5 The notion of the complex function of a circuit




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Chapter 5. Frequency characteristics of circuits
5.1. Complex function of a circuit

5.1.1. The notion of the complex function of a circuit
The application of exponential functions allows us to introduce the notion of the complex function of a circuit which is used for describing circuits not containing independent sources of energy.

In a common case, the signal xout at the output of an electric circuit (circuit reaction) is connected with the input signal xin (action) by relation

(5.1)

The harmonic functions xin, xout can be represent as images by complex numbers simiear to

(5.2)

Replace in (5.1) the real quantities xin ,xout with their complex instantaneous quantities according to relation (5.2):

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The ration of reaction to action when they are given as an exponential function of imaginary frequency j is called the complex function of a circuit (CFC) K(j):
(5.3)
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That is, the complex function of a circuit determines the circuit reaction to the action of exponential function of the imaginary frequency j in the state established.

With the help of the complex function, the theory of analytical functions of the complex variable is linked to the analysis of electric circuits.

Under analyzing of electric circuits, we set up an equation system in which it is always possible to identify two main quantities of action and reaction, for example, current in one branch and voltage in the another.
Fig. 5.1
Consider the types of complex functions of a circuit. Fig. 5.1 shows a diagram of the passive , linear circuit N with the marked input (1-1)and output (2-2) terminals. The diagram indicates input and output currents and voltages and also the load impedance Zh connected to the output terminals. There are input and transfer complex functions of a circuit.

The input complex function of a circuit is the relation of complex images of voltage and current acting on the terminals of the circuit.

They are as follows :

- complex input impedance

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- complex input admittance

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It’s obvious that

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The transfer complex function of a circuit is the relation of complex images of voltage and current acting on different pairs of circuit terminals.

They are as follows :

- complex voltage transfer ratio

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- complex current transfer ratio

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- complex transfer impedance

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- complex transfer admittance

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It’s evident that

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^ 5.1.2. Connection between complex functions and circuit parameters
Fig 5.2 shows a diagram of the passive linear circuit N with two marked pairs of K-K’ ,1-1’ terminals. The voltage source , the EMF of Which is equal to Ein , and the internal resistance is equal to Zint is connected to the terminals l-l’. Find the ratio of currents and voltages in this circuit. Write down the equation system a ecovding to the method of loop currents. Mark out at terminals l-l’ – the l-th loop with the loop current t1 , at terminals K-K’-the K-th loop with the loop current Ik. Indicate directions of loop tracing. And as a result we get.

(5.4)

Fig.5.2
Here N is the total number of independent circuit loops , see Fig. 5.2. Since the circuit is passive, the loop EMF Ein is contained only in the K-th line (5.12) of the system ,the other loops do not contain no independent energy sources. The resistances Zji , Zjk are the internal and mutual resistances of the loops. The currents I1,I2,.. Ik ,…, In are the loop currents.

For the k-th loop

, (5.5)

where Z’kk is the internal resistance of the k-th loop without taking into account resistance Zint. That is Z’kk is the resistance of the circuit N on the side of terminals K-K’.

For the k-th loop we can write the following to Kirchoff’s voltage law:

. (5.6)

From equation (5.6) we have relation

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Let us consider the k-th equation in system of equations (5.4)

(5.7)
From expressions (5.5) and (5.6) we have

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Then equation (5.7) can be rewritten as
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or

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So, system of equations (5.4) is:
(5.8)

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According to Cramer’s rule the solution (5.8) for the l’s current is
, (5.9)

where  is a determinant of the system for the matrix of loop resistances; l is a determinant of  in which the l-th column is replaced by the column of absolute terms – loop EMF.

Expand the determinant 1 by elements of the l-th column. Taking into account that absolute term Uk is only in the k-th line we get

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Here k1 is an algebraic adjunct to an element of the k-th line and l-th column; Mk1 – is an element minor of the k-th line and l-th column obtained by deleting the k-th line and l-th column from the determinant .

Then according to (5.9) we have expression
(5.10)

Similarly
(5.11)
Voltage across load Z1is
(5.12)

From (5.10)-(5.12) we can define the circuit complex functions. From (5.11) we get the complex input impedance and complex input admittance

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From (5.10) we get the complex transfer admittance

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From (5.11) , (5.12) we get the complex transfer impedance

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From (5.12) we get the complex transfer coefficient by voltage

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From (5.10), (5.11) we get the complex transfer coefficient by current

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Thus , the circuit complex functions can be expressed by elements of the loop resistances matrix compiled for the passive circuit N and the load resistance Z1.

A determinant of the N-th order  is
. (5.13)

Here : q = 1 , 2, … , N- the number of lines of a determinant ;  , r,…, v are the permutations from N! of all possible permutations from the number 1 , 2 , … , N. The number q is at the sometime the number of inversions in each permutation. It is obvious that the number of terms in (5.13) is also equal to N!

Any of the loop resistances in (5.13) Zl , Z2r , … , ZNv can imaginary frequency j
(5.14)

where = 1 , 2 , … ,N.

It is evident that the determinant  that consists of that loop resistances Zi , Z2r , … , ZNv is also the rational function of the imaginary frequency j. The same can be said about the determinants kl , kk. Therefore , the complex functions of the circuit are the rational functions of the imaginary frequency j. Coefficients in the circuit complex functions according to (5.14) are determined only by the circuit parameters r , L , C and , consequently , are real coefficients.

