The harmonic answer in the systems to feedback

1) System to minimal phase:

The zeroes and the poles of the function of transfer to open cycle F(s) are all in the left semiplan.

 

2) System to regular stability:

It is a comprised stable system for k between 0 and a value advanced limit, if the gain grows ulteriorly, the system becomes unstable.

3) Pulsation of attraversamento w_t :

It is the pulsation for which the diagram of Bode of the modules intersects the straight horizontal correspondent to a gain of 0 dB.

 

4) Margin of phase m_j :

If the phase of the system in correspondence to the attraversamento pulsation w_t is greater of â?"180°, the difference comes said phase margin, an ideal value for the stability is 30° £ m_j £ 60°.

 

5) w_-p :

It is the pulsation for which the phase of the system crosses the straight horizontal j = -180°.

 

6) Margin of gain m_g :

If the gain of the system in correspondence to the pulsation w_-p he is smaller of 0 dB, the difference comes called gain margin, an ideal value for the stability è 3dB £ m_g £ 5dB.

 

7) Criterion of Bode:

In reference to a system to minimal phase to regular stability, it is had that the system turns out stable if the gain margin is 3dB £ m_g £ 5dB and the margin of phase è 30° £ m_j £ 60°.

 

8) Effect of the increase of the gain on the diagrams of Bode:

The diagram of makes remains immutato while the diagram of the modules trasla towards the high that it implies that, in the case of a system that you respect the criterion of Bode, the pulsation w_t is approached the w_-p , therefore are reduced is the phase margin that the margin of gain and the closed loop system is approached the stability limit.

 

9) Describe the diagram of Nichols:

Implicit representation in the flat phase-module of the logarithm of the harmonic answer is one. The phase is had on the abscissas that is worth 0 to the skillful end and â?"360° to the left end. On the formers there is the module in dB for which the 0dB it is found to half of the axis. In short for every frequency law on the diagram of Bode the module and the phase and them filler in a point of the diagram of Nichols.

 

10) Condition of stability in the representation of Nichols :

Once that has been traced the function of transfer to cycle opened on the diagram of Nichols is had that the phase margin is characterized from the intersection of the same one with the axis of the abscissas while the gain margin is characterized from the intersection with the axis of the formers, in particular the system turns out to be stable if such intersections finds both in 4° the quadrant.

 

11) Diagram of Nyquist:

It is an implicit representation in natural coordinates of the function of transfer to opened cycle, if of it it has on the abscissas the real part and the formers the imaginary part, the usefullness of such representation is tied only to the application of the criterion of stability of Nyquist.

 

12) Criteria for the construction of the diagram of Nyquist:

It is well to hold to mind the courses of the diagrams of Bode of module and phase, in particular for an approximate construction is sufficient to know module and phase is for w®0 that for w®¥.

In particular for w®0 the difference between the number of poles h ₂ is important and the number of zeroes h₁ in the origin, is had:

while for w®¥ the difference between the number of poles n and the number of zeroes m is important, it is had:

 

13) critical Point on the diagram of Nyquist:

It is the point that on the polar plan assumes coordinates (-1, j0) and therefore phase of â?"180° corresponds on the diagrams of Bode to one and a unitary gain.

 

14) Margin of gain on the diagram of Nyquist:

Numerically it corresponds to the distance from the origin of the intersection of the diagram of Nyquist with the real axis, on condition that it happens intuitivamente between (0, 0) and (-1, j0), instead since the gain margin characterizes the distance from the instability and this begins in point (-1, j0) then the gain margin is just the distance between this point and the intersection with the real axis of the diagram of Nyquist.

 

15) Margin of phase on the diagram of Nyquist:

Negative is the included angle between the real axle shaft and the intersection of the diagram of Nyquist with the unitary circle, on condition that this intersection happens in the 2° and 3° the quadrant.

