#1

More hub material.

The purpose of this thread is to discuss mathematical manipulations and comparisons of sound intensities using the decibel scale. It can be very useful for a musician to get an approximate idea of how loud an amplifier can be when paired with a speaker (or set of speakers). You can use these equations to compare the nominal output of two different amplifiers or speakers, so if you are having trouble being heard in the band you can get an idea of what difference equipment can make.

Since we are getting ready to perform some math that uses decibels as input, I think it would be to our benefit to get some confusing aspects of decibels out of the way. As mentioned in the Principles of Sound thread, we use decibels because of the enormous dynamic range of values we are faced with and when we use a decibel scale this makes the numbers much more reasonable to deal with. But decibels are not just used ‘for calculating amp output’; decibels actually have quite a few abstract relationships with power, voltage, and loudness.

When we say we want a +3 dB increase in the audio signal, this means we will need twice as much power. Well twice as much power means twice as much voltage and loudness right? No, actually it doesn’t. A +3 dB only requires 1.4 times the voltage and only provides 1.23 times the loudness.

You should be asking: “What does this mean to me?” To put this in terms guitarist can appreciate, if you double your amp’s power output from 10 watts to 20 watts you will get a +3 dB increase, but this only provides 1.23 times the volume. To go more extreme, if you increase your amp’s power 100 times from 10 watts to 1000 watts you will get a +20 dB increase and your amp will be 4 times louder.

Below is a list of decibel increases associated with a ratio increase of the pertinent parameter.

+dB Change – Voltage – Power – Loudness

3 – 1.4X – 2X – 1.23X

6 – 2 – 4 – 1.52

10 – 3.16 – 10 – 2

20 – 10 – 100 - 4

40 – 100 – 10,000 - 16

As a side note, in the section below labeled “Adding Coherent Acoustic Signals” the resulting output increase can be a bit enigmatic if you don’t consider the attributes of decibels discussed above. Normally, when adding two acoustic sources you only add their power output; but when you are adding two Coherent Sources you actually add the voltages of the two sources together which results in a +3 dB boost to the signal over using a single speaker.

An amplifier’s output is rated in watts (which is a unit of energy conversion). A speaker’s loudness is called the sensitivity of the speaker, it is rated in decibels and the reference is taken at 1 meter with 1 watt of power supplied to the speaker. So we will use these calculations to determine nominal output:

Where P1 is the amp’s nominal output. So a 50 watt amp paired with a speaker of 100 dB sensitivity would be plugged in as such:

Nominal Output = (10*LOG10(50)) + 100 = 116.99 dB

So the nominal output of the 1x12 cabinet with a 50 watt head would be 116.99 dB.

When adding acoustic signals from two different sources that don’t share attributes we will use the formula:

Where S1 is first source, S2 is the second source. You can actually keep extending this formula to include as many sources as you like.

We must first calculate the nominal output for each source, let’s say S1 is a 25 watt amp and S2 is a 35 watt amp. We will once again assume a speaker with a 100 dB sensitivity for each amp to make the math easier.

S1 = (10*LOG10(25)) + 100 = 113.979

S2 = (10*LOG10(35)) + 100 = 115.441

So let’s plug this into our function

Incoherent Output = 10*LOG10(10^(113.979/10)+10^(115.441/10) = 117.781 dB

http://www.sengpielaudio.com/calculator-spl.htm

Due to coherent signal summing, a 2x12 cabinet would be added together differently than two 1x12 amplifiers projecting different signals

Where S1 is the signal output and N is the number of speakers producing the signal. Remember, sound power is divided evenly among the speakers so each speaker in a 2x12 should be ‘seeing’ 25 watts from a 50 watt amp, so S1partial will get calculated with 25 watts. We will once again assume a 100 dB speaker so that we get:

S1partial = (10*LOG10(25)) + 100 = 113.979 dB

Coherent Output = 113.979 + 20 log(2) = 120 dB

http://www.engineeringtoolbox.com/adding-decibel-d_63.html

For a more in depth explanation of summing coherent audio signals and how this phenomenon manifests itself in real world environments then please see the Mutual Coupling thread.

It is convenient to be able to see one dB output compared to another. Let’s go over how we may do that. Suppose you are running a 50 watt head with a single speaker with a 101 dB sensitivity (S1). The other guitarist in your band runs a 100 watt head with a 2x12 cabinet with speakers that have a 98 dB sensitivity (S2). We are going to calculate both of your nominal outputs and see which one is louder.

S1 = (10*LOG10(50)) + 101 = 117.990

So S1’s total output is 117.99 dB. Let’s calculate S2’s output, remember that power is distributed equally among speakers so this will be a two-part problem.

S2partial = (10*LOG10(50)) + 98 = 114.990 dB

S2 Coherent Output = 114.99 + 20 log(2) = 121.010 dB

S2’s total output will be 121.01 dB. To find out how much louder S2 is than S1 we will simply subtract the values:

121.010 – 117.990 = 3.021 dB

So S2 is 3.021 dB louder than S1. Is that a lot louder or a little bit louder? What does 3.021 dB mean to us?

