Diagram of Bode

The diagram of Bode is a means of representing the frequential behavior of a system. It allows a simplified graphic resolution, in particular for the study of the transfer functions transfer of analogical systems. It is used for the properties of margin of profit, strokes of phase, continuous profit, Band-width, rejection of the disturbances and stability of the systems.

Definition

The diagram of Bode of a frequential system of answer $H\; (J\; \backslash \; Omega)\; \backslash$ is composed of two layouts:

• profit (or amplitude) in Decibel S (dB). Its value is calculated starting from $20\; \backslash \; log\_\; \{10\}\; \{(|H\; (J\; \backslash \; Omega)|)\}\backslash$.
• the phase in degree, given by $\backslash \; arg\; \{(H\; (J\; \backslash \; Omega))\}\backslash$

The scale of the pulsations is logarithmic curve and is expressed in rad/s (radian a second). The logarithmic scale allows a very readable layout, because mainly made up of linear sections.

Asymptotic layout of the analogical systems

Let us take an unspecified transfer transfer function which is written in the following way:

$H\; (p)\; =\; \backslash \; alpha\; p^q\; \backslash \; frac\; \{\backslash \; prod\_\; \{k=1\}\; ^K\; \backslash \; left\; (1+2\; \backslash \; xi\_k\; \backslash \; frac\; \{p\}\; \{\backslash \; omega\_k\}\; +\; \backslash \; left\; (\backslash \; frac\; \{p\}\; \{\backslash \; omega\_k\}\; \backslash \; right)\; ^2\; \backslash \; right)\; \backslash \; prod\_\; \{l=1\}\; ^L\; \backslash \; left\; (1+\; \backslash \; frac\; \{p\}\; \{\backslash \; omega\_l\}\; \backslash \; right)\}\{\backslash \; prod\_\; \{m=1\}\; ^M\; \backslash \; left\; (1+2\; \backslash \; xi\_m\; \backslash \; frac\; \{p\}\; \{\backslash \; omega\_m\}\; +\; \backslash \; left\; (\backslash \; frac\; \{p\}\; \{\backslash \; omega\_m\}\; \backslash \; right)\; ^2\; \backslash \; right)\; \backslash \; prod\_\; \{n=1\}\; ^N\; \backslash \; left\; (1+\; \backslash \; frac\; \{p\}\; \{\backslash \; omega\_n\}\; \backslash \; right)\}$

where $\backslash \; alpha\; \backslash \; in\; \backslash \; mathbb\; R\; \backslash ;\; \backslash \; Q\; \backslash \; in\; \backslash \; mathbb\; Z\; \backslash ;\; \backslash \; \backslash \; omega\_k,\; \backslash \; omega\_l,\; \backslash \; omega\_m,\; \backslash \; omega\_n\; \backslash \; in\; \backslash \; mathbb\; R^*\; \backslash ;\; \backslash \; \backslash \; xi\_k,\; \backslash \; xi\_m\; \backslash \; in\; \backslash \; mathbb\; R\; \backslash$

Although a transfer transfer function can be written in several ways, it is in a way described above that they should be written:

• the constant terms of the elementary polynomials of the first and the second degree must be worth $1$. For that to use the constant $\backslash \; alpha$.
• the terms in $p$ of the elementary polynomials of the first and the second degree must be with the numerator. (see the rewriting of the function High-pass below)

It is noticed that the module of $H\; (p)\; \backslash$ is equal to the sum of the modules of the elementary terms because of the Logarithme. The same applies to the phase, this time because of the function argument. This is why one initially will be interested in the diagrams of Bode of the elementary terms.

First order systems

Low-pass

• Definition

That is to say the transfer transfer function:

$H\; (p)\; =\; \backslash \; frac\; \{1\}\; \{1+\; \backslash \; frac\; \{p\}\; \{\backslash \; omega\_0\}\}\; \backslash$

The pulsation $\backslash \; omega\_0\; \backslash$ is called pulsation of cut.

• Traced asymptotic

For $\backslash \; Omega\; \backslash \; L\; \backslash \; omega\_0,\; \backslash \; H\; (J\; \backslash \; Omega)\; \backslash \; approx\; 1\; \backslash$ thus $|H\_\; \{dB\}\; (J\; \backslash \; Omega)|=0\; \backslash$ and $\backslash \; arg\; \{(H\; (J\; \backslash \; Omega))\}=0^\backslash circ\backslash$.

For $\backslash \; Omega\; \backslash \; gg\; \backslash \; omega\_0,\; \backslash \; H\; (J\; \backslash \; Omega)\; \backslash \; approx\; -\; J\; \backslash \; frac\; \{\backslash \; omega\_0\}\; \{\backslash \; Omega\}\; \backslash$ thus $|H\_\; \{dB\}\; (J\; \backslash \; Omega)|=-20\; \backslash \; log\_\; \{10\}\; (\backslash \; Omega)\; +20\; \backslash \; log\_\; \{10\}\; (\backslash \; omega\_0)\; \backslash$ and $\backslash \; arg\; \{(H\; (J\; \backslash \; Omega))\}=-90^\backslash circ\backslash$.

