Tech Design.....

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Below is a screen shot of a Excel spreadsheet that simulates the response of a driver to an abruptly

switched on sine wave input through solution of the differential equation governing the driver's

motion. The spreadsheet computes the response of a linear system and a similar system that

optionally includes the nonlinearity of the box compliance (air compression/expansion) and/or

suspension compliance. Upon completion of the simulation the harmonic distortion in the nonlinear

solution is extracted. The suspension compliance is modeled generically.

To run the spreadsheet you enter the driver data (in red) and the code computes the driver

parameters in black and the purple data. Then enter the simulation input. If Fsim = Fb then the actual

cone excursion comes out very close to the input value unless you include the nonlinear suspension

compliance and the input value for excursion at Fb is significantly greater than Xmax. Changing the

desired Qtc will also change Fb and you should make appropriated changes to Fsim, as desired. The

nonlinear effects can be switched on or off. To include them in the nonlinear simulation enter "yes".

The "Actual excursion @ Fsim" is the max excursion that is computed in the simulation. In the linear

case this will be close to the value entered for the excursion at Fb when Fsim = Fb. When Fsim is

greater than Fb the actual excursion will increase to the input value/Qtc. Xref (purple) is the

normalizing length scale used in the plot of displacement vs. time. The compliance shape factor

controls the steepness of the reduction in suspension compliance as x exceeds Xmax. It should not

be entered less than 2. While not visible in the screen shoot, the variation of the suspension with

displacement is shown in the complete display.

If you turn off the air nonlinearity you will see that you can generate only odd order harmonics. This is

a result of the symmetry in the suspension compliance about the driver's rest position at x=0. Turning

off the suspension nonlinearity and only considering the air compliance nonlinearity shows that the

air nonlinearity can generate both even and odd order HD. Entering a value for Qtc very slightly

larger the Qts will result in a very large box volume and also result in very little air nonlinearity. If Vbox

comes out negative the values entered for Qtc is below Qts and should be corrected. If you care to

model a free air result or an open backed dipole you can turn off the air nonlinearity and examine

only the effect of the suspension nonlinearity. As noted, in this case only odd order HD is generated

at potentially high excursion. Since dipole woofers often operate near the linear excursion limits at

higher SPL if they do not poses sufficient cone area, there is a real possibility of significant odd

order distortion in such woofer systems. In real cases where the suspension compliance may exhibit

some asymmetry there is a possibility of even order HD as well. The even order HD can be

eliminated, or reduced, by using two woofers in a back configuration, but the odd order distortion can

not.

The plots shown below are top right, SS HD. Next down is driver (cone) velocity vs. time. Additional

plots presented in the complete spread sheet include blow ups of the velocity vs. time for the first and

last 4 cycles, displacement vs. time, variation in the nonlinear air compliance over the last simulated

cycle, and suspension and total system compliance over the last cycle (the same cycle used for the

HD calculation). Please note the correct axis labels for each color as noted in the plot title. Also

presented is a plot of the suspension compliance vs. displacement where you can see the affect of

the shape factor.

One interesting thing to do is to run the code at fairly high frequency, like 1k Hz, and look at the

displacement. Notice the way the driver responds to a sudden turn on of a sine wave when initially at

rest, and the differences in behavior with and without nonlinear effects. Distortion is not exactly

meaning full when you run these higher frequency cases since the solution does not always reach

steady state after the 30 cycles simulated. This can be verified by the behavior of the nonlinear error

since it fails to become periodic. But you will notice that as the frequency rises the distortion goes

down even while the excursion is held constant. This is correct because above resonance the motion

is mass controlled and the effect of the compliance on the driver motion is diminished. So at higher

frequency distortion is primarily from motor nonlinearity (BL nonlinearity) and other sources.

A generic model for nonlinear BL will be added in the future.

The spreadsheet can be downloaded by clicking on the appropriate button for a zip file (1meg) or an

rar file (0.5 meg). Have fun with it. Contact me with questions.

