CN106227925A - A kind of symbolic analysis method of discontinuous mode fractional order switch converters - Google Patents
A kind of symbolic analysis method of discontinuous mode fractional order switch converters Download PDFInfo
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Abstract
nullThe invention discloses a kind of symbolic analysis method of discontinuous mode fractional order switch converters,The method combines the principle of harmonic balance,By changer being converted to differential operator about differentiating of state variable fractional-order,And all differential operators are merged into diagonal angle sign matrix,Thus the process solving calculus of non-integral rank computing is converted into matrix operations and process that linear equation (group) solves,Compare the more existing analysis method of among Matlab/Simulinks setting up Oustaloup wave filter approximate model conventional for fractional order switch converters,The inventive method is except can analytically analytic transformation device state variable ripple peak-to-peak value size、The energy-storage travelling wave tube order change impact on changer duty,Analytic solutions steady-state period of state variable can also be obtained rapidly,And may be used for analyzing the harmonic components of state variable.
Description
Technical field
The present invention relates to modeling and the analysis field of fractional order switch converters, refer in particular to a kind of discontinuous current mode mould
The symbolic analysis method of formula (DCM, Discontinuous-conduction Mode) fractional order switch converters.
Background technology
Past has with analysis method for the modeling that switch converters is conventional: model based on State-space Averaging Principle, from
Dissipate iteration map model, piecewise linear model based on circuit theory (KCL, KVL) and combine harmonic balance and method of perturbation
Equivalent small parameter method, the analysis object of these methods is all the switch converters on integer rank, and i.e. the electric capacity in changer, inductance are all
It is the element on integer rank, but existing list of references 1 " Westerlund S.Dead Matter Has Memory [M]
.Kalmar, Sweden:Causal Consulting, 2002, Chap.7. " and list of references 2 " Podlubny
I.Fractional Differential Equations[M].San Diego:Academic Press,1999,Chap.2.”
Pointing out that actually electric capacity and inductance are the most all fractional orders, this is accomplished by setting up corresponding mark order mode for changer
Type.
Existing list of references 3 " Wang Faqiang, Ma Xikui. Boost under discontinuous mode based on fractional calculus
The modeling of changer and analysis [J]. Chinese science: science of technology, 2013,43 (4) .368-374 " consider inductance and electric capacity simultaneously
Fractional order characteristic, initially set up under discontinuous mode (DCM, Discontinuous-conduction Mode)
The space State Average Model of fractional order Boost, and non-integral order frequency domain based on Oustaloup wave filter approaches micro-
Integral algorithm, establishes phantom (as shown in Figure 1, 2) under Matlab/Simulink environment, converts fractional order DC-DC
The operating characteristic that device changes with order has carried out preliminary analysis.According to this thinking, prior art (such as list of references 4 " king
To send out strong, Ma Xikui. under continuous current mode pattern, the fractional order modeling and simulation of Boost analyzes [J]. Acta Physica Sinica,
2011,60 (7) .070506-1 070506-8 " etc.) have studied respectively under continuous current mode and pseudo-continuous work mode
Fractional order switch converters, Fig. 3 and Fig. 4 is the capacitance voltage obtained by the model set up in Fig. 1 and Fig. 2 respectively and inductance electricity
Stream waveform, existing technology is to show that by the way of setting up modularization model in Matlab/Simulink fractional order switchs
The operating characteristic of changer, and by the way of simulation waveform, show the ripple of stable state downconverter state variable;This method
Analytic solutions steady-state period of state variable can not be obtained, it is difficult to analytically analyze ripple peak-to-peak value size.
Summary of the invention
It is an object of the invention to the shortcoming overcoming prior art with not enough, it is provided that a kind of discontinuous mode
The symbolic analysis method of (DCM, Discontinuous-conduction Mode) fractional order switch converters, it is possible to quickly obtain
Fractional order switch converters state variable analytic solutions steady-state period under DCM state must be operated in.