In a reference mark logarithmic curve, $|H\_\; \{dB\}\; (J\; \backslash \; Omega)|\backslash$ results in a slope of -20dB/Décade or -6dB/Octave. One also speaks about slope -1. The asymptotic diagram of Bode of the module is thus summarized with two linear sections.

• Traced real

in $\backslash \; omega\_0\; \backslash$, $|H\; (J\; \backslash \; omega\_0)|=\; \backslash \; frac\; \{1\}\; \{1+j\}$ is $|H\_\; \{dB\}\; (J\; \backslash \; omega\_0)|=20\; \backslash \; log\_\; \{10\}\; (\backslash \; sqrt\; \{2\})\; =10\; \backslash \; log\_\; \{10\}\; (2)$: the curve passes 3dB in lower part of the asymptote.

High-pass

That is to say the transfer transfer function:

$H\; (p)\; =\; \backslash \; frac\; \{1\}\; \{1+\; \backslash \; frac\; \{\backslash \; omega\_0\}\; \{p\}\}\; =\; \backslash \; frac\; \{\backslash \; frac\; \{p\}\; \{\backslash \; omega\_0\}\}\; \{1+\; \backslash \; frac\; \{p\}\; \{\backslash \; omega\_0\}\}$

The layout is obtained by taking the opposite of the module in dB and the phase of the low-pass one.

Systems of the second order

Low-pass

• Definition

That is to say the transfer transfer function:

$H\; (p)\; =\; \backslash \; frac\; \{1\}\; \{1+2\; \backslash \; xi\; \backslash \; frac\; \{p\}\; \{\backslash \; omega\_0\}\; +\; \backslash \; left\; (\backslash \; frac\; \{p\}\; \{\backslash \; omega\_0\}\; \backslash \; right)\; ^2\}\; \backslash$

The pulsation $\backslash \; omega\_0\; \backslash$ is called own pulsation and $\backslash \; xi\; \backslash$ is damping.

• Traced asymptotic

For $\backslash \; Omega\; \backslash \; L\; \backslash \; omega\_0\; \backslash \; H\; (J\; \backslash \; Omega)\; \backslash \; approx\; 1\; \backslash$ thus $|H\_\; \{dB\}\; (J\; \backslash \; Omega)|=0\; \backslash$ and $\backslash \; arg\; \{(H\; (J\; \backslash \; Omega))\}=0^\backslash circ\backslash$.

For $\backslash \; Omega\; \backslash \; gg\; \backslash \; omega\_0\; \backslash \; H\; (J\; \backslash \; Omega)\; \backslash \; approx\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; omega\_0\}\; \{\backslash \; Omega\}\; \backslash \; right)\; ^2\; \backslash$ thus $|H\_\; \{dB\}\; (J\; \backslash \; Omega)|=-40\; \backslash \; log\_\; \{10\}\; (\backslash \; Omega)\; +40\; \backslash \; log\_\; \{10\}\; (\backslash \; omega\_0)\; \backslash$ and $\backslash \; arg\; \{(H\; (J\; \backslash \; Omega))\}=-180^\; \backslash \; circ\; \backslash \; times\; \backslash \; operatorname\; \{sign\; (\backslash \; omega\_0\; \backslash \; xi)\}\backslash$.

In a reference mark logarithmic curve, $|H\_\; \{dB\}\; (J\; \backslash \; Omega)|\backslash$ results in a slope of -40dB/Décade or -12dB/Octave. One also speaks about slope -2. The asymptotic diagram of Bode of the module is thus summarized with two linear sections.

• Traced real

When , the system presents a resonance of module $|H\; (J\; \backslash \; Omega)|\_\; \{max\}\; =\; \backslash \; frac\; \{1\}\; \{2\; \backslash \; xi\; \backslash \; sqrt\; \{1\; \backslash \; xi^2\}\}\; \backslash$ in $\backslash \; omega=\; \backslash \; omega\_0\; \backslash \; sqrt\; \{1-2\; \backslash \; xi^2\}\; \backslash$.

High-pass

August 1st

The layout is obtained by taking the opposite of the module in dB and the phase of the low-pass one.

return to the general case

As we pointed out higher, one could add all the diagrams with Bode of the elementary terms to obtain the diagram of the transfer transfer function $H\; (p)\; \backslash$.

However, when this transfer transfer function is complicated, it is easier progressively to take into account the contributions of each term while making grow the pulsation $\backslash \; Omega\; \backslash$.

