The harmonic answer

1) Conditions of Dirichlet:

Draft of the conditions under which periodic function exists the development in series of Fourier of one f(t):

to)       in every period of f(t) us it must be an ended number of maximums, minimums and discontinuity

b)       f(t) it must be univoca

c)       f(t) must be sommabile

 

2) Development in series of Fourier in trigonometrical shape:

essendo is had

 

3) Development in series of Fourier in exponential shape :

essendo

 

4) Integral of Fourier :

it corresponds to the antitransformed one of Fourier.

 

5) Transformed of Fourier of one function f(t):

the function f(t) also being able not to be periodic must however to satisfy the conditions of Dirichlet and the integral must exist for every value of w .

 

6) Phantom and phantom of energy of the f(t) :

The transformed one of Fourier of f(t) is a complex number and it can be written in the shape where A(w) comes called phantom of the f(t) while To²(w) comes called phantom of energy of the f(t).

 

7) Property of change of the temporal scale :

it is observed that to an expansion in the dominion of the time contraction in the dominion of the frequency corresponds one.

 

8) Property of translation in the time :

it is observed that the translation in the dominion of the time maintains the sign.

 

9) Property of translation in the frequency :

it is observed that the translation in the dominion of the frequency alters the sign.

 

10) Property of differentiation in the time :

 

11) Property of differentiation in the frequency :

 

12) Property of convoluzione in the time :

 

13) Property of convoluzione in the frequency :

14) Formula of Parseval:

where To²(w) it is the energy phantom while the integral to first member represents just the energy of marks them.

 

15) Relations between impulsive answer, harmonic answer, function of transfer and transformed of Laplace and Fourier :

The impulsive answer is the antitransformed one of Laplace of the transfer function and coincides also with the antitransformed one of Fourier of the harmonic answer.

 

16) Usefullness of the harmonic answer:

to)       it is of easy relief experiences them, enough in fact to subject to the system sinusoidi to various frequency

b)       it is variable complex function of real therefore can easy be represented.

 

17) harmonic Answer and its relation with the transfer function :

Draft of the value of the function of W(s) transfer calculated in s = jw that one sinusoide corresponds to considering like income and to neglect the transitory course.

18) Decade and eighth :

A variation of a decade is had when from a frequency f₀ passes to one frequenza f₁ = 10 f₀ .

A variation is had of eighth when from a frequency f₀ passes to one frequenza f₁ = 2 f₀ .

It is had that a variation of 20 dB for decade corresponds to one variation of 6 dB for eighth.

 

19) Decibel :

The introduction of the Decibel is justified having to render to delineate the scale of the amplitudes in the diagrams of Bode in how much is to delineate also the scale of is made, and this could not be made since the logarithm comes inserted in order to decompose the G(jw) in the sum and difference of various factors that were multiplies or uniforms to you, ha .

 

20)  :

0.3

 

21) Representation of the function G₀ :

It corresponds in module to the expressed static gain in dB and is therefore a straight horizontal while the phase is equal a 0° oppure -180° to second that the static gain is positive or negative.

 

22) Representation of the G function_1N(jw) :

It represents a zero in the origin, gives place for the having module to straight a slope 20dB/decade that it intersects the axis of the abscissas for w = 1 while the phase is characterized from a straight horizontal that intersects the formers for j = 90°.

 

23) Representation of the G function_1D(jw) :

It represents a pole in the origin, gives place for the having module to straight a slope of -20dB/decade equivalents -6dB/ottava while the phase is characterized from a straight horizontal that intersects the formers for j = -90°.

 

24) Diagrams of Bode in the case of zeroes or poles in the origin with greater variety of one :

The slope is equal to the variety multiplied for ±20dB/decade while in the diagram of the slope is made is equal to the variety multiplied for ±90°. In both the cases are relati you to the zeroes and - they are relati to you to the poles.

 

25) Point of breach :

A point of breach in correspondence to is had where t it is the relative constant of time to the pole .

 

26) Representation of the G function_2N(jw) in module and relati characteristic errors to you:

Draft of a zero real one, the asymptotic diagram of the module is worth 0dB before the breach point while after it knows them with equal slope to 20dB/decade, the error of the approximation is of 3dB in the point of breach, 1dB one eighth before and one after the breach point e 0.1dB one decade before and one decade after the breach point.

 

27) Representation of the G function_2N(jw) in phase and relati characteristic errors to you :

The asymptotic diagram of is made is constant and is worth 0° until to a decade before the breach point, he is constant and it is worth 90° beginning from a decade after the breach point while it introduces one before equal slope to 45°/decade in the interval comprised between one decade and one decade after the breach point.

 

28) Representation of the G function_2D(jw) :

Draft of a real pole, the asymptotic diagram of the module is worth 0dB before the breach point while after it comes down with equal slope to -20dB/decade, the error of the approximation is of -3dB in the point of breach, -1dB one eighth before and one after the breach point e -0.1dB one decade before and one decade after the breach point.

 

29) System to minimal phase :

A system is said to minimal phase when poles and zeroes of the function of transfer to open cycle are all in the skillful semiplan.

 

30) Roots of the factor trinomio in relation to the damping coefficient :

ž real roots are had coinciding

ž conjugated complex real roots are had

ž imaginary roots are had

 

31) Sign of zeroes and poles in correspondence to coefficients of damping denied to you :

Draft of zeroes and poles to positive real part.

 

32) Tracciamento of the diagram of the modules for factor trinomio G_3N(jw):

For inferior pulsations to the breach pulsation, the module is worth 0dB while to leave from the pulsation of breach in ahead the module it grows of 40dB/decade.

 

33) exact Representation of the function G₃(jw):

the curve intersects the axis of the abscissas after w_n therefore is always over to the asymptotic diagram

the curve intersects the axis of the abscissas before w_n

the curve does not intersect the axis of the abscissas therefore is always under to the asymptotic diagram

it is observed moreover that for always sovraelongazione is had.

 

34) Tracciamento of the diagram of is made for factor trinomio G_3N(jw):

the phase is worth 0° for inferior pulsations to the resonance frequency, comes down slowly until being worth 90° in correspondence of the same one while for advanced frequencies it knows them until being worth 180°.

the phase is worth 0° for inferior pulsations to the resonance frequency, dopodichè is one abrupt discontinuity that door the phase to be worth 180°

 

35) Modality of calculation of the phase begins them :

He is equal to the product of the difference between the number of poles and the number of zeroes in the origin multiplied for â?"90°. To the found value it must be added â?"180° in the case that the static gain is negative.

 

36) Modality of calculation of the final phase :

He is equal to the product of the difference between the number of poles and the number of zeroes multiplied for â?"90°, where between the poles the zeroes to positive real part must be considered also while between the zeroes the poles to positive real part must be considered also. To the found value it must be added â?"180° in the case that the static gain is negative.

 

37) Modality of calculation of the slope begins them :

He is equal to the product of the difference between the number of poles in the origin and the number of zeroes in the multiplied origin

for â?"20dB/decade.

 

38) Modality of calculation of the final slope :

He is equal to the product of the difference between the number of poles and the number of zeroes multiplied for â?"20dB/decade.

 

39) Effect on the diagram of Bode of one zero or pole to positive real part :

The effect on the module is indifferent from the effect produced from a zero or from a pole to negative real part while the effect on is made it is opposite therefore a zero to positive real part determines a negative slope of â?"45°/decade while a pole to positive real part determines one positive slope of 45°/decade.

 

40) Effect on the diagram of Bode of a brace of complex poles conjugates to you :

Slope in the module of â?"40dB/decade and one determine one discontinuity in the phase of â?"180°.