Chapter 4
PRACTICAL EXERCISE
Working with Decibels
Power Decibels
Without a Calculator
When working with decibels for power signals, you can use a rule of thumb called the “3
and 10 Rule” to help you solve some level conversions without the aid of the calculator.
This rule states that every time you double a power level there is a 3 dB increase.
Conversely, cutting a power level in half will result in a 3 dB decrease. Multiply a power
level by 10 and you will get a 10 dB increase. Dividing a power level by 10 will result in a
10 dB decrease.
We can prove the “3 and 10 Rule” mathematically based on the definition of power
difference decibels (or power decibels for short), denoted Power
.
Power
where
( )
is the original power and
is the power we are comparing to the original power.
First we show that each time you double the power, you increase the power by about 3 dB:
(
)
Now we show that each time you multiply the power by 10, you increase the power by
about 10 dB:
(
)
Combining the two observations, we have the following:
(
), Power
(
)
Digital Sound and Music
Working with Decibels
( )
PRACTICAL EXERCISE
Page 1
This material is based on work supported by the National Science Foundation under CCLI Grant DUE 0717743, Jennifer Burg PI, Jason Romney, Co-PI.
(
)
(
)
Using these basic rules, you can do many simple level conversions in your head without a
calculator. Consider the following questions:
Question 1: How many dB greater than 4 watts is 64 watts?
We can solve this problem using the “3” part of the “3 and 10 rule”:
. Thus,
Having shown that
we know that
.
Thus,
Answer: 64 watts is about 12 dB greater than 4 watts.
Question 2: How many dB greater than 10 watts is 1000 watts?
We can solve this problem using the “10” part of the “3 and 10 rule”:
Thus,
Having shown that
.
we know that
.
Thus,
Answer: 1000 watts is about 20 dB greater than 10 watts.
Question 3: How many dB greater than 1 watt is 80 watts?
We can solve this problem using both parts of the “3 and 10 rule”:
Digital Sound and Music
Working with Decibels
PRACTICAL EXERCISE
Page 2
This material is based on work supported by the National Science Foundation under CCLI Grant DUE 0717743, Jennifer Burg PI, Jason Romney, Co-PI.
Observe that
. Thus,
(
Having shown that
(
)
)
we know that
Thus,
If you prefer, you can do this calculation in your head stepwise this way:
Answer: 80 watts is 19 dB greater than 1 watt.
Question 4: How many dB less than 1 watt is 1 milliwatt (0.001 watts)?
We can solve this problem using the “10” part of the “3 and 10 rule”:
. Thus,
Having shown that
we know that
Thus,
(
)
Answer: 1 milliwatt is 30 dB less than 1 watt.
Question 5: How many dB less than 1 watt is 4 milliwatts (0.004 watts)?
Observe that
Having shown that
Thus,
Digital Sound and Music
Working with Decibels
(
).
(
)
(
we know that
)
PRACTICAL EXERCISE
Page 3
This material is based on work supported by the National Science Foundation under CCLI Grant DUE 0717743, Jennifer Burg PI, Jason Romney, Co-PI.
Answer: 4 milliwatts is 24 dB less than 1 watt.
With a Calculator
If you have a calculator, you can perform more complicated conversions using the formula
for power decibels. P0 is the power level you started with and P1 is the power level you end
up with.
Power
( )
Question 1: How many dB greater than 63 watts is 700 watts?
Power
(
)
Answer: 700 watts is 10.46 dB greater than 63 watts.
Question 2: How many dB greater than 3,000 watts is 10,000 watts?
Power
(
)
Answer: 10,000 watts is 5.23 dB greater than 3,000 watts.
Question 3: You want to purchase a new power amplifier for the sound system in your
car. You want an 800 watt amplifier but can only afford a 350 watt amplifier. How
many dB of sound are you going to sacrifice using the smaller amplifier?
Power
(
)
Answer: The 350 watt amplifier is only about 3.59 dB quieter than the 800 watt
amplifier. 3 dB is smallest difference in level that average listeners can perceive. In
other words, the difference in level between these two amplifiers will be barely
perceptible to the average listener.