Obtain the circuit complex functions according to Fig. 5.2 with the method of node voltages. Set the nodes K’ and l’ be connected between themselves. Assume the point of their connection to be the basis node. The voltages Uk , Ul will be considered as node voltages. Write a system of node equations according to circuit in Fig. 5.2:
(5.15)

Here N is the total number of independent nodes. Because the circuit is passive , the loop current Yint , Eint only in the k-th line of the system (5.33) is presented, where Yint is the internal conductance of the current source connected with k-th node.

The remaining nodes don’t have an energy source. The conductances Yjj , Yjk are intrinsic conductors and transfer conductance of the nodes. The voltages U1 , U2 , … UN are node voltages.

For the k-th node

(5.16)
where Y’kk is the intrinsic conduction of the k-th node without due regard for conductance Yint. That is Y’kk is the conductance of the N circuit the side of terminals k-k’.

For the k-th node , we can write the following according to Kirchoff’s current law:

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where

(5.17)

Set us consider the k-th equation in (5.15)

(5.18)

Here with taking into account (5.16) and (5.17) we have:

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Then , (5.18) one can rewritten in such way

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or

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Now the system of equations (5.15) have the following form:
(5.19)
Solution of the system of equations (5.19) for l-th voltage is:

(5.20)
where  is a determinant of the system of equations for the matrix of loop conductivities; l is a determinant of  in which the l-th column is replaced with the column of absolute terms – nodes’ currents.

Expand the determinant 1 by elements of the l-th column. Taking into account that absolute term Uk is only in the k-th line we get

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Then according to formula (5.20)
(5.21)

Similarly
(5.22)

Current in resistance Zl is:

,(5.23)

where Yl = 1/Zl is a conductivity of resistance.

From the formulae (5.21) and (5.23) we get the CFC, from formula (5.22) we find

complex input impedance and complex input admittance:
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From the formula (5.21) we get the complex transfer impedance:
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From the formulae (5.22) – (5.23) we get the complex transfer admittance:
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From the formula (5.23) we get the complex transfer current coefficient:
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From the formulae (5.21) and (5.22) we get the complex transfer voltage coefficient:
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Thus , the circuit complex functions CFC can be expressed by elements of the loop resistances matrix compiled for the passive circuit N and the load conductance Y1.

A determinant  of the N-th order is:

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Determination of node admittance is provided as in equation (5.14):

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Comparing the expressions for CFC getting with the Contour Current Method (CCM) and Node Voltage Method (NVM) we find relations:

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^ 5.2.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.



The quantities R() , X() , φ() , K() 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  - 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() , R() when the frequency  is changed from 0 up to  or , in a common case , from ”-” up to . Each value of frequency has a value R() which is laid off along the abscissa and X(w) is laid off along the ordinate axis. The values X() and R() define a point on the plane X(), R().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 =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() values appear and the R() values decrease. So, at  = 1 we have that X(1)<0 and R(1)2>1, we get the following point of the characteristic in the same way, and so on. At  =  the functions X()=0 and R()=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 <0.In this case in Fig.5.4 there appears a dotted curve which is symmetrical to the characteristic at >0.

APFC can be built also by calculating the module K() and argument φ() at each value of the frequency . Tying off the segment K() in the direction φ() 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() describes a curve which is called the frequency locus. The arrow on the curve indicates the direction in which the frequency  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(j)= α() + jβ() - 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() 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() 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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^ 5.3. Frequency characteristics of simplest circuits.

5.3.1. Frequency characteristics of first-order circuits
Consider the simplest rC-circuit (Fig.5.5.a).Let us define the complex voltage transfer ratio:
(5.26)
Fig.5.5
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Fig.5.6
Fig.5.7
If τ is denoted as time constant: τ = l/r, than

(5.27)
The expression (5.84) coincides with (5.71) for the rC-circuit in Fig.5.5.a. Consequently, the frequency characteristics of the circuits in Fig.5.8.a and 5.5.a are identical.

The analysis shows that the frequency characteristics on the circuits in Fig.5.8.b and 5.5.b are also identical.
^ 5.3.2. Frequency characteristics of second- and third-order circuits
Consider the second-order circuit in Fig.5.9.The matrix of node conductances for this circuit is as follows:
(5.28)
Fig. 5.8 Fig.5.9

Hence, the algebraic adjuncts 13 and 11 are:

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Here M13 and M11 are the minor-determinants obtained from the matrix determinant (5.28) by deleting the first line and third column and the first line and first column, respectively, that is
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The voltage transfer ratio in (5.53) is:



(5.29)

Here

(5.30)
(5.31)
where τ = rC is a time constant.

The diagrams of AFC and APFC are given in Fig.5.10.a and Fig. 5.10.b respectively.
Fig. 5.10
Consider the second-order circuit in Fig. 5.11 (the Wien bridge).The matrix of node conductances for the circuit is as follows:

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Fig.5.11 Fig. 5.12
Hence, the algebraic adjuncts 12 and 11 are:

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The voltage transfer ratio in will be

(5.32)
Here

(5.33)
(5.34)
When comparing (5.33) and (5.34)with (5.30), (5.31), we see that these expressions are identical and, consequently, the circuit frequency characteristics in Fig.5.9 and 5.11 coincide

Consider the third-order circuit in Fig. 5.2 (double T-form bridge).

The matrix of node conductances for this circuit is as follows:

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Hence, the algebraic adjuncts 14, 11 are:

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The voltage transfer ratio equals

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Here

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Phase angle equal
The diagrams of AFC and APHFC are represented in Fig. 5.13.a and Fig. 5.13.b.


Fig.5.13

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