CN103701464A - Fractional order element converter - Google Patents

Abstract

The invention discloses a fractional order element converter, and belongs to the technical field of theory of electrical engineering. The fractional order element converter is structurally and functionally characterized in that a minimum number of active integrated devices and existing circuit elements are connected according to a designed circuit topological structure, and the existing circuit elements in a circuit are configured reasonably, so that fractional order conversion, order conversion and element value conversion can be realized. The fractional order element converter has the characteristic of comprehensive and flexible functions, and the following functions can be realized: firstly, conversion of fractional order element types can be realized to obtain a fractional order element of a needed type; secondly, conversion of fractional order element orders can be realized to obtain a fraction order element of a needed order; thirdly, conversion of fractional order element values can be realized to obtain a fraction order element of a needed element value. By using the fractional order element converter, an integer order element can be obtained through the fractional order element.

Description

A kind of fractional order element converter

Technical field

The invention belongs to electrical theory technical field, particularly a kind of fractional order element converter.

Background technology

At present, the widely used circuit element of people is all thought for integer rank element, i.e. integer rank capacity cell, inductance element, resistive element.But these elements nonideal integer order, all show certain non-integer order characteristic, is also fractional order characteristic.Therefore, draw fractional order element and more can describe exactly the characteristic of circuit element.In addition, a common fractional order element can be described a complicated high-order integer rank system, thereby uses fractional order element, can realize more compactly circuit.Fractional order element comprises fractional order inductance element and fractional order capacity cell.Fractional order element has long-term memory effect, and the terminal voltage of fractional order inductance element is all relevant with the historical electric current of the fractional order element of flowing through; The electric current of the fractional order of flowing through capacity cell is all relevant with the historical voltage of fractional order electric capacity.The feature of fractional order element is the straight line that is expressed as a constant-slope on Bode diagram, and phase-frequency characteristic is steady state value, so this class device is also referred to as permanent phase element.Fractional order element due to than integer rank element many order degree of freedom, to circuit design, brought greater flexibility, a large amount of uses of fractional order element in integrated circuit are inexorable trends.The realization of fractional order element at present also rests on laboratory stage, and operational is only the fractional order element of some specific order.

Select the fractional order element of dissimilar, different orders and different elements value can make circuit design and large-scale circuit integrated more flexible and easy, but at present available fractional order component type is limited and still do not have a circuit with the mutual translation function of fractional order element.Therefore, realize a kind of translation circuit that is applicable to flexible conversion between fractional order element and there is very important Practical significance and value.In actual applications, can utilize this translation circuit to carry out suitable conversion to the fractional order element of easily buying on market according to design needs, obtain being difficult for manufacturing, fractional order object component not easy of integration, thereby reduce design complexity, improve design efficiency.

Summary of the invention

The problem existing for above-mentioned prior art, the present invention proposes a kind of fractional order element converter, it is characterized in that, structure and the function of this fractional order element converter are: the active integrated device and the existing circuit element that utilize minimum number, according to designed circuit topological structure, connect, by the existing circuit element in reasonable disposition circuit, realize the conversion of the conversion of fractional order classification, order conversion and component value;

Circuit topological structure comprises four circuit structures of realizing this function:

Topological structure one: the W that the X end of I AD844 is connected to II AD844 by existing element 3 holds, and Y holds ground connection, and Z end is the input of circuit, W end is connected to the Y end of II AD844; The X end of II AD844 is by existing element 1 ground connection, and Y end is connected with the W end of I AD844, and Z end is by existing element 2 ground connection, and the X that W end is connected to I AD844 by existing element 3 holds;

Topological structure two: the X end of I AD844 is by existing element 3 ground connection, and Y end is connected to the W end of II AD844, and Z end is input, and the X that W end is connected to II AD844 by existing element 1 holds; The W that the X end of II AD844 is connected to I AD844 by existing element 1 holds, and Y holds ground connection, and Z end is by existing element 2 ground connection, and W end is connected to the Y end of I AD844;

Topological structure three: the X end of I AD844 is by existing element 1 ground connection, and Y end is connected to the Z end of II AD844, and is input, and Z end is by existing element 2 ground connection, and the X that W end is connected to II AD844 by existing element 3 holds; The W that the X end of II AD844 is connected to I AD844 by existing element 3 holds, and Y holds ground connection, and Z end is connected to the Y end of I AD844, and W end is unsettled;

Topological structure four: the X end of I AD844 is input, Y end is connected to the W end of II AD844, and Z end is by existing element 1 ground connection, and the X that W end is connected to II AD844 by existing element 2 holds; The W that the X end of II AD844 is connected to I AD844 by existing element 2 holds, and Y holds ground connection, and Z end is by existing element 3 ground connection, and W end is connected to the Y end of I AD844.

