What is Decibel Scale Easy Formula

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.

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The Decibel Scale

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

  1. log P1 P2 = log P1 + log P2
  2. log P1 /P2 = log P1 − log P2
  3. log Pn = n log P
  4. 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,

decibel scale
(1)

The decibel (dB) provides us with a unit of less magnitude. It is 1/10th of bel and is given by

decibel scale
(2)

When P1 = P2, there is no change in power and the gain is 0 dB. If P2 = 2P1, the gain is

decibel scale
(3)

and when P2 = 0.5P1, the gain is

decibel scale
(4)

Equations.(3) and (4) show another reason why logarithms are greatly used: The logarithm of the reciprocal of a quantity is simply negative 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 is shown in Figure.(1).

decibel scale
Figure 1. Voltage-current relationships for a four-terminal network.

If P1 is the input power, P2 is the output (load) power, R1 is the input resistance, and R2 is the load resistance, then P1 = 0.5V12/R1 and P2 = 0.5V22/R2, and Equation.(2) becomes

decibel scale
(5)
decibel scale
(6)

For the case when R2 = R1, a condition that is often assumed when comparing voltage levels, Equation.(6) becomes

decibel scale
(7)

Instead, if P1 = I12R1 and P2 = I22R2, for R1 = R2, we obtain

decibel scale
(8)

Two things are important to note from Equations.(2), (5), and (8):

  1. That 10 log is used for power, while 20 log is used for voltage or current, because of the square relationship between them (P = V2/R = I2R).
  2. 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.

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