What is Decibel Scale Easy Formula

decibel scale

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.

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.