Described active integrated device is commercial one chip amplifier AD844, and the function of realization is operating amplifier with current feedback.

Described minimum number is 2, and wherein, I AD844 is the active integrated device in left side in circuit topology, and II AD844 is the source integrated device on right side in circuit topology.

Described existing circuit element, according to the difference of circuit practical function, adopt integer rank electric capacity, inductance, resistive element and be equivalent to circuit or existing fractional order electric capacity, the inductance element of integer rank electric capacity, inductance, resistive element and be equivalent to the circuit that has fractional order electric capacity, inductance element.

The invention has the beneficial effects as follows: this invention has complete function feature flexibly, can be achieved as follows function: the first, realize the conversion of fractional order element classification, obtain the fractional order element of required type; The second, realize the conversion of fractional order element order, obtain the fractional order element of required order; The 3rd, realize the conversion of fractional order elements value, obtain the fractional order element of required component value.Use this invention also can obtain integer rank element by fractional order element.

Accompanying drawing explanation

Fig. 1 is the schematic diagram of operating amplifier with current feedback (CFOA);

Fig. 2 is fractional order element converter circuit topological structure one;

Fig. 3 is fractional order element converter circuit topological structure two;

Fig. 4 is fractional order element converter circuit topological structure three;

Fig. 5 is fractional order element converter circuit topological structure four;

Fig. 6 is the principle key diagram of fractional order element converter topology structure one.

Embodiment

Below in conjunction with accompanying drawing, the present invention is described in more detail.

Structure and the function of this fractional order element converter are: the active integrated device and the existing circuit element that utilize minimum number, according to designed circuit topological structure, connect, by the element in reasonable disposition circuit, realize the conversion of the conversion of fractional order classification, order conversion and component value.

The active integrated device of using in the middle of this converter is commercial one chip amplifier AD844, and the function of realization is operating amplifier with current feedback (CFOA).

CFOA is that a kind of high resistant voltage homophase is inputted and low-resistance electric current is inputted, the amplifier of low-resistance Voltage-output, and its circuit symbol as shown in Figure 1.The port identity equation of current feedback amplifier is

$\left[\begin{array}{c}{u}_x\\ i_y\\ i_z\\ u_w\end{array}\right]=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 1& 0\\ 0& 1& 0& 0\end{array}\right]\left[\begin{array}{c}{u}_y\\ u_z\\ i_x\\ i_w\end{array}\right]$

This converter when described in minimum number be 2, for convenience of describing, hereinafter referred to as left side in circuit topology be I AD844, right side be II AD844.

This converter when described in existing circuit element, according to the difference of circuit practical function, adopt integer rank electric capacity, inductance, resistive element and be equivalent to circuit or existing fractional order electric capacity, the inductance element of integer rank electric capacity, inductance, resistive element and be equivalent to the circuit that has fractional order electric capacity, inductance element.

This converter when described in circuit topological structure comprise four circuit structures of realizing this function, the circuit that is two active integrated devices and existing circuit element is connected.

Fig. 2 is fractional order element converter circuit topological structure one; In described topological structure one, the W that the X of I AD844 end is connected to II AD844 by existing element 3 holds, and Y holds ground connection, and Z end is the input of circuit, and W end is connected to the Y end of II AD844; The X end of II AD844 is by existing element 1 ground connection, and Y end is connected with the W end of I AD844, and Z end is by existing element 2 ground connection, and the X that W end is connected to I AD844 by existing element 3 holds.

Fig. 3 is fractional order element converter circuit topological structure two; In described topological structure two, the X of I AD844 end is by existing element 3 ground connection, and Y end is connected to the W end of II AD844, and Z end is input, and the X that W end is connected to II AD844 by existing element 1 holds; The W that the X end of II AD844 is connected to I AD844 by existing element 1 holds, and Y holds ground connection, and Z end is by existing element 2 ground connection, and W end is connected to the Y end of I AD844.

Fig. 4 is fractional order element converter circuit topological structure three; In described topological structure three, the X of I AD844 end is by existing element 1 ground connection, and Y end is connected to the Z end of II AD844, and is input, and Z end is by existing element 2 ground connection, and the X that W end is connected to II AD844 by existing element 3 holds; The W that the X end of II AD844 is connected to I AD844 by existing element 3 holds, and Y holds ground connection, and Z end is connected to the Y end of I AD844, and W end is unsettled.