Now that we performed that fancy math above, we get some answer that really doesn’t mean anything to us. Most people are not fluent in the language of decibels and decibels have different relations to power, intensity and loudness. It would be most useful if we could take these decibel values and put it in a simpler form, like a ratio that compares loudness of the two amplifiers.

Now, believe it or not, I am actually quite weak at mathematics but I have managed to derive a formula to suit our purposes (I don’t claim that this is the most elegant or simplest form of the equation, so if you know a simpler formula or can reduce this one please let me know and I’ll happily replace this clunky one)

So if we take our previous result comparing the two amplifiers (S2 was 3.021 dB louder than S1), then we can get a simple ratio of loudness

Ratio Louder = 2^(Log10(10^(3.021/10))) = 1.233

The nominal output of amplifier S2 is 1.233 times louder than the nominal output of amplifier S1. That is a number we can draw better conclusions from.

I should note that these values are calculated off of nominal ratings of the speaker and the amplifier. Considerations such as amplifiers being driven below or above nominal rating are not reflected in the above mathematics. Most contemporary amps come equipped with a master volume knob, so just because you have a 100 watt amp does not mean it is creating a 110+ dB output; that fact that most musicians run their amp below nominal ratings should be taken into consideration.

The purpose of this thread is to discuss mathematical manipulations and comparisons of sound intensities using the decibel scale. It can be very useful for a musician to get an approximate idea of how loud an amplifier can be when paired with a speaker (or set of speakers). You can use these equations to compare the nominal output of two different amplifiers or speakers, so if you are having trouble being heard in the band you can get an idea of what difference equipment can make.

**Decibels**Since we are getting ready to perform some math that uses decibels as input, I think it would be to our benefit to get some confusing aspects of decibels out of the way. As mentioned in the Principles of Sound thread, we use decibels because of the enormous dynamic range of values we are faced with and when we use a decibel scale this makes the numbers much more reasonable to deal with. But decibels are not just used ‘for calculating amp output’; decibels actually have quite a few abstract relationships with power, voltage, and loudness.

When we say we want a +3 dB increase in the audio signal, this means we will need twice as much power. Well twice as much power means twice as much voltage and loudness right? No, actually it doesn’t. A +3 dB only requires 1.4 times the voltage and only provides 1.23 times the loudness.

You should be asking: “What does this mean to me?” To put this in terms guitarist can appreciate, if you double your amp’s power output from 10 watts to 20 watts you will get a +3 dB increase, but this only provides 1.23 times the volume. To go more extreme, if you increase your amp’s power 100 times from 10 watts to 1000 watts you will get a +20 dB increase and your amp will be 4 times louder.

Below is a list of decibel increases associated with a ratio increase of the pertinent parameter.

+dB Change – Voltage – Power – Loudness

3 – 1.4X – 2X – 1.23X

6 – 2 – 4 – 1.52

10 – 3.16 – 10 – 2

20 – 10 – 100 - 4

40 – 100 – 10,000 - 16

As a side note, in the section below labeled “Adding Coherent Acoustic Signals” the resulting output increase can be a bit enigmatic if you don’t consider the attributes of decibels discussed above. Normally, when adding two acoustic sources you only add their power output; but when you are adding two Coherent Sources you actually add the voltages of the two sources together which results in a +3 dB boost to the signal over using a single speaker.

**Calculating Sound Output From a Loudspeaker and Amplifier (One amp into one speaker math)**An amplifier’s output is rated in watts (which is a unit of energy conversion). A speaker’s loudness is called the sensitivity of the speaker, it is rated in decibels and the reference is taken at 1 meter with 1 watt of power supplied to the speaker. So we will use these calculations to determine nominal output:

**Nominal Output = (10*LOG10(P1)) + sensitivity**Where P1 is the amp’s nominal output. So a 50 watt amp paired with a speaker of 100 dB sensitivity would be plugged in as such:

Nominal Output = (10*LOG10(50)) + 100 = 116.99 dB

So the nominal output of the 1x12 cabinet with a 50 watt head would be 116.99 dB.

**Adding Incoherent Acoustic Signals, Two Separate Guitar Amps with Separate Speakers Receiving Different Signals**When adding acoustic signals from two different sources that don’t share attributes we will use the formula:

**Incoherent Output = 10*LOG10(10^(S1/10)+10^(S2/10))**Where S1 is first source, S2 is the second source. You can actually keep extending this formula to include as many sources as you like.

We must first calculate the nominal output for each source, let’s say S1 is a 25 watt amp and S2 is a 35 watt amp. We will once again assume a speaker with a 100 dB sensitivity for each amp to make the math easier.

