CN113408126B - Decoupling method for solving transient solution of fractional order very high frequency resonant converter - Google Patents

Decoupling method for solving transient solution of fractional order very high frequency resonant converter Download PDF

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CN113408126B
CN113408126B CN202110674123.3A CN202110674123A CN113408126B CN 113408126 B CN113408126 B CN 113408126B CN 202110674123 A CN202110674123 A CN 202110674123A CN 113408126 B CN113408126 B CN 113408126B
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陈艳峰
江心怡
张波
丘东元
肖文勋
谢帆
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South China University of Technology SCUT
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Abstract

The invention discloses a decoupling method for solving a transient solution of a fractional order very high frequency resonant converter, which comprises the steps of decoupling a state variable into a transient main oscillation component and a steady-state ripple component, solving the two components by respectively adopting a nonlinear circuit equivalent substitution method and a numerical analysis fusion method, and finally superposing the two components to obtain an approximate transient analysis solution of the state variable of the fractional order very high frequency resonant converter. The transient solution of the fractional order VHF resonant converter can be quickly obtained only by analyzing the nonlinear equivalent circuit of the converter and combining the steady state solution in the transient solution process of the method, and the method is suitable for analyzing the transient process of the converter.

Description

Decoupling method for solving transient solution of fractional order very high frequency resonant converter
Technical Field
The invention relates to the field of modeling and analysis of fractional order very high frequency resonant converters, in particular to a decoupling method for solving a transient solution of a fractional order very high frequency resonant converter.
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 increase the power density and the transient response speed, and on the other hand, the influence of the parasitic parameters on the converter becomes non-negligible.
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 (schroedingyu. fractional calculus and fractional control [ M ]. beijing: scientific press, 2018.1) to make modeling analysis of fractional system possible. 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 vacancy of theoretical analysis of the existing fractional order very high frequency resonant converter, provides a decoupling method for solving a transient solution of the fractional order very high frequency resonant converter, and can quickly obtain a transient analytical solution of a state variable of the fractional order very high frequency resonant converter.
In order to realize 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 γ X＝A(δ (1) (t)，δ (2) (t))X+BU in (1)
in the formula (I), the compound is shown in the specification,
Figure GDA0003619691360000021
for the state variable matrix, the superscript T represents the transpose of the matrix,
Figure GDA0003619691360000022
i LMR 、i Lr respectively representing the through-flow inductance
Figure GDA0003619691360000023
L MR 、L r Steady state current value of u CF 、u CMR 、u Cr 、
Figure GDA0003619691360000024
Respectively represent capacitance C F 、C MR 、C r And
Figure GDA0003619691360000025
steady state voltage values at both ends, the superscripts alpha and beta being inductances
Figure GDA0003619691360000026
And a capacitor
Figure GDA0003619691360000027
Fractional order of (d); delta γ A fractional order differential matrix represented as X, and a superscript gamma representing the fractional order matrix in a specific form
Figure GDA0003619691360000028
Wherein n is 1 To n 7 Is the fractional order of the state variable; when n is 1 ＝n 2 ＝...＝n 7 When the value is 1, the converter is converted into an integer-order circuit; b is a routing-only circuit elementThe elements forming a control matrix, U in To include an input DC voltage V in The 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:
Figure GDA0003619691360000029
Figure GDA00036196913600000210
wherein T is a time variable, T s Represents a duty cycle; delta. for the preparation of a coating (1) (t) ═ 1 denotes a duty cycle of D 1 Switch tube S T Conduction, delta (2) (t) ═ 1 denotes a duty cycle of D 2 Diode S D Is turned on by D 3 Denotes S T And S D Simultaneous turn-off of the occupied time and period T s The ratio of (a) to (b); by D 4 Denotes S T And S D The time and period T of simultaneous conduction s The ratio of (A) to (B); d 2 And D 1 、D 3 、D 4 The following relationships exist: d 2 ＝1+D 4 -D 1 -D 3 ；
Due to the diode S D Constant conduction during transient state, and a capacitor C r Reactance of
Figure GDA0003619691360000031
Much smaller than the capacitance
Figure GDA0003619691360000032
Reactance of (2)
Figure GDA0003619691360000033
Setting equivalent capacitance
Figure GDA0003619691360000034
Make it approximate to
Figure GDA0003619691360000035
In the formula
Figure GDA0003619691360000036
Respectively representing capacitances