Fig. 5 is fractional order element converter circuit topological structure four; In described topological structure four, the X of I AD844 end is input, and Y end is connected to the W end of II AD844, and Z end is by existing element 1 ground connection, and the X that W end is connected to II AD844 by existing element 2 holds; The W that the X end of II AD844 is connected to I AD844 by existing element 2 holds, and Y holds ground connection, and Z end is by existing element 3 ground connection, and W end is connected to the Y end of I AD844.

Be illustrated in figure 6 the analysis circuit of fractional order element converter topology structure one, wherein the commercial AD844 of active device realizes CFOA function.For convenience of specification and analysis, the existing element 1,2,3 in circuit topology is all with impedance form Z _αrepresent.When existing element is different, its corresponding Z _αalso different.

By the port identity of CFOA device, can obtain circuit equation and be

$\left\{\begin{array}{c}{U}_2\left(s\right)=U_1\left(s\right)=U_{\mathrm{IN}}\left(s\right)\\ I_3=I_2=\frac{U_2\left(s\right)}{Z_{{\mathrm{\&alpha;}}_1}}\\ U_3\left(s\right)=I_3{Z}_{{\mathrm{\&alpha;}}_2}\\ U_4\left(s\right)=U_3\left(s\right)\\ U_5\left(s\right)=0\\ I_{\mathrm{IN}}={\mathrm{<figure-callout\; id="5"\; label="I"\; filenames="HDA0000437183150000012.png"\; state="\{\{state\}\}">I</figure-callout>}}_5=\frac{U_4}{Z_{{\mathrm{\&alpha;}}_3}}\end{array}\right.$

Thereby can obtain impedance, be: $Z_{\mathrm{IN}}=\frac{U_{\mathrm{IN}}\left(s\right)}{I_{\mathrm{IN}}}=\frac{Z_{{\mathrm{\&alpha;}}_1}{Z}_{{\mathrm{\&alpha;}}_3}}{Z_{{\mathrm{\&alpha;}}_2}}.$

The first, realize the type conversion of fractional order element.Selecting existing element 1,3 is integer rank resistive elements,

select the fractional order capacity cell that existing element 2 is α for order,

the element obtaining by fractional order element converter is fractional order inductance element

completed the conversion of fractional order component type.

The second, realize the order conversion of fractional order element.Selecting existing element 1,2,3 to be respectively order is β ₁, β ₂, β ₃fractional order element,

fractional order element after being converted is that order is β ₁+ β ₂-β ₃fractional order element, choose reasonable parameter value, can determine whether order when conversion carry out classification conversion.

The 3rd, realize the conversion of the component value of fractional order element.Select existing element 1,2,3 for order is all α, the different fractional order element of component value only,

now can complete the conversion of fractional order elements value.Fractional order element after conversion is for being worth for V ₁v ₂/ V ₃order is the similar fractional order element of α.

So far, the function that the described fractional order element of topological circuit one converter is realized the conversion of fractional order element classification, order, component value has carried out description detailed.Other three kinds of topological structures can be realized the function of fractional order element conversion equally, and its derivation is identical with the derivation of topological circuit one.

It will be understood by those skilled in the art that actual hardware facility can not possess ideal characterisitics.But can make actual fractional order element converter in wide as far as possible frequency band, realize the function of fractional order conversion by hardware debug.Now the circuit topology one under non-ideality is analyzed.

In actual CFOA model, its anti-phase input resistance R _iN-representative value is 20～50 Ω, homophase input resistance R _iN+representative value is 10M Ω, output resistance R _orepresentative value is 15 Ω, gain impedance end resistance R _trepresentative value is 3M Ω, capacitor C _trepresentative value is 3～5pF.As shown in Figure 6, while considering CFOA actual characteristic, circuit equation can be expressed as

$\left\{\begin{array}{c}{I}_{2x}(R_{\mathrm{IN}-}+Z_{{\mathrm{\&alpha;}}_1})=U_{2y}\\ U_{2z}(\frac{1}{R_T}+\frac{1}{Z_{{\mathrm{\&alpha;}}_2}}+\mathrm{j\&omega;}{C}_T)=I_{2x}\\ I_{1x}(R_o+Z_{{\mathrm{\&alpha;}}_3}+R_{\mathrm{IN}-})=U_{2z}\\ U_{2y}=\frac{R_{\mathrm{IN}+}}{R_{\mathrm{IN}+}+R_o}{U}_{2z}\\ I_{1z}=I_{1x}+U_{1z}(\frac{1}{R_T}+\mathrm{j\&omega;}{C}_T)\end{array}\right.$

