1. A decoupling method for solving a transient solution of a fractional order very high frequency resonant converter is characterized by comprising the following steps:

s1, analyzing the working principle of the converter, and writing a converter steady-state differential equation;

s2, decoupling the state variable of the converter into a transient main oscillation component and a steady-state ripple component; wherein, the transient main oscillation component is calculated by establishing a nonlinear equivalent circuit of the transient process converter, and the steady-state ripple component is calculated by using the steady-state differential equation of the step S1;

and S3, taking the solution obtained by superposing the transient main oscillation component on the steady-state ripple component as the transient solution of the state variable of the converter.

2. The decoupling method of claim 1 wherein said decoupling method comprises the steps of: in step S1, a steady-state differential equation is established for the fractional order very high frequency resonant converter:

Δ^γX＝A(δ⁽¹⁾(t),δ⁽²⁾(t))X+BU_in (1)

in the formula (I), the compound is shown in the specification,for the state variable matrix, the superscript T represents the transpose of the matrix,i_LMR、i_Lrrespectively representing the through-flow inductanceL_MR、L_rAt a steady-state current value of u_CF、u_CMR、u_Cr、Respectively represent capacitances C_F、C_MR、C_rAndsteady state voltage values at both ends, the superscripts alpha and beta being inductancesAnd a capacitorFractional order of (d); delta^γA fractional order differential matrix represented as X, and a superscript gamma representing the fractional order matrix in the specific formWherein n is₁To n₇Is the fractional order of the state variable; when n is₁＝n₂＝…＝n₇When the value is 1, the converter is converted into an integer-order circuit; b is a control matrix composed of circuit elements only, U_inTo include an input DC voltage V_inThe input matrix of (2); a is a signal containing a switching function delta⁽¹⁾(t)、δ⁽²⁾Coefficient matrix of (t), δ⁽¹⁾(t)、δ⁽²⁾(t) meets the following definition:

wherein T is a time variable, T_sRepresents a duty cycle; delta⁽¹⁾(t) 1 and δ⁽²⁾(t) 1 represents a duty ratio of D₁、D₂Switch tube S_TDiode S_DIs turned on by D₃Denotes S_TAnd S_DSimultaneous turn-off of the occupied time and period T_sThe ratio of (A) to (B); by D₄Denotes S_TAnd S_DThe occupied time and period T of simultaneous conduction_sThe ratio of (A) to (B); d₂And D₁、D₃、D₄The following relationships exist: d₂＝1+D₄-D₁-D₃；

Due to the diode S_DConstant conduction during transient state, and a capacitance C_rReactance ofMuch smaller than the capacitanceReactance ofSetting equivalent capacitanceMake it approximate toIn the formulaRespectively representing capacitancesC_r、The capacitance value of (2).

3. A method of solving for a transient solution for a fractional order very high frequency resonant converter according to claim 1, characterized by: in step S2, the specific process of decoupling the converter state variables into the main transient oscillation component and the steady-state ripple component is as follows:

s21, establishing a non-linear equivalent circuit of the transient process converter, and calculating to obtain a transient main oscillation component;

the resonance period of the converter is far shorter than the duration time of the transient process, and a non-time-varying controlled source is used for replacing the main switch and the parallel elements thereof by utilizing the principle of a high-frequency network averaging method; according to the power supply serial-parallel connection simplification rule, a serial circuit of a voltage source and a current source is simplified into a current source, and a voltage source and current source parallel circuit is simplified into a voltage source; through the simplification, a nonlinear equivalent circuit of the converter is obtained;

when the load of the nonlinear equivalent circuit is open, the input impedance of the nonlinear equivalent circuit is Z(s), s is a variable of a complex frequency domain, and an expression of Z (j omega) is obtained when s is j omega, wherein omega is a variable of the frequency domain, and j is an imaginary part unit; according to the definition of the integer order circuit on the series resonance, the impedance is in pure resistance characteristic, the imaginary part of Z (j omega) is zero, and the resonance frequency and the transient duration time of the transient process are calculated;

column writes the state equation of the nonlinear equivalent circuit:

in which p is a differential operator, i.e.The superscripts alpha and beta are respectively inductancesAnd a capacitorFractional order of (u)_COutputting instantaneous voltage value, i, for non-linear equivalent circuit_LFor the flowing-through inductance in a non-linear equivalent circuitAnd L_rInstantaneous current value of a₁、a₂、a₃、a₄、b₁、b₂、b₃For a constant coefficient related to a specific circuit parameter, the analytic solution of the transient main oscillation component of the fractional order very high frequency resonant converter is:

where t is a time variable, Γ represents a gamma function, y₁And y₂For intermediate variables, switching tubes S_TThe transient main oscillation component u of the voltage at two ends is calculated by using a superposition theorem:

u＝λ₁·V_in+λ₂·u_C (5)

in the formula, V_inRepresenting the input DC voltage, λ₁、λ₂Is a constant system composed of specific circuit elementsCounting;

s22, solving a converter steady-state differential equation, and calculating to obtain a steady-state ripple component;

according to the solution process of the kalman filtering technique, an observation equation is added on the basis of the differential equation of step S1, that is:

