# Bode diagram

A Bode diagram is a representation of two function graphs: One graph shows the amount ( amplitude gain ), the other the argument (the phase shift ) of a complex-valued function as a function of frequency. This type of representation is named after Hendrik Wade Bode , who used these diagrams in his work at Bell Laboratories in the 1930s.

Example of a Bode diagram

Bode diagrams are used for the representation of linear time-invariant systems (LZI) in the field of electronics / electrical engineering , control engineering and mechatronics as well as in impedance spectroscopy .

A Bode diagram describes the relationship between a harmonic excitation (" sinusoidal oscillation ") at an input of the system and the associated output signal in the steady state , i.e. H. for . For a complete description of an LZI system with inputs and outputs, you need diagrams. ${\ displaystyle t \ to \ infty}$${\ displaystyle n}$${\ displaystyle m}$${\ displaystyle n \ cdot m}$

## classification

The Bode diagram is used to represent the transmission behavior of a dynamic system, also called frequency response or frequency response . Other diagram forms to describe dynamic systems, such as B. the Nyquist diagram (frequency response locus ) or the pole-zero diagram , on the other hand, serve other purposes, the two mentioned about stability considerations. Like the other diagrams, the Bode diagram is derived and calculated from mathematical system descriptions using differential equations .

## Characteristic properties

• On the x-axes ( abscissa ) the frequency resp. Angular frequency represented logarithmically. This shows the behavior over a large frequency range at a glance.
• On the y-axis ( ordinate ) of the first graph, the amplitude gain, i.e. the amount of the frequency response, is shown in decibels or in logarithmic scaling. This graph is called the amplitude response .
• The phase shift, i.e. the argument of the frequency response, is plotted linearly on the y-axis of the second graph. This graph is called the phase response .

The amplitude and phase response are plotted on top of each other so that the gain and phase of a frequency are vertically above each other.

Due to the logarithmic scaling of the amplitude response, Bode diagrams have the advantage that complex Bode diagrams can be created from (additive) overlaying of simple partial diagrams. This corresponds to a series connection of transmission elements. For this purpose, the complex function is broken down into sub-functions of the first and second order by factoring. By plotting the gain logarithmically , the multiplication of the partial functions becomes the addition of their amplitude responses. The phase responses are additively superimposed without logarithmic scaling.

Transfer function designation Amplitude response Phase response Bode diagram
${\ displaystyle K}$ P element ${\ displaystyle 20 \ cdot \ lg {| K |}}$ ${\ displaystyle 0 {\ text {if}} K \ geq 0, - \ pi {\ text {if}} K <0}$
Bode diagram of a P element (K = 2)
${\ displaystyle {\ frac {s} {\ omega _ {0}}}}$ D link +20 dB / decade, 0 dB at ${\ displaystyle \ omega _ {0}}$ constant at ${\ displaystyle + {\ frac {\ pi} {2}}}$
Bode diagram of a D-link (K = 2)
${\ displaystyle - {\ frac {s} {\ omega _ {0}}}}$ +20 dB / decade, 0 dB at ${\ displaystyle \ omega _ {0}}$ constant at ${\ displaystyle - {\ frac {\ pi} {2}}}$
${\ displaystyle {\ frac {\ omega _ {0}} {s}}}$ I-link −20 dB / decade, 0 dB at ${\ displaystyle \ omega _ {0}}$ constant at ${\ displaystyle - {\ frac {\ pi} {2}}}$
Bode diagram of an I link (K = 2)
${\ displaystyle - {\ frac {\ omega _ {0}} {s}}}$ −20 dB / decade, 0 dB at ${\ displaystyle \ omega _ {0}}$ constant at ${\ displaystyle + {\ frac {\ pi} {2}}}$
${\ displaystyle 1 + {\ frac {s} {\ omega _ {0}}}}$ PD link Kink at , then +20 dB / decade ${\ displaystyle \ omega _ {0}}$ from 0 to over two decades, at${\ displaystyle + {\ frac {\ pi} {2}}}$${\ displaystyle + {\ frac {\ pi} {4}}}$${\ displaystyle \ omega _ {0}}$
${\ displaystyle {\ frac {1} {1 + {\ frac {s} {\ omega _ {0}}}}}}$ PT1 element Kink at , then −20 dB / decade ${\ displaystyle \ omega _ {0}}$ from 0 to over two decades, at${\ displaystyle - {\ frac {\ pi} {2}}}$${\ displaystyle - {\ frac {\ pi} {4}}}$${\ displaystyle \ omega _ {0}}$
Bode diagram of a PT 1 link (K = 2, T = 1)
${\ displaystyle {\ frac {1} {1 + 2d {\ frac {s} {\ omega _ {0}}} + {\ frac {s ^ {2}} {\ omega _ {0} ^ {2} }}}}}$ PT2 element Kink at , then −40 dB / decade ${\ displaystyle \ omega _ {0}}$ from 0 to over two decades with a compression depending on d${\ displaystyle - \ pi}$
Bode diagram of a PT 2 link (K = 2, T = 1, d = 0.2; 1; 5)

