How to cheat with dBs ... an intuitive approach
Not copyrighted. 1999 Chris Scott
Decibel relationships are relatively simple, yet many engineers forget that approximate calculations can be done mentally. We're so quick to hit the calculator LOG button that we sometimes fail to get a conceptual feel for the unit. When the calculator is left back at the office and these dB units come up close approximations can still be made. Decibels, or dBs are very useful units, - one-tenth of a bel, (nobody uses bels anymore) allowing both small and large ratios to be expressed in a meaningful way with manageable numbers. In audio and RF work dBs are de rigeur. A simple method is presented that allows mental calculation. While not quite as accurate as the full logarithmic machinery:
10 or 20 times the log(10) of the ratio
this approach does well enough for most work, and promotes a clearer relational understanding. We will start by memorizing three pairs of numbers which apply to voltage (and current). Later on, with practice, these will be seen to relate via ohms law to power as well:
For VOLTAGE
6 dB = multiply or divide the ratio by 2
10 db = multiply or divide the ratio by 3.33
20 db = multiply or divide the ratio by 10
Try to memorize these. Let's now prove that the numbers
in the left column add and subtract, and the numbers in the right column
multiply and divide:
Example 1:
An amplifier is said to have 26 dB of gain. If the input is 1 volt, what is the output?
Put down the calculator - this is easy:
Since 26 dB has at least 20 dB, we know that the output will be at least multiplied by 10 or at least ten volts.
So we subtract the 20 dB from the 26 dB and are left with 6 dB to "get rid of."
We then multiply the ten volts by 2 (6 dB) to "get rid of" the remaining six.
The amplifier output is 20 volts.
Example 2:
A microphone is said to produce - 60 dBV output. How many volts is this?
First, let's review that for a dB term to be meaningful, it must represent either a CHANGE, as in example 1, or it must be RELATIVE to an identifiable quantity, as in this example. - 60 dBV means less than (because of the sign) one VOLT (the identifiable quantity is V) by 60 dB.
So we simply divide ONE VOLT by the calculated ratio:
20... 20... 20... = 60 (dB) = 10 times 10 times 10 = 1000
One volt divided by 1000 = .001 Volt or one millivolt (mV).
Now let us try another:
Example 3:
What is the Voltage ratio represented by 16 dB?
10 dB equals a 3.33 multiplier. A minor additional simplification
would be to call it 3.
6 dB equals a 2 multiplier.
3 x 2 = 6... a ratio of 6.
(If we want to be more accurate, use 3.33 * 2 equals a 6.66 ratio)
Our eyes have now glazed over, and a yawn is beginning to form... but if you cannot see a pattern developing, you have no soul.
That's it.
The rest is simply practicing these relationships.
If we need to reverse the calculation, from a known ratio to dB, we just divide or multiply the ratio quantity at each reduction step:
Last example:
What is the Decibel expression for a Voltage ratio of 10,000 ?
Hmm... looks like ten (20 dB) will divide into it well...
10,000 / 10 = 1,000 (20 dB)
1,000 / 10 = 100 (40 dB)
100 / 10 = 10 (60 dB)
10 / 10 = (end) (80 dB)
The answer is 80 dB.
POWER
If the ratios are POWER ratios like WATTS, the following chart applies:
3 dB = multiply or divide by 2
6 dB = multiply or divide by 4
10 db = multiply or divide by 10
20 db = multiply or divide by 100
Why the different constants for power? Because power, instead of changing linearly with voltage, changes with the square of voltage. Ohms law proves it: [P = V^2 / R]. Take some time to play with the numbers and you will see how the tables are consistent with Ohm's formula.
Here are some common Decibel unit expressions.
dBV = dB change relative to a VOLT
dBW = dB change relative to a WATT
dBmv = dB change relative to a MILLIVOLT - common in CATV
dBm = dB change relative to a MILLIWATT - common in RF
dBK = dB change relative to a KILOWATT - common in broadcast
dBu = dB change relative to a MICROVOLT - " "
dBc = dB change relative to CARRIER (voltage or power)