in the formula (I), the compound is shown in the specification,for the state variable matrix, the superscript T represents the transpose of the matrix, wherei_LMR、i_LrRespectively representing the through-flow inductanceL_MR、L_rAt a steady-state current value of u_CF、u_CMR、u_Cr、Respectively represent capacitances C_F、C_MR、C_rAnda steady state voltage value at both ends; superscripts alpha and beta are inductancesAnd a capacitorFractional order of (d); delta^γA fractional order differential matrix represented as X, and a superscript gamma representing the fractional order matrix in the specific formWherein n is₁To n₇Is the fractional order of the state variable; when n is₁＝n₂＝…＝n₇When the value is 1, the converter is converted into an integer-order circuit; b is a control matrix composed of circuit elements only, U_inTo include an input DC voltage V_inThe input matrix of (2); y is an observation matrix of X; v is observed white noise with mean 0 and variance R; h is a 7-order identity matrix used for selecting state variables; a is a signal containing a switching function delta⁽¹⁾(t)、δ⁽²⁾Coefficient matrix of (t), δ⁽¹⁾(t)、δ⁽²⁾(t) meets the following definition:

wherein T is a time variable, T_sRepresents a duty cycle; delta⁽¹⁾(t) 1 and δ⁽²⁾(t) 1 represents a duty ratio of D₁、D₂Switch tube S_TDiode S_DIs turned on by D₃Denotes S_TAnd S_DSimultaneous turn-off of the occupied time and period T_sThe ratio of (A) to (B); by D₄Denotes S_TAnd S_DThe occupied time and period T of simultaneous conduction_sThe ratio of (A) to (B); d₂And D₁、D₃、D₄Has the following relationship of₂＝1+D₄-D₁-D₃；

Respectively solving the problem of the switching tube S flowing through in a continuous state_TDiode S_DCurrent i of_ST(t)、i_SD(t) nonlinear function:

i_SD(t)＝δ⁽²⁾(t)·i_Lr(t) (7.2)

the formula (6) is obtained through a discretization process:

wherein the subscript k denotes the sample value of the corresponding matrix at the kh-th moment, X_k、Y_kAnd V_kRespectively representing the value of the state variable, the observed value of the state variable and the variance of the observed value of the state variable at the kh-th moment, X_k-1、X_k-cRespectively representing the variable values of the state at the (k-1) th moment and the (k-c) h moment, wherein h represents the step length, and c is an intermediate variable; g_dAnd C are coefficient matrixes formed by specific circuit parameters after discretization; gamma ray_kA matrix of fractional orders at the kh-th time, in particular as Where N ═ (1,2, …,7) denotes the nth state vector, N_NRepresenting the order of the nth state variable; the calculation process of the fractional Kalman filtering is as follows:

1) estimated value X of state variable X at kh moment_k|k-1Predicted value X from time (k-1) h_k-1|k-1Calculating to obtain:

wherein, U_in,k-1The input matrix at the (k-1) h moment;

2) estimated value P of error covariance at kh moment_k|k-1Predicted value P from (k-1) h_k-1|k-1Calculating to obtain:

wherein, P_k-c|k-cCovariance matrix representing the (k-c) h timePredicted value of (a), gamma₁And gamma_cA fractional order matrix representing h and ch time instants;

3) filter gain matrix K at time kh_kComprises the following steps:

wherein R is_kThe mark-1 represents the inverse matrix of the matrix for the variance at the kh moment;

4) predicted value X of sampling point of state variable X at kh moment_k|kComprises the following steps:

X_k|k＝X_k|k-1+K_k(Y_k-HX_k|k-1)；

5) predicted value P of error covariance at kh-th moment_k|kComprises the following steps: p_k|k＝(I-K_kH)P_k|k-1；

Wherein I represents an identity matrix;

calculating to obtain semiconductor switch current in discrete state, and determining current i by Fourier series fitting_ST(t)、i_SD(t) a non-linear expression; and then i will be_ST(t)、i_SD(t) replacement of the switching function δ of the steady-state differential equation in step S1⁽¹⁾(t)、δ⁽²⁾(t) and adding a new switching function delta⁽³⁾(t)、δ⁽⁴⁾(t) denotes a switching tube S_TAnd diode S_DThe common state of (1):

wherein, delta⁽³⁾(t) 1 and δ⁽⁴⁾(t) 1 represents S_TAnd S_DSimultaneously off and simultaneously on; rearranging the steady state differential equation of the converter into an expression form suitable for equivalent small parameter method calculation, comprising the following steps:

G₀(p^α,p^β,p)X+G₁f⁽¹⁾(X，E₁)+G₂f⁽²⁾(X，E₂)+G₃f⁽³⁾(X，E₃)＝U (9)

in the formula, p^α、p^βAnd p represents differential operators of order alpha, beta and integer, respectively, i.e. Input matrix U, G₀(p^α,p^β,p)、G₁、G₂、G₃All are coefficient matrices composed of circuit elements; f. of^(q)A nonlinear vector function matrix of the state variable X related to the excitation matrix E, q is a correlation coefficient with a circuit working mode, and q is 1,2 and 3;

the state variable X, the input matrix U, the excitation matrix E and the switching function delta^(q)And a non-linear vector function matrix f^(q)Expressed in the form of a series of sums of the main part and small quantities of the remainder of each order:

wherein ε is a small number of marksⁱThe specific numerical value of the small quantity epsilon in the operation process is 1; x₀Is the main part of X, with εⁱMultiplied by X_iAn ith order correction quantity of X; n represents the calculation accuracy of a small quantity, and the larger the value is, the more accurate the calculation result is; in the same way, the method for preparing the composite material,U₀、δ₀andis E^(q)U, delta and f^(q)The main part of (a) is,U_i、δ_iand f_i ^(q)Is E^(q)U, delta and f^(q)The ith correction amount of (1);is f_i ^(q)Neutralization of X_iThe terms having the same frequency distribution are,is f_i ^(q)The remainder of (2), including_iTerms having different frequency distributions; after arrangement, an equivalent mathematical model of the ultrahigh frequency converter is described by an equivalent small parameter method combined with fractional Kalman filtering, and the method comprises the following steps:

an approximate expression for a periodic steady state solution with the state variables expressed exponentially is as follows:

in the formula, ω_sIs the angular frequency of the fractional order very high frequency resonant converter; direct current component X_DC＝M₀Is the steady state primary oscillation component of the converter state variable; x_acFor steady state ripple components: m₁Is the magnitude vector of the fundamental wave, M_mIs the magnitude vector of the mth harmonic; re (-) and Im (-) denote the real and imaginary parts of the complex number, respectively.