For achieving the above object, technical scheme provided by the present invention is: a kind of discontinuous mode fractional order is opened
Close the symbolic analysis method of changer, comprise the following steps:
1) mathematical model of fractional order switch converters is set up
It is operated in the system mode of fractional order DC-DC converter under discontinuous mode DCM state to be described as:
In formula, x=[iLm vCm]TThe state variable of expression system, including the electric current i on m-th inductanceLm, m-th electric capacity
On voltage vCm, p represents the order of system state variables corresponding on corresponding inductance L, electric capacity C-element, A0And B0Table respectively
Show the coefficient matrix not affected by switch function, A1B1And A3B3Represent the coefficient matrix affected by switch function respectively;
Switch function δ(1)And δ(3)It is defined as:
Wherein, changer dutycycle D when open loop works1And D3For fixed value, meanwhile, non-linear is made to be divided into:
f(q)=δ(q)(Aqx+Bq)
τ=ω t, wherein
Then the calculus computing for state variable is converted to the algebraic operation about integro-differential operator, i.e.Owing to changer existing multiple state variable, therefore corresponding for each state variable integro-differential operator is merged into
Differential operator diagonal angle sign matrixIn matrix κ, these elements of α, β are for representing different conditions variable
Fractional calculus order, when L, C are integer rank, κ=-I, I are unit matrix, and +/-number therein represents respectively to shape
State variable is quadratured/differential;By being converted to the algebraic operation about differential operator by differentiating, it is possible to fractional order is switched
The mathematical model of changer is as follows:
In formula (3), G0For all G comprising differential operator diagonal angle sign matrix κkiThe column matrix of composition, k ∈ EirRepresent and work as
Overtone order k in front i-th rank correction, same after the definition of i, k,From GkiShape
Formula can embody the fractional-order impact on state variable analytic solutions;
By state variable x and switch function δ(q)All expand into the form of principal part and in a small amount remainder sum:
Above formula is substituted into f(q)=δ(q)(Aqx+Bq), merge identical order remainder in a small amount:
Wherein:
In formula, useRepresent the principal part of described state variable x the i-th rank correction, useRepresent described state variable x
The remainder of i rank correction is in a small amount;
According to principle of harmonic balance, by described state variable x and switch function δ(q)Expansion (4) in principal part and the i-th rank
It is as follows that remainder does Fourier expansion in a small amount:
Wherein akiRepresent the amplitude of the k subharmonic composition of the i-th rank correction, described switch function δ(q)Expansion coefficient table
Reaching formula is:
Wherein
According to principle of harmonic balance, coefficient expansion (8) is substituted into Fourier expansion formula (7), solving state variable successively
Main oscillations component and each rank correction;
2) main oscillations component is sought
First, the main oscillations component of solving state variable, usual main oscillations comprises only DC quantity, therefore is set to:
x0=a00
=[I00 V00]T (9)
Work as k=0, i.e. G0=G00=A0, x0In substitution formula (6)Substitute in (4) formula again:
G00·x0+b0(A1x0+B1)+c0(A3x0+B3)+B0=0 (10)
The main oscillations component of transducer status variable is tried to achieve by formula (10):
3) each rank correction is sought
According to main oscillations component remainder R1In the harmonic components that contains, if the first-order correction form of state variable is as follows:
Wherein, a11=[I11 V11]T, c.c represents conjugation item, same afterwards;By the harmonic wave in the first-order correction of state variable
Composition understandsk∈E1r, substitute into f in formula (6)1, obtain first-order correction table
Reach formula:
Gk1·x1+(b0A1x1+b1B1+b1A1x0+b1B1)+(c0A3x1+c1B3+c1A3x0+c1B3)+B0=0 (13)
It is obtained in that about harmonic amplitude a according to formula (13)01And ak1System of linear equations;
Parameter is substituted into the expression formula of gained current order correction, if each harmonic amplitude phase of current order correction
In comparison, first-order correction is less than an order of magnitude, then be not required to do the correction of higher order, otherwise, continue according to said process and continue
Seek the correction of higher order time;
4) main oscillations component and each rank correction are added, it is thus achieved that analytic solutions steady-state period about state variable are expressed
Formula.
