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:

- log
*P*_{1}*P*_{2}= log*P*_{1}+ log*P*_{2} - log
*P*_{1}*/P*= log_{2}*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 *decibel *(dB) 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} = 2*P*_{1}, the gain is

and when *P*_{2} = 0*.*5*P*_{1}, the gain is

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).

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*.*5*V*_{1}^{2}*/R*_{1} and *P*_{2} = 0*.*5*V*_{2}^{2}*/R*_{2}, and Equation.(2) becomes

For the case when *R*_{2} = *R*_{1}, a condition that is often assumed when comparing voltage levels, Equation.(6) 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.(2), (5), and (8):

- 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.