4. A method of solving for a transient solution for a fractional order very high frequency resonant converter according to claim 1, characterized by: in step S3, the specific process of solving the transient state solution of the state variable of the fractional order vhf resonant converter is as follows:

steady state ripple component X_acSuperposed with the transient main oscillation component, the transient solution of the fractional order very high frequency resonant converter state variable is as follows:

in the formula i_lf、i_lrAre respectively a current flowing inductorL_rOf the transient current value u_coutIs a capacitorTransient voltage values at both ends; u. of_COutputting instantaneous voltage value, i, for non-linear equivalent circuit_LFor the flowing-through inductance in a non-linear equivalent circuitAnd L_rU is the nonlinear equivalent circuit switching tube S_TThe instantaneous voltage main oscillation component at both ends; i.e. i_LF.ac、i_Lr.acRespectively representing the through-flow inductanceL_rOf the steady-state current ripple component u_Cout.acRepresenting capacitanceA steady state voltage ripple component at both ends; i.e. during transient analysis, u_CFIn a D₁T_sTimeHas an average value of zero, wherein D₁Is shown in the switch tube S_TDuty ratio of (1), T_sRepresents a duty cycle; considering the influence of high frequency sub-nets, when delta⁽¹⁾(t) 0, the oscillation envelope of u should satisfy the relationship X- σ X, where X represents the transient envelope, X represents the steady state envelope, and the proportionality coefficient σ u/| u | where | u | represents the mode length of u; at the switch S_TWhen conducting, the capacitor C_FThe instantaneous values of the voltages at the two terminals are:

u_cf≈(u+σu_CF.ac)δ⁽¹⁾ (12)

u_cfis a switch tube S_TInstantaneous voltage value of both ends u_CF,acIs a switch tube S_TSteady state voltage ripple value, delta, at both ends⁽¹⁾To show a switch tube S_TThe switching function of (1).

Background

The fractional order very high frequency resonant converter generally refers to a power electronic converter with the working frequency of 30MHz to 300MHz, and has a wide prospect in the fields of aerospace and the like, so that the understanding of the relation among the working characteristics, reliability and parameters of the fractional order very high frequency resonant converter is more and more important. However, the ultra-high operating frequency can reduce the volume of the energy storage element and improve the power density and the transient response speed on one hand, and can also make the influence of the parasitic parameters on the converter non-negligible on the other hand.

In recent years, the research results of modeling inductance and capacitance show that: in real life, ideal integral-order inductance and capacitance do not exist, inductance and capacitance models established by utilizing a fractional-order calculus theory can more accurately reflect the characteristics of elements in a very high frequency working environment (a Tan journey, a Beam aspiration San, fractional-order modeling and analysis of a Boost converter under an inductance current pseudo-continuous mode [ J ]. Physics report, 2014(7): 070502-1-070502-10.). Scholars both at home and abroad have also developed a series of toolboxes for fractional calculus calculation (xue fixed space. fractional calculus and fractional control [ M ]. beijing: scientific press, 2018.1) to make possible the modeling analysis of fractional systems. Therefore, the fractional order element is used for establishing an equivalent model of the very high frequency resonant converter, the working mechanism of the very high frequency resonant converter is analyzed, the influence of parasitic parameters is further analyzed, and further the circuit parameters are optimized and the reliability analysis is further performed.

Disclosure of Invention

The invention aims to fill the gap of theoretical analysis of the existing fractional order very high frequency resonant converter, provides a decoupling method for solving the transient solution of the fractional order very high frequency resonant converter, and can quickly obtain the transient analytical solution of the state variable of the fractional order very high frequency resonant converter.

In order to achieve the purpose, the technical scheme provided by the invention is as follows: a decoupling method for solving a transient solution of a fractional order very high frequency resonant converter comprises the following steps:

s1, analyzing the working principle of the converter, and writing a converter steady-state differential equation;

s2, decoupling the state variable of the converter into a transient main oscillation component and a steady-state ripple component; wherein, the transient main oscillation component is calculated by establishing a nonlinear equivalent circuit of the transient process converter, and the steady-state ripple component is calculated by using the steady-state differential equation of the step S1;

and S3, taking the solution obtained by superposing the transient main oscillation component on the steady-state ripple component as the transient solution of the state variable of the converter.

Further, in step S1, a steady-state differential equation is established for the fractional order very high frequency resonant converter:

Δ^γX＝A(δ⁽¹⁾(t),δ⁽²⁾(t))X+BU_in (1)

in the formula (I), the compound is shown in the specification,for the state variable matrix, the superscript T represents the transpose of the matrix,i_LMR、i_Lrrespectively representing the through-flow inductanceL_MR、L_rAt a steady-state current value of u_CF、u_CMR、u_Cr、Respectively represent capacitances C_F、C_MR、C_rAndsteady state voltage values at both ends, the superscripts alpha and beta being inductancesAnd a capacitorFractional order of (d); delta^γA fractional order differential matrix represented as X, and a superscript gamma representing the fractional order matrix in the specific formWherein n is₁To n₇Is the fractional order of the state variable; when n is₁＝n₂＝…＝n₇When the value is 1, the converter is converted into an integer-order circuit; b is a control matrix composed of circuit elements only, U_inTo include an input DC voltage V_inThe input matrix of (2); a is a signal containing a switching function delta⁽¹⁾(t)、δ⁽²⁾Coefficient matrix of (t), δ⁽¹⁾(t)、δ⁽²⁾(t) meets the following definition:

wherein T is a time variable, T_sRepresents a duty cycle; delta⁽¹⁾(t) 1 and δ⁽²⁾(t) 1 represents a duty ratio of D₁、D₂Switch tube S_TDiode S_DIs turned on by D₃Denotes S_TAnd S_DSimultaneous turn-off of the occupied time and period T_sThe ratio of (A) to (B); by D₄Denotes S_TAnd S_DThe occupied time and period T of simultaneous conduction_sThe ratio of (A) to (B); d₂And D₁、D₃、D₄The following relationships exist: d₂＝1+D₄-D₁-D₃；

