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WOOFER SPEED Woofer Speed
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Adire Audio
®
All Rights Reserved. Visit Adire Audio at http://www.adireaudio.com
WOOFER SPEED
There's a common misconception out there that heavy woofers must be "slow", and light woofers must be
"fast". If a woofer A's moving mass is higher than woofer B's, then woofer A is probably going to be
sloppy, or slow and inaccurate. Can't keep up with the bass line. Woofer A simply can't respond as fast as
woofer B.
There's also this concept that the "acceleration factor" (BL/Mms) is an indicator of woofer speed/transient
response. High BL, combined with a low Mms, should give great transient response, right?
Well, on surface these might sound like logical assumptions. However, they are in fact incorrect! More to
the point, moving mass has precious little to do with woofer speed or signal response! And we'll prove it...
Go back to good old Newtonian physics... We're going to start with the famous law:
F=ma (eq 1)
Or force equals mass times acceleration. Anyone who's been through introductory physics (or watched a
bit of PBS) should be familiar with the equation above. It's pretty much the bedrock equation of Newtonian
physics - it's number 2 of the Big 3 Newtonian equations (the first being about interia, and the third being
about complementary/opposed actions).
Now, let's look at a loudspeaker... What do we have? We have a coil of wire which creates an alternating
magnetic field which interacts with the static magnetic field in the gap (see our page on DVC subs for a bit
more information on this). How does the alternating magnetic field of the coil come into being? Well, the
magnetic field is created by passing a current through the voice coil. As the current through the voice coil
changes, the magnetic field created by the coil changes. This field interacts with the static magnetic field
of the permanent magnet, and you get a force - the cone moves in and out. Just like two permanent
magnets will attract/repel each other depending upon how they are oriented. And a bigger current means
more force. Just like bigger/stronger permanent magnets mean stronger attraction/repulsion.
Additionally, what if we make the field of the permanent magnet stronger? Well, that's call the B field in
the gap. Increase B, we increase the force as well. Or, what if we could somehow make the magnetic field
from the voice coil stronger? We can - increase the number of turns of the voice coil (increase the Length
of the wire in the gap). Guess what - in both cases, we increased the BL of the speaker (yes, this is the BL
of the T/S parameters - now you know where it comes from!).
Now, let's go back to equation 1... Let's define each of the terms in that equation so we know what we're
talking about:
F= Force
m = mass (moving mass)
a = acceleration Woofer Speed
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Adire Audio
®
All Rights Reserved. Visit Adire Audio at http://www.adireaudio.com
So, what is the Force (NO Star Wars jokes, please!). From above, we see that the force on the cone is the
motor force factor (BL) times the current. So, let's rewrite equation 1 in these terms that are applicable to
the loudspeaker:
BLi = ma (eq 2)
So, the Motor Force Factor BL times the current i equals the moving mass of the driver m times the
acceleration of the driver a. Note that we have italicized i and a. There's a reason for it!
Now, back to the original question - transient response of a driver. What is transient response? Simply a
measure of how fast the driver can respond to the input signal. That means that - inherently - there is a time
dependency on the driver. How much TIME elapses before the driver responds. So, let's look at equation
2, and cancel out all terms that are not based on time. After all, if a quantity is time-invariant, then it won't
affect time-variant effects like transient response (think of this as a simple offset).
Or, to put it in an analogy, does the 1/4 mile time of a vehicle depend upon where you start? No, the
distance over which you measure is still 1/4 mile. So whether you start in front of your house and go
straight 1/4 mile, or motor on down to the dragstrip and go 1/4 mile doesn't really affect the car's 1/4 mile
time. The 1/4 mile time is strictly dependent on how fast the car can accelerate from a dead stop over a 1/4
mile length.
So, looking at equation 2 we see that:
BL is time invariant, assuming small excursions (assume an ideal motor with a flat BL curve; I
know, most drivers don't have that, but assume that it does, like our XBL2 enabled motors). So
BL is essentially a constant.
i is the current into the driver (we used italics to indicate a parameter that is time-variant). This is
the music, or test tone sweep, or whatever signal is coming from the amp. It's an AC waveform so
by definition it changes with time.
m is mass. Well, if the moving mass of the driver is changing as you operate, you've got big
problems! The weight of the cone, dustcap, former, voice coil, surround, and spider are pretty
much fixed. The don't change either. So m is essentially a constant.
a is the acceleration. This is what we're after. After all, the rate of change of acceleration IS the
transient response - it's what dictates how fast the driver can change speed, which also means it
dictates how fast the driver can move from position to position. And note that it's in italics, too.
After all, acceleration in the time-variant parameter we care about here!
So, let's rewrite equation 2, and replace the time-invariant parameters with a simple "C" to indicate a
constant (a parameter that does not change with time):
Ci = Ca (eq 3) or i :: a (eq 4)
(note: the "::" symbol is the mathematical symbol for proportionality; that is, i is
proportional to a).
Interesting! This says that the change in acceleration of a driver - how fast it can change position - is
strictly a function of the current through the driver. In fact, if you could make the current change infinitely
fast, then the driver would accelerate infinitely fast, and we'd have infinite transients - zero time to change
between states. Infinite frequency response.
So, now that we know that current is the driving force (pun somewhat intended) behind driver acceleration
changes, let's look at what limits how fast we can ramp current through the driver. Because if we are not
restricted in how fast we can change the current, then we are not restricted in how fast the driver can
accelerate - transient response is not limited at all. Woofer Speed
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Copyright
©
Adire Audio
®
All Rights Reserved. Visit Adire Audio at http://www.adireaudio.com
So, back to that loudspeaker model... A loudspeaker is a coil of wire wound on a former that attaches to the
cone. The current flows through the coil, creating an alternating magnetic field that interacts with the static
magnetic field of the permanent magnet. So, what could limit current flow? Well, what does a voice coil
look like?
How about an inductor? You know, those coils of wire (hey, isn't that what a voice coil is) that you use in
crossovers? Guess what - a voice coil IS an inductor! In fact, an inductor stores its energy in the magnetic
field (as opposed to a capacitor which stores its energy in the electric field). It is this magnetic field of the
voice coil "inductor" that interacts with the permanent magnet field we talked about above. Hey, a
loudspeaker is an inductor hung on the end of a cone in a static magnetic field!
So, what about an inductor will alter the way current flows? Well, inductors don't like to have the current
flowing through them change. They like to hold the current constant. They will allow you to change the
current flowing in them, but the bigger the inductor (or, the higher the measured inductance) the longer it
will hold the current before it starts to change (I'll leave it to the reader to go research inductance on their
own, to learn why this happens).
So, the voice coil is an inductor. And we see that inductors don't like to change current. But we also see
from equation 4 above that we need to change the current if we want to change the acceleration. So, the
voice coil doesn't want us to change the current. How good is it at holding the current? Depends upon the
inductance! The higher the inductance of the driver, the longer it can hold the current flowing through it.
Which means the more time elapses before it starts to respond to the amplifier's applied voltage. Which
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