Figure GDA0003619691360000037
C r 、
Figure GDA0003619691360000038
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:
Figure GDA0003619691360000039
in the formula, p is a differential operator,
Figure GDA00036196913600000310
the superscripts alpha and beta are respectively inductances
Figure GDA00036196913600000311
And a capacitor
Figure GDA00036196913600000312
Fractional order of (u) C Outputting instantaneous voltage value, i, for non-linear equivalent circuit L For the current-through inductance in a non-linear equivalent circuit
Figure GDA00036196913600000313
And L r Instantaneous current value of a 1 、a 2 、a 3 、a 4 、b 1 、b 2 、b 3 For 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:
Figure GDA0003619691360000041
Figure GDA0003619691360000042
where t is a time variable, Γ represents a gamma function, y 1 And y 2 For intermediate variables, switching the tube S T The transient main oscillation component u of the voltage at the two ends is calculated by using the superposition theorem:
u＝λ 1 ·V in +λ 2 ·u C (5)
in the formula, V in Representing the input DC voltage, λ 1 、λ 2 Is 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:
Figure GDA0003619691360000043
in the formula (I), the compound is shown in the specification,
Figure GDA0003619691360000044
for the state variable matrix, superscript T denotes the transposition of the matrix, where
Figure GDA0003619691360000045
i LMR 、i Lr Respectively representing the through-flow inductance
Figure GDA0003619691360000046
L MR 、L r Steady state current value of u CF 、u CMR 、u Cr 、
Figure GDA0003619691360000047
Respectively represent capacitance C F 、C MR 、C r And
Figure GDA0003619691360000048
the steady state voltage values at both ends; superscripts alpha and beta are inductances
Figure GDA0003619691360000049
And a capacitor
Figure GDA00036196913600000410
Fractional order of (d); delta of γ A fractional order differential matrix represented as X, and a superscript gamma representing the fractional order matrix in a specific form
Figure GDA00036196913600000411
Wherein n is 1 To n 7 Is the fractional order of the state variable; when n is 1 ＝n 2 ＝...＝n 7 When the value is 1, the converter is converted into an integer-order circuit; b is a control matrix composed of circuit elements only, U in To include an input DC voltage V in The input matrix of (a); y is an observation matrix of X; v is an observed white noise with a mean of 0 and a variance of R; h is 7-order identity matrix for state changeSelection of an amount; 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:
Figure GDA0003619691360000051
Figure GDA0003619691360000052
wherein T is a time variable, T s Represents a duty cycle; delta. for the preparation of a coating (1) (t) ═ 1 denotes a duty cycle of D 1 Switch tube S T Conduction, delta (2) (t) ═ 1 denotes a duty cycle of D 2 Diode S D Is turned on by D 3 Denotes S T And S D Simultaneous turn-off of the occupied time and period T s The ratio of (A) to (B); by D 4 Denotes S T And S D The occupied time and period T of simultaneous conduction s The ratio of (A) to (B); d 2 And D 1 、D 3 、D 4 The following relationships exist: d 2 ＝1+D 4 -D 1 -D 3 ；
Respectively solving the problem of S flowing through the switch tube in a continuous state T Diode S D Current i of ST (t)、i SD (t) nonlinear function:
Figure GDA0003619691360000053
i SD (t)＝δ (2) (t)·i Lr (t) (7.2)
the formula (6) is obtained through a discretization process:
Figure GDA0003619691360000054
wherein, subscript k represents the sampling value of the corresponding matrix at the kh moment, Xk, Y k And V k Respectively represent the firstThe value of the state variable at the time kh, the observed value of the state variable and the variance of the observed value of the state variable, X k-1 、X k-c Respectively representing the variable values of the state at the (k-1) h th time and the (k-c) h th time, wherein h represents the step length, and c is an intermediate variable; g d And C is a coefficient matrix formed by specific circuit parameters after discretization; gamma ray c A fractional order matrix at the ch-th time, specifically expressed as
Figure GDA0003619691360000055
Figure GDA0003619691360000061
Where N ═ 1, 2., 7) denotes the nth state vector, N N Representing 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-1 Predicted value X from time (k-1) h k-1k-1 Calculating to obtain:
Figure GDA0003619691360000062
wherein, U in，k-1 The input matrix at the (k-1) h moment;
2) estimated value P of error covariance at kh moment k|k-1 Predicted value P from (k-1) h k-1|k-1 Calculating to obtain:
Figure GDA0003619691360000063
wherein, P k-c|k-c The predicted value, γ, of the covariance matrix at time (k-c) h 1 And gamma c A fractional order matrix representing h and ch time instants;
3) filter gain matrix K at time kh k Comprises the following steps:
Figure GDA0003619691360000064
wherein R is k The 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|k Comprises 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 moment k|k Comprises the following steps: p k|k ＝(I-K k H)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 will i 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 T And diode S D The common state of (1):