Due to the diode S_DConstant conduction during transient state, and a capacitance C_rReactance X of_CrMuch smaller than the capacitanceReactance ofSetting equivalent capacitanceMake it approximate toIn the formulaRespectively representing capacitancesC_r、The capacitance value of (2).

Further, in step S2, the specific process of decoupling the converter state variable into the main transient oscillation component and the steady-state ripple component is as follows:

s21, establishing a non-linear equivalent circuit of the transient process converter, and calculating to obtain a transient main oscillation component;

the resonance period of the converter is far shorter than the duration time of the transient process, and a non-time-varying controlled source is used for replacing the main switch and the parallel elements thereof by utilizing the principle of a high-frequency network averaging method; according to the power supply serial-parallel connection simplification rule, a serial circuit of a voltage source and a current source is simplified into a current source, and a voltage source and current source parallel circuit is simplified into a voltage source; through the simplification, a nonlinear equivalent circuit of the converter is obtained;

when the load of the nonlinear equivalent circuit is open, the input impedance of the nonlinear equivalent circuit is Z(s), s is a variable of a complex frequency domain, and an expression of Z (j omega) is obtained when s is j omega, wherein omega is a variable of the frequency domain, and j is an imaginary part unit; according to the definition of the integer order circuit on the series resonance, the impedance is in pure resistance characteristic, the imaginary part of Z (j omega) is zero, and the resonance frequency and the transient duration time of the transient process are calculated;

column writes the state equation of the nonlinear equivalent circuit:

in which p is a differential operator, i.e.The superscripts alpha and beta are respectively inductancesAnd a capacitorFractional order of (u)_COutputting instantaneous voltage value, i, for non-linear equivalent circuit_LFor the flowing-through inductance in a non-linear equivalent circuitAnd L_rInstantaneous current value of a₁、a₂、a₃、a₄、b₁、b₂、b₃For a constant coefficient related to a specific circuit parameter, the analytic solution of the transient main oscillation component of the fractional order very high frequency resonant converter is:

where t is a time variable, Γ represents a gamma function, y₁And y₂For intermediate variables, switching tubes S_TThe transient main oscillation component u of the voltage at two ends is calculated by using a superposition theorem:

u＝λ₁·V_in+λ₂·u_C (5)

in the formula, V_inRepresenting the input DC voltage, λ₁、λ₂Is a constant coefficient formed by specific circuit elements;

s22, solving a converter steady-state differential equation, and calculating to obtain a steady-state ripple component;

according to the solution process of the kalman filtering technique, an observation equation is added on the basis of the differential equation of step S1, that is:

in the formula (I), the compound is shown in the specification,for the state variable matrix, the superscript T represents the transpose of the matrix, wherei_LMR、i_LrRespectively representing the through-flow inductanceL_MR、L_rAt a steady-state current value of u_CF、u_CMR、u_Cr、Respectively represent capacitances C_F、C_MR、C_rAnda steady state voltage value at both ends; superscripts alpha and beta are inductancesAnd a capacitorFractional order of (d); delta^γA fractional order differential matrix represented as X, and a superscript gamma representing the fractional order matrix in the specific formWherein n is₁To n₇Is the fractional order of the state variable; when n is₁＝n₂＝…＝n₇When the value is 1, the converter is converted into an integer-order circuit; b is a control matrix composed of circuit elements only, U_inTo include an input DC voltage V_inThe input matrix of (2); y is an observation matrix of X; v is observed white noise with mean 0 and variance R; h is a 7-order identity matrix used for selecting state variables; a is a signal containing a switching function delta⁽¹⁾(t)、δ⁽²⁾Coefficient matrix of (t), δ⁽¹⁾(t)、δ⁽²⁾(t) meets the following definition:

wherein T is a time variable, T_sRepresents a duty cycle; delta⁽¹⁾(t) 1 and δ⁽²⁾(t) 1 represents a duty ratio of D₁、D₂Switch tube S_TDiode S_DIs turned on by D₃Denotes S_TAnd S_DSimultaneous turn-off of the occupied time and period T_sThe ratio of (A) to (B); by D₄Denotes S_TAnd S_DThe occupied time and period T of simultaneous conduction_sThe ratio of (A) to (B); d₂And D₁、D₃、D₄Has the following relationship of₂＝1+D₄-D₁-D₃；

Respectively solving the problem of the switching tube S flowing through in a continuous state_TDiode S_DCurrent i of_ST(t)、i_SD(t) nonlinear function:

i_SD(t)＝δ⁽²⁾(t)·i_Lr(t) (7.2)

the formula (6) is obtained through a discretization process:

wherein the subscript k denotes the sample value of the corresponding matrix at the kh-th moment, X_k、Y_kAnd V_kRespectively representing the value of the state variable, the observed value of the state variable and the variance of the observed value of the state variable at the kh-th moment, X_k-1、X_k-cRespectively representing the variable values of the state at the (k-1) th moment and the (k-c) h moment, wherein h represents the step length, and c is an intermediate variable; g_dAnd C are coefficient matrixes formed by specific circuit parameters after discretization; gamma ray_kA matrix of fractional orders at the kh-th time, in particular as Where N ═ (1,2, …,7) denotes the nth state vector, N_NRepresenting the order of the nth state variable; the calculation process of the fractional Kalman filtering is as follows:

1) estimated value X of state variable X at kh moment_k|k-1Predicted value X from time (k-1) h_k-1|k-1Calculating to obtain:

wherein, U_in,k-1The input matrix at the (k-1) h moment;

2) estimated value P of error covariance at kh moment_k|k-1Predicted value P from (k-1) h_k-1|k-1Calculating to obtain:

wherein, P_k-c|k-cDenotes the predicted value of the covariance matrix at time (k-c) h, gamma₁And gamma_cA fractional order matrix representing h and ch time instants;

3) filter at kh-th momentGain matrix K_kComprises the following steps:

wherein R is_kThe mark-1 represents the inverse matrix of the matrix for the variance at the kh moment;

4) predicted value X of sampling point of state variable X at kh moment_k|kComprises the following steps:

X_k|k＝X_k|k-1+K_k(Y_k-HX_k|k-1)；

5) predicted value P of error covariance at kh-th moment_k|kComprises the following steps: p_k|k＝(I-K_kH)P_k|k-1；

Wherein I represents an identity matrix;

calculating to obtain semiconductor switch current in discrete state, and determining current i by Fourier series fitting_ST(t)、i_SD(t) a non-linear expression; and then i will be_ST(t)、i_SD(t) replacement of the switching function δ of the steady-state differential equation in step S1⁽¹⁾(t)、δ⁽²⁾(t) and adding a new switching function delta⁽³⁾(t)、δ⁽⁴⁾(t) denotes a switching tube S_TAnd diode S_DThe common state of (1):

wherein, delta⁽³⁾(t) 1 and δ⁽⁴⁾(t) 1 represents S_TAnd S_DSimultaneously off and simultaneously on; rearranging the steady state differential equation of the converter into an expression form suitable for equivalent small parameter method calculation, comprising the following steps:

G₀(p^α,p^β,p)X+G₁f⁽¹⁾(X，E₁)+G₂f⁽²⁾(X，E₂)+G₃f⁽³⁾(X，E₃)＝U (9)

in the formula, p^α、p^βAnd p represents differential operators of order alpha, beta and integer, respectively, i.e. Input matrix U, G₀(p^α,p^β,p)、G₁、G₂、G₃All are coefficient matrices composed of circuit elements; f. of^(q)A nonlinear vector function matrix of the state variable X related to the excitation matrix E, q is a correlation coefficient with a circuit working mode, and q is 1,2 and 3;

the state variable X, the input matrix U, the excitation matrix E and the switching function delta^(q)And a non-linear vector function matrix f^(q)Expressed in the form of a series of sums of the main part and small quantities of the remainder of each order:

wherein ε is a small number of marksⁱThe specific numerical value of the small quantity epsilon in the operation process is 1; x₀Is the main part of X, with εⁱMultiplied by X_iAn ith order correction quantity of X; n represents the calculation accuracy of a small quantity, and the larger the value is, the more accurate the calculation result is; in the same way, the method for preparing the composite material,U₀、δ₀andis E^(q)U, delta and f^(q)The main part of (a) is,U_i、δ_iandis E^(q)U, delta and f^(q)The ith correction amount of (1);is composed ofNeutralization of X_iThe terms having the same frequency distribution are,is composed ofThe remainder of (2), including_iTerms having different frequency distributions; after arrangement, an equivalent mathematical model of the ultrahigh frequency converter is described by an equivalent small parameter method combined with fractional Kalman filtering, and the method comprises the following steps:

an approximate expression for a periodic steady state solution with the state variables expressed exponentially is as follows:

in the formula, ω_sIs the angular frequency of the fractional order very high frequency resonant converter; direct current component X_DC＝M₀Is the steady state primary oscillation component of the converter state variable; x_acFor steady state ripple components: m₁Is the magnitude vector of the fundamental wave, M_mIs the magnitude vector of the mth harmonic; re (-) and Im (-) denote the real and imaginary parts of the complex number, respectively.

Further, in step S3, the specific process of solving the fractional order vhf resonant converter state variable transient solution is as follows:

steady state ripple component X_acSuperposed with the transient main oscillation component, the transient solution of the fractional order very high frequency resonant converter state variable is as follows:

in the formula i_lf、i_lrAre respectively a current flowing inductorL_rOf the transient current value u_coutIs a capacitorTransient voltage values at both ends; u. of_COutputting instantaneous voltage value, i, for non-linear equivalent circuit_LFor the flowing-through inductance in a non-linear equivalent circuitAnd L_rU is the nonlinear equivalent circuit switching tube S_TThe instantaneous voltage main oscillation component at both ends; i.e. i_LF.ac、i_Lr.acRespectively representing the through-flow inductanceL_rOf the steady-state current ripple component u_Cout.acRepresenting capacitanceA steady state voltage ripple component at both ends; i.e. during transient analysis, u_CFIn a D₁T_sAverage value over time is zero, where D₁Is shown in the switch tube S_TDuty ratio of (1), T_sRepresents a duty cycle; considering the influence of high frequency sub-nets, when delta⁽¹⁾When (t) is 0, the oscillation envelope of u should satisfy the relationship X- σ X, where X represents the transient stateAn envelope, X denotes the steady-state envelope, and the proportionality coefficient σ is u/| u | where | u | denotes the mode length of u; at the switch S_TWhen conducting, the capacitor C_FThe instantaneous values of the voltages at the two terminals are:

u_cf≈(u+σu_CF.ac)δ⁽¹⁾ (12)

u_cfis a switch tube S_TInstantaneous voltage value of both ends u_CF,acIs a switch tube S_TSteady state voltage ripple value, delta, at both ends⁽¹⁾To show a switch tube S_TThe switching function of (1).

