Working with dBsDefinition of deciBel (dB)Representing Values as dBsRepresenting Powers using dBsRepresenting Power Ratios using dBsSignal PropagationRounding Decibel ValuesWorking 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: ()1010log=dBNN 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. 1Representing 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): lin dB1.00E-06 -600.001 -300.01 -200.1 -100.5 -3102310 10100 201000 301.00E+06 60values 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: 1010dBNN = 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: ()()() ()10 1010 10200 10log 200 10log 100 210log 100 10log 2 20 3 23 dB→=×=+=+= Below are some more examples of using this multiplicative effect to quickly find dB values: ()()() ()()()10 1010 10610 1050 10log 10 10log 5 10 7 17 dB110log 10 10 log 5 10 7 17 dB504.0E06 10 log 10 10 log 4 60 6 66 dB→+=+=→− − =− − =−→+=+= 2Representing 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: dBWdBWWWP 799.6)5(log1015log101010≅=== 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): ()dBmmWWP 375000log1015log101010=== and we can relate the power in dBW to power in dBm rather simply: ()30)1000(log10)5(log1010005log101100015log10)1100015(log101010101010+=+=⋅===dBWPmWmWWWWmWmWWdBmP 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: ()()()()()()10 1010 1010 101W1W 10log dBW 10log 1 dBW 0 dBW1W1W1W 10log dBW 10log 1000 dBm 30 dBm1mW20 mW20 mW 10log dBW 10log 20 dBm 13 dBm1mW→==→==→== 3Another 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 Ω resistance. Suppose we know a signals rms voltage. Given the resistance over which the voltage is developed, the power would be equal to 2rmsvR. It turns out that many times we ignore the resistance value in the calculation, calculating the power developed across a 1 Ω resistance. Thus P = vrms2. We can represent this signal using dB units and a reference of W as follows: 21010log1rmsdBWVWPdW=BW. Now, we could reason that the 1 W reference is just (1 Vrms)2, and rewrite ()22_10 1010log 20log11rms rmsdB
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