Deriving the relation between speaker sound pressure level and input power from amplifier

[SYRINX]

Would love some help with a question I have been struggling with. I feel that I must be misunderstanding something about decibel scales or speaker mechanisms.

In discussions about how much amplifier power

(in watts, W) is required to achieve a desired sound pressure level (SPL)

(in dB SPL) from a speaker with a sensitivity of

(also in dB SPL) at the reference listening position of 1 meter, one finds the following equation:

But where does this equation come from? It is not clear to me why a 1 dB increase in input power level from the amplifier should result in exactly 1 dB increase in output SPL from the speaker, which is what this equation says. I would think that the relation between speaker output SPL and input power level would be dependent on the specific material properties of each speaker so that the relation would need to be measured for each speaker or derived from a computational model of the speaker.

The only thing I can think is that we assume the speaker's output sound power to be the same as its input power, and we know (see e.g. this reference) that for a given listening distance

and speaker directivity factor

, the sound pressure level is related to the sound power level by

so that a 1 dB increase in input power level => a 1 dB increase in output sound power level => a 1 dB increase in output sound pressure level. But if that's the case, I don't know why we can assume that a speaker's output sound power level is equal to its input power level.

Would like to know where I'm going wrong here. Thanks for any help you can provide.

 

[Blumlein 88]

Q the directivity factor is the cause for your confusion. Whatever the Q for a speaker is will not change with power level. What the latter equation is showing you is the SPL change of an omni vs a speaker that is not omnidirectional. If a speaker is directional for a given power, then the SPL at some points will be greater than the SPL for an omnidirectional speaker for the total power put into the air. It will still be true, that for any given measuring point, with any given Q factor, that you see a linear increase in SPL at that measuring point.

For Q = 1 (full sphere propagation) the sound power level is equal to sound pressure level or intensity level at the distance of r = 0.2821 m from the source.

BTW, looks like your first post so welcome to ASR.

Maybe think of it this way. If you put power into a speaker that is omnidirectional, and you measure 90 db SPL at 1 meter away you will get that measurement at any location. Now imagine a speaker that only put sound out into 1/2 of the sphere. Feeding it the same power as before on one side of the speaker you will measure more than 90 db SPL because the same power is being radiated into less space. On the other side of the speaker you would measure nothing as no power is radiated that way. Not saying speakers are exactly like this, but that is the reason you need a Q factor in that latter equation.

This link also explains it pretty well in regards to Q.

What is Directivity?

 

[SYRINX]

Hi @Blumlein 88 , thanks for your reply. I am not so worried about the second equation relating sound pressure level to sound power level and the Q factor. I merely put that equation in to explain my reasoning about where I think the first equation relating sound pressure level to input power from the amplifier comes from. That is the equation I am confused about; I'm not sure why we can say that a 1 dB increase in input power level from an amplifier creates a 1 dB increase in output SPL from the speaker without first modeling or measuring it to determine that this relation holds.

 

[NTK]

Hi @Blumlein 88 , thanks for your reply. I am not so worried about the second equation relating sound pressure level to sound power level and the Q factor. I merely put that equation in to explain my reasoning about where I think the first equation relating sound pressure level to input power from the amplifier comes from. That is the equation I am confused about; I'm not sure why we can say that a 1 dB increase in input power level from an amplifier creates a 1 dB increase in output SPL from the speaker without first modeling or measuring it to determine that this relation holds.

dB is a ratio in the logarithmic (base 10) scale. An addition of 1 dB means, when used for power, it is multiplied by 10^(1/10) = 1.259x.

Therefore an increase in 1 dB in the input power means the power is increased by 1.259x. For a linear system, the output will also increase by 1.259x, which is 1 dB.

 

[SYRINX]

dB is a ratio in the logarithmic (base 10) scale. An addition of 1 dB means, when used for power, it is multiplied by 10^(1/10) = 1.259x.

Therefore an increase in 1 dB in the input power means the power is increased by 1.259x. For a linear system, the output will also increase by 1.259x, which is 1 dB.

Thanks @NTK, this is helpful, it was the linearity of the speaker that I was unsure about. However, even though output power increases proportionally to input power, the sound pressure level is a root power quantity, not itself a power quantity, while the input power is a power quantity. So why is it that 1 dB increase in the input power level leads to a 1 dB increase in the sound pressure level?