Compared with the prior art, the invention has the following advantages and beneficial effects:

1. in the modeling of the fractional order very high frequency resonant converter, a transient solution can be estimated by combining a steady state solution which is easy to obtain with a nonlinear equivalent circuit, so that the calculation amount can be greatly reduced.

2. Continuous and unified modeling of the converter is realized by adopting a continuous nonlinear function to fit the discrete function of the switching device branch.

3. The analytic solution of the transient solution of the fractional order very high frequency resonant converter is solved, the transient process of the converter can be qualitatively and quantitatively analyzed, and the influence of the fractional order energy storage element order on the transient process is described.

4. The analytic solution of the transient process is obtained by approximately superposing the steady-state ripple component and the transient main oscillation component, the transient process can be analyzed from the transient process time scale and the resonance period time scale, and a plurality of time scale view angles are provided for the research of the fractional order VHF resonant converter.

Drawings

Fig. 1 is a schematic diagram of a fractional order vhf resonant converter and its non-linear equivalent circuit in an embodiment of the present invention.

FIG. 2a shows a converter pass-through inductor L according to an embodiment of the present invention_FThe transient current waveform of (1).

FIG. 2b shows a converter pass-through inductor L according to an embodiment of the present invention_rThe transient current waveform of (1).

FIG. 2C shows an exemplary embodiment of a converter C_FTwo terminal transient voltage waveform diagrams.

FIG. 2d is a diagram of a transient output voltage waveform of the converter in accordance with an embodiment of the present invention.

Fig. 3 is a flowchart of the steps of the decoupling method for solving the transient solution of the fractional order vhf resonant converter according to the present invention.

Detailed Description

The present invention will be described in further detail with reference to examples and drawings, but the present invention is not limited thereto.

As shown in fig. 3, the decoupling method for solving the transient solution of the fractional order very high frequency resonant converter provided by the present embodiment includes the following steps;

s1, analyzing the working principle of the converter, and writing a converter steady-state differential equation; wherein, establishing a steady state differential equation for the fractional order VHF resonant converter is as follows:

Δ^γX＝A(δ⁽¹⁾(t),δ⁽²⁾(t))X+BU_in (1)

in the formula (I), the compound is shown in the specification,for the state variable matrix, the superscript T represents the transpose of the matrix,i_LMR、i_Lrrespectively representing the through-flow inductanceL_MR、L_rAt a steady-state current value of u_CF、u_CMR、u_Cr、Respectively represent capacitances C_F、C_MR、C_rAndsteady state voltage values at both ends, the superscripts alpha and beta being inductancesAnd a capacitorFractional order of (d); delta^γA fractional order differential matrix represented as X, and a superscript gamma representing the fractional order matrix in the specific formWherein n is₁To n₇Is the fractional order of the state variable; when n is₁＝n₂＝…＝n₇When the value is 1, the converter is converted into an integer-order circuit; b is a control matrix composed of circuit elements only, U_inTo include an input DC voltage V_inThe input matrix of (2); a is a signal containing a switching function delta⁽¹⁾(t)、δ⁽²⁾Coefficient matrix of (t), δ⁽¹⁾(t)、δ⁽²⁾(t) meets the following definition:

wherein T is a time variable, T_sRepresents a duty cycle; delta⁽¹⁾(t) 1 and δ⁽²⁾(t) 1 represents a duty ratio of D₁、D₂Switch tube S_TDiode S_DIs turned on by D₃Denotes S_TAnd S_DSimultaneous turn-off of the occupied time and period T_sThe ratio of (A) to (B); by D₄Denotes S_TAnd S_DThe occupied time and period T of simultaneous conduction_sThe ratio of (A) to (B); d₂And D₁、D₃、D₄Has the following relationship of₂＝1+D₄-D₁-D₃；

Due to the diode S_DConstant conduction during transient state, and a capacitance C_rReactance X of_CMuch smaller than the capacitanceReactance ofSetting equivalent capacitanceMake it approximate toIn the formulaRespectively representing capacitancesC_r、The capacitance value of (2).

S2, decoupling the state variable of the converter into a transient main oscillation component and a steady-state ripple component; wherein, the transient main oscillation component is calculated by establishing a nonlinear equivalent circuit of the transient process converter, and the steady-state ripple component is calculated by using the steady-state differential equation of the step S1; the specific process of decoupling and dividing the converter state variable into the transient main oscillation component and the steady-state ripple component is as follows:

s21, establishing a non-linear equivalent circuit of the transient process converter, and calculating to obtain a transient main oscillation component;

the resonance period of the converter is far shorter than the duration time of the transient process, and a non-time-varying controlled source is used for replacing the main switch and the parallel elements thereof by utilizing the principle of a high-frequency network averaging method; according to the power supply serial-parallel connection simplification rule, a serial circuit of a voltage source and a current source is simplified into a current source, and a voltage source and current source parallel circuit is simplified into a voltage source; through the simplification, a nonlinear equivalent circuit of the converter is obtained;

when the load of the nonlinear equivalent circuit is open, the input impedance of the nonlinear equivalent circuit is Z(s), s is a variable of a complex frequency domain, and an expression of Z (j omega) is obtained when s is j omega, wherein omega is a variable of the frequency domain, and j is an imaginary part unit; according to the definition of the integer order circuit on the series resonance, the impedance is in pure resistance characteristic, the imaginary part of Z (j omega) is zero, and the resonance frequency and the transient duration time of the transient process are calculated;