The present invention compared with prior art, has the advantage that and beneficial effect:
By the solution formula of institute of the present invention extracting method it can be seen that use this method to ask fractional order switch converters state to become
Analytic solutions steady-state period of amount, are equivalent to the complex process solving calculus of non-integral rank computing is converted into matrix operations and asks
The process of linear equation (group), as long as setting up such as the fractional order switch converters shape of formula (3) matrix form according to circuit theory
State equation, then substitutes into each rank correction formula by coefficient expressions, except plus and minus calculation and just disappears unit by simple Matrix Multiplication
Can obtain about fractional order transducer status variable stable state solution's expression.Compare over pure mathematics field propose all kinds of
The method for solving of fractional calculus equation, the solution procedure of institute of the present invention extracting method combines the feature of switch converters, keeps away
Having opened thoroughly discussing for fractional calculus principle of operation, the solution of gained has obvious physical significance, according to using this
The form of the steady state solution that invention institute extracting method obtains, can be clearly seen that the harmonic components that state variable is comprised, is conducive to
To this quasi-converter launch deeper into analysis.
Accompanying drawing explanation
Fig. 1 is that the open loop set up based on Oustaloup filter method in Matlab/Simulink in list of references 3 divides
Number rank Boost phantom.
Fig. 2 is the Oustaloup filter subsystem based on fractional order frequency domain approximation method of encapsulation in Fig. 1.
Fig. 3 is the Cycle by Cycle simulation waveform of open loop fractional order Boost phantom output capacitance voltage in Fig. 1.
Wherein, abscissa is the time of emulation, and vertical coordinate represents capacitance voltage value.
Fig. 4 is open loop fractional order Boost phantom inductive current Cycle by Cycle simulation waveform in Fig. 1.Wherein, horizontal
Coordinate is the time of emulation, and vertical coordinate represents inductor current value.
Fig. 5 a is presently disclosed method and method inductance when the exponent number of inductance Yu electric capacity is 0.8 in list of references 3
The simulation result comparison diagram of current waveform.
Fig. 5 b is presently disclosed method and method electric capacity when the exponent number of inductance Yu electric capacity is 0.8 in list of references 3
The simulation result comparison diagram of voltage alternating component waveform.
Fig. 5 c is presently disclosed method and method electric capacity when the exponent number of inductance Yu electric capacity is 0.8 in list of references 3
The simulation result comparison diagram of voltage waveform.
Detailed description of the invention
Below in conjunction with specific embodiment, the invention will be further described.
The symbolic analysis method of the discontinuous mode fractional order switch converters described in the present embodiment, including following
Step:
1) mathematical model of fractional order switch converters is set up
It is operated in the system mode of fractional order DC-DC converter under discontinuous mode DCM state to be described as:
In formula, x=[iLm vCm]TThe state variable of expression system, including the electric current i on m-th inductanceLm, m-th electricity
Voltage v in appearanceCm, p represents the order of system state variables corresponding on corresponding inductance L, electric capacity C-element, A0And B0Respectively
Represent the coefficient matrix not affected by switch function, A1B1And A3B3Represent the coefficient matrix affected by switch function respectively;
Switch function δ(1)And δ(3)It is defined as:
Wherein, changer dutycycle D when open loop works1And D3For fixed value, meanwhile, non-linear is made to be divided into:
f(q)=δ(q)(Aqx+Bq)
τ=ω t, wherein
Then the calculus computing for state variable is converted to the algebraic operation about integro-differential operator, i.e.Owing to changer existing multiple state variable, therefore corresponding for each state variable integro-differential operator is merged into
Differential operator diagonal angle sign matrixIn matrix κ, these elements of α, β are for representing different conditions variable
Fractional calculus order, when L, C are integer rank, κ=-I, I are unit matrix, and +/-number therein represents respectively to shape