Figure GDA0003619691360000071
Figure GDA0003619691360000072
wherein, delta (3) (t) 1 and δ (4) (t) 1 represents S T And S D Simultaneously off and simultaneously on; rearranging the steady state differential equation of the converter into an expression form suitable for equivalent small parameter method calculation, wherein the expression form comprises the following steps:
G 0 (p α ，p β ，p)X+G 1 f (1) (X，E 1 )+G 2 f (2) (X，E 2 )+G 3 f (3) (X，E 3 )＝U (9)
in the formula, p α 、p β And p represents differential operators of order alpha, order beta and order integer respectively,
Figure GDA0003619691360000073
Figure GDA0003619691360000074
input matrix U, G 0 (p α ，p β ，p)、G 1 、G 2 、G 3 All 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 switch function delta are combined (q) And a non-linear vector function matrix f (q) The series form of the sum of the main part and the small quantity of the remainder of each level is used for representing:
Figure GDA0003619691360000075
Figure GDA0003619691360000076
Figure GDA0003619691360000077
wherein ε is a small number of symbols i The ith order small quantity is expressed, and the specific numerical value of the small quantity epsilon in the operation process is 1; x 0 Is the main part of X, with ε i Multiplied by X i An ith order correction quantity of X; n represents the calculation accuracy of a small amount, and the larger the value is, the more accurate the calculation result is; in the same way, the method has the advantages of,
Figure GDA0003619691360000078
U 0 、δ 0 、
Figure GDA0003619691360000079
are each E (q) 、U、δ、f (q) The main part of (a) is,
Figure GDA00036196913600000710
U i 、δ i 、f i (q) are respectively E (q) 、U、δ、f (q) The ith correction amount of (1);
Figure GDA00036196913600000711
is f i (q) Neutralization of X i The terms having the same frequency distribution are,
Figure GDA00036196913600000712
is f i (q) The remainder of (2) including i Terms 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 order Kalman filtering, and the equivalent mathematical model comprises the following steps:
Figure GDA0003619691360000081
an approximate expression for a periodic steady state solution with the state variables expressed exponentially is as follows:
Figure GDA0003619691360000082
in the formula, omega s Is the angular frequency of the fractional order very high frequency resonant converter; direct current component X DC ＝M 0 Is the steady state primary oscillation component of the converter state variable; x ac For steady state ripple components: m 1 Is the magnitude vector of the fundamental wave, M m Is the magnitude vector of the mth harmonic; re (-) and Im (-) denote real and imaginary parts of the complex numbers, respectively.
Further, in step S3, a detailed process of solving the transient solution of the state variable of the fractional order very high frequency resonant converter is as follows:
steady state ripple component X ac Superposed with the transient main oscillation component, the transient solution of the fractional order very high frequency resonant converter state variable is as follows:
Figure GDA0003619691360000083
in the formula i lf 、i lr Are respectively a current flowing inductor
Figure GDA0003619691360000084
L r Of the transient current value u cout Is a capacitor
Figure GDA0003619691360000085
Transient voltage values at both ends; u. of C Outputting instantaneous voltage value, i, for non-linear equivalent circuit L For the flowing-through inductance in a non-linear equivalent circuit
Figure GDA0003619691360000086
And L r The instantaneous current value of (a); i.e. i LF.ac 、i Lr.ac Respectively representing the through-flow inductance
Figure GDA0003619691360000087
L r Of the steady-state current ripple component u Cout.ac Representing capacitance
Figure GDA0003619691360000088
A steady state voltage ripple component at both ends; during transient analysis, u CF In a D 1 T s Average value over time is zero, where D 1 Is shown in the switch tube S T Duty cycle of (d), T s Represents 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', δ (1) Is a switching tube S T Where X represents the transient envelope, X' represents the steady state envelope, and the scaling factor σ is u/| u | where | u | represents the modular length of u; at the switch S T When conducting, the capacitor C F Instantaneous values of the voltages at the two ends are:
u cf ≈(u+σu CF.ac )δ (1) (12)
u cf is a switching tube S T Instantaneous electricity at both endsPressure value u CF，ac Is a switching tube S T Steady state voltage ripple value, delta, at both ends (1) To show a switch tube S T The switching function of (a).
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 branch circuit of the switching device.
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 order of the fractional order energy storage element 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 very high frequency resonant converter.
Drawings
Fig. 1 is a schematic diagram of a fractional order very high frequency 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 F The transient current waveform of (1).
FIG. 2b shows a converter pass-through inductor L according to an embodiment of the present invention r The transient current waveform of (2).
