陕西理工学院毕业论文（设计）

This leads to the characteristic polynomial

According to [3] we denote the coefficients and D1 by for regions DO

Usually the eigenvalues of a given system are calculated from (2). Here we have the inverse problem- to calculate the coefficients (3) from the given eigenvalues in each region. This can be done using Vieta's formulas :

where pi and vi(i = 1, . . . , n) are the eigenvalues in DO and D1 respectively. C. Network Function

Our goal is to determine the structure of the two-terminal linear network and to calculate all parameters of the circuit shown in Fig. I(b). By simply applying Kirchhoff s voltage law to the circuit in the complex domain we obtain

where G, and Gb denote the small-signal conductance corresponding to the slope of the v-i-characteristics of the piecewise linear two-terminal element and s is the complex frequency s = o + jw. Assuming the complex admittance function of the linear network to be

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陕西理工学院毕业论文（设计）

and

If the circuit in Fig. l(b) has to realize the state equation (l), (8) and (9) have to correspond to the characteristic polynomials ((2), with respect to (3)). This will be achieved by comparising the coefficients of both groups of equations. D. Design Algorithm

The algorithm to design a network which realizes the given sets of eigenvalues will consist of the following two steps:

Step one: Determine the coefficients ( b 0 . . . ~ - 1 , a0 ...,-I) of the polynomial of the two-terminal function, and also the parameters G, and Gb by comparing it with the characteristic polynomial resulting from the given set of eigenvalues. Design the network which realizes the two-terminal function. Step two:

Note that we use admittance functions in developing our algorithm. Using impedance functions is also possible and leads to the dual network, as will be shown at the end of this chapter.

To carry out the first step we have to convert (8) and (9) into the form of the characteristic equation (2) for comparison. Multiplying (8) and (9) with the denominator polynomial we obtain:

First decide how the order of the denominator and numerator polynomials has to be chosen. Since the order of the denominator and numerator of a two-terminal network-function can only differ by 1 at maximum, we have three possible cases:

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陕西理工学院毕业论文（设计）

In the following we choose the Case 3, for this is the only one, which does not include restrictions to the choice of the component values (especially G,, Gb and K) of the circuit and hence can lead to a canonical structure. Then, (10) and (11) can be written in the form

TABLE I

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陕西理工学院毕业论文（设计）

Comparing the coefficients of (15) and (16) with those of the characteristic polynomial (2) Table I, we obtain the system of (17)-(20) shown in Table I.

These are 2 . n equations for the 2 . n + 2 unknown values ai(i = O , . - . , n - 2), bi(i = O , . . . ,n - l), Ga, Gb and K. Hence to clearly determine all unknown values, two additional equations are required. The definition of them allows to introduce additional conditions conceming the desired form of the network. To get the first additional equation we carry out the first step of polynomial division in (7) thus separating a parallel capacitor to be the first network element seen from the input (see (21) in Table I). C1 is the normalized value of the parallel capacitor. Since one circuit element parameter can be chosen arbitrarily because of the impedance scaling property, we set it for convenience and simplicity as in (22) in Table I. There are different ways to determine the second additional equation (for instance (23) in Table I). Two of these possibilities are shown in the table below. If all necessary 2n + 2 equations are determined, the unknown values can be calculated as follows: Subtracting (18) from (17) and (20) from (19) gives

Substituting (24) into (25) we obtain all ai (i = 0, 1, . . . , n - 2)in (26) in Table I. The kind of the structure we will obtain depends on thechosen way shown in Table I. Note that there is the restriction

which also arises in [l] and [3]. However p,-l # qn-l is a singular situation and can be eliminated by perturbing one of the eigenvalues without substantially changing the behaviour of the system . We give examples of both proposed ways in Sections 111 and IV.

Closing this chaDter we mention that there exists a dual network to every circuit realization outlined above. Then instead of (8) and (9), we have where Rap,. denotes the slope of the i-v-characteristic of the piecewise linear one-port.

111. SYNTHESIS OF CANONICAL CHUA CIRCUITS

In this chapter we take the first possibility in Table I. First we determine the coefficients and the parameters of the non-linearity. Substituting (26) for i = n - 2 into (23):

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