Power Ratios
Sometimes you don't have two power levels to compare, but you know the dB difference
you’d like to achieve. In these situations you need to figure out the power ratio for the dB
Digital Sound and Music
Working with Decibels
PRACTICAL EXERCISE
Page 4
This material is based on work supported by the National Science Foundation under CCLI Grant DUE 0717743, Jennifer Burg PI, Jason Romney, Co-PI.
difference you need and then multiply that ratio by a known power level. The following
equation can be used for this purpose:
Question 1: You currently have a 300 watt amplifier in your sound system. You would
like to purchase an amplifier that is 13 dB louder. How many watts do you need?
Answer: You will need a 6000 watt amplifier (rounded up from 5986) to get another 13
dB out of your sound system. In this situation, it might be less expensive to get a more
sensitive loudspeaker or figure out a way to send your 300 watt amplifier a line-level
voltage signal that is 13 dB higher.
Question 2: If a loudspeaker can generate 60 dBSPL at 1 meter with 2 watts of power,
how much power would be required to get 98 dBSPL at 1 meter from the loudspeaker?
Answer: The power amplifier would need to put 2000 watts into the loudspeaker in
order to generate 98 dBSPL at 1 meter.
Energy Decibels
Without A Calculator
Any time you work with voltage levels (dBu and dBV) or pressure levels (dBSPL), you use
the formulas for energy difference decibels (or energy decibels for short, and sometimes
called voltage decibels, since sound pressure is often measured in volts). With energy
decibels, you can do some conversions in your head just like you do with power decibels.
This time the rule of thumb is called the “6 and 20 Rule.” Doubling a voltage will result in a
6 dB increase. Multiplying a voltage by 10 will result in a 20 dB increase. Distance changes
Digital Sound and Music
Working with Decibels
PRACTICAL EXERCISE
Page 5
This material is based on work supported by the National Science Foundation under CCLI Grant DUE 0717743, Jennifer Burg PI, Jason Romney, Co-PI.
work reciprocally. Doubling the distance a sound wave travels in the air will result in a 6
dB decrease in dBSPL. Multiplying by 10 the distance a sound wave travels in the air will
result in a 20 dB decrease in dBSPL.
We can prove the “6 and 20 Rule” mathematically based on the definition of voltage (or
energy) difference decibels, which we denote as Voltage
.
( )
where is the original voltage and
voltage.
is the voltage we are comparing to the original
Below we show that each time you double the voltage, you increase the energy by
approximately 6 dB:
Now we show that each time you multiply the voltage by 10, you increase the energy by
about 20 dB:
Combining the two observations, we have the following:
(
(
)
(
)
)
We can also show the “6 and 20 Rule” for distance differences, based on the equation below.
Notice that the original distance, , is in the numerator and the new distance, , is in the
denominator.
(
where
is the original distance and
distance.
)
is the distance we are comparing to the original
First we show that each time you double the distance, you decrease the energy by
approximately 6 dB:
Digital Sound and Music
Working with Decibels
PRACTICAL EXERCISE
Page 6
This material is based on work supported by the National Science Foundation under CCLI Grant DUE 0717743, Jennifer Burg PI, Jason Romney, Co-PI.
Now we show that each time you increase the distance by a factor or 10, you decrease the
energy by approximately 20 dB:
To prove the full “6 and 20 Rule,” we do the following:
,
( )
(
)
Question 1: How many dB greater than 2 volts is 10 volts?
(
)
Answer: 10 volts is 14 dB greater than 2 volts
Question 2: How many dB greater than 1 volt is 100 volts?
Answer: 100 volts is 40 dB greater than 1 volt.
Question 3: If a loudspeaker measures 100 dB SPL at 3 feet, how loud will it be at 30
feet?
(
)
Answer: This loudspeaker will produce 80 dB SPL at 30 feet.