FIG. 2C shows an exemplary embodiment of a converter C F Two 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 γ X＝A(δ (1) (t)，δ (2) (t))X+BU in (1)
in the formula (I), the compound is shown in the specification,
Figure GDA0003619691360000101
for the state variable matrix, the superscript T represents the transpose of the matrix,
Figure GDA0003619691360000102
i LMR 、i Lr respectively representing the through-flow inductance
Figure GDA0003619691360000103
L MR 、L r At a steady-state current value of u CF 、u CMR 、u Cr 、
Figure GDA0003619691360000104
Respectively represent capacitance C F 、C MR 、C r And
Figure GDA0003619691360000105
steady state voltage values at both ends, the superscripts alpha and beta being inductances
Figure GDA0003619691360000106
And a capacitor
Figure GDA0003619691360000107
Fractional order of (d); delta of γ A fractional order differential matrix represented as X, and a superscript gamma representing the fractional order matrix in a specific form
Figure GDA0003619691360000108
Wherein n is 1 To n 7 Is the fractional order of the state variable; when n is 1 ＝n 2 ＝...＝n 7 When the value is 1, the converter is converted into an integer-order circuit; b is a control matrix composed of circuit elements only, U in To include an input DC voltage V in The 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:
Figure GDA0003619691360000109
Figure GDA00036196913600001010
wherein T is a time variable, T s Represents a duty cycle; delta. for the preparation of a coating (1) (t) ═ 1 denotes a duty cycle of D 1 Switch tube S T Conduction, delta (2) (t) ═ 1 denotes a duty cycle of D 2 Diode S D Is turned on by D 3 Denotes S T And S D Simultaneous turn-off of the occupied time and period T s The ratio of (a) to (b); by D 4 Denotes S T And S D The time and period T of simultaneous conduction s The ratio of (a) to (b); d 2 And D 1 、D 3 、D 4 The following relationships exist: d 2 ＝1+D 4 -D 1 -D 3 ；
Due to the diode S D Constant conduction during transient state, and a capacitance C r Reactance of (2)
Figure GDA0003619691360000111
Much smaller than the capacitance
Figure GDA0003619691360000112
Reactance of (2)
Figure GDA0003619691360000113
Setting equivalent capacitance
Figure GDA0003619691360000114
Make it approximate to
Figure GDA0003619691360000115
In the formula
Figure GDA0003619691360000116
Respectively representing capacitances
Figure GDA0003619691360000117
C r 、
Figure GDA0003619691360000118
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; calculating a transient main oscillation component by establishing a nonlinear equivalent circuit of the transient process converter, and calculating a steady-state ripple component 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 a main switch and a parallel element thereof by utilizing the principle of a high-frequency network averaging method; according to the power supply series-parallel connection simplification rule, a series 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-circuited, 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 by making s equal to 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:
Figure GDA0003619691360000119
in the formula, p is a differential operator,
Figure GDA0003619691360000121
superscripts alpha and beta are inductances
Figure GDA0003619691360000122
And a capacitor
Figure GDA0003619691360000123
Fractional order of (u) C Outputting instantaneous voltage value, i, for non-linear equivalent circuit L For the current-through inductance in a non-linear equivalent circuit
Figure GDA0003619691360000124
And L r Instantaneous current value of (a) 1 、a 2 、a 3 、a 4 、b 1 、b 2 、b 3 For 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:
Figure GDA0003619691360000125
Figure GDA0003619691360000126
where t is a time variable, Γ represents a gamma function, y 1 And y 2 For intermediate variables, switching tubes S T The transient main oscillation component u of the voltage at two ends is calculated by using a superposition theorem:
u＝λ 1 ·V in +λ 2 ·u C (5)
in the formula，V in Representing the input DC voltage, λ 1 、λ 2 Is 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:
Figure GDA0003619691360000127
in the formula (I), the compound is shown in the specification,
Figure GDA0003619691360000128
for the state variable matrix, superscript T denotes the transposition of the matrix, where
Figure GDA0003619691360000129
i LMR 、i Lr Respectively representing the through-flow inductance
Figure GDA00036196913600001210
L MR 、L r Steady state current value of u CF 、u CMR 、u Cr 、
Figure GDA00036196913600001211
Respectively represent capacitance C F 、C MR 、C r And
Figure GDA00036196913600001212
a steady state voltage value at both ends; superscripts alpha and beta are inductances
Figure GDA00036196913600001213
And a capacitor
Figure GDA00036196913600001214
Fractional order of (d); delta γ A fractional order differential matrix represented as X, and a superscript gamma representing the fractional order matrix in a specific form
Figure GDA0003619691360000131
Wherein n is 1 To n 7 Is the fractional order of the state variable; when n is 1 ＝n 2 ＝...＝n 7 When the value is 1, the converter is converted into an integer-order circuit; b is a control matrix composed of circuit elements only, U in To include an input DC voltage V in The 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:
Figure GDA0003619691360000132
Figure GDA0003619691360000133
wherein T is a time variable, T s Represents a duty cycle; delta. for the preparation of a coating (1) (t) ═ 1 denotes a duty cycle of D 1 Switch tube S T Conduction, delta (2) (t) ═ 1 denotes a duty cycle of D 2 Diode S D Conducting with D 3 Denotes S T And S D Simultaneous turn-off of the occupied time and period T s The ratio of (A) to (B); by D 4 Denotes S T And S D The occupied time and period T of simultaneous conduction s The ratio of (a) to (b); d 2 And D 1 、D 3 、D 4 The following relationships exist: d 2 ＝1+D 4 -D 1 -D 3 ；
Respectively solving the problem of S flowing through the switch tube in a continuous state T Diode S D Current i of ST (t)、i SD (t) nonlinear function:
Figure GDA0003619691360000134
i SD (t)＝δ (2) (t)·i Lr (t) (7.2)
the formula (6) is obtained through a discretization process:
Figure GDA0003619691360000135
wherein, subscript k represents the sampling value of the corresponding matrix at the kh moment, X k 、Y k And V k Respectively representing the variance, X, of the state variable value, state variable observed value and state variable observed value at the kh-th time k-1 、X k-c Respectively representing the variable values of the state at the (k-1) h th time and the (k-c) h th time, wherein h represents the step length, and c is an intermediate variable; g d And C is a coefficient matrix formed by specific circuit parameters after discretization; gamma ray c A fractional order matrix at the ch-th time, expressed specifically as
Figure GDA0003619691360000141
Figure GDA0003619691360000142
Where N ═ 1, 2., 7) denotes the nth state vector, N N Representing 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-1 The predicted value X at the (k-1) h-th time k-1|k-1 Calculating to obtain:
Figure GDA0003619691360000143
wherein, U in，k-1 An input matrix at the (k-1) h moment;
2) estimated value P of error covariance at kh moment k|k-1 Predicted value P from (k-1) h k-1|k-1 Calculating to obtain:
Figure GDA0003619691360000144
wherein, P k-c|k-c Denotes the predicted value of the covariance matrix at time (k-c) h, gamma 1 And gamma c A fractional order matrix representing h and ch time instants;
3) filter gain matrix K at the kh-th instant k Comprises the following steps:
Figure GDA0003619691360000145
wherein R is k For the variance of the kh moment, the superscript-1 represents to calculate the inverse matrix of the matrix;
4) predicted value X of sampling point of state variable X at kh moment k|k Comprises 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|k Comprises the following steps: p k|k ＝(I-K k H)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 will i ST (t)、i SD (t) replacing the switching function δ of the steady-state differential equation in step S1 (1) (t)、δ (2) (t) and adding a switching function delta (3) (t)、δ (4) (t) denotes a switching tube S T And diode S D The common state of (1):
Figure GDA0003619691360000151
Figure GDA0003619691360000152
wherein, delta (3) (t) 1 and δ (4) (t) 1 represents S T And S D Simultaneously turned off and simultaneously turned on; rearrangement of formula (1) toThe expression forms calculated by the equivalent small parameter method are as follows:
G 0 (p α ，p β ，p)X+G 1 f (1) (X，E 1 )+G 2 f (2) (X，E 2 )+G 3 f (3) (X，E 3 )＝U (9)
in the formula, p α 、p β And p respectively represent differential operators of the order alpha, beta and integer,
Figure GDA0003619691360000153
Figure GDA0003619691360000154
input matrix U, G 0 (p α ，p β ，p)、G 1 、G 2 、G 3 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 switch function delta are combined (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:
Figure GDA0003619691360000155
Figure GDA0003619691360000156
Figure GDA0003619691360000157
wherein ε is a small number of marks i The specific numerical value of the small quantity epsilon in the operation process is 1; x 0 Is the main part of X, and ε i Multiplied by X i An ith order correction quantity of X; n represents a small number of calculationsThe higher the value is, the more accurate the calculation result is; in the same way, the method for preparing the composite material,
Figure GDA0003619691360000161
U 0 、δ 0 、
Figure GDA0003619691360000162
are each E (q) 、U、δ、f (q) The main part of (a) is,
Figure GDA0003619691360000163
U i 、δ i 、f i (q) are respectively E (q) 、U、δ、f (q) The ith correction amount of (1);
Figure GDA0003619691360000164
is f i (q) Neutralization of X i The terms having the same frequency distribution are,
Figure GDA0003619691360000165
is f i (q) The remainder of (2), including i Terms 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:
Figure GDA0003619691360000166
an approximate expression for a periodic steady state solution with the state variables expressed exponentially is as follows:
Figure GDA0003619691360000167
in the formula, ω s Is the angular frequency of the fractional order very high frequency resonant converter; direct current component X DC ＝M 0 Is the steady state primary oscillation component of the converter state variable; x ac For steady state ripple components: m 1 Is a magnitude vector of the fundamental wave, M m Is the m-th timeMagnitude vectors of harmonics; re (-) and Im (-) denote the real and imaginary parts of the complex number, respectively.
