thanks to Jakob Dyreby for helping me to wrap my head around the simulation part of this posting. I did the math, but only after he pointed me in the right direction.)

Post script

Every once in a while, someone will send me a link to a YouTube page that shows an electron microscope “video” of a needle tracking a groove in a vinyl record. If you listen to the explanation of that video, he explains that it’s not really a video. It’s a series of photographs that he took, one by one, and then assembled into a video.

This means that, in that video, the needle isn’t really behaving like it does in real life when the vinyl is moving underneath it.

Imagine setting up a video camera on the side of the road, next to a small speed bump, and making a video of a car driving over it. You’d see that, as the car drives by, the wheels move up into the wheel wells and the car doesn’t get pushed upwards as much, since some of the vertical movement caused by the speed bump is “taken up” by the car’s springs and shock absorbers.

If, instead, you set up a camera, and got the car to move forwards 5 cm and stop – and you take a photo, then the car moves forwards another 5 cm and stops – and you take another photo, and the you repeat this until the car is out of the frame – and then you assemble all of those photos into a video, it would look very different. The car would not remain horizontal when the wheels are on the speed bump because the springs and shock absorbers wouldn’t be compressed at all.

That video is like the second “video” of the car. Of course, it’s still interesting, and it’s well-explained, so no one is playing any tricks on you. But it’s not a video of what actually happens…

Transistors vs. Tubes/Valves: A good little reminder

I was leafing through some old editions of Wireless World magazine this week and came across an article in the July, 1968 issue called “Computing Distortion: Method for low-power transistor amplifiers” by L. B. Arguimbau and D.M. Fanger.

I was immediately intrigued by the first sentence, which read:

Unlike those of thermionic valves, the non-linearities in junction transistors for low collector currents are highly uniform and predictable, hardly differing from one transistor to another.

Now, as an “audio professional”, I’m very used to seeing the “±” sign in data sheets. Any production line of anything has some tolerance limits within which the product will fall.

For example, the (on-axis, where applicable) magnitude response of a loudspeaker or headphone is typically spec’ed something like ± 3 dB within some frequency range. This would mean that, at some frequency within that range, when measured under identical conditions, two “identical” products (e.g. with the same brand and model name) might be as much as 6 dB apart.

For different devices and components inside those devices, the tolerance values are different.

This is why, for example, when I read that someone says “headphone model A has more bass than headphone model B”, I know that if you included the missing information, it would actually read “my sample of headphone model A has more bass than my sample of headphone model B”.

However, when it comes down to the component level, I’m used to seeing tighter tolerances. Of course, if you save money on resistors, they might be within 20% of the stated value. However, if I look at the specs of a decent DAC (which, in my case, is a chip that would be used inside a product – not a big DAC-in-a-box that sits on your desk), I’m used to seeing numbers like < ±1 dB within pragmatically usable frequency ranges.

Since I’m only a young person, I’ve only really worked with transistor-based equipment, both when I worked in studios and also since I started working in home audio. So, I’ve always taken it for granted, and never even considered that the distortion characteristics of a transistor would vary from one to another. This is because, as the article from 1968 states: they don’t… much…

However, I’ve never thought about the (now obvious) possibility that two “identical” tubes/valves will have different distortion behaviour, even at low levels, due to manufacturing differences.

So, the next time someone tells you that this tube amp is better than that tube amp (which I translate in my head to actually mean “I prefer the sound of this tube amp over the sound of that tube amp” since “better” is multi-dimensional with different weightings of the different dimensions by person), remind them that the full sentence should be:

“I prefer My sample of this tube amp with the tubes that are currently in it to that tube amp with the tubes that are currently in it.”

Wow

This article, from The Gramophone magazine, August 1932 foretells the future of turntables with platters driven by electric motors. Note that, to test this particular one, they increased what we would today call the “tracking force” to 3.5 pounds (about 1.6 kg) on the outside groove of a 10″ record without reducing the speed. Try that on a turntable today…

Sad to see a familiar mantra here though: “the motor is remarkably efficient, very well made and ridiculously expensive.”

