Working with dBs When EEs monitor a signal passing through a system most often they are concerned with its power level as it propagates through the various components and subsystems Normally tracking the power level involves multiplication or division of gains and losses representing the action of the various components of the system on the signal Many times the signal power can vary over a wide range of values several orders of magnitude which makes representing the signal power at different points a bit inconvenient The mathematical computations involving multiplication and division of large numbers can be cumbersome In order to make the math easier we often use a logarithmic scale to represent values There are two chief advantages in working with logarithmic scales 1 Multiplication becomes addition log a x b log a log b and log a b log a log b 2 Scales are compressed if we have values ranging over several orders of magnitude the plot scale is large for linear representations while relatively compressed for logarithmic representation of values Definition of deciBel dB The logarithmic scale most often used is one in which the values are represented in deciBels dBs often written as decibels Here the decibel is defined as a logarithmic representation of a unit less quantity normally a ratio of two powers The logarithmic base or radix used for dBs is 10 N dB 10 log10 N where N refers to the numerical value being represented and NdB refers to the value in decibels dBs Note that other definitions of dBs exist but this is the definition used for representing electrical signal powers and therefore it is the only valid definition as far as we are concerned There are a number of things to note about this definition First there is a multiplier of 10 in front of the logarithm That is simply part of the definition which makes the numbers work out more intuitively However it is important to remember that the multiplier is 10 only for quantities involving power Next the logarithm is base 10 only Other radixes may be used to represent values but not for dBs Finally note that the argument of the logarithm is a unit less value This last point is very important in our discussion here 1 Representing Values as dBs There are a number of values for which it will be good to know the dB representation in order to make life easier and to understand the lingo Simply plug the values in the dB expression to create this table verify a few in your head values lin 1 00E 06 0 001 0 01 0 1 0 5 1 2 10 100 1000 1 00E 06 dB 60 30 20 10 3 0 3 10 20 30 60 Note that number values less than 1 produce negative dB values and number values greater than 1 produce positive dB values The dB value of 1 is zero dB and the dB value of 0 is undefined but can be approximated by 99 dB To convert dB values back to linear values simply invert the definition of decibels as follows N 10 N dB 10 Be sure to verify a few of the table entries using this relationship as well You must feel very comfortable with the relationship between numerical values and dB representations before moving on One of the advantages mentioned above was that multiplication was easier using logarithms Let s try it using the table above The value 2 converts to 3 dB and 100 converts to 20 dB What should the value 200 convert to Using multiplication we see that 200 2 x 100 So when converting to dBs we find 200 10 log10 200 10 log10 100 2 10 log10 100 10 log10 2 20 3 23 dB Below are some more examples of using this multiplicative effect to quickly find dB values 50 10 log10 10 10 log10 5 10 7 17 dB 10 log10 10 10 log10 5 10 7 17 dB 50 4 0E06 10 log10 106 10 log10 4 60 6 66 dB 1 2 Representing Powers using dBs There are two ways we typically use dBs to represent powers normally average powers and to represent power ratios We must be fluent in both and how to combine the two Lets look first at how we use deciBels to represent power values or express powers in terms of corresponding voltages or currents Here we need to recall one important point from the definition of dBs the value we represent is to be a unit less quantity a power ratio To create this unit less ratio we express the power relative to some standard or reference power For example suppose we wish to represent 5 W in terms of deciBels First consider the use of a power reference of 1W 10 log 5 6 99 dBW 7 dBW P 10 log10 5W 10 1W Here the appended W to the dB unit reminds us that this is a decibel representation of a power relative to 1 W A more common measure is dBm or power relative to 1 mW note that the W is missing in this unit 10 log 5000 37 dBm P 10 log10 5W 10 1 mW and we can relate the power in dBW to power in dBm rather simply P dBm 10 log10 5W 1000 mW 5W 1000 mW 10 log10 1 mW 1W 1W 1 mW 10 log10 5 1000 10 log10 5 10 log10 1000 P dBW 30 So 5 W may be represented by 7 dBW or 37 dBm values separated by 30 dB or a factor of 1000 Do not think of dBW and dBm as different units They both are dBs the W and m suffixes are there to remind us of the power reference A couple of examples would be nice 1W dBW 10 log 1 dBW 0 dBW 1W 10 log 1W dBW 10 log 1000 dBm 30 dBm 1mW 20 mW 10 log 20 mW dBW 10 log 20 dBm 13 dBm 1mW 1W 10 log10 1W 10 10 10 10 10 3 Another dB unit used to represent a power level is the dBV or dBmV which is a power level referenced back to an equivalent rms voltage level which would produce that power given a 1 resistance Suppose we know a signals rms voltage Given the resistance over which the voltage is developed the power v 2 would be equal to rms It turns out that many times we ignore the resistance R value in the calculation calculating the power developed across a 1 resistance Thus P vrms2 We can represent this signal using dB units and a reference of W as follows V 2 W PdBW 10 log10 rms dBW W 1 Now we could reason that the 1 W reference is just 1 Vrms 2 and rewrite v 2 PdB X 10 log10 rms 2 1Vrms v dBV dBW 20 log10 rms 1V rms Here we have used the fact that log x2 2 log x The new unit dBV is still a dB measure of power but the suffix V reminds …
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