State variable is quadratured/differential;By being converted to the algebraic operation about differential operator by differentiating, it is possible to fractional order is switched
The mathematical model of changer is as follows:
In formula (3), G0For all G comprising differential operator diagonal angle sign matrix κkiThe column matrix of composition, k ∈ EirRepresent and work as
Overtone order k in front i-th rank correction, same after the definition of i, k,From GkiShape
Formula can embody the fractional-order impact on state variable analytic solutions;
By state variable x and switch function δ(q)All expand into the form of principal part and in a small amount remainder sum:
Above formula is substituted into f(q)=δ(q)(Aqx+Bq), merge identical order remainder in a small amount:
Wherein:
In formula, useRepresent the principal part of described state variable x the i-th rank correction, useRepresent described state variable x
The remainder of i rank correction is in a small amount;
According to principle of harmonic balance, by described state variable x and switch function δ(q)Expansion (4) in principal part and the i-th rank
It is as follows that remainder does Fourier expansion in a small amount:
Wherein akiRepresent the amplitude of the k subharmonic composition of the i-th rank correction, described switch function δ(q)Expansion coefficient table
Reaching formula is:
Wherein
According to principle of harmonic balance, coefficient expansion (8) is substituted into Fourier expansion formula (7), solving state variable successively
Main oscillations component and each rank correction;
2) main oscillations component is sought
First, the main oscillations component of solving state variable, usual main oscillations comprises only DC quantity, therefore is set to:
x0=a00
=[I00 V00]T (9)
Work as k=0, i.e. G0=G00=A0, x0In substitution formula (6)Substitute in (4) formula again:
G00·x0+b0(A1x0+B1)+c0(A3x0+B3)+B0=0 (10)
The main oscillations component of transducer status variable is tried to achieve by formula (10):
3) each rank correction is sought
According to main oscillations component remainder R1In the harmonic components that contains, if the first-order correction form of state variable is as follows:
Wherein, a11=[I11 V11]T, c.c represents conjugation item, same afterwards;By the harmonic wave in the first-order correction of state variable
Composition understandsk∈E1r, substitute into f in formula (6)1, obtain first-order correction table
Reach formula:
Gk1·x1+(b0A1x1+b1B1+b1A1x0+b1B1)+(c0A3x1+c1B3+c1A3x0+c1B3)+B0=0 (13)
It is obtained in that about harmonic amplitude a according to formula (13)01And ak1System of linear equations;
Parameter is substituted into the expression formula of gained current order correction, if each harmonic amplitude phase of current order correction
In comparison, first-order correction is less than an order of magnitude, then be not required to do the correction of higher order, otherwise, continue according to said process and continue
Seek the correction of higher order time;
4) main oscillations component and each rank correction are added, it is thus achieved that analytic solutions steady-state period about state variable are expressed
Formula.
The symbolic analysis method using above-mentioned discontinuous conduct mode fractional order switch converters below for instantiation is entered
Row operation, for the fractional order Boost of open loop, its state variable x=[iL vC]T, it is contemplated that the mark of energy-storage travelling wave tube
Rank characteristic and inductor loss, state equation is as follows:
Form described by corresponding (1), it is known that
B1=[00]T、Differential operator Matrix
When use list of references " Wang Faqiang, Ma Xikui. under discontinuous mode based on fractional calculus
The modeling of Boost with analyze [J]. Chinese science: science of technology, 2013,43 (4), pp.368-374 " in parameter time,
There are Boost switch periods fs=25kHZ, input voltage E=24V, inductance L=4mH, inductive resistance loss RL=0 Ω,
Electric capacity C=100 μ F, loads R=50 Ω, power taking sense exponent number α=0.8, electric capacity exponent number β=0.8.
The main oscillations component of fractional order Boost, first-order correction and second order correction is asked according to step above,
Now due in second order correction the Amplitude Ration main oscillations component of each harmonic much smaller, therefore do not continue to seek higher order correction,
Fractional order Boost is as follows through two rank analytic solutions forms revised steady-state period:
In formula, Re (aik) represent take aikReal part, Im (aik) represent take aikImaginary part, xdcAnd xacRepresent that state becomes respectively
The direct current component of amount and of ac.