Historical Context

This episode of 99 Percent Invisible tells the story of the Recording Ban of 1942, the impact on the rise of modern jazz music, and the parallels with the debates between artists and today’s streaming services. It’s worth the 50 minutes and 58 seconds it takes to listen to this!

At the end of that episode, the ban on record manufacture is mentioned, almost as an epilogue. This page from the January, 1949 issue of RCA’s “Radio Age” magazine discusses the end of that ban.

Interestingly, that same issue of the magazine has an article that introduces a new recording format: 7-inch records operating at 45 revolutions per minute! The article claims that the new format is “distortion free” and “noise-free”, stating that this “new record and record player climax more than 10 years of research and refinement in this field by RCA.”

Fibre needles

Reading through some old magazines again…

This time, it’s The Gramophone magazine from October, 1930. In the editorial, Compton Mackenzie says

What caught my eye was the discussion of gramophone needles made of “hard wood”, and also the prediction that “the growth of electrical recording steps … to grapple with that problem of wear and tear.”

The fact that electrical (instead of mechanical) recording and playback was seen as a solution to “wear and tear” reminded me of my first textbook in Sound Recording where “Digital Audio” was introduced only within the chapter on Noise Reduction.

Later in that same issue, there is a little explanation of the “Electrocolor” and “Burmese” needles.

The March 1935 issue raises the point of wear vs. fidelity in the Editorial (which starts by comparing players with over-sized horns).

I like the comment about having to be in the “right mood” for Ravel. Some things never change.

What’s funny is that, now that I’ve seen this, I can’t NOT see it. There are advertisements for fibre, thorn, and wood needles all over the place in 1930s audio magazines.

Vinyl info and calculators

This is just a collection of information about turntables and vinyl for anyone wanting to dig deeper into It (which might mean that it’s just for me…). I’ll keep adding to this (and completing it) as time goes by.

Glossary

Cantilever

  • The rod or arm that connects the stylus on one end to the “motor” on the other.

Effective Length

  • The straight-line distance between the pivot point of the tonearm and the top of the stylus

Equivalent Mass

  • definition to come

Flutter

  • Higher-frequency modulation of the audio frequency caused by changes in the groove speed. These may be the result of changes in problems such as unstable motor speed, variable compliance on a belt, issues with a spindle bearing, drive wheel eccentricity, and other issues.
  • Flutter describes a modulation in the groove speed ranging from 6 to 100 times a second (6 Hz to 100 Hz).

Frequency Drift

  • Very long-term (or low-frequency) changes in the audio frequency, typically caused by slow changes in the platter rotation speed.
  • Typically, changes with a modulation frequency of less than 0.5 Hz (a period of no less than 2 seconds) are considered to be frequency drift. Faster changes are labelled “Wow”

Groove

  • The v-shaped track pressed into the surface of the vinyl record, in which the stylus sits

Linear Tracking

  • A tonearm that moves linearly, following a path that is parallel to the radius line traced by the stylus. This (in theory) ensures that the tracking error is always 0º, however, in practice this error is merely small.

Modulation Width

  • The distance measured on a line through the spindle from the start of the modulated groove to the end of the modulated groove. This is approximately 3″ or 76 mm.

Mounting Distance

  • The distance between the spindle and the pivot point of the tonearm.

Needle

  • Also known as the stylus. The point that is placed in the groove of the vinyl record.
    Some persons distinguish between the “stylus” (to indicate the chisel on the mastering lathe that creates the groove in the master record), and the “needle” (to indicate the portion of the pickup on a turntable that plays the signal).

Null Radius

  • The radius (distance between the spindle and the stylus) where the tracking error is 0º. A typically-designed and correctly installed radial tracking tonearm has two null radii (see this posting).

Offset Angle

  • The angle between the axis of the stylus and a line drawn between the tonearm pivot and the stylus. See the line diagram below.

