Explaining Loudspeaker Behavior: Current Carrying Coils Equation

[juicebox12]

Hi there,
I'm doing an assignment on current carrying coils, and I am in quick need of an equation that is relevant to loudspeakers. I am not the brightest physics student (especially when it comes to magnetism and electricity), but if I work hard at a concept I can eventually get it.

Anyway, I need an equation that is relevant to how a loudspeaker works, in relation to current carrying coils. Can anyone help me out? An explanation of what the formula tells us and how I can use it to explain behaviour in the loudspeaker is necessary as well.

Thanks for your help, much appreciated!

 

[OlderDan]

Hi there,
I'm doing an assignment on current carrying coils, and I am in quick need of an equation that is relevant to loudspeakers. I am not the brightest physics student (especially when it comes to magnetism and electricity), but if I work hard at a concept I can eventually get it.

Anyway, I need an equation that is relevant to how a loudspeaker works, in relation to current carrying coils. Can anyone help me out? An explanation of what the formula tells us and how I can use it to explain behaviour in the loudspeaker is necessary as well.

Thanks for your help, much appreciated!

It may be a bit oversimplified, but a speaker is just a coil of wire in a magnetic field produced by a permanent magnet. A force will result if a current runs through the coil. If the current alternates direction, the force alternates direction. Current alternating at some frequency will result in motion at that frequency, causing the membrane of the speaker to oscillate at that frequency causing the air to vibrate at that frequency. The equations that describe such motion are the equations of forced harmonic motion, which have some complicated details, but are very similar to the equations for all harmonic motion. If a current of the form

$$i = Icos \omega t$$

is run through the coil, there will be a resulting motion of the form

$$x = A cos \omega t$$

 

[roger]

It may be a bit oversimplified, but a speaker is just a coil of wire in a magnetic field produced by a permanent magnet. A force will result if a current runs through the coil. If the current alternates direction, the force alternates direction. Current alternating at some frequency will result in motion at that frequency, causing the membrane of the speaker to oscillate at that frequency causing the air to vibrate at that frequency. The equations that describe such motion are the equations of forced harmonic motion, which have some complicated details, but are very similar to the equations for all harmonic motion. If a current of the form

$$i = Icos \omega t$$

is run through the coil, there will be a resulting motion of the form

$$x = A cos \omega t$$

But why is the displacement of the cone inversely proportional to the frequency ?

 

[OlderDan]

But why is the displacement of the cone inversely proportional to the frequency ?

There is both a qualitative answer and a quantitative answer to this question. If the current did not change direction the force would always be in the same direction and would push the coil to an extreme position, likely ruining the speaker. Most output circuits to drive loudspeakers contain some sort of protention against direct currents. For a low frequency, the current stays in the same direction for a long time, giving the coil a lot of time to move in one direction before the current reverses and starts to push the coil the opposite way. For a high frequency, the force persists in one direction for a much shorter time, so the coil does not get a chance to move very far before the force direction reverses.

Think about a mass on a spring where you apply an oscillating force. If the applied force changes very slowly, the mass will move very slowly and the spring force will always be about equal to the applied force. The mass will just follow the applied force. But if you change the applied force very rapidly, the spring never gets a chance to stretch much and the mass is pulled back to the central position by the applied force rather than the spring. The effect of the spring becomes almost of no consequence and the mass follows the force, but lags behind it. Much of the time the force is in one direction while the mass is moving in the other direction. The mass never gets to move very far.

The mathematics is a bit complex, but the result is that the amplitude is more or less inversely proportional to the frequency of the applied force. The equation for the displacement of a driven oscillator can be found here

http://hyperphysics.phy-astr.gsu.edu/hbase/oscdr.html#c2

Look at the panel titled "Underdamped Driven Oscillator". That big expression for x(t) contains a complicated first term that is called a transient term that can be ignored in this context. The second terms shows the frequency dependent amplitude times the cosine that has the frequency of the driving force with a phase shift that also depends on the frequency.

 

[roger]

Why would it ruin the speaker ?

But I thought that the actual applied voltage is a constant, so even if it were alternating at a lower rate, wouldn't the same voltage and therefore displacement be achieved ?

( meaning if f(x)=sinx , then just f(1/2x) etc. )

 

[OlderDan]

Why would it ruin the speaker ?

But I thought that the actual applied voltage is a constant, so even if it were alternating at a lower rate, wouldn't the same voltage and therefore displacement be achieved ?

( meaning if f(x)=sinx , then just f(1/2x) etc. )

Speakers are not designed to withstand displacements beyond certain limits. A direct current would keep pushing the coil in the same direction until something strong enough to counter the force stopped the motion, or perhaps got cracked or torn from the excess strain. A continuous field produced by the coil with a direct current might also affect the magnetic properties of the permanent magnet. Furthermore, during normal operation a dc component in the current is undesireable because it would shift the equilibrium postion of the coil, potentially distorting the sound waves produced by the alternating portion of the current.

In what sense do you think the applied voltage is constant? I don't understand your point about f(x) = sinx and f(1/2x). What is x?

And no, the same displacement would not be achieved for the same applied voltage. The displacement would depend on the frequency for the reasons already stated. You need to think about what it means to rapidly change the direction of a force applied to an object. A force will always accelerate an object. An acceleration that only lasts for a very short time is not going to have much effect on the velocity of the object, or its resulting displacement. The same magnitude of force with a persistent direction will have a much bigger impact on the velocity and resulting displacement of that object.

 

[roger]

Speakers are not designed to withstand displacements beyond certain limits. A direct current would keep pushing the coil in the same direction until something strong enough to counter the force stopped the motion, or perhaps got cracked or torn from the excess strain. A continuous field produced by the coil with a direct current might also affect the magnetic properties of the permanent magnet. Furthermore, during normal operation a dc component in the current is undesireable because it would shift the equilibrium postion of the coil, potentially distorting the sound waves produced by the alternating portion of the current.

In what sense do you think the applied voltage is constant? I don't understand your point about f(x) = sinx and f(1/2x). What is x?

And no, the same displacement would not be achieved for the same applied voltage. The displacement would depend on the frequency for the reasons already stated. You need to think about what it means to rapidly change the direction of a force applied to an object. A force will always accelerate an object. An acceleration that only lasts for a very short time is not going to have much effect on the velocity of the object, or its resulting displacement. The same magnitude of force with a persistent direction will have a much bigger impact on the velocity and resulting displacement of that object.

My point was that sin 1/2x is just half the original frequency, but nevertheless, the amplitude is still 1.

So I just wondered why as you say, the speaker doesn't have a chance to reach the applied voltage as frequency increases ?

 

[OlderDan]

My point was that sin 1/2x is just half the original frequency, but nevertheless, the amplitude is still 1.

So I just wondered why as you say, the speaker doesn't have a chance to reach the applied voltage as frequency increases ?

I never said that it doesn't have a chance to reach applied voltage. I said it never has a chance to move very far before the force reverses direction.

For a high frequency, the force persists in one direction for a much shorter time, so the coil does not get a chance to move very far before the force direction reverses.

The change in momentum of an object is equal to the impulse applied. Impulse is the force times the duration time of that force. When the force is constantly changing the impulse can be approximated by breaking time into small intervals and multiplying by the average force for each interval, then adding all the force*time products. In calculus, the sum is done in the limit as the time intervals approach zero and is called an integral. If you look at two sinusoidal forces of the same amplitude, where one is twice the frequency of the other, the impulse in one direction from one half cycle for the lower frequency will be twice the impulse from the higher frequency. Greater impulse will produce greater momentum, velocity, and kinetic energy, resulting in greater amplitude.