The dutycycle of each mode is:
aikExpression formula as follows:
Parameter is substituted into formula (15) and (16) can obtain analytic solutions steady-state period and are:
By Symbolic Analysis Method of the present invention and list of references 3 method therefor state variable waveform ratio when stable state
Relatively, as shown in Fig. 5 a, 5b, 5c, analogous diagram uses parameter in list of references 3.But due to mistake stabilization time when resistance is 500 Ω
Long, therefore it is taken as 50 ohm.When the exponent number of inductance Yu electric capacity is 0.8.Presently disclosed method and method in list of references 3
Simulation result contrast verification figure, in figure, dotted line is the time-domain simulation results according to list of references 3 the set up model of method, and solid line is
The Numerical Simulation Results of the method that the present invention proposes, Fig. 5 a is inductive current waveform, and Fig. 5 b is the AC compounent ripple of capacitance voltage
Shape, Fig. 5 c is the capacitance voltage waveform that superposition DC component is later.
It can be seen that two correlation curve matchings of inductive current and capacitance voltage obtain very well, illustrate that the present invention is carried
The method gone out is effective.By analytic solutions formula it can be seen that use this method to seek fractional order switch converters state variable
Steady-state period, analytic solutions, were equivalent to the complex process solving calculus of non-integral rank computing is converted into matrix operations and asks linear
The process of equation (group), as long as setting up such as the fractional order changer of formula (3) form, then substituting into coefficient expressions each rank and repairing
Positive quantity formula, can be obtained by the table about fractional order transducer status variable steady state solution by simple matrix operations and the unit that disappears
Reach formula, be can clearly be seen that the harmonic components in state variable by this expression formula, by the expression formula of harmonic amplitude coefficient,
Can be seen that the impact on transducer status variable of the energy-storage travelling wave tube order.
Embodiment described above is only the preferred embodiments of the invention, not limits the practical range of the present invention with this, therefore
The change that all shapes according to the present invention, principle are made, all should contain within the scope of the present invention.
Claims (1)
1. the symbolic analysis method of a discontinuous mode fractional order switch converters, it is characterised in that include following
Step:
1) mathematical model of fractional order switch converters is set up
It is operated in the system mode of fractional order DC-DC converter under discontinuous mode DCM state to be described as:
In formula, x=[iLm vCm]TThe state variable of expression system, including the electric current i on m-th inductanceLm, on m-th electric capacity
Voltage vCm, p represents the order of system state variables corresponding on corresponding inductance L, electric capacity C-element, A0And B0Represent respectively not
The coefficient matrix affected by switch function, A1B1And A3B3Represent the coefficient matrix affected by switch function respectively;
Switch function δ(1)And δ(3)It is defined as:
Wherein, changer dutycycle D when open loop works1And D3For fixed value, meanwhile, non-linear is made to be divided into:
f(q)=δ(q)(Aqx+Bq)
τ=ω t, wherein
Then the calculus computing for state variable is converted to the algebraic operation about integro-differential operator, i.e.
