Feb 19th 1997

Distortion due to Non-Linear Effects of Air

Problem:

Some people are taking advantage of the fact that the compliance of the air in the box is a linear function of cone position. This is not true. Air does not have a linear restoring force and therefore there is distortion that results from this non-linearity. The real question is how much distortion is there, and is it significant?

Solution:

For adiabatic compression inside the box we have the equality:

where P is pressure, V is volume and y is the adiabatic compression constant for our gas (1.4 for air). Now if we find a 'k', called the change in volume, such that (1+k)*V1=V2 we can adjust our formula:

If P1 is the equilibruium pressure (101.5 KPa) and P2 is the pressure inside the box, we can relate the excursion, x to k by k = A*x/Vb. Where A is the area of the cone and Vb is the volume of the box. We can also relate the applied force on the cone, F, by F/A = P1-P2. Sub these two into (3) to get:

And we will use the computer to find the Taylor series expansion so we have x as a function of F.

If F is an applied force; F = K*sin(w*t)

A Fourier analysis gives us the amplitude of the first and second harmonics. This can be done easily if we see a little trig identity involving (sin(x))^2.

The second harmonic over the first results in the ratio between the two:

We will replace K. We assume the frequency is well below resonance, so the force on the cone is maximum when the excursion is at its maximum.

We can approximate the distortion of a NHT 1259 (A=.05067 m^2) in a 1.5 cubic foot box(.04248 m^3). We will take the worst case scenario where the driver is moving +/- 1 cm at a frequency well below resonance. This way the distortion due to the air is dominant.

The max force needed to bring the cone out(or in) this far is approximatly:

We sub this into (9) to get:

Thanks to Tom Danley; we should mention that this is amplitude distortion, not power distortion which is the square of this figure.

Remember, this is the distortion only due to the nonlinear effects of the air. Taylor's theorem says it is only an approximation, but it is a good one.

It is up to the individual to determine if this is a significant amount of distortion.