Overhang

  • The difference between the Effective Length of the tonearm and the Mounting Distance. This value is used in some equations for calculating the Tracking Error.

Pickup, Electromagnetic

  • Includes three general types: Moving Coil, Moving Magnet, and Variable Reluctance (aka Moving Iron). These produce an output proportional to the velocity of the stylus movement.

Pickup, Piezoelectic

  • Produces an output proportional to the displacement of the stylus.

Pitch

  • The density of the groove count per distance in lines per inch or lines per mm. The pitch can vary from disc to disc, or even within a single track, according to the requirements of the mastering.

Radial Tracking

  • A tonearm that rotates on a pivot point with the stylus tracing a circular path around that pivot.

Radius

  • The distance between the centre of the vinyl disc and the pickup stylus.

RIAA

  • A pre-emphasis / de-emphasis filter designed to fill two functions.
    • The first is a high-frequency attenuation de-emphasis that reduces the playback system’s sensitivity to surface noise. This requires a reciprocal high-frequency pre-emphasis boost.
    • The second is a low-frequency attenuation pre-emphasis that maintains a constant modulation amplitude at lower frequencies to avoid over-excursion of the playback stylus. This requires a reciprocal low-frequency de-emphasis boost.
  • The first of the two plots below, show the theoretical (black lines) and typical (red) response of the pre-emphasis filter. The second of the two plots shows the de-emphasis filter response.

Side Thrust

  • definition to come

Skating Force

  • definition to come

Spindle

  • The centre of the platter around which the record rotates

Stylus

  • Also known as the needle. The point that is placed in the groove of the vinyl record.
    Some persons distinguish between the “stylus” (to indicate the chisel on the mastering lathe that creates the groove in the master record), and the “needle” (to indicate the portion of the pickup on a turntable that plays the signal).

Stylus, Bonded vs. Nude

  • Although the tip of the stylus is typically made of diamond today, in lower-cost units, that diamond tip is mounted or bonded to a metal pin (typically steel, aluminium, or titanium) which is, in turn, connected to the cantilever (the long “arm” that connects back to the cartridge housing). This bonded design is cheaper to manufacture, but it results in a high mass at the stylus tip, which means that it will not move easily at high frequencies.
  • In order to reduce mass, the metal pin is eliminated, and the entire stylus is made of diamond instead. This makes things more costly, but reduces the mass dramatically, so it is preferred if the goal is higher sound performance. This design is known as a nude stylus.

Tracking Error

  • The angle between the tangent to the groove and the alignment of the stylus. In a perfect system, the stylus would align with the tangent to the groove at all radii (distances from the spindle), since this matches the angular rotation of the cutting head when the master was made on a lathe. A linear tracking arm minimises this error. A radial tracking arm can be designed to have two radii with no tracking error (each called a “Null Radius”) but will have some measurable tracking error at all other locations on the disk.
  • One side-effect of tracking error is distortion of the audio signal, typically calculated and expressed as a 2nd-harmonic distortion on a sinusoidal audio signal. However, higher order distortion and intermodulation artefacts also exist.

Warp Wow

  • A modulation of the frequency of the audio signal caused by vertical changes in the vinyl surface (a warped record). This typically happens at a lower frequency, which is why it is “warp wow” and not “warp flutter”.

Wow

  • Low-frequency modulation of the audio frequency caused by changes in the groove speed. These may be the result of changes in problems such as rotation speed of the platter, discs with an incorrectly-placed centre hole, or vertical changes in the surface of the vinyl, and other issues.
  • Wow is a modulation in the groove speed ranging from once every 2 seconds to 6 times a second (0.5 Hz to 6 Hz). Note that, for a turntable, the rotational speed of the disc is within this range. (At 33 1/3 RPM: 1 revolution every 1.8 seconds is equal to approximately 0.556 Hz.)

