Diagram of Bode

The diagram of Bode is a means of representing the frequential behaviour of a system. It allows a simplified graphic resolution, in particular for the study of the transfer functions of analogue systems. It is used for the properties of margin of profit, strokes of phase, continuous profit, Band-width, rejection of the disturbances and system stability. The diagram of Bode owes its name in Hendrik Wade Bode.
Definition
The diagram of Bode of a system of frequential answer T (jω) is composed of two layouts:
profit in decibels (dB). Its value is calculated from 20 log10 (|T(jω)|)
the phase in degree, given by arg (T (jω))
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.

Diagram of Bode of the passive low-pass filter of command 1. In red dotted lines, the linear approximation.
Asymptotic layout of the analogue systems
Let us take an unspecified transfer function which is written in the following way:
H (p) = αpq * {ΠKk = 1 (1 + 2ξk p⁄ωk + (p⁄ωk)²) ΠLl = 1 (1 + p⁄ωl)} ⁄ {ΠMm = 1 (1 + 2ξm p⁄ωm + (p⁄ωm)²) ΠNn = 1 (1 + p⁄ωn)}

α ∈ ℜ ; q ∈ Z ; ωk, ωl, ωm, ωn ∈ ℜ* ; ξk, ξm ∈ ℜ
Although a 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.Pour that to use the constant α.
The terms out of p of the elementary polynomials of the first and the second degree must be with the numerator.
It is noticed that the module of H(p) is equal to the sum of the modules of the elementary terms because of the logarithm. 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

Diagram of Bode of a filter passes low (system of the 1st command)
Definition
That is to say the transfer function:
H (p) = 1 ⁄ 1 + (p/ω0)
The pulsation ω0 is called pulsation of cut.
Asymptotic layout
ω « ω0, H(jω) ≈ 1
|HdB (jω)| = 0 and arg (H(jω)) = 0°
ω » ω0,H(jω) ≈ - j*ω0 ⁄ ω
|HdB (jω)| = - 20 log100) et arg (H(jω)) = - 90°
In a reference mark logarithmic curve, |HdB (jω)| results in a slope of -20dB ⁄ decade or -6dB ⁄ octave.On also speaks about slope -1. The asymptotic diagram of Bode of the module is thus summarised with two linear sections.
Real layout
ω0,H (jω0) = 1 ⁄ 1 + j
that is to say
Hdb(jω0)| = - 20 log10 (√2) = - 10 log10 (2)
the curve passes 3dB in lower part of the asymptote.
High-pass

Diagram of Bode of a filter passes high (system of the 1st command)
That is to say the transfer function
H(p) = 1 ⁄ 1 + (ω0 ⁄ p) = (p ⁄ ω0) ⁄ 1 + (p ⁄ (ω0)
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 command
Low-pass
Definition
A system of the second command of the type passes low is characterised by a transfer function of the type
H(p) = H0 ⁄ 1 + 2ξ*(p ⁄ ω0) + (p ⁄ ω0
H0 is the static profit
The pulsation ω0 is called own pulsation
and ξ is damping.
Asymptotic layout and real Curve
In this part one takes the static profit H0 is equal to 1
The asymptotic layout depends on the value of damping
Three cases are distinguished
ξ › 1
The poles of the transfer function are real and negative for reasons of stability, and the system breaks up into a product of two transfer functions of the 1st command. That is to say p1 and p2 real poles of the transfer function:
H (p) = 1 ⁄ 1 + 2ξ*(p ⁄ ω0) + (p ⁄ ω0)² = 1 ⁄ (1 + p ⁄ p1)*(1 + p ⁄ p2)

Diagram of Bode of a system of command two with a damping equal to 5.5 en ω0 = 1
. The system breaks up then in the shape of a product of first order systems.
ξ = 1
The poles are real, negative and equal (double pole). If p0 is a double pole of the transfer function, one obtains:
H (p) = 1 ⁄ 1 + 2ξ * (p ⁄ ω0) + (p ⁄ ω0)² = 1 ⁄ (1 + p ⁄ ω0
For
ω » ω0 H (jω) ≈ (ω0 ⁄ ω)²
thus
|HdB (jω) | = - 40 log10ω + 40 log10 ω0
and
arg (H(jω)) = - 180° * signe (ω0ξ)
In a reference mark logarithmic curve, |HdB (jω)| results in a slope of -40dB/decade or -12dB ⁄ octave. One also speaks about slope -2. The asymptotic diagram of Bode of the module is thus summarised with two sections linéaires.ξ < 1
The asymptotic diagram is the same one as in the preceding case
The poles of the transfer function complex and are combined, with negative real part
When ξ < √2 ⁄ 2, the system presents a resonance
The maximum of the module of the transfer function is then |H (jω)|max = 1 ⁄ 2ξ√(1 - ξ²)
in
ω0 √(1 - 2ξ²)
The pulsation ωR corresponding to the maximum is thus always lower than ω0

