η0 - The reference or "power available" efficiency of the driver, in percent.
The expression ρ/2πc can be replaced by the value 5.445×10^-4 m^2*s/kg for dry air at 25 °C. For 25 °C air with 50% relative humidity the expression evaluates to 5.365×10^-4.
A version more easily calculated with typical published parameters is:
The expression 4π^2/c^3 can be replaced by the value 9.523×10^–7 s³/m³ for dry air at 25 °C. For 25 °C air with 50% relative humidity the expression evaluates to 9.438×10^–7.
From the efficiency, we may calculate sensitivity, which is the sound pressure level a speaker produces for a given input:
A speaker with an efficiency of 100% (1.0) would output a watt of energy for every watt input. Considering the driver as a point source in an infinite baffle, at one meter this would be distributed over a hemisphere with area 2π m² for an intensity of (1/(2π))=0.159154 W/m², which gives an SPL of 112.02 dB.
SPL at 1 meter for an input of 1 watt is then: dB(1 Watt) = 112.02+ 10*log(η0)
SPL at 1 meter for an input of 2.83 Volts is then: dB(2.83V) = dB(1 Watt) + 10*log(8/Re) = 112.02+ 10*log(η0) + 10*log(8/Re)
The expression ρ/2πc can be replaced by the value 5.445×10^-4 m^2*s/kg for dry air at 25 °C. For 25 °C air with 50% relative humidity the expression evaluates to 5.365×10^-4.
A version more easily calculated with typical published parameters is:
The expression 4π^2/c^3 can be replaced by the value 9.523×10^–7 s³/m³ for dry air at 25 °C. For 25 °C air with 50% relative humidity the expression evaluates to 9.438×10^–7.
From the efficiency, we may calculate sensitivity, which is the sound pressure level a speaker produces for a given input:
A speaker with an efficiency of 100% (1.0) would output a watt of energy for every watt input. Considering the driver as a point source in an infinite baffle, at one meter this would be distributed over a hemisphere with area 2π m² for an intensity of (1/(2π))=0.159154 W/m², which gives an SPL of 112.02 dB.
SPL at 1 meter for an input of 1 watt is then: dB(1 Watt) = 112.02+ 10*log(η0)
SPL at 1 meter for an input of 2.83 Volts is then: dB(2.83V) = dB(1 Watt) + 10*log(8/Re) = 112.02+ 10*log(η0) + 10*log(8/Re)
Perhaps explain what you want to achieve, since the question you want answered seems to be a moving target.
It sounds like you want frequency response calculations - you may find them (and a ready made spreadsheet) at www.diysubwoofers.org
It sounds like you want frequency response calculations - you may find them (and a ready made spreadsheet) at www.diysubwoofers.org
Ah, you are rediscovering Hoffman's Iron Law, then?
Efficiency = k*Vb*Fc^3 (or use efficiency equation above)
Without going through a whole derivation and achieving a true relation, you could substitute some curve fit relations for vented box such as:
Vb=20*Vas*Qts^3.3
Fc=.42*Fs*Qts^-0.96
to back out some parameters.
Sealed box can be done analytically with
Qtc/Qts=Fc/Fs=sqrt(Vas/Vb+1)
I think you will find you are always bouncing off of constraints you need to place on Fs, Vas, Qes, Qts, etc to make a realizable driver. You often end up with a Fs of 3 and a Vas of 10000 and a Qts of 0.13 or some other such nonsense.
Efficiency = k*Vb*Fc^3 (or use efficiency equation above)
Without going through a whole derivation and achieving a true relation, you could substitute some curve fit relations for vented box such as:
Vb=20*Vas*Qts^3.3
Fc=.42*Fs*Qts^-0.96
to back out some parameters.
Sealed box can be done analytically with
Qtc/Qts=Fc/Fs=sqrt(Vas/Vb+1)
I think you will find you are always bouncing off of constraints you need to place on Fs, Vas, Qes, Qts, etc to make a realizable driver. You often end up with a Fs of 3 and a Vas of 10000 and a Qts of 0.13 or some other such nonsense.
