# Bode plot

The Bode diagram is a special function graph and consisting of a graph of the amount (amplitude gain) and the argument ( phase shift ) of a complex transfer function. This type of display is named after Hendrik Wade Bode, who used these diagrams in his work at Bell Laboratories in the 1930s.

Bode diagrams find their application in the representation of linear time-invariant systems in the field of electronics / electrical engineering, control engineering and mechatronics as well as in the impedance spectroscopy.

A Bode diagram describes the stationary reaction at an output of a system to a harmonic excitation ( " sine wave " ) at an input of the system. For a complete description of an LTI system with n inputs and m outputs so you need n by m plots.

## Classification

The Bode plot is used to display the transmission behavior of a dynamic system, even frequency response or frequency response called. Other diagram types for describing dynamic systems, such as the Nyquist plot (frequency response locus ) or the pole - zero plot, on the other hand serve other purposes, the two said about the stability analysis. The Bode diagram, as are the other graphs derived and calculated from the mathematical description of system equations.

## Characteristics

- On the x -axis (abscissa), the frequency, respectively. Angular frequency logarithmic. This is at a glance the performance over a wide frequency range can be seen.
- On the y- axis (ordinate) of the first graph, the gain of the amplitude, ie, the complex magnitude of the transfer function in decibels - and thus also logarithmic - illustrated. This graph is called the amplitude response.
- On the y- axis of the second graph, the phase shift, so the complex argument of the transfer function is applied linearly. This graph is called phase transition.

Amplitude and phase response are applied over one another, so that the gain and phase of a frequency are vertically above one another.

Since the gain is plotted in decibels, Bode diagrams have the advantage that complex Bode diagrams of ( additive ) superposition can be created from simple part diagrams. This corresponds to a series connection of transmission links. To this end, the complex function is decomposed by factoring in partial transfer functions of the first and second order. By applying the logarithmic gain is obtained by multiplying the transfer function, the addition of their amplitude responses in decibels. The phase transitions are superimposed additively and without logarithmic scaling.

The statement from 0 to x in two decades is valid only approximately. However, the statement is often accurate enough. Using the example of a PT1 system:

## Illustrate the advantages of a logarithmic representation

A simple low-pass filter, for example, an RC element, forming a so-called PT1 system.

K is obtained here from the ratio of output to input at low speed. If the cut-off frequency, or cutoff frequency fE reached, the real part of the denominator is equal to its imaginary part. This results in at that point a phase shift and a gain of.

The a formula determined values of the cut-off frequency can be relatively easily read from the linear banded chart out yet. However, at least in more complex systems, it is to work more meaningful in a double logarithmic Bode plot.

In the Bode diagram of the function curve can be idealized shown with straight lines. In this example, the idealized curve is increased by 3 dB to better be distinguishable. At the intersection of the horizontal with the sloping straight line is the cutoff frequency. The real function has already fallen here to -3 dB. If the system has proportional action, the gain, K = 0 dB = 1 here, on the Y - axis (s very small) can be read.

From the slope of the phase characteristic and may identify a system. In a PT1 system above fE is the slope. A doubling of frequency thus leads to halve (-6 dB ) of the amplitude corresponding to the tenfold increase in frequency reduces the gain to one-tenth, ie -20 dB. The phase at fE is -45 ° and -90 ° for them.

PT1 two systems connected in series, the result is a system with a damping PT2 D> 1, above the first cut-off frequency is the slope of -1:1, -2:1 cutoff frequency according to the second ( see the top Bode diagram with phase)., The two cut-off frequencies far enough apart, the phase of the cutoff frequency at -45 ° and the second -135 °.

An oscillatory PT2S system (for example, RLC resonant circuit ) can be represented by a complex pole or as a second order polynomial. Above the cutoff frequency, the slope is -2:1. The phase is in the cutoff -90 ° and seeks the infinite against -180 °. There occurs a resonance magnification as a function of D.

For integrators, called I- systems, there is no horizontal line segment for small frequencies. It immediately goes wrong with a slope -1:1.

Named according to a differentiator D system, the slope immediately 1:1.

For the integration or differentiation time can be read. This can also be viewed as a reinforcement (systems basically have only P, I or D behavior).