 

16) complete Diagram of Nyquist :

It is the diagram of Nyquist for w that it varies from -¥ to ¥ and remembering that the Transformed one of Fourier of the function of transfer to open cycle is a function that has equal real part and uneven imaginary part, the diagram for w that it varies from -¥ to 0 can be obtained for simple symmetry regarding the real axis of the diagram that is had for w that it varies from 0 to ¥ , the complete diagram introduces of the discontinuities in the case is of the poles in the origin or of the imaginary poles.

 

17) Consideration on the poles in the origin in the diagrams of Nyquist :

The poles in the origin give place to of the discontinuities, that is 0^- it does not correspond more with 0 , can be gone around the pole in hour sense or counter-clockwise sense, that it is reflected on the diagram of Nyquist in a curve that is closed to the infinite which connects 0 e 0^- by means of a number of arch of 180° equal to the order of the pole in the origin and therefore to the type of the system, in particular is had that if the pole comes that is gone around in hour sense…(considering it stable) the closing to the infinite happens in counter-clockwise sense.

 

18) Consideration on the imaginary poles in the diagrams of Nyquist :

They are worth the same considerations made for the poles in the origin, with the difference that in this case of the brace of imaginary poles, can be decided to go around some in hour sense and in counter-clockwise sense, and us must be remembered that if the encircling happens in s in a sense, the closing to the infinite on the diagram of Nyquist happens in opposite sense.

The value of the angle for which the asymptote desume from the diagrams of Bode of the phase is had.

 

19) Relation between the function of transfer to open cycle F(s) and the characteristic equation Q_c(s):

The poles of the F(s) coincide with the poles of the characteristic equation Q_c(s).

 

20) Relation between the function of transfer closed cycle W(s) and the characteristic equation Q_c(s):

The poles of the W(s) coincide with the zeroes of the characteristic equation Q_c(s).

 

21) Criterion of Nyquist :

The number of poles to positive real part presents is:

where N is the number of spins in hour sense completed from a carrier applied in point (-1, j0) and that with the other end it covers the complete diagram of once single Nyquist while P₀ is the number of poles to positive real part presents in the function of transfer to open cycle F(s).

Clearly affinchè the closed loop system is stable must be ₀Z = 0 and therefore .

 

22) Passage from the harmonic answer to cycle opened to the harmonic answer by means of Nyquist :

The lines to constant module must be traced and on the same sheet the diagram of Nyquist, the curve of Nyquist represents therefore the harmonic answer to open cycle if they are considered like coordinated those orthogonal ones while it represents the harmonic answer if the curvilinear coordinates are considered.

 

23) Passage to the cycle closed on the diagrams of Nichols :

On the diagram of Nichols that is obtained from the diagram of Bode, the places to module are traced and constant phase obtaining in such a way the paper of Nichols, obtains therefore a graded curve in w such that if read second the orthogonal coordinates represents the harmonic answer to open cycle while if read second the curvilinear coordinates represents the harmonic answer.

 

24) Typology of measures of the harmonic answer:

to)       a sinusoidale generator to variable frequency can be used, for which it must be maintained to constant the amplitude and be measured the its answer to the various frequencies if not confronting it with it marks them of income by means of an oscilloscope to 2 traces which to obtain also information approximately the phase.

b)       it is sended to the system marks them sinusoidale to pulsation w, to follow a block to unitary feedback is placed having in direct chain a block with trasferenza to goes them of which ₁ is captured marks themV(s) and in cascade a block with trasferenza to goes them of which one are captured mark them V₂(s). Calculating antitransformed of these the two tensions for entire multiples k (…generally k = 1) of the T period of the sinusoidale income, it is had

e

 

25) Like estimating if a system is to minimal phase through the measure of the harmonic answer:

Enough to observe that they cannot be present poles to positive real part in how much would have divergence, famous instead the presence of zeroes to positive real part in fact in vhf we have ourselves that for the module they are behaved like the other zeroes while for the phase they are behaved like the poles.

 

26) Compensator :