CN106227925A - A kind of symbolic analysis method of discontinuous mode fractional order switch converters - Google Patents

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

A kind of symbolic analysis method of discontinuous mode fractional order switch converters

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=[i_Lm v_Cm]^TThe state variable of expression system, including the electric current i on m-th inductance_Lm, m-th electric capacity On voltage v_Cm, 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)(A_qx+B_q)

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 κ_kiThe column matrix of composition, k ∈ E_irRepresent and work as Overtone order k in front i-th rank correction, same after the definition of i, k,From G_kiShape 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)(A_qx+B_q), 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 a_kiRepresent 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∈E_1r, substitute into f in formula (6)₁, obtain first-order correction table Reach formula:

G_k1·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 a_k1System 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=[i_Lm v_Cm]^TThe state variable of expression system, including the electric current i on m-th inductance_Lm, m-th electricity Voltage v in appearance_Cm, 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)(A_qx+B_q)

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 κ_kiThe column matrix of composition, k ∈ E_irRepresent and work as Overtone order k in front i-th rank correction, same after the definition of i, k,From G_kiShape 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)(A_qx+B_q), 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 a_kiRepresent 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∈E_1r, substitute into f in formula (6)₁, obtain first-order correction table Reach formula:

G_k1·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 a_k1System 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=[i_L v_C]^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 R_L=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 (a_ik) represent take a_ikReal part, Im (a_ik) represent take a_ikImaginary part, x_dcAnd x_acRepresent that state becomes respectively The direct current component of amount and of ac.

The dutycycle of each mode is:

a_ikExpression 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:

$\frac{d^{p}x}{{\mathrm{dt}}^p}=(A_0+B_0)+{\mathrm{\δ}}^{\left(1\right)}(A_{1}x+B_1)+{\mathrm{\δ}}^{\left(3\right)}(A_{3}x+B_3)---\left(1\right)$

In formula, x=[i_Lm v_Cm]^TThe state variable of expression system, including the electric current i on m-th inductance_Lm, on m-th electric capacity Voltage v_Cm, 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:

${\mathrm{\&delta;}}^{\left(1\right)}=\left\{\begin{array}{cc}1,& 0<t\&le;D_{1}T\\ 0,& D_{1}T<t\&le;T\end{array}\right.---\left(2.a\right)$

${\mathrm{\&delta;}}^{\left(3\right)}=\left\{\begin{array}{cc}0,& 0<t\&le;(1-D_3)T\\ 1,& (1-D_3)T<t\&le;T\end{array}\right.---\left(2.b\right)$

Wherein, changer dutycycle D when open loop works₁And D₃For fixed value, meanwhile, non-linear is made to be divided into:

f^(q)=δ^(q)(A_qx+B_q)

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:

$\left\{\begin{array}{c}{G}_0\left(\mathrm{\&kappa;}\right)x+f^{\left(1\right)}+f^{\left(3\right)}+B_0=0\\ G_0\left(\mathrm{\&kappa;}\right)=A_0-\mathrm{\&kappa;}\end{array}\right.---\left(3\right)$

In formula (3), G₀For all G comprising differential operator diagonal angle sign matrix κ_kiThe column matrix of composition, k ∈ E_irRepresent current Overtone order k in the correction of i rank, same after the definition of i, k,From G_kiForm 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:

$\left\{\begin{array}{c}x=x_0+\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{\mathrm{\&epsiv;}}^i{x}_i\\ {\mathrm{\&delta;}}^{\left(1\right)}={\mathrm{\&delta;}}_0^{\left(1\right)}+\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{\mathrm{\&epsiv;}}^i{\mathrm{\&delta;}}_i^{\left(1\right)}\\ {\mathrm{\&delta;}}^{\left(3\right)}={\mathrm{\&delta;}}_0^{\left(3\right)}+\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{\mathrm{\&epsiv;}}^i{\mathrm{\&delta;}}_i^{\left(3\right)}\end{array}\right.---\left(4\right)$

Above formula is substituted into f^(q)=δ^(q)(A_qx+B_q), merge identical order remainder in a small amount:

