Amplitude-phase, frequency, logarithmic frequency characteristics. problem solving examples

The frequency characteristics describe the transfer properties of the links (elements) and systems in the steady-state harmonic mode, caused by the external harmonic effect.

The dependence of the ratio of the amplitudes of the output and input signals on the frequency is called the amplitude frequency response (AFC). It is denoted by A (?) (Fig. 5.1).

Fig.5.1. Amplitude frequency response

Fig. 5.2. Phase frequency response

Fig. 5.3. Amplitude and phase response

The dependence of the phase shift on the frequency is called the phase frequency response (phase response), and it is denoted by? (?) (Fig.5.2). The frequency response shows how well the element transmits signals of different frequencies. The phase response shows what kind of lag or advance of the input signal in phase the element creates at different frequencies.

The amplitude frequency and phase frequency characteristics can be combined into one common: amplitude-phase frequency response (AFC) or (AFC) (Fig. 5.3). It is denoted by W (jw) and is a function of a complex variable whose modulus is equal to A (?) , And the argument is equal to? (?).

AFC W (jw), as well as any complex value, can be represented in the exponential:

(5.1)

or in algebraic form:

(5.2)

where Re (w) is the projection of the vector W (jw) onto the real axis, and lm (w) is the projection of the vector W (jw) onto the imaginary axis.

Re (w) is called the real frequency response, and lm (w) is the imaginary frequency response.

AFH is an Fourier transform image of an impulse transition function:

(5.3)

The inverse Fourier transform of the AFC will give a pulse transient function:

(5.4)

An analytical expression for the AFC of a particular element can be obtained from its transfer function by substituting p = jw:

(5.5)

The most common is the amplitude-phase frequency response (AFC), which is formally determined from the transfer function by replacing p with the argument jw .

(5.6)

(5.7)

(5.8)

Often W (jw), R (jw), K (jw) is called the complex transfer coefficient.

Amplitude-phase frequency response is a function of the complex variable jw and can be expressed in terms of module and phase. For example, for the AFC link K ( jw ) the module will be K ( jw ) = K ( p ) phase ( p ( w ) = argK ( jw )). AFC can be expressed in the following forms:

- algebraic form:

K ( jw ) = P ( jw ) + jQ ( w ), (5.9)

where P ( jw ) is the real part , jQ ( w ) is the imaginary part;

- indicative form:

(5.10)

Where:

In equality

(5.11)

the real part of the complex function K ( jw ), called the real frequency response (VCHH);

- the imaginary part of the complex function K ( jw ) called imaginary frequency response (MFC).

In equality

(5.12)

the module of the complex function K ( jw ) is called the amplitude-frequency characteristic (AFC). The amplitude-frequency characteristic of the AFC is the dependence of the system gain on the frequency of the input harmonic signal:

(5.13)

where Ay ( w ) is the output signal; Ax ( w ) is the input signal.

Therefore, the frequency response is equal to the ratio of the amplitudes of the forced component of the output and input signals.

In equality (5.12) - the argument of the complex function K ( jw ), called the phase frequency response (phase response).

Therefore, the phase response is equal to the phase difference of the forced component of the output and input signals.

The amplitude K ( w ) and phase frequency? (W) characteristics determine the stimulated component of the output signal of the system when it is acted upon by a harmonic input signal specified in a scalar form. So, if the system is affected by the harmonic input signal x (t) of the form, then the forced component of the output signal is determined accordingly. where

(5.14)

The hodograph of the frequency response K ( jw ) is called the trajectory of the point K ( jw ) on the complex plane when the frequency w changes from 0 to? . The hodograph point corresponding to w = 0 is called the beginning of the hodograph, and the point corresponding to w =? end point.

The hodograph is constructed by pairs of real characteristics. P ( w ), Q ( w ) or K ( w ),? (W) .

Logarithmic frequency characteristics