thadman,
If by the equation for "sensitivity vs frequency," you mean the equation that gives you the sound pressure at 1m from the speaker with a constant voltage input as a function of frequency, then yes, there is such an equation. If the following assumptions are made, the equation below describes just this:
- radiation into half-space
- sealed enclosure on the rear
- low frequencies (so the radiation is omnidirectional, the radiation impedance looks like a mass, the cone acts as one lumped mass, and the air in the box acts as one lumped spring)
- small signal (speaker is linear)
Parameters:
Mm: effective moving mass (including air out front)
Rm: lumped mechanical resistance of moving parts
Cm: lumped compliance of the suspension
Bl: force factor
Le: voice coil inductance
Re: voice coil resistance
Sd: piston area
Vb: rear enclosure volume
constants:
rho: density of air (~1.2 kg/m^3)
gamma: heat capacity ratio (~1.4)
Po: rest pressure of air (~10^5 N/m^2)
variables:
s: complex frequency (s = i*w = i*2*pi*f where w is angular frequency and f is frequency)
Then the sound pressure at 1 meter, with 1V of input is:
p = rho*Bl*Sd*s^2 / [2*pi * (Le*s + Re) * (Mm*s^2 + Bl^2*s/(Le*s + Re) + 1/Cm + Sd^2*gamma*Po/Vb]
Note this is a complex transfer function. The magnitude is obtained by taking the magnitude of this expression. Also, if you want to display it in dB you need to take the ratio of the magnitude to reference pressure (2*10^-5 N/m^2) and then take 20*log10 of that.
This is the equation that describes the LF behavior of the speaker in a sealed box. If you search around the net you'll find the relationship between T/S parameters and the parameters above- they are the same thing, just described a bit differently. The reference efficiency, Hoffman's iron law, etc all come from this. This is the equation from which you might be able to obtain
"an equation that defines the optimal driver parameters and box volume for a given bandwidth (ie highest sensitivity over that bandwidth)"
I can't tell precisely what you mean by the last statement, but whatever it is, you might be able to get it from this.
There is a corresponding equation for vented enclosures. I can figure that out and put it on here if you want it.
If by the equation for "sensitivity vs frequency," you mean the equation that gives you the sound pressure at 1m from the speaker with a constant voltage input as a function of frequency, then yes, there is such an equation. If the following assumptions are made, the equation below describes just this:
- radiation into half-space
- sealed enclosure on the rear
- low frequencies (so the radiation is omnidirectional, the radiation impedance looks like a mass, the cone acts as one lumped mass, and the air in the box acts as one lumped spring)
- small signal (speaker is linear)
Parameters:
Mm: effective moving mass (including air out front)
Rm: lumped mechanical resistance of moving parts
Cm: lumped compliance of the suspension
Bl: force factor
Le: voice coil inductance
Re: voice coil resistance
Sd: piston area
Vb: rear enclosure volume
constants:
rho: density of air (~1.2 kg/m^3)
gamma: heat capacity ratio (~1.4)
Po: rest pressure of air (~10^5 N/m^2)
variables:
s: complex frequency (s = i*w = i*2*pi*f where w is angular frequency and f is frequency)
Then the sound pressure at 1 meter, with 1V of input is:
p = rho*Bl*Sd*s^2 / [2*pi * (Le*s + Re) * (Mm*s^2 + Bl^2*s/(Le*s + Re) + 1/Cm + Sd^2*gamma*Po/Vb]
Note this is a complex transfer function. The magnitude is obtained by taking the magnitude of this expression. Also, if you want to display it in dB you need to take the ratio of the magnitude to reference pressure (2*10^-5 N/m^2) and then take 20*log10 of that.
This is the equation that describes the LF behavior of the speaker in a sealed box. If you search around the net you'll find the relationship between T/S parameters and the parameters above- they are the same thing, just described a bit differently. The reference efficiency, Hoffman's iron law, etc all come from this. This is the equation from which you might be able to obtain
"an equation that defines the optimal driver parameters and box volume for a given bandwidth (ie highest sensitivity over that bandwidth)"
I can't tell precisely what you mean by the last statement, but whatever it is, you might be able to get it from this.
There is a corresponding equation for vented enclosures. I can figure that out and put it on here if you want it.
The answer
Thanks Rybaudio for the kind response.
It seems there are some errors unfortunately.
For completeness, here is a very helpful document by Francis Deck:
http://personalpages.tds.net/~fdeck/bass/speaker2.pdf
He slogged through quite a lot of algebra and kindly shared the thought process for other souls in search of answers.
Thanks Rybaudio for the kind response.
It seems there are some errors unfortunately.
For completeness, here is a very helpful document by Francis Deck:
http://personalpages.tds.net/~fdeck/bass/speaker2.pdf
He slogged through quite a lot of algebra and kindly shared the thought process for other souls in search of answers.
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