Digital Sound and Music
Working with Decibels
PRACTICAL EXERCISE
Page 7
This material is based on work supported by the National Science Foundation under CCLI Grant DUE 0717743, Jennifer Burg PI, Jason Romney, Co-PI.
With a Calculator
Using a calculator, you can perform more complicated dB conversions with voltage
levels. When working with voltage levels, you multiply by 20 the base 10 log of the
voltage you ended up with divided by the voltage you started with. When working
with dB loss over distance level you use the same formula, but you divide the
distance you started with by the distance you ended up with.
If you know the dB difference and one of the voltage values, you can find the voltage
ratio of that dB difference and multiply that by the voltage you started with to find
the voltage level for that dB difference. Here are the equations again:
( )
(
)
Also note in the following questions that dBV and dBu are both used for voltage
decibels, but with different reference voltages. The reference voltage for dBV is 1 V
while the reference voltage for dBu is 0.775 V.
Question 1: How many dB greater than 2 volts is 63 volts?
(
)
Answer: 63 volts is 30 dB greater than 2 volts.
Question 2: Professional line level is +4 dBu. O dBu = 0.775 volts. How many volts is
professional line level?
Answer: Professional line level is 1.23 volts
Question 3: If a loudspeaker generates 113 dBSPL at 1 meter, how loud will it be at 30
meters?
Digital Sound and Music
Working with Decibels
PRACTICAL EXERCISE
Page 8
This material is based on work supported by the National Science Foundation under CCLI Grant DUE 0717743, Jennifer Burg PI, Jason Romney, Co-PI.
(
)
Answer: The loudspeaker will generate 83.4 dBSPL at 30 meters
Working with Power and Voltage Decibels
Sometimes there will be situations where you will need to perform calculations involving
both energy decibels and power decibels. For example, you might need to choose an
appropriate power amplifier that will allow your loudspeaker to achieve a certain SPL level
at a known distance. So you’re trying to figure out what difference in power will result in a
certain value in voltage.
Question: The sensitivity of a loudspeaker is 97 dBSPL at 1 meter with 1 watt of power.
How powerful does the amplifier need to be in order for the loudspeaker to generate
100 dBSPL at 10 meters away?
First, calculate the dB loss for 10 meters. This is an important step because you don’t really
know what the loudspeaker is capable of at 10 meters, but you know what it can do at 1
meter. So if you need 100 dBSPL at 10 meters, how loud does it need to be at 1 meter? You
can do this one in your head since we’re dealing with a multiple of 10 (a loss of 20 dB).
(
)
If the loudspeaker needs to be 100 dB SPL at 10 meters, it will need to be 20 dB louder at 1
meter since you will lose 20 dB by traveling that distance through the air.
We know the loudspeaker can do 97 dBSPL with only 1 watt. Before we can figure out how
many watts is needed for 120 dBSPL, we need to figure out the difference between these
two values.
In other words, we need 23 dB more than the 97 dBSPL that we get with only 1 watt. Now
let’s find the power ratio of 23 dB.
Digital Sound and Music
Working with Decibels
PRACTICAL EXERCISE
Page 9
This material is based on work supported by the National Science Foundation under CCLI Grant DUE 0717743, Jennifer Burg PI, Jason Romney, Co-PI.
Now we can take this power ratio for 23 dB and multiply it by the number of watts we
already know about. In this case, that’s 1 watt.
Answer: You will need a 200 watt amplifier in order to generate 100 dBSPL at 10
meters with this loudspeaker.
Working with Decibel Reference Levels
Technically, decibels are merely a measurement of the difference between two values. As
discussed in Chapter 4, there are several reference levels that have been agreed upon in the
professional sound community. When working with sound, you may need to convert a
decibel value from its scale in one reference level into its position relative to a different
reference level. The most common conversion is between dBu and dBV. Professional grade
equipment uses dBu in its specifications while consumer grade equipment used dBV.
However, consumer and professional grade equipment are often interconnected in sound
systems. Let’s begin by simply defining some known reference levels.
Question 1: Nominal operating line level for consumer grade equipment is 10 dBV.