Regards,

John k....

switched on sine wave input through solution of the differential equation governing the driver's

motion. The spreadsheet computes the response of a linear system and a similar system that

optionally includes the nonlinearity of the box compliance (air compression/expansion) and/or

suspension compliance. Upon completion of the simulation the harmonic distortion in the nonlinear

solution is extracted. The suspension compliance is modeled generically.

To run the spreadsheet you enter the driver data (in red) and the code computes the driver

parameters in black and the purple data. Then enter the simulation input. If Fsim = Fb then the actual

cone excursion comes out very close to the input value unless you include the nonlinear suspension

compliance and the input value for excursion at Fb is significantly greater than Xmax. Changing the

desired Qtc will also change Fb and you should make appropriated changes to Fsim, as desired. The

nonlinear effects can be switched on or off. To include them in the nonlinear simulation enter "yes".

The "Actual excursion @ Fsim" is the max excursion that is computed in the simulation. In the linear

case this will be close to the value entered for the excursion at Fb when Fsim = Fb. When Fsim is

greater than Fb the actual excursion will increase to the input value/Qtc. Xref (purple) is the

normalizing length scale used in the plot of displacement vs. time. The compliance shape factor

controls the steepness of the reduction in suspension compliance as x exceeds Xmax. It should not

be entered less than 2. While not visible in the screen shoot, the variation of the suspension with

displacement is shown in the complete display.

If you turn off the air nonlinearity you will see that you can generate only odd order harmonics. This is

a result of the symmetry in the suspension compliance about the driver's rest position at x=0. Turning

off the suspension nonlinearity and only considering the air compliance nonlinearity shows that the

air nonlinearity can generate both even and odd order HD. Entering a value for Qtc very slightly

larger the Qts will result in a very large box volume and also result in very little air nonlinearity. If Vbox

comes out negative the values entered for Qtc is below Qts and should be corrected. If you care to

model a free air result or an open backed dipole you can turn off the air nonlinearity and examine

only the effect of the suspension nonlinearity. As noted, in this case only odd order HD is generated

at potentially high excursion. Since dipole woofers often operate near the linear excursion limits at

higher SPL if they do not poses sufficient cone area, there is a real possibility of significant odd

order distortion in such woofer systems. In real cases where the suspension compliance may exhibit

some asymmetry there is a possibility of even order HD as well. The even order HD can be

eliminated, or reduced, by using two woofers in a back configuration, but the odd order distortion can

not.

The plots shown below are top right, SS HD. Next down is driver (cone) velocity vs. time. Additional

plots presented in the complete spread sheet include blow ups of the velocity vs. time for the first and

last 4 cycles, displacement vs. time, variation in the nonlinear air compliance over the last simulated

cycle, and suspension and total system compliance over the last cycle (the same cycle used for the

HD calculation). Please note the correct axis labels for each color as noted in the plot title. Also

presented is a plot of the suspension compliance vs. displacement where you can see the affect of

the shape factor.

One interesting thing to do is to run the code at fairly high frequency, like 1k Hz, and look at the

displacement. Notice the way the driver responds to a sudden turn on of a sine wave when initially at

rest, and the differences in behavior with and without nonlinear effects. Distortion is not exactly

meaning full when you run these higher frequency cases since the solution does not always reach

steady state after the 30 cycles simulated. This can be verified by the behavior of the nonlinear error

since it fails to become periodic. But you will notice that as the frequency rises the distortion goes

down even while the excursion is held constant. This is correct because above resonance the motion

is mass controlled and the effect of the compliance on the driver motion is diminished. So at higher

frequency distortion is primarily from motor nonlinearity (BL nonlinearity) and other sources.

A generic model for nonlinear BL will be added in the future.

The spreadsheet can be downloaded by clicking on the appropriate button for a zip file (1meg) or an

rar file (0.5 meg). Have fun with it. Contact me with questions.

Regards,

John k....

Download zip file (1 meg)

Download rar file (0.5 meg)