The statement from 0 to x in 2 decades only applies approximately. However, the statement is often precise enough. Using the example of a PT 1 system:

${\ displaystyle F (s) = {\ frac {1} {{\ frac {s} {\ omega _ {0}}} + 1}}}$
${\ displaystyle \ phi (0 {,} 1 \ cdot \ omega _ {0}) = - \ arctan 0 {,} 1 = 5 {,} 75 ^ {\ circ}}$
${\ displaystyle \ phi (10 \ cdot \ omega _ {0}) = - \ arctan 10 = 84 {,} 3 ^ {\ circ}}$

## Illustration of the advantages of a logarithmic representation

Example of an amplitude curve for a low-pass filter

A simple low pass , for example an RC element , forms a so-called PT 1 system.

${\ displaystyle F (s) = K {\ frac {1} {1 + T_ {1} s}}}$

${\ displaystyle K}$results from the ratio of the output variable to the input variable at a low frequency. If the corner frequency or limit frequency is reached, the real part of the denominator is equal to its imaginary part. This results in a phase shift of and a gain of: ${\ displaystyle f _ {\ text {E}}}$${\ displaystyle \ textstyle - {\ frac {\ pi} {4}}}$

${\ displaystyle {\ frac {1} {\ sqrt {2}}} \ approx -3 \, \ mathrm {dB} \ approx 0 {,} 71}$

The formula-based values ​​of the corner frequency can still be read relatively easily from the linearly divided diagram. However, with more complex systems at the latest, it makes more sense to work in the double logarithmic Bode diagram.

In the Bode diagram, the course of the function can also be idealized with straight lines. In this example, the idealized curve is raised by +3 dB in order to be better distinguishable. The corner frequency is at the intersection of the horizontal and the descending straight line. The real function has already dropped by −3 dB here. If the system shows proportional behavior, the gain can be read off , here , on the Y-axis ( very small). ${\ displaystyle K = 0 \, \ mathrm {dB} = 1}$${\ displaystyle s}$

A system can be identified on the basis of the gradient and the phase progression. With a PT 1 system, the gradient above is −1: 1. A doubling of the frequency leads to a halving (−6 dB) of the amplitude, and a tenfold increase in the frequency reduces the gain to a tenth, i.e. −20 dB. The phase at is −45 ° and for it is −90 °. ${\ displaystyle f _ {\ text {E}}}$${\ displaystyle f _ {\ text {E}}}$${\ displaystyle f \ rightarrow \ infty}$

If two PT 1 systems are connected in series, the result is a PT 2 system with damping . Above the first corner frequency, the slope is −1: 1, after the second corner frequency −2: 1 (see top Bode diagram with phase). If the two corner frequencies are far enough apart, the phase is −45 ° at the corner frequency and −90 ° at the second. ${\ displaystyle D> 1}$

Example of an amplitude curve for a low-pass filter
Example of a phase response of a low pass

A PT 2S system capable of oscillation (for example RLC oscillating circuit) can be represented with a complex pole or as a second-order polynomial . Above the corner frequency, the slope is −2: 1. The phase is −90 ° at the corner frequency and tends towards −180 ° at infinity. A resonance increase occurs depending on. ${\ displaystyle D}$

${\ displaystyle F (s) = K {\ frac {1} {1 + {\ frac {2Ds} {\ omega _ {0}}} + {\ frac {s ^ {2}} {{\ omega _ { 0}} ^ {2}}}}}}$

With integrators, called I-systems, there is no horizontal straight line segment for small frequencies. It starts immediately with a slope of −1: 1.

${\ displaystyle F (s) = {\ frac {1} {T _ {\ text {I}} \ cdot s}}}$

Correspondingly, with a differentiator, called a D system, the slope is immediately +1: 1.

${\ displaystyle F (s) = T _ {\ text {D}} \ cdot s}$

The integration or differentiation time constant can be read off for. This can also be viewed as reinforcement (systems basically only have P, I or D behavior). ${\ displaystyle \ omega _ {0} = 1s ^ {- 1}}$