# Bode plot

A Bode plot is a plot of the magnitude and of the phase of a transfer function using logarithmic scales for both the frequency and the magnitude. It turns out that using such scales, the drawing of the plot can be done quickly by hand with a very good aproximation and in a very simple way. Additionally, the use of logarithmic scales allows us to represent a wider range of values for the gain and for the frequency.

## How is it drawn?

The first step is to write the transfer function in normal form, id est, as a division of factorized polynomials. This form is benefitial for this purpose, because as we are taking the logarithm of the magnitude, all those factors will separate in terms, so we have but draw the different terms separately and finally draw their sum. The phase plot also benefits from having the transfer function written in normal form, as the angle of a product of complex factors is the sum of their respective angles.

Until this lesson is completed, an excellent guide to drawing Bode plots, written in Spanish, can be found here.

### First order factors

#### Magnitude

The plot can be assumed to coincide with the asymptotes until a decade prior the transition frequency, and from a decade after that frequency.

#### Phase

The plot can be assumed to coincide with the asymptotes until two decades prior the transition frequency, and from two decades after that frequency.

The slope at the transition frequency is 66º per decade. It is found by derivating the arctangent with respect to the decimal logarithm.

### Second order factors

There are no greater order factors resulting from factorizing a polynomial. This is a consequence of Algebra's fundamental theorem.