 

[boxerfan88]

dB is just a ratio.

Let's see if the below example helps ...

Assume:

• Speaker sensitivity is 90db SPL @ 1m @ 1watt
• Amplifier gives out 0.1watt if given input of 10dBm
• The chain is linear, no losses, no compression.

Theory:

• Input (10dBm) -> amplifier -> Output (0.1 watt) -> speaker -> 80dBSPL @1m
• If we increase input to 20dBm (ie. 10dB higher), amplifier should output 1watt (10dB higher), which should get you 90dBSPL of sound pressure @ 1m.
• Input (+10dB) -> amplifier output (+10dB) -> speaker output (+10dB)

If you have a calibrated measurement mic (eg. UMIK) you can experiment and test this yourself.

• use UMIK & REW
• place the mic around 1m from speaker
• set preamp/DAC volume to -40dB (or something comfortable but not too loud)
• play a 1kHz sine tone
• use REW to measure the SPL and note it down
• increase preamp/DAC volume by 10dB
• measure the SPL again
• SPL should go up by 10dB

 

[Blumlein 88]

If you look near the end of Erin's speaker reviews he tests for differing response at different levels of output. There are sometimes changes that are somewhat significant. Most speakers don't change a whole lot in level. In essence if you increase input, does the output increase in step with it. Sometimes over some frequency ranges a speakers output may not be in lockstep with input. Usually it is not a big issue.

Erin's Audio Corner

The Sound of Science

www.erinsaudiocorner.com

 

[NTK]

Thanks @NTK, this is helpful, it was the linearity of the speaker that I was unsure about. However, even though output power increases proportionally to input power, the sound pressure level is a root power quantity, not itself a power quantity, while the input power is a power quantity. So why is it that 1 dB increase in the input power level leads to a 1 dB increase in the sound pressure level?

The rule-of-thumb for dB is that it is always (or at least should be) about a ratio of powers. That's why, when you are working on electricity, you use the formulas :

since power is proportional to voltage squared or current squared.

It is similar for sound pressure. Below is from Wikipedia. You can see that sound intensity (acoustic power per area) is proportional to the square of the pressure amplitude, and it uses the 20 log10 form of the formula to give the "sound pressure level", in dB SPL, which is a ratio of quantities that are proportional to power.

 

[thulle]

So why is it that 1 dB increase in the input power level leads to a 1 dB increase in the sound pressure level?

That's not what you're doing tho, you're increasing the signal level, the voltage. Doubling voltage through same resistance gives four times the power. Squared.

 

[terryforsythe]

Expanding on what @NTK provided, perhaps the following example will help. SPL is proportional to the cone acceleration, which is proportional to the current through the voice coil. Using the above equations, if we increase the power by 3dB, we increase the voice coil current by 1.41, which increases the cone acceleration by 1.41. Thus, p/p0 = 1.41. Plug that into the SPL equation and you get 3dB.

 

[SYRINX]

The rule-of-thumb for dB is that it is always (or at least should be) about a ratio of powers. That's why, when you are working on electricity, you use the formulas :
View attachment 438392
View attachment 438395
since power is proportional to voltage squared or current squared.

It is similar for sound pressure. Below is from Wikipedia. You can see that sound intensity (acoustic power per area) is proportional to the square of the pressure amplitude, and it uses the 20 log10 form of the formula to give the "sound pressure level", in dB SPL, which is a ratio of quantities that are proportional to power.
View attachment 438394

Thanks again @NTK , it was that factor of 2 difference in the conversion to dB between SPL and the amplifier power that was tripping me up. As I look back at your first post, I realize I have been overthinking the whole matter. I see now that for any linear system, any linear function of the output will also change proportionally to the input, and when expressed in dB, this change will be equal to the change in input. Thanks all for your input, I appreciate your patience.

 

[SYRINX]

Expanding on what @NTK provided, perhaps the following example will help. SPL is proportional to the cone acceleration, which is proportional to the current through the voice coil. Using the above equations, if we increase the power by 3dB, we increase the voice coil current by 1.41, which increases the cone acceleration by 1.41. Thus, p/p0 = 1.41. Plug that into the SPL equation and you get 3dB.

Thanks so much @terryforsythe , this is exactly the sort of derivation from first principles that I was looking for. It all makes sense now.