Disk size limits

Outside starting diameter

  • 7″ discs
    • 6.78″, +0.06″, -0.00″
    • 172.2 mm, +1.524 mm, – 0.0 mm
  • 10″ discs
    • 9.72″, +0.06″, -0.00″
    • 246.9 mm, +1.524 mm, – 0.0 mm
  • 12″ discs
    • 11.72″, +0.06″, -0.00″
    • 297.7 mm, +1.524 mm, – 0.0 mm

Start of modulated pitch diameter

  • 7″ discs
    • 6.63″, +0.00″, -0.03″
    • 168.4 mm, +0.0 mm, – 0.762 mm
  • 10″ discs
    • 9.50″, +0.00″, -0.03″
    • 241.3 mm, +0.0 mm, – 0.762 mm
  • 12″ discs
    • 11.50″, +0.00″, -0.03″
    • 292.1 mm, +0.0 mm, – 0.762 mm

Minimum inside diameter

  • 7″ discs
    • 4.25″
    • 107.95 mm
  • 10″ discs
    • 4.75″
    • 120.65 mm
  • 12″ discs
    • 4.75″
    • 120.65 mm

Lockout Groove diameter

  • 7″ discs
    • 3.88″, +0.00, -0.08
    • 98.552 mm, +0.0 mm, -2.032 mm
  • 10″ discs
    • 4.19″, +0.00, -0.08
    • 106.426 mm, +0.0 mm, -0.762 mm
  • 12″ discs
    • 4.19″, +0.00, -0.08
    • 106.426 mm, +0.0 mm, -0.762 mm

Unmodulated (silent) groove width

  • 2 mil minimum, 4 mil maximum
  • 0.0508 mm minimum, 0.1016 mm maximum

Modulated groove depth

  • 1 mil minimum, 5 mil maximum
  • 0.0254 mm minimum, 0.127 mm maximum
  • The figure below shows the typical, minimum, and maximum groove depths, drawn to scale (with a 13 µm spherical stylus)

Signal levels

  • A typical standard reference level is a velocity of 35.4 mm/sec on one channel.
  • This means that a monophonic signal (identical signal in both channels) with that modulation will have a lateral (side-to-side) velocity of 50 mm/sec.

Calculators and Measurements

Conversion

  • 1 mil = 1 “thou” = 1/1000 inch
  • Lengthmm = Lengthmil * 127/5000

Revolutions per Second

  • RevolutionsPerSecond = RevolutionsPerMinute / 60
  • e.g.
    • 0.556 Rev/Sec @ 33 1/3 RPM
    • 0.75 Rev/Sec @ 45 RPM
    • 1.3 Rev/Sec @ 78 RPM

Seconds per revolution

  • SecondsPerRevolution = 60 / RevolutionsPerMinute
  • e.g.
    • 1.8 Sec/Rev @ 33 1/3 RPM
    • 1.333 Sec/Rev @ 45 RPM
    • 0.769 Sec/Rev @ 78 RPM

Modulation Width

  • MaximumModulationWidth = (StartOfModulatedPitch – MinimumInsideDiameter) / 2
  • e.g. for a 12″ disc
    • (292.1 mm – 120.65 mm) / 2 = 85.725 mm
    • Typically approximately 3″ = 76 mm

Pitch (assuming constant pitch)

  • (RunningTime * RPM) / ModulationWidth
  • e.g.
    • (20 minutes * 33.333 RPM) / 76 mm = 8.77 lines per mm
    • (20 minutes * 33.333 RPM) / 3″ = 222.22 lines per inch

Groove Width

  • GrooveWidthInMil = (1000 / PitchInLinesPerInch + 1) / 2
  • e.g.
    • (1000 / 222 LPI + 1) / 2 = 2.75 mil = 2.75 x 10-3 inches = 0.07 mm

Angular Frequency (of the audio)

  • abbreviated ω (unit: radians per second)
  • ω = 2 * π * FrequencyHz

Angular Speed of Rotation (of of the disk)

  • commonly abbreviated ωr (unit: radians per second)
  • ωr = 2 * π * RevolutionsPerSecond