Subscript 1,2 represents respectively I AD844 and II AD844, and subscript x, y, z represents respectively X, Y, the Z end of AD844.By above equation, arrange and obtain being transformed under non-ideal characteristic

$Z_{{\mathrm{\&alpha;}}_{\mathrm{ch}}}=\frac{U_{1z}}{I_{1z}}=\frac{1}{\frac{1}{R_T}+\mathrm{j\&omega;}{C}_T+\frac{R_{\mathrm{in}+}}{(R_{\mathrm{in}+}+R_o)(R_{\mathrm{in}-}+Z_{{\mathrm{\&alpha;}}_1})(R_o+R_{\mathrm{in}-}+Z_{{\mathrm{\&alpha;}}_3})}(\frac{1}{Z_{{\mathrm{\&alpha;}}_2}}+\frac{1}{R_T}+\mathrm{j\&omega;}{C}_T)}$

In the situation that introducing non-ideal characteristic, it is complicated that the characteristic of fractional order element converter becomes, thereby only in limited frequency band, can realize previously described desirable fractional order element converter characteristic.But by hardware debug and Rational choice element, can in wide as far as possible frequency band, realize mapping function.

The above, be only preferably embodiment of the present invention, but protection model of the present invention

Enclose and be not limited to this, be anyly familiar with those skilled in the art in the technical scope that the present invention discloses, the variation that can expect easily or replacement, within all should being encompassed in protection scope of the present invention.Therefore, protection scope of the present invention should be as the criterion with the protection range of claim.

Claims (4)

1. a fractional order element converter, it is characterized in that, structure and the function of this fractional order element converter are: the active integrated device and the existing circuit element that utilize minimum number, according to designed circuit topological structure, connect, by the existing circuit element in reasonable disposition circuit, realize the conversion of the conversion of fractional order classification, order conversion and component value;

Circuit topological structure comprises four circuit structures of realizing this function:

Topological structure one: the W that the X end of I AD844 is connected to II AD844 by existing element 3 holds, and Y holds ground connection, and Z end is the input of circuit, W end is connected to the Y end of II AD844; The X end of II AD844 is by existing element 1 ground connection, and Y end is connected with the W end of I AD844, and Z end is by existing element 2 ground connection, and the X that W end is connected to I AD844 by existing element 3 holds;

Topological structure two: the X end of I AD844 is by existing element 3 ground connection, and Y end is connected to the W end of II AD844, and Z end is input, and the X that W end is connected to II AD844 by existing element 1 holds; The W that the X end of II AD844 is connected to I AD844 by existing element 1 holds, and Y holds ground connection, and Z end is by existing element 2 ground connection, and W end is connected to the Y end of I AD844;

Topological structure three: the X end of I AD844 is by existing element 1 ground connection, and Y end is connected to the Z end of II AD844, and is input, and Z end is by existing element 2 ground connection, and the X that W end is connected to II AD844 by existing element 3 holds; The W that the X end of II AD844 is connected to I AD844 by existing element 3 holds, and Y holds ground connection, and Z end is connected to the Y end of I AD844, and W end is unsettled;

Topological structure four: the X end of I AD844 is input, Y end is connected to the W end of II AD844, and Z end is by existing element 1 ground connection, and the X that W end is connected to II AD844 by existing element 2 holds; The W that the X end of II AD844 is connected to I AD844 by existing element 2 holds, and Y holds ground connection, and Z end is by existing element 3 ground connection, and W end is connected to the Y end of I AD844.

2. a kind of fractional order element converter according to claim 1, is characterized in that, described active integrated device is commercial one chip amplifier AD844, and the function of realization is operating amplifier with current feedback.

3. a kind of fractional order element converter according to claim 1, is characterized in that, described minimum number is 2, and wherein, I AD844 is the active integrated device in left side in circuit topology, and II AD844 is the source integrated device on right side in circuit topology.

4. a kind of fractional order element converter according to claim 1, it is characterized in that, described existing circuit element, according to the difference of circuit practical function, adopt integer rank electric capacity, inductance, resistive element and be equivalent to circuit or existing fractional order electric capacity, the inductance element of integer rank electric capacity, inductance, resistive element and be equivalent to the circuit that has fractional order electric capacity, inductance element.

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