$f^{\left(q\right)}=f_0^{\left(q\right)}+{\mathrm{\&epsiv;f}}_1^{\left(q\right)}+{\mathrm{\&epsiv;}}^2{f}_2^{\left(q\right)}+...+{\mathrm{\&epsiv;}}^n{f}_n^{\left(q\right)}---\left(5\right)$

Wherein:

$\left\{\begin{array}{c}{f}_0^{\left(q\right)}={\mathrm{\&delta;}}_0^{\left(q\right)}(A_q{x}_0+B_q)=f_{0m}^{\left(q\right)}+R_1^{\left(q\right)}\\ f_1^{\left(q\right)}={\mathrm{\&delta;}}_0^{\left(q\right)}{x}_1+{\mathrm{\&delta;}}_1^{\left(q\right)}(A_q{x}_0+B_q)=f_{1m}^{\left(q\right)}+R_2^{\left(q\right)}\\ f_2^{\left(q\right)}=A_q({\mathrm{\&delta;}}_0^{\left(q\right)}{x}_2+{\mathrm{\&delta;}}_1^{\left(q\right)}{x}_1)+{\mathrm{\&delta;}}_2^{\left(q\right)}(A_q{x}_0+B_q)=f_{2m}^{\left(q\right)}+R_3^{\left(q\right)}\\ ...\end{array}\right.---\left(6\right)$

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:

$\left\{\begin{array}{c}{x}_i=a_{0i}+\underset{k\&Element;E_{ir}}{\mathrm{\&Sigma;}}(a_{ki}{e}^{jk\mathrm{\&tau;}}+{\stackrel{\&OverBar;}{a}}_{ki}{e}^{-jk\mathrm{\&tau;}})\\ {\mathrm{\&delta;}}_i^{\left(1\right)}=b_0+\underset{k\&Element;E_{ir}}{\mathrm{\&Sigma;}}(b_{ki}{e}^{jk\mathrm{\&tau;}}+{\stackrel{\&OverBar;}{b}}_{ki}{e}^{-jk\mathrm{\&tau;}})\\ {\mathrm{\&delta;}}_i^{\left(3\right)}=c_0+\underset{k\&Element;E_{ir}}{\mathrm{\&Sigma;}}(c_{ki}{e}^{jk\mathrm{\&tau;}}+{\stackrel{\&OverBar;}{c}}_{ki}{e}^{-jk\mathrm{\&tau;}})\end{array}\right.---\left(7\right)$

Wherein a_kiRepresent the amplitude of the k subharmonic composition of the i-th rank correction, described switch function δ^(q)Expansion coefficient expressions For:

$\begin{array}{c}\left\{\begin{array}{c}{b}_0=\frac{1}{T}\underset{0}{\overset{T}{\&Integral;}}{\mathrm{\&delta;}}^{\left(1\right)}dt=D_1\\ b_{ki}=\frac{1}{2}\left({\mathrm{\&alpha;}}_{ki}-{\mathrm{j\&beta;}}_{ki}\right)\end{array}\right.\\ \left\{\begin{array}{c}{c}_0=\frac{1}{T}\underset{0}{\overset{T}{\&Integral;}}{\mathrm{\&delta;}}^{\left(3\right)}dt=D_3\\ c_{ki}=\frac{1}{2}\left({\mathrm{\&alpha;}}_{ki}-{\mathrm{j\&beta;}}_{ki}\right)\end{array}\right.\end{array}---\left(8\right)$

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):

$\begin{array}{c}{x}_0=a_{00}={\left[\begin{array}{cc}{I}_{00}& V_{00}\end{array}\right]}^T\\ =-G_{00}^{-1}\&CenterDot;\&lsqb;b_0\left(A_1{x}_0+B_1\right)+c_0\left(A_3{x}_0+B_3\right)+B_0\&rsqb;\end{array}---\left(11\right)$

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:

$x_1=a_{01}+\underset{k\&Element;E_{1r}}{\&Sigma;}(a_{k1}{e}^{jk\mathrm{\&tau;}}+c.c)---\left(12\right)$

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∈E_1r, substitute into f in formula (6)₁, obtain first-order correction expression formula:

G_k1·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 a_k1System 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.