Displacement Amplitude

  • DisplacementAmplitudePeak = Velocitypeak / ω
  • e.g.
    • 50 mm/sec / (2 * pi * 1000 Hz) = 0.008 mm (peak)

Groove Speed

  • 2 * π * Radius * RevolutionsPerSecond
  • e.g.
    • at the Start of modulated pitch on a 12″ disk turning at 33 1/3 RPM
      • 2 * π * (292.1 mm / 2) * (33.333 / 60) = 509.8 mm/sec
    • at the Minimum inside diameter on a 12″ disk turning at 33 1/3 RPM
      • 2 * π * (120.65 mm / 2) * (33.333 / 60) = 210.6 mm/sec
  • The plot below shows the groove speeds of 12″ 33 1/3 RPM and 7″ 45 RPM for all possible radii.

Wavelength

  • GrooveSpeed / Frequency
  • e.g.
    • 20 Hz at the Start of modulated pitch on a 12″ disk turning at 33 1/3 RPM
      • 509.8 / 20 = 25.5 mm
    • 20 kHz at the Start of modulated pitch on a 12″ disk turning at 33 1/3 RPM
      • 509.8 / 20000 = 0.0255 mm
    • The plot below shows wavelengths of 4 different frequencies for 12″ 33 1/3 RPM records (the longer curves) and 7″ 45 RPM records (the shorter curves)

Tracking Error

  • Tracking Error =
    OffsetAngle – asin ((EffectiveLength2 + Radius2 – MountingDistance2) / (2 * EffectiveLength * Radius))
  • see this posting for an explanation and example

Distortion caused by tracking error

  • Equation is for calculating percentage of second-harmonic distortion of a laterally-modulated monophonic sinusoidal audio signal
  • DistortionPercent = 100 * (PeakVelocity * tan(TrackingError)) / (GrooveSpeed)
  • see this posting for an explanation and examples

Wow and Flutter

  • Typically measured with a 3150 Hz sinusoidal tone, played from the vinyl surface
  • This signal is then de-modulated to determine its change over time. That modulation is then filtered through the response shown below which approximates human sensitivity to frequency modulation of an audio signal. More detailed information is given below
  • The AES6-2008 standard, which is the currently accepted method of measuring and expressing the wow and flutter specification, uses a “2σ” or “2-Sigma” method, which is a way of looking at the peak deviation to give a kind of “worst-case” scenario. In this method, the tone is played from a disc and captured for as long a time as is possible (or feasible). Firstly, the average value of the actual frequency of the output is found (in theory, it’s fixed at 3,150 Hz, but this is never true). Next, the short-term variation of the actual frequency over time is compared to the average, and weighted using the filter shown above. The result shows the instantaneous frequency variations over the length of the captured signal, relative to the average frequency (however, the effect of very slow and very fast changes have been reduced by the filter). Finally, the standard deviation of the variation from the average is calculated, and multiplied by 2 (hence “2-Sigma”, or “two times the standard deviation”), resulting in the value that is shown as the specification. The reason two standard deviations is chosen is that (in the typical case where the deviation has a Gaussian distribution) the actual Wow & Flutter value should exceed this value no more than 5% of the time.

References

All of these are available online. Some of them require you to purchase them (or be a member of an organisation).