Owing to changer existing multiple state variable, therefore corresponding for each state variable integro-differential operator is merged into differential operator pair
Angle sign matrixIn matrix κ, these elements of α, β are micro-long-pending for the fractional order representing different conditions variable
Sublevel, when L, C are integer rank, κ=-I, I are unit matrix, and +/-number therein represents respectively to state variable quadrature
Point/differential;By being converted to the algebraic operation about differential operator by differentiating, it is possible to by the number of fractional order switch converters
Model is as follows:
In formula (3), G0For all G comprising differential operator diagonal angle sign matrix κkiThe column matrix of composition, k ∈ EirRepresent current
Overtone order k in the correction of i rank, same after the definition of i, k,From GkiForm can
Embody the fractional-order impact on state variable analytic solutions;
By state variable x and switch function δ(q)All expand into the form of principal part and in a small amount remainder sum:
Above formula is substituted into f(q)=δ(q)(Aqx+Bq), merge identical order remainder in a small amount:
Wherein:
In formula, useRepresent the principal part of described state variable x the i-th rank correction, useRepresent described state variable x the i-th rank
The remainder of correction is in a small amount;
According to principle of harmonic balance, by described state variable x and switch function δ(q)Expansion (4) in principal part and the i-th rank remainder
Do Fourier expansion in a small amount as follows:
Wherein akiRepresent the amplitude of the k subharmonic composition of the i-th rank correction, described switch function δ(q)Expansion coefficient expressions
For:
Wherein
According to principle of harmonic balance, coefficient expansion (8) is substituted into Fourier expansion formula (7), the master of solving state variable successively
Oscillating component and each rank correction;
2) main oscillations component is sought
First, the main oscillations component of solving state variable, usual main oscillations comprises only DC quantity, therefore is set to:
x0=a00
=[I00 V00]T (9)
Work as k=0, i.e. G0=G00=A0, x0In substitution formula (6)Substitute in (4) formula again:
G00·x0+b0(A1x0+B1)+c0(A3x0+B3)+B0=0 (10)
The main oscillations component of transducer status variable is tried to achieve by formula (10):
3) each rank correction is sought
According to main oscillations component remainder R1In the harmonic components that contains, if the first-order correction form of state variable is as follows:
Wherein, a11=[I11 V11]T, c.c represents conjugation item, same afterwards;Can by the harmonic components in the first-order correction of state variable
Knowk∈E1r, substitute into f in formula (6)1, obtain first-order correction expression formula:
Gk1·x1+(b0A1x1+b1B1+b1A1x0+b1B1)+(c0A3x1+c1B3+c1A3x0+c1B3)+B0=0 (13)
It is obtained in that about harmonic amplitude a according to formula (13)01And ak1System of linear equations;
Parameter is substituted into the expression formula of gained current order correction, if each harmonic amplitude of current order correction compares
Upper first-order correction is less than an order of magnitude, then be not required to do the correction of higher order, otherwise, continue according to said process and continue to ask more
The correction of high order;
4) main oscillations component and each rank correction are added, it is thus achieved that about analytic solutions expression formula steady-state period of state variable.
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CN106874548A (en) * | 2017-01-10 | 2017-06-20 | 华南理工大学 | A kind of method that inverter is analyzed based on duplex treatments |
CN106909711A (en) * | 2017-01-11 | 2017-06-30 | 华南理工大学 | A kind of method for seeking fractional order CCM switch converters transient silutions |
CN114970368A (en) * | 2022-06-14 | 2022-08-30 | 中国长江三峡集团有限公司 | Efficiency optimization method and device for double active bridges, electronic equipment and storage medium |
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Cited By (6)
Publication number | Priority date | Publication date | Assignee | Title |
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CN106874548A (en) * | 2017-01-10 | 2017-06-20 | 华南理工大学 | A kind of method that inverter is analyzed based on duplex treatments |
CN106874548B (en) * | 2017-01-10 | 2020-04-28 | 华南理工大学 | Method for analyzing inverter based on double Fourier transform |
CN106909711A (en) * | 2017-01-11 | 2017-06-30 | 华南理工大学 | A kind of method for seeking fractional order CCM switch converters transient silutions |
CN106909711B (en) * | 2017-01-11 | 2020-04-28 | 华南理工大学 | Method for solving transient solution of fractional order CCM switching converter |
CN114970368A (en) * | 2022-06-14 | 2022-08-30 | 中国长江三峡集团有限公司 | Efficiency optimization method and device for double active bridges, electronic equipment and storage medium |
CN114970368B (en) * | 2022-06-14 | 2023-07-07 | 中国长江三峡集团有限公司 | Efficiency optimization method and device for double active bridge, electronic equipment and storage medium |
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