S3, taking a solution obtained after the transient main oscillation component is superposed with the steady-state ripple component as a 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 ac Superposed with the transient main oscillation component, the transient solution of the fractional order very high frequency resonant converter state variable is as follows:
Figure GDA0003619691360000168
in the formula i lf 、i lr Are respectively a flowing-through inductor
Figure GDA0003619691360000171
L r Transient current value of (u) cout Is a capacitor
Figure GDA0003619691360000172
Transient voltage values at both ends; u. of C Outputting instantaneous voltage value, i, for non-linear equivalent circuit L For the current-through inductance in a non-linear equivalent circuit
Figure GDA0003619691360000173
And L r The instantaneous current value of (a); i.e. i LF.ac 、i Lr.ac Respectively representing the through-flow inductance
Figure GDA0003619691360000174
L r Of the steady-state current ripple component u Cout.ac Representing capacitance
Figure GDA0003619691360000175
A steady state voltage ripple component at both ends; during transient analysis, u CF At a D 1 T s Average value over time is zero, where D 1 Is shown in the switching tube S T Duty ratio of (1), T s Represents a duty cycle; consider thatInfluence of high frequency sub-network, when (1) When (t) is equal to 0, the oscillation envelope of u should satisfy the relationship X ═ σ X', δ (1) Is a switch tube S T Where X represents the transient envelope, X' represents the steady state envelope, and the scaling factor σ is u/| u | where | u | represents the modular length of u; at the switch S T When conducting, the capacitor C F The instantaneous values of the voltages at the two terminals are:
u cf ≈(u+σu CF.ac )δ (1) (12)
u cf is a switching tube S T Instantaneous voltage value of both ends u CF，ac Is a switch tube S T Steady state voltage ripple value, delta, at both ends (1) To show a switching tube S T The switching function of (1).
In this embodiment, the operating frequency f s At 30MHz, and input DC voltage V in A15V fractional order VHF resonant Boost converter is shown in FIG. 1, where S is T Denotes a main switch, S D Representing parameters of the elements of the diode
Figure GDA0003619691360000176
L MR ＝75nH，L r ＝111nH，C F ＝100pF，C MR ＝95pF，C r ＝220pF，
Figure GDA0003619691360000177
R33.3 omega, where alpha and beta are the order of inductance and capacitance, and a switch tube S T And diode S D Are 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:
Figure GDA0003619691360000181
obtaining a steady-state ripple component of the fractional order vhf resonant converter according to step S22:
Figure GDA0003619691360000182
wherein τ is ω s t，
Figure GDA0003619691360000183
Finally, the instantaneous solution of the fractional order very high frequency resonant converter is obtained according to step S3:
Figure GDA0003619691360000184
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 PSIM circuit simulation. It can be found 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.

Claims (3)

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; 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, wherein the specific process 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 a main switch and a parallel element thereof by utilizing the principle of a high-frequency network averaging method; according to the power supply series-parallel connection simplification rule, a series 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:
Figure FDA0003628129530000011
in the formula, p is a differential operator,
Figure FDA0003628129530000012
the superscripts alpha and beta are respectively inductances
Figure FDA0003628129530000013
And a capacitor
Figure FDA0003628129530000014
Fractional order of (u) C Outputting instantaneous voltage value, i, for non-linear equivalent circuit L For the flowing-through inductance in a non-linear equivalent circuit
Figure FDA0003628129530000015
And L r Instantaneous current value of a 1 、a 2 、a 3 、a 4 、b 1 、b 2 、b 3 Is a constant related to a specific circuit parameterAnd (3) the analytic solution of the transient main oscillation component of the fractional order very high frequency resonant converter is as follows:
Figure FDA0003628129530000021
Figure FDA0003628129530000022
where t is a time variable, Γ represents a gamma function, y 1 And y 2 For intermediate variables, switching the tube S T The transient main oscillation component u of the voltage at the two ends is calculated by using the superposition theorem:
u＝λ 1 ·V in +λ 2 ·u C (5)
in the formula, V in Representing the input DC voltage, λ 1 、λ 2 Is 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 technology, an observation equation is added on the basis of the differential equation of the step S1:
Figure FDA0003628129530000023
in the formula (I), the compound is shown in the specification,
Figure FDA0003628129530000024
for the state variable matrix, superscript T denotes the transposition of the matrix, where
Figure FDA0003628129530000025
i LMR 、i Lr Respectively representing the through-flow inductance
Figure FDA0003628129530000026
L MR 、L r At a steady-state current value of u CF 、u CMR 、u Cr 、
Figure FDA0003628129530000027