column writes the state equation of the nonlinear equivalent circuit:

in which p is a differential operator, i.e.Superscripts alpha and beta are inductancesAnd a capacitorFractional order of (u)_COutputting instantaneous voltage value, i, for non-linear equivalent circuit_LFor the flowing-through inductance in a non-linear equivalent circuitAnd L_rInstantaneous current value of a₁、a₂、a₃、a₄、b₁、b₂、b₃For a constant coefficient related to a specific circuit parameter, the analytic solution of the transient main oscillation component of the fractional order very high frequency resonant converter is:

where t is a time variable, Γ represents a gamma function, y₁And y₂For intermediate variables, switching tubes S_TThe transient main oscillation component u of the voltage at two ends is calculated by using a superposition theorem:

u＝λ₁·V_in+λ₂·u_C (5)

in the formula, V_inRepresenting the input DC voltage, λ₁、λ₂Is a constant coefficient formed by specific circuit elements;

s22, solving a converter steady-state differential equation, and calculating to obtain a steady-state ripple component;

according to the solution process of the kalman filtering technique, an observation equation is added on the basis of the differential equation of step S1, that is:

in the formula (I), the compound is shown in the specification,for the state variable matrix, the superscript T represents the transpose of the matrix, wherei_LMR、i_LrRespectively representing the through-flow inductanceL_MR、L_rAt a steady-state current value of u_CF、u_CMR、u_Cr、Respectively represent capacitances C_F、C_MR、C_rAnda steady state voltage value at both ends; superscripts alpha and beta are inductancesAnd a capacitorFractional order of (d); delta^γA fractional order differential matrix denoted as X,the superscript gamma represents a fractional order matrix, and is specifically in the form ofWherein n is₁To n₇Is the fractional order of the state variable; when n is₁＝n₂＝…＝n₇When the value is 1, the converter is converted into an integer-order circuit; b is a control matrix composed of circuit elements only, U_inTo include an input DC voltage V_inThe input matrix of (2); y is an observation matrix of X; v is observed white noise with mean 0 and variance R; h is a 7-order identity matrix used for selecting state variables; a is a signal containing a switching function delta⁽¹⁾(t)、δ⁽²⁾Coefficient matrix of (t), δ⁽¹⁾(t)、δ⁽²⁾(t) meets the following definition:

wherein T is a time variable, T_sRepresents a duty cycle; delta⁽¹⁾(t) 1 and δ⁽²⁾(t) 1 represents a duty ratio of D₁、D₂Switch tube S_TDiode S_DIs turned on by D₃Denotes S_TAnd S_DSimultaneous turn-off of the occupied time and period T_sThe ratio of (A) to (B); by D₄Denotes S_TAnd S_DThe occupied time and period T of simultaneous conduction_sThe ratio of (A) to (B); d₂And D₁、D₃、D₄Has the following relationship of₂＝1+D₄-D₁-D₃；

Respectively solving the problem of the switching tube S flowing through in a continuous state_TDiode S_DCurrent i of_ST(t)、i_SD(t) nonlinear function:

i_SD(t)＝δ⁽²⁾(t)·i_Lr(t) (7.2)

the formula (6) is obtained through a discretization process:

wherein the subscript k denotes the sample value of the corresponding matrix at the kh-th moment, X_k、Y_kAnd V_kRespectively representing the value of the state variable, the observed value of the state variable and the variance of the observed value of the state variable at the kh-th moment, X_k-1、X_k-cRespectively representing the variable values of the state at the (k-1) th moment and the (k-c) h moment, wherein h represents the step length, and c is an intermediate variable; g_dAnd C are coefficient matrixes formed by specific circuit parameters after discretization; gamma ray_kA matrix of fractional orders at the kh-th time, in particular as Where N ═ (1,2, …,7) denotes the nth state vector, N_NRepresenting the order of the nth state variable; the calculation process of the fractional Kalman filtering is as follows:

1) estimated value X of state variable X at kh moment_k|k-1Predicted value X from time (k-1) h_k-1|k-1Calculating to obtain:

wherein, U_in,k-1The input matrix at the (k-1) h moment;

2) estimated value P of error covariance at kh moment_k|k-1Predicted value P from (k-1) h_k-1|k-1Calculating to obtain:

wherein, P_k-c|k-cDenotes the (k-c)Prediction of covariance matrix at time h, gamma₁And gamma_cA fractional order matrix representing h and ch time instants;

3) filter gain matrix K at time kh_kComprises the following steps:

wherein R is_kThe mark-1 represents the inverse matrix of the matrix for the variance at the kh moment;

4) predicted value X of sampling point of state variable X at kh moment_k|kComprises the following steps:

X_k|k＝X_k|k-1+K_k(Y_k-HX_k|k-1)；

5) predicted value P of error covariance at kh-th moment_k|kComprises the following steps: p_k|k＝(I-K_kH)P_k|k-1；

Wherein I represents an identity matrix;

calculating to obtain semiconductor switch current in discrete state, and determining current i by Fourier series fitting_ST(t)、i_SD(t) a non-linear expression; and then i will be_ST(t)、i_SD(t) replacement of the switching function δ of the steady-state differential equation in step S1⁽¹⁾(t)、δ⁽²⁾(t) and adding a new switching function delta⁽³⁾(t)、δ⁽⁴⁾(t) denotes a switching tube S_TAnd diode S_DThe common state of (1):

wherein, delta⁽³⁾(t) 1 and δ⁽⁴⁾(t) 1 represents S_TAnd S_DSimultaneously off and simultaneously on; the rearrangement formula (1) is an expression form suitable for equivalent small parameter method calculation, and comprises the following steps:

