Components of complex functions and circuit frequency

Characteristics

Circuit complex functions CFC as any complex number can be represented in the algebraic, exponential and trigonometric forms:

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where

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-

- is the real part of a complex function ;

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-

- is the imaginary part of a circuit complex function ;

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-

- is the argument of a circuit complex function.

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-

- is the module of a circuit complex function.

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The quantities R(w) , X(w) , φ(w) , K(w) depend on the frequency and are called as a real (RFC) , imaginary (IFC) , phase (PFC) and amplitude (AFC) frequency characteristics respectively. They are plotted against the frequency w - along to the abscissa axis and the frequency characteristic – along to the ordinate axis.

For example , for a r , L , C series circuit the typical AFC and PFC have the forms according to Fig. 5.3.a and Fig. 5.3. b.

Fig. 5.3

The amplitude – frequency (AFC) and phase – frequency (PFC) characteristics can be united into one amplitude – phase – frequency characteristic (APFC). It is plotted in relation to the coordinates X(w) , R(w) when the frequency w is changed from 0 up to µ or , in a common case , from ”-µ” up to µ. Each value of frequency has a value R(w) which is laid off along the abscissa and X(w) is laid off along the ordinate axis. The values X(w) and R(w) define a point on the plane X(w), R(w).The geometric location of the points which are obtained which different values of W make the APFC plot. Fig 5.4 shows an APFC plot for a circuit of the second order. Here at w=0 the value X(0)=0.Since R(0)=0,the point of the characteristic is on the abscissa. When the frequency is increased the X(w) values appear and the R(w) values decrease. So, at w = w₁ we have that X(w₁)<0 and R(w₁)<R(0). At w₂>w₁, we get the following point of the characteristic in the same way, and so on. At w = µ the functions X(µ)=0 and R(µ)=0, and the characteristic point coincides with the beginning of the coordinates axes.

Fig.5.4

Quite often APFC is supplemented by a curve at w<0.In this case in Fig.5.4 there appears a dotted curve which is symmetrical to the characteristic at w>0.

APFC can be built also by calculating the module K(w) and argument φ(w) at each value of the frequency w. Tying off the segment K(w) in the direction φ(w) we get the characteristic point.

At the end of the segment an arrow is placed and we say that with frequency changing the vector K(w) describes a curve which is called the frequency locus. The arrow on the curve indicates the direction in which the frequency w increases. Sometimes the frequency locus is called also the Nyquist diagram.

In the theory of electric filters for the circuit complex functions the exponential form of notation is used.

(5.24)

Here χ(jw)= α(w) + jβ(w) - is the complex coefficient of distribution.

Taking the logarithm from expression (5.24) we get:

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or

(5.25)

The value α(w) is called the logarithmic amplitude-frequency characteristic (ALFC).If it is calculated through the natural logarithm, then the value is measured in Napiers. If α(w) is calculated through the decimal logarithm, then the unit of measurement is called the Bell (B):

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Because

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then

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Date: 2015-12-18; view: 1023