  • “Tracking Angle in Phonograph Pickups”
    B. B. Bauer. Electronics magazine, March 1945
  • “Minimising Pickup Tracking Error”
    Dr. John D. Seagrave, Audiocraft Magazine, December 1956, January 1957, and August 1957
  • “Understanding Phono Cartridges”
    S.K. Pramanik, Audio magazine, March 1979
  • “Tonearm Geometry and Setup Demystified”
    Martin D. Kessler and B.V.Pisha, Audio magazine, January 1980
  • “Understanding Tonearms”
    S.K. Pramanik, Audio magazine, June 1980
  • “Analytic Treatment of Tracking Error and Notes on Optimal Pick-up Design”
    H.G.Baerwald, Journal of the Society of Motion Picture Engineers, December 1941
  • “Pickup Arm Design”
    J.K. Stevenson, Wireless World magazine, May 1966, and June 1966
  • “The Optimum Pivot Position on Tonearm”
    S. Takahashi et. al., Audio Engineering Society Preprint no. 1390 (61st Convention, November 1978)
  • “Audible Effects of Mechanical Resonances in Turntables”
    Brüel and Kjær Application Note (1977)
  • “Basic Disc Mastering”; “
    Larry Boden (1981)
  • “Cartridge / Arm / Turntable Followup: Loose Ends and New Developments”
    The Audio Critic, 1:43 (Spring/Fall, 1978)
  • “Have Tone Arm Designers Forgotten Their High-School Geometry?”
    The Audio Critic, 1:31 (Jan./Feb. 1977).
  • “How the Stereo Disc Works”
    Radio-Electronics, (July 1958)
  • “Manual of Analogue Sound Restoration Techniques”
    Peter Copeland (2008)
  • “On the Mechanics of Tonearms”
    Dick Pierce (2005)
  • “Reproduction of Sound in High-Fidelity and Stereo Phonographs”
    Edgar Villchur (1966)
  • Journal of the Audio Engineering Society (www.aes.org)
    • “Centennial Issue: The Phonograph and Sound Recording After One-Hundred Years”
      Vol. 25, No. 10/11 (Oct./Nov. 1977)
    • “Factors Affecting the Stylus / Groove Relationship in Phonograph Playback Systems”
      C.R. Bastiaans; Vol. 15 Issue 4 (Oct. 1967)
    • “Further Thoughts on Geometric Conditions in the Cutting and Playing of Stereo Disk”
      C.R. Bastiaans; Vol. 11 Issue 1 (Jan. 1963)
    • “Record Changers, Turntables, and Tone Arms-A Brief Technical History”
      James H. Kogen; Vol. 25 (Oct./Nov. 1977)
    • “Some Thoughts on Geometric Conditions in the Cutting and Playing of Stereodiscs and Their Influence on the Final Sound Picture”
      Ooms, Johan L., Bastiaans, C. R.; Vol. 7 Issue 3 (Jul. 1959)
    • “The High-Fidelity Phonograph Transducer”
      B.B. Bauer; Vol. 25 Issue 10/11 (Nov. 1977)
  • DIN Standards
    • 45 500: Hi-Fi Technics: Requirements for Disk Recording Reproducing Equipment
    • 45 507: Measuring Apparatus for Frequency Variations in Sound Recording Equipment
    • 45 538: Definitions for Disk Record Reproducing Equipment
    • 45 539: Disk Record Reproducing Equipment: Directives for Measurements, Markings, and Audio Frequency, Connections, Dimensions of Interchangeable Pickups, Requirements of Playback Amplifiers
    • 45 541: Frequency Test Record St 33 and M 33 (33 1/3 rev/min; Stereo and Mono)
    • 45 542: Distortion Test Record St 33 and St 45 (33 1/3 or 45 rev/min; Stereo)
    • 45 543: Frequency Response and Crosstalk Test Record
    • 45 544: Rumble Measurement Test Record St 33 and M 33 (33 1/3 rev/min; Stereo and Mono)
    • 45 545: Wow and Flutter Test Records, 33 1/3 and 45 rev/min
    • 45 546: Stereophonic Disk Record St 45 (45 rpm)
    • 45 547: Stereophonic Disk Record St 33 (33 1/3 rpm)
    • 45 548 Aptitude for Performance of Disk Record Reproducing Equipment
    • 45 549: Tracking Ability Test Record
  • IEC Publications
    • 98: Recommendations for Lateral-Cut Commercial and Transcription Disk Recordings
    • 98: Processed Disk Records and Reproducing Equipment
    • 386: Method of Measurement of Speed Fluctuations in Sound Recording and Reproducing Equipment