G₀(p^α,p^β,p)X+G₁f⁽¹⁾(X，E₁)+G₂f⁽²⁾(X，E₂)+G₃f⁽³⁾(X，E₃)＝U (9)

in the formula, p^α、p^βAnd p represents differential operators of order alpha, beta and integer, respectively, i.e. Input matrix U, G₀(p^α,p^β,p)、G₁、G₂、G₃All are coefficient matrices composed of circuit elements; f. of^(q)A nonlinear vector function matrix of the state variable X related to the excitation matrix E, q is a correlation coefficient with a circuit working mode, and q is 1,2 and 3;

the state variable X, the input matrix U, the excitation matrix E and the switching function delta^(q)And a non-linear vector function matrix f^(q)Expressed in the form of a series of sums of the main part and small quantities of the remainder of each order:

wherein ε is a small number of marksⁱThe specific numerical value of the small quantity epsilon in the operation process is 1; x₀Is the main part of X, with εⁱMultiplied by X_iAn ith order correction quantity of X; n represents the calculation accuracy of a small quantity, and the larger the value is, the more accurate the calculation result is; in the same way, the method for preparing the composite material,U₀、δ₀andis E^(q)U, delta and f^(q)The main part of (a) is,U_i、δ_iandis E^(q)U, delta and f^(q)The ith correction amount of (1);is composed ofNeutralization of X_iThe terms having the same frequency distribution are,is f_i ^(q)The remainder of (2), including_iTerms having different frequency distributions; after arrangement, an equivalent mathematical model of the ultrahigh frequency converter is described by an equivalent small parameter method combined with fractional Kalman filtering, and the method comprises the following steps:

an approximate expression for a periodic steady state solution with the state variables expressed exponentially is as follows:

in the formula, ω_sIs the angular frequency of the fractional order very high frequency resonant converter; direct current component X_DC＝M₀Is the steady state primary oscillation component of the converter state variable; x_acFor steady state ripple components: m₁Is the magnitude vector of the fundamental wave, M_mIs the magnitude vector of the mth harmonic; re (-) and Im (-) denote the real and imaginary parts of the complex number, respectively.

S3, taking the solution obtained after the transient main oscillation component is superposed with the steady-state ripple component as the transient solution of the state variable of the converter; the specific process of solving the state variable transient solution of the fractional order very high frequency resonant converter is as follows;

steady state ripple component X_acSuperposed with the transient main oscillation component, the transient solution of the fractional order very high frequency resonant converter state variable is as follows:

in the formula i_lf、i_lrAre respectively a current flowing inductorL_rOf the transient current value u_coutIs a capacitorTransient voltage values at both ends; u. of_COutputting instantaneous voltage value, i, for non-linear equivalent circuit_LFor the flowing-through inductance in a non-linear equivalent circuitAnd L_rU is the nonlinear equivalent circuit switching tube S_TThe instantaneous voltage main oscillation component at both ends; i.e. i_LF.ac、i_Lr.acRespectively representing the through-flow inductanceL_rOf the steady-state current ripple component u_Cout.acRepresenting capacitanceA steady state voltage ripple component at both ends; i.e. during transient analysis, u_CFIn a D₁T_sAverage value over time is zero, where D₁Is shown in the switch tube S_TDuty ratio of (1), T_sTo representA work cycle; considering the influence of high frequency sub-nets, when delta⁽¹⁾(t) 0, the oscillation envelope of u should satisfy the relationship X- σ X, where X represents the transient envelope, X represents the steady state envelope, and the proportionality coefficient σ u/| u | where | u | represents the mode length of u; at the switch S_TWhen conducting, the capacitor C_FThe instantaneous values of the voltages at the two terminals are:

u_cf≈(u+σu_CF.ac)δ⁽¹⁾ (12)

u_cfis a switch tube S_TInstantaneous voltage value of both ends u_CF,acIs a switch tube S_TSteady state voltage ripple value, delta, at both ends⁽¹⁾To show a switch tube S_TThe switching function of (1).

In this embodiment, the operating frequency f_s30MHz, and input DC voltage V_inA15V fractional order VHF resonant Boost converter is shown in FIG. 1, where S is_TDenotes a main switch, S_DRepresenting parameters of the diode, elementsL_MR＝75nH，L_r＝111nH，C_F＝100pF，C_MR＝95pF，C_r＝220pF，R33.3 omega, where alpha and beta are the order of inductance and capacitance, and switch tube S_TAnd diode S_DAre all ideal elements.

Obtaining a solution of the fractional order very high frequency resonant converter simplified equivalent circuit according to the step S21, that is, a converter transient main oscillation component:

obtaining a steady-state ripple component of the fractional order vhf resonant converter according to step S22:

where τ ═ ω_st，Finally, the instantaneous solution of the fractional order very high frequency resonant converter is obtained according to step S3:

the voltage-current curves obtained by the method of the present invention are respectively compared with the corresponding curves obtained by the simulation of the PSIM circuit, as shown in FIG. 2a, FIG. 2b, FIG. 2c, and FIG. 2 d. In the figure, the solid line is the waveform obtained by the invention, and the dotted line is the waveform obtained by the simulation of the PSIM circuit. It can be seen from the figure that the method of the present invention can embody the voltage variation, and the fitting error of the current waveform is small, thus, the method of the present invention is demonstrated to be effective.

The above embodiments are preferred embodiments of the present invention, but the present invention is not limited to the above embodiments, and any other changes, modifications, substitutions, combinations, and simplifications which do not depart from the spirit and principle of the present invention should be construed as equivalents thereof, and all such changes, modifications, substitutions, combinations, and simplifications are intended to be included in the scope of the present invention.