Respectively represent capacitances C F 、C MR 、C r And
Figure FDA0003628129530000028
a steady state voltage value at both ends; superscripts alpha and beta are inductances
Figure FDA0003628129530000029
And a capacitor
Figure FDA00036281295300000210
Fractional order of (d); delta γ X is a fractional order differential matrix of X, and the superscript gamma is a fractional order matrix in a specific form
Figure FDA00036281295300000211
Wherein n is 1 To n 7 Is the fractional order of the state variable; when n is 1 ＝n 2 ＝…＝n 7 When the value is 1, the converter is converted into an integer-order circuit; b is a control matrix composed of circuit elements only, U in To include an input DC voltage V in The 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:
Figure FDA0003628129530000031
Figure FDA0003628129530000032
wherein T is a time variable, T s Represents a duty cycle; delta (1) (t) ═ 1 denotes a duty cycle of D 1 Switch tube S T Conduction, delta (2) (t) ═ 1 denotes a duty cycle of D 2 Diode S D Conducting with D 3 Denotes S T And S D Simultaneous turn-off of the occupied time and period T s The ratio of (a) to (b); by D 4 Denotes S T And S D The time and period T of simultaneous conduction s The ratio of (A) to (B); d 2 And D 1 、D 3 、D 4 Has the following relationship of 2 ＝1+D 4 -D 1 -D 3 ；
Respectively solving the problem of S flowing through the switch tube in a continuous state T Diode S D Current i of ST (t)、i SD (t) nonlinear function:
Figure FDA0003628129530000033
i SD (t)＝δ (2) (t)·i Lr (t) (7.2)
the formula (6) is obtained through a discretization process:
Figure FDA0003628129530000034
wherein, subscript k represents the sampling value of the corresponding matrix at the kh moment, X k 、Y k And V k Respectively representing the variance, X, of the state variable value, state variable observed value and state variable observed value at the kh-th time k-1 、X k-c Respectively representing the variable values of the state at the (k-1) h th time and the (k-c) h th time, wherein h represents the step length, and c is an intermediate variable; g d And C are coefficient matrixes formed by specific circuit parameters after discretization; gamma ray c A fractional order matrix at the ch-th time, expressed specifically as
Figure FDA0003628129530000035
Figure FDA0003628129530000041
Where N ═ (1,2, …,7) denotes the nth state vector, N N Representing 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-1 Predicted value X from time (k-1) h k-1|k-1 Calculating to obtain:
Figure FDA0003628129530000042
wherein, U in,k-1 An input matrix at the (k-1) h moment;
2) estimated value P of error covariance at kh moment k|k-1 Predicted value P from (k-1) h k-1|k-1 Calculating to obtain:
Figure FDA0003628129530000043
wherein, P k-c|k-c The predicted value, γ, of the covariance matrix at time (k-c) h 1 And gamma c A fractional order matrix representing h and ch time instants;
3) filter gain matrix K at the kh-th instant k Comprises the following steps:
Figure FDA0003628129530000044
wherein R is k For the variance of the kh moment, the superscript-1 represents to calculate the inverse matrix of the matrix;
4) predicted value X of sampling point of state variable X at kh moment k|k Comprises 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 moment k|k Comprises the following steps: p k|k ＝(I-K k H)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) replacing 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 T And diode S D The common state of (1):
Figure FDA0003628129530000051
Figure FDA0003628129530000052
wherein, delta (3) (t) 1 and δ (4) (t) 1 represents S T And S D Simultaneously 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:
G 0 (p α ,p β ,p)X+G 1 f (1) (X，E 1 )+G 2 f (2) (X，E 2 )+G 3 f (3) (X，E 3 )＝U (9)
in the formula, p α 、p β And p respectively represent differential operators of alpha | order, beta order and integer order,
Figure FDA0003628129530000053
Figure FDA0003628129530000054
input matrix U, G 0 (p α ,p β ,p)、G 1 、G 2 、G 3 All are coefficient matrices composed of circuit elements; f. of (q) Is a state variable X and an excitation matrixE, a related nonlinear vector function matrix, q is a correlation coefficient with a circuit working mode, and q is 1,2, 3;
the state variable X, the input matrix U, the excitation matrix E and the switch function delta are combined (q) And a non-linear vector function matrix f (q) The series form of the sum of the main part and the small quantity of the remainder of each level is used for representing:
Figure FDA0003628129530000055
Figure FDA0003628129530000056
Figure FDA0003628129530000057
wherein ε is a small number of symbols i The ith order small quantity is expressed, and the specific numerical value of the small quantity epsilon in the operation process is 1; x 0 Is the main part of X, with ε i Multiplied by X i An ith order correction quantity of X; n represents the calculation accuracy of a small amount, and the larger the value is, the more accurate the calculation result is; in the same way, the method has the advantages of,
Figure FDA0003628129530000058
U 0 、δ 0 、
Figure FDA0003628129530000059
are each E (q) 、U、δ、f (q) The main part of (a) is,
Figure FDA00036281295300000510
U i 、δ i 、
Figure FDA00036281295300000511
