Decibel formula helps us in calculating the decibel based on the power gain. We can also use this to convert between decibel scale and power gain.

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**Decibel Scale**

Decibel, measured in dB is the unit we use to measure the ratio between two quantities. For now, we use it to measure the ratio between the amounts of electric power.

One decibel is equivalent to 10 times the logarithm of power ratio. Since decibel is commonly used in sound or loudness measurement, we can say that the loudness or intensity of sound is 10 log 10 (P_{1}/P_{2}) measured in decibels.

The P_{1} and P_{2} are the intensity of two sounds. It will increase the intensity by about 3 dB the moment we double the sound intensity.

The decibel scale will be the basic thing before we learn about ‘Bode Plot’. It is not always easy to get a quick plot of the magnitude and phase of the transfer function as we did above.

A more systematic way of obtaining the frequency response is to use Bode plots.

Before we begin to construct Bode plots, we should take care of two important issues: the use of logarithms and decibels in expressing gain.

**Decibel Formula**

Since Bode plots are based on logarithms, it is important that we keep the following properties of logarithms in mind:

- log P
_{1}P_{2}= log P_{1}+ log P_{2} - log P
_{1}/P_{2}= log P_{1}− log P_{2} - log P
^{n}= n log P - log 1 = 0

In communications systems, the gain is measured in bels. Historically, the bel is used to measure the ratio of two levels of power or power gain G; that is,

The db formula provides us with a unit of less magnitude. It is 1/10th of bel and is given by

When P_{1} = P_{2}, there is no change in power and the gain is 0 dB. If P_{2} = 2P_{1}, the gain is

and when P_{2} = 0.5P_{1}, the gain is

Decibel equations above show another reason why logarithms are greatly used: The logarithm of the reciprocal of a quantity is simply negative of the logarithm of that quantity.

Alternatively, the gain G can be expressed in terms of voltage or current ratio. To do so, consider the network shown below.

If P_{1} is the input power, P_{2} is the output (load) power, R_{1} is the input resistance, and R_{2} is the load resistance, then P_{1} = 0.5V_{1}^{2}/R_{1} and P_{2} = 0.5V_{2}^{2}/R_{2}, and it becomes

For the case when R_{2} = R_{1}, a condition that is often assumed when comparing voltage levels, equation above becomes

Instead, if P_{1} = I_{1}^{2}R_{1} and P_{2} = I_{2}^{2}R_{2}, for R_{1} = R_{2}, we obtain

Two things are important to note from equations above:

- That 10 log is used for power, while 20 log is used for voltage or current, because of the square relationship between them (P = V
^{2}/R = I^{2}R). - That the dB value is a logarithmic measurement of the ratio of one variable to another of the same type. Therefore, it applies in expressing the transfer function H in Equations in Transfer Function before.

With this in mind, we now apply the concepts of logarithms and decibels to construct Bode plots.