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The Decibel 

When we deal with electronic circuits that operate in some fashion on ac signals, we generally have a certain amount of signal power coming in to the circuit, and a different amount of signal power leaving the circuit. An amplifier will have a power output greater than its power input, while an attenuator will have less power going out than coming in. The actual amount of power being handled is important, of course, but so is the ratio of output to input power, which expresses the power gain or loss of the circuit.
But both the absolute power level and the relative gain or loss may vary widely, over several orders of magnitude. Therefore we turn to logarithms to compress the range and enable us to handle these variations more easily.
The primary unit describing a signal power ratio in this fashion is the bel, and is calculated as follows:
bels = log_{10}  p_{out} 
p_{in} 
Unfortunately, the bel is a rather large unit. It is more convenient to deal in units of about 0.1 Bel instead. In addition, a change of 0.1 Bel in audio volume level is just barely audible to the human ear. Therefore, we want to use units of 0.1 Bel simply for practical reasons. The prefix "deci" designates "onetenth," so we assign the name decibels to the new unit.
Since the decibel is equal to 1/10 bel, it takes 10 decibels to make up one bel. Accordingly,
decibels = 10 log_{10}  p_{out} 
p_{in} 
In many situations we are actually concerned with signal voltage gain or loss, rather than signal power. To accomplish this, we turn to the basic expression, p = v²/r. Therefore, if we keep the circuit resistance the same for input and output, we get:
decibels  =  10 log_{10}  v²_{out}/r 
v²_{in}/r  
=  10 log_{10}  v²_{out}  
v²_{in}  
=  20 log_{10}  v_{out}  
v_{in} 
Since we can also say that p = i²r, current ratios are expressed in the same way as voltage ratios. However, this is not used nearly as often as voltage ratios.
One of the very useful features of using decibels (abbreviated db) to express voltage gain or loss in a circuit comes when circuits are cascaded so that a given signal passes through one after another. We could note the gain or attenuation factor of each circuit and multiply them all together. However, their gains or attenuations will probably already be expressed in db (positive for gain, negative for attenuation). Therefore, we can simply add these individual db factors together, and then convert to actual gain or loss only as the last step, if it is even needed there. This simplifies the math involved and helps to reduce the liklihood of errors.


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