are respectively E (q) 、U、δ、f (q) The ith correction amount of (1);
Figure FDA00036281295300000512
is f i (q) Neutralization of X i The terms having the same frequency distribution are,
Figure FDA00036281295300000513
is f i (q) The remainder of (2), including i Terms 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:
Figure FDA0003628129530000061
an approximate expression for a periodic steady state solution with the state variables expressed exponentially is as follows:
Figure FDA0003628129530000062
in the formula, ω s Is the angular frequency of the fractional order very high frequency resonant converter; direct current component X DC ＝M 0 Is the steady state primary oscillation component of the converter state variable; x ac For steady state ripple components: m is a group of 1 Is the magnitude vector of the fundamental wave, M m Is the magnitude vector of the mth harmonic; re (-) and Im (-) denote real and imaginary parts of the complex number, respectively;
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 γ X＝A(δ (1) (t),δ (2) (t))X+BU in (1)
in the formula (I), the compound is shown in the specification,
Figure FDA0003628129530000063
is a state variable matrix, the superscript T represents the transposition of the matrix,
Figure FDA0003628129530000064
i LMR 、i Lr respectively representing the through-flow inductance
Figure FDA0003628129530000065
L MR 、L r At a steady-state current value of u CF 、u CMR 、u Cr 、
Figure FDA0003628129530000066
Respectively represent capacitance C F 、C MR 、C r And
Figure FDA0003628129530000067
steady state voltage values at both ends, the superscripts alpha and beta being inductances
Figure FDA0003628129530000068
And a capacitor
Figure FDA0003628129530000069
Fractional order of (d); delta of γ X is a fractional order differential matrix of X, and the superscript gamma is a fractional order matrix in a specific form
Figure FDA0003628129530000071
Wherein n is 1 To n 7 Is the fractional order of the state variable; when n is 1 ＝n 2 ＝…＝n 7 When the value is 1, the converter is converted into an integer-order circuit; b is a control matrix composed of circuit elements only, U in To include an input DC voltage V in The input matrix of (a); 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:
Figure FDA0003628129530000072
Figure FDA0003628129530000073
wherein T is a time variable, T s Represents a duty cycle; delta (1) (t) ═ 1 denotes a duty cycle of D 1 Switch tube S T Conduction, delta (2) (t) ═ 1 denotes a duty cycle of D 2 Diode S D Is turned on by D 3 Denotes S T And S D Simultaneous turn-off of the occupied time and period T s The ratio of (A) to (B); by D 4 Denotes S T And S D The occupied time and period T of simultaneous conduction s The ratio of (a) to (b); d 2 And D 1 、D 3 、D 4 The following relationships exist: d 2 ＝1+D 4 -D 1 -D 3 ；
Due to the diode S D Constant conduction during transient state, and a capacitor C r Reactance of
Figure FDA0003628129530000074
Much smaller than the capacitance
Figure FDA0003628129530000075
Reactance of
Figure FDA0003628129530000076
Setting equivalent capacitance
Figure FDA0003628129530000077
Make it approximate to
Figure FDA0003628129530000078
In the formula
Figure FDA0003628129530000079
Respectively representing capacitances
Figure FDA00036281295300000710
C r 、
Figure FDA00036281295300000711
The capacitance value of (2).
3. A method of solving for a fractional order very high frequency resonant converter transient solution as defined in claim 1, characterized in that: in step S3, the specific process of solving the transient solution of the state variable of the fractional order very high frequency resonant converter is as follows:
steady state ripple component X ac Superposed with the transient main oscillation component, the transient solution of the fractional order very high frequency resonant converter state variable is as follows:
Figure FDA00036281295300000712
in the formula i lf 、i lr Are respectively a flowing-through inductor
Figure FDA00036281295300000713
L r Transient current value of (u) cout Is a capacitor
Figure FDA00036281295300000714
Transient voltage values at both ends; u. u C Outputting instantaneous voltage value, i, for non-linear equivalent circuit L For the flowing-through inductance in a non-linear equivalent circuit
Figure FDA0003628129530000081
And L r The instantaneous current value of (a); i all right angle LF.ac 、i Lr.ac Respectively representing the through-flow inductance
Figure FDA0003628129530000082
L r Of the steady-state current ripple component u Cout.ac Representing capacitance
Figure FDA0003628129530000083
A steady state voltage ripple component at both ends; during transient analysis, u CF At a D 1 T s Average value over time is zero, where D 1 Is shown in the switch tube S T Duty ratio of (1), T s Represents 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, δ (1) Is a switching tube S T Where X denotes the transient 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 T When conducting, the capacitor C F The instantaneous values of the voltages at the two terminals are:
u cf ≈(u+σu CF.ac )δ (1) (12)
u CF,ac is a switch tube S T Steady state voltage ripple value, delta, at both ends (1) To show a switch tube S T The switching function of (1).
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