At the beginning, when $\backslash \; Omega\; \backslash \; rightarrow\; 0\; \backslash$, the asymptote of the module is a line of slope Q (q*20dB/Décade) and the phase is constant with $q\; \backslash \; times\; 90^\; \backslash \; circ\; \backslash$. Thereafter, with each time one meets a pulsation, one modifies the layout according to the following procedure:

• For $\backslash \; omega=\; \backslash \; omega\_k\; \backslash$ one adds +2 with the slope of the module (+40dB/Décade) and $180^\; \backslash \; circ\; \backslash \; times\; \backslash \; operatorname\; \{sign\; (\backslash \; omega\_k\; \backslash \; xi\_k)\}\backslash$ with the phase.
• For $\backslash \; omega=\; \backslash \; omega\_l\; \backslash$ one adds +1 with the slope of the module (+20dB/Décade) and $90^\; \backslash \; circ\; \backslash \; times\; \backslash \; operatorname\; \{sign\; (\backslash \; omega\_l)\}\backslash$ with the phase.
• For $\backslash \; omega=\; \backslash \; omega\_m\; \backslash$ one adds -2 with the slope of the module (- 40dB/Décade) and $-180^\; \backslash \; circ\; \backslash \; times\; \backslash \; operatorname\; \{sign\; (\backslash \; omega\_m\; \backslash \; xi\_m)\}\backslash$ with the phase.
• For $\backslash \; omega=\; \backslash \; omega\_n\; \backslash$ one adds -1 with the slope of the module (- 20dB/Décade) and $-90^\; \backslash \; circ\; \backslash \; times\; \backslash \; operatorname\; \{sign\; (\backslash \; omega\_n)\}\backslash$ with the phase.

layout of the numerical systems

Limitation of the field of the pulsations

We have this time a function of tranfert $G\; (Z)\; =\; \backslash \; mathcal\; \{Z\}\; \backslash \; \{G\; (N)\; \backslash \}\; \backslash$ of a discrete system.

To obtain its diagram of Bode, it is necessary to evaluate the function on the circle unit.

In other words, $z=e^\; \{2\; \backslash \; pi\; J\; \backslash \; naked\}\; \backslash$ with $\backslash \; naked\; \backslash \; in\; \backslash \; left$ (one obtains the complete circle by symmetry).

If the discrete system were obtained starting from the Échantillonnage at the period T of a continuous system, then $z=e^\; \{J\; \backslash \; Omega\; T\}\; \backslash$ with $\backslash \; Omega\; \backslash \; in\; \backslash \; left$.

Moreover, the relations $|G\; (Z)|\_\; \{z=e^\; \{2\; \backslash \; pi\; J\; \backslash \; naked\}\}\; \backslash$ and $\backslash \; operatorname\; \{arg\; (G\; (Z)\; \_\; \{z=e^\; \{2\; \backslash \; pi\; J\; \backslash \; naked\}\})\}\backslash$ is not rational in $\backslash \; naked\; \backslash$. By consequence, the study of the layout is complicated and requires average data processing.

Bilinear transformation

However, there exists an application making it possible to be reduced to the continuous case:

$z=\; \backslash \; frac\; \{\backslash \; frac\; \{2\}\; \{T\}\; +w\}\; \{\backslash \; frac\; \{2\}\; \{T\}\; -\; W\}\; \backslash$

or the reciprocal function $w=\; \backslash \; frac\; \{2\}\; \{T\}\; \backslash \; frac\; \{z-1\}\; \{z+1\}\; \backslash$

It is about a Transformée of Möbius.

This transformation makes correspond the secondary axis $w=j\; \backslash \; Omega\; \backslash$ of the continuous field with the circle unit $z=e^\; \{J\; \backslash \; Omega\; T\}\; \backslash$ of the discrete field with $\backslash \; omega=\; \backslash \; frac\; \{2\}\; \{T\}\; \backslash \; operatorname\; \{arctan\; \backslash \; left\; (\backslash \; frac\; \{T\; \backslash \; Omega\}\; \{2\}\; \backslash \; right)\}\backslash$.

However, when $\backslash \; Omega\; T\; \backslash \; L\; 1$, one has $\backslash \; Omega\; \backslash \; approx\; \backslash \; Omega\; \backslash$, in which case one finds oneself in the continuous case of a fraction rational to study. One can then bring back to a traditional study analogical systems on $\backslash \; Omega\; \backslash \; in\; \backslash \; left$ by knowing that the values of the diagram close to $\backslash \; omega=\; \backslash \; frac\; \{\backslash \; pi\}\; \{T\}\; \backslash$ are sullied with an error.

See too

• Automatic
• Cybernetic
• Transformed of Laplace
• Transformed into Z
• low-pass Filter
• high-pass Filter
• Diagram of Nyquist

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