S1 = (10*LOG10(25)) + 100 = 113.979

S2 = (10*LOG10(35)) + 100 = 115.441

So let’s plug this into our function

Incoherent Output = 10*LOG10(10^(113.979/10)+10^(115.441/10) = 117.781 dB

http://www.sengpielaudio.com/calculator-spl.htm

**Adding Coherent Acoustic Signals (One amp into two speakers)**Due to coherent signal summing, a 2x12 cabinet would be added together differently than two 1x12 amplifiers projecting different signals

**Coherent Output = S1 + 20 log10(N)**Where S1 is the signal output and N is the number of speakers producing the signal. Remember, sound power is divided evenly among the speakers so each speaker in a 2x12 should be ‘seeing’ 25 watts from a 50 watt amp, so S1partial will get calculated with 25 watts. We will once again assume a 100 dB speaker so that we get:

S1partial = (10*LOG10(25)) + 100 = 113.979 dB

Coherent Output = 113.979 + 20 log(2) = 120 dB

http://www.engineeringtoolbox.com/adding-decibel-d_63.html

For a more in depth explanation of summing coherent audio signals and how this phenomenon manifests itself in real world environments then please see the Mutual Coupling thread.

**Comparing Decibels**It is convenient to be able to see one dB output compared to another. Let’s go over how we may do that. Suppose you are running a 50 watt head with a single speaker with a 101 dB sensitivity (S1). The other guitarist in your band runs a 100 watt head with a 2x12 cabinet with speakers that have a 98 dB sensitivity (S2). We are going to calculate both of your nominal outputs and see which one is louder.

S1 = (10*LOG10(50)) + 101 = 117.990

So S1’s total output is 117.99 dB. Let’s calculate S2’s output, remember that power is distributed equally among speakers so this will be a two-part problem.

S2partial = (10*LOG10(50)) + 98 = 114.990 dB

S2 Coherent Output = 114.99 + 20 log(2) = 121.010 dB

S2’s total output will be 121.01 dB. To find out how much louder S2 is than S1 we will simply subtract the values:

121.010 – 117.990 = 3.021 dB

So S2 is 3.021 dB louder than S1. Is that a lot louder or a little bit louder? What does 3.021 dB mean to us?

**Converting Decibels to Ratios**Now that we performed that fancy math above, we get some answer that really doesn’t mean anything to us. Most people are not fluent in the language of decibels and decibels have different relations to power, intensity and loudness. It would be most useful if we could take these decibel values and put it in a simpler form, like a ratio that compares loudness of the two amplifiers.

Now, believe it or not, I am actually quite weak at mathematics but I have managed to derive a formula to suit our purposes (I don’t claim that this is the most elegant or simplest form of the equation, so if you know a simpler formula or can reduce this one please let me know and I’ll happily replace this clunky one)

**Ratio Louder = 2^(Log10(10^(dBm/10)))**So if we take our previous result comparing the two amplifiers (S2 was 3.021 dB louder than S1), then we can get a simple ratio of loudness

Ratio Louder = 2^(Log10(10^(3.021/10))) = 1.233

The nominal output of amplifier S2 is 1.233 times louder than the nominal output of amplifier S1. That is a number we can draw better conclusions from.

**End Note**I should note that these values are calculated off of nominal ratings of the speaker and the amplifier. Considerations such as amplifiers being driven below or above nominal rating are not reflected in the above mathematics. Most contemporary amps come equipped with a master volume knob, so just because you have a 100 watt amp does not mean it is creating a 110+ dB output; that fact that most musicians run their amp below nominal ratings should be taken into consideration.

*Last edited by gumbilicious at May 14, 2013,*

#2

This is good work, and well presented. It's simple enough to understand without being condescending, and precise enough to provide valid output should someone care to use it.

Don't expect to be thanked for it when we use it later to debunk amp myths that arise on occasion. At that point your credibility, parentage, patriotism and sexual preference will all probably be questioned!

No good deed goes unpunished.

Don't expect to be thanked for it when we use it later to debunk amp myths that arise on occasion. At that point your credibility, parentage, patriotism and sexual preference will all probably be questioned!

No good deed goes unpunished.

#3

This is good work, and well presented. It's simple enough to understand without being condescending, and precise enough to provide valid output should someone care to use it.

Don't expect to be thanked for it when we use it later to debunk amp myths that arise on occasion. At that point your credibility, parentage, patriotism and sexual preference will all probably be questioned!

No good deed goes unpunished.

agreed, also the audience for this type of thread is pretty small, but the concepts it addresses (like handling dB's) can be enlightening to a slightly more technical reader who desires clarification. applying decibels to actual problems and comparing results can be useful for understanding the whole concept. it's helped me anyway.

311 had asked me to post more hub material so it can get exposed to the public for a bit before being submitted to the hub. mutual coupling is next.

*Last edited by gumbilicious at May 14, 2013,*

#4

311 had asked me to post more hub material so i can get exposed to the public for a bit before being submitted to the hub. mutual coupling is next.

So if I understand this right, your are going to expose yourself to the public, submit to the hub, then move on to mutual coupling?

And this was all instigated by 311?

Yeah yeah yeah, immature....so sue me!

#5