How many volts does 10 dBV represent?
First, find the Voltage Ratio of 10 dB:
Now multiply this voltage ratio by the reference level for 0 dBV (1 volt):
Answer: 10 dBV represents an actual voltage level of 0.316 volts. This means that
unity gain for consumer equipment is when the signal is at 0.316 volts.
Question 2: Nominal operating line level for professional grade equipment is +4 dBu.
How many volts does +4 dBu represent?
First, find the voltage ratio of +4 dB:
Digital Sound and Music
Working with Decibels
PRACTICAL EXERCISE
Page 10
This material is based on work supported by the National Science Foundation under CCLI Grant DUE 0717743, Jennifer Burg PI, Jason Romney, Co-PI.
Now multiply this voltage ratio by the reverence level for 0 dBu (0.775 volts):
Answer: +4 dBu represents an actual voltage level of 1.23 volts. This means that unity
gain for professional equipment is when the signal is at 1.23 volts.
As you can see, there is a difference in nominal operating voltage level for consumer and
professional grade equipment. This means that if you connect a consumer grade device to a
professional grade device, you’ll need to compensate for the difference in level using a
preamplifier in order to maintain an optimized gain structure.
Consider the situation where you’re connecting two CD players to a mixing console. One CD
player is a professional grade model, and the other is a consumer grade model. These two
CD players will be generating their analog output signal at different levels. You can
compensate for this using the preamplifiers in the mixing console. Let’s see how much
you’ll need to compensate:
Question 3: What’s the dB difference between professional line level and consumer line
level?
We know from the previous two questions that professional line level is 1.23 volts and
consumer line level is 0.316 volts. Let’s just figure out the dB difference between these two
voltage levels:
(
)
We can’t really hear changes less than 1 dB, and it would be difficult to dial in a fraction of a
dB on an analog preamp. So it would be appropriate to simply round this down to 12 dB.
Answer: The dB difference between professional and consumer line level is 12 dB. In
other words, you’ll have to turn up the preamplifier on the consumer CD player by 12
dB to get the level in the mixing console to match the professional CD player.
Digital Sound and Music
Working with Decibels
PRACTICAL EXERCISE
Page 11
This material is based on work supported by the National Science Foundation under CCLI Grant DUE 0717743, Jennifer Burg PI, Jason Romney, Co-PI.
Practice Problems
1. If you want to replace a 5 watt power amplifier with an amplifier that was 7 dB
more powerful, how many watts will you need?
2. When looking to purchase a new power amplifier, you find two power amplifiers
with similar features. The more expensive amplifier is 900 watts, and the less
expensive amplifier is only 200 watts. How many more dB will your money buy with
the 900 watt amplifier?
3. An EAW UB12se model loudspeaker has a sensitivity of 89 dBSPL at 1watt/1meter.
Its peak dBSPL is specified at 116.5 dBSPL at 1 meter. What is the peak power
handling of this loudspeaker?
4. For this same UB12se loudspeaker, how many dBSPL will you get at 1 meter with a
50 watt power amplifier?
5. What power would be 18 dB greater than 3 watts?
6. What voltage would be 14 dB greater than 19 volts?
7. 0 dBu = 0.775 volts. 0 dBV = 1 volt. What is the dB difference between the two?
8. If a loudspeaker can produce 102 dBSPL at 5 feet, how loud will it be at 25 feet?
9. For this same loudspeaker, how loud will it be if you then move 7 feet closer from 25
feet?
10. If a loudspeaker’s sensitivity is 98 dB 1w/1m, how loud will it be at 23 feet away
with 300 watts of power?
11. If a loudspeaker’s sensitivity is 105 dB 1w/1m, how much power will be needed to
generate 98 dBSPL at 35 feet?
Digital Sound and Music
Working with Decibels
PRACTICAL EXERCISE
Page 12
This material is based on work supported by the National Science Foundation under CCLI Grant DUE 0717743, Jennifer Burg PI, Jason Romney, Co-PI.