Diagram of Bode of a system of command two with a damping equal to 0.8 and ω0 = 1

Diagram of Bode of a system of command two with a damping equal to 0.3 and ω0
The system presents an overpressure.
High-pass
H (p) = (p ⁄ ω0)² ⁄ 1 + (2ξ*(p ⁄ ω0)) + (p ⁄ ω0
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 function H (p).
However, when this transfer function is complicated, it is easier progressively to take into account the contributions of each term while making grow the pulsation ω.
At the beginning, when ω → 0, the asymptote of the module is a line of slope Q (q * 20dB ⁄ decade) and the phase is constant with Q * 90°. Thereafter, with each time one meets a pulsation, one modifies the layout according to the following procedure:
For ω = ωk one adds +2 with the slope of the module ( + 40dB ⁄ decade) and 180° * sign (ωkξk with the phase.
For ω = ωl one adds +1 with the slope of the module ( + 20dB ⁄ decade) and 90° * sign (ωl) with the phase.
For ω = ωm one adds -2 with the slope of the module ( - 40dB ⁄ decade) and -180° * signe (ωmξm) with the phase.
For ω = ωn one adds -1 with the slope of the module ( - 20dB ⁄ decade) and -90° * signe (ωn) with the phase.
Layout of the number systems
Limitation of the field of the pulsations
We have this time a transfer function G (Z) = Z {G (N)} 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 = e2πjv with v ∈ [0;½] (one obtains the complete circle by symmetry).
If the discrete system were obtained starting from sampling at the period T of a continuous system, then z = ejωT with ω ∈ [0;π ⁄ T]
Moreover, relations |G (z)| ∈ = e2πjv and arg (G(z)z = e2πjv) are not rational in v.Par 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 = (2 ⁄ T) + ω ⁄ (2 ⁄ T) - ω
or the reciprocal function ω = 2 z - 1 ⁄ Tz + 1, It acts of a transformation of Möbius.
This transformation makes correspond the secondary axis ω = jΩ
continuous field with the circle unit z = ejwT
discrete field with ω = 2 ⁄ T arctan (TΩ ⁄ 2)

Diagram of Black

The diagram of Black is a graph used automatically to study a system. It represents, in a semi-logarithmic reference mark, the profit in decibels according to the phase, according to a curve parameterised by the pulsation or the frequency. This diagram combines in two diagrams of Bode.
It is usual to trace in the plan of Black the abacus of Nichols, one speaks then about diagram of Black-Nichols. This abacus makes it possible to trace the graph of the transfer function in closed-loop (FTBF) with unit return starting from the graph of the transfer function in open loop (FTBO). These two transfer functions check the relation indeed:
FTBF (jω) = FTBO (jω) ⁄ 1 + FTBO (jω)
or
FTBF (jω) and FTBO (jω) are complex numbers whose module represents the profit, and the argument represents the phase, according to the pulsation ω.

Diagram of Nyquist

The diagram of Nyquist is a graph used automatically to evaluate the stability of a system in closed-loop. It represents, in the complex plan, the harmonic answer of the corresponding loop system open. The phase is the angle and the module the distance from the point in the beginning. Just like the diagram of Nichols, the diagram of Nyquist combines the two types of diagram of Bode, modulates and phase, in only one. The diagram of Nyquist owes its name in Harry Nyquist.
The diagram of Nyquist is very useful for the study of the stability EBSB of the loop systems open to negative feedback, thanks to the theorem of Nyquist.
The system is stable in closed-loop with unit return in negative feedback on the input if the critical point (- 1,0) is left with the left of the curve plotted for a pulsation varying of 0 ad infinitum.

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