CN110580384B - Nonlinear modeling method for simultaneously solving multi-scale state variables of switching converter - Google Patents

Nonlinear modeling method for simultaneously solving multi-scale state variables of switching converter Download PDF

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CN110580384B
CN110580384B CN201910763950.2A CN201910763950A CN110580384B CN 110580384 B CN110580384 B CN 110580384B CN 201910763950 A CN201910763950 A CN 201910763950A CN 110580384 B CN110580384 B CN 110580384B
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陈艳峰
蒋宗祺
张波
丘东元
谢帆
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South China University of Technology SCUT
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Abstract

The invention discloses a nonlinear modeling method for simultaneously solving multi-scale state variables of a switching converter, which is characterized in that a switching tube and a diode are equivalent by using mechanism models thereof in the modeling process, and a discrete switching function for describing the on-off characteristics of a device is fitted by adopting a continuous nonlinear periodic function to obtain a uniform nonlinear continuous mathematical model for describing the state variables of a switching converter circuit level and a device level; and then solving by using a nonlinear analysis method, an approximate analytical expression of the steady-state period solution of the state variables of the switching converter circuit level and the device level can be obtained at the same time, namely the steady-state period analytical solution of the multi-scale state variables of the switching converter can be obtained. The invention realizes the unified modeling of the combination of the device-level scale and the circuit-level scale of the switching converter, and realizes the continuous unified modeling of the converter by adopting a continuous nonlinear periodic function to fit the traditional discrete switching function for describing the on-off characteristics of the device.

Description

Nonlinear modeling method for simultaneously solving multi-scale state variables of switching converter
Technical Field
The invention relates to the technical field of modeling and analysis of a switching converter, in particular to a nonlinear modeling method for simultaneously solving multi-scale state variables of the switching converter.
Background
The switching converter is a nonlinear time-varying system, the analytical solution for solving the dynamic characteristic of the switching converter is complex, and the existing nonlinear analysis methods comprise a state space averaging method, a circuit averaging method, a generalized state space averaging method, an equivalent small parameter method and the like. Among them, the state space averaging method only considers the approximation of the system under the low frequency characteristic, and ignores the high frequency dynamic characteristic of the system, and thus cannot be used for analyzing the ripple of the converter waveform. The three-terminal switching device model rule which is commonly used in the circuit averaging method needs to know the direct-current steady-state characteristic of the converter, and has certain limitation. The computational analysis process of the generalized state space averaging method is relatively complex. The equivalent small parameter method is an analysis method combining a disturbance method and a harmonic balance method, and has the advantages of more accurate analysis result and relatively simple analysis process.
State variables with different time scales exist in the switching converter, and typically, device-level state variables reflecting microscopic characteristics of devices and circuit-level state variables reflecting macroscopic characteristics of the converter exist; and state variables of different time scales may influence each other. Therefore, in order to accurately describe the operating characteristics of a switching converter, it is necessary to establish a mathematical model that can describe both the state variables of the converter device and the circuit. However, the existing modeling method of the switching converter only analyzes the state variables of the circuit level, ignores the dynamic characteristics of the switching device, and only adopts a discrete switching function to describe the on-off state of the switching device, so that the established mathematical model cannot reflect the influence of the state variables of the device level on the circuit operation, and the guiding significance of the analysis result on the actual converter parameter design is not great.
Disclosure of Invention
The invention aims to overcome the defects of the prior art and provides a nonlinear modeling method for simultaneously solving multi-scale state variables of a switching converter, which can obtain steady-state period analytic solutions of the state variables of different scales of the converter.
In order to achieve the purpose, the technical scheme provided by the invention is as follows: the nonlinear modeling method for simultaneously solving the multi-scale state variables of the switching converter is characterized in that a switching tube and a diode are equivalent by using mechanism models thereof in the modeling process, and a discrete switching function for describing the on-off characteristics of a device is fitted by adopting a continuous nonlinear periodic function to obtain a uniform nonlinear continuous mathematical model for describing the state variables of a circuit level and a device level of the switching converter; solving by using a nonlinear analysis method, an approximate analytical expression of the steady state period solution of the state variables of the switching converter circuit level and the device level can be obtained at the same time, namely the steady state period analytical solution of the multi-scale state variables of the switching converter can be obtained; which comprises the following steps:
s1, the MOSFET of the switching device is equivalent by using a simplified device mechanism model; equating the diode device with a nonlinear resistor, wherein the volt-ampere characteristic of the nonlinear resistor is described by a PN junction current equation of the diode;
s2, fitting a discrete switching function of the gate drive signal of the switching device by using a continuous nonlinear periodic function;
s3, establishing a multi-scale unified nonlinear mathematical model of the switching converter described by a differential equation;
s4, expanding the nonlinear periodic function of the step S2 into Fourier series;
s5 obtaining approximate linear equivalent mathematical model of switching converter
Decomposing the nonlinear mathematical model established in the step S3 by using a nonlinear analysis method to obtain an approximate linear equivalent mathematical model of the nonlinear mathematical model, wherein the equivalent mathematical model is an equation set consisting of a series of linear equations and comprises an equation for solving the main component of the state variable of the system and a plurality of equations for solving the correction quantity of each order of the state variable;
s6, obtaining approximate analytical expression of multi-scale state variable steady-state period solution of switching converter
And (4) solving each equation in the equivalent mathematical model in the step (S5) step by using a harmonic balance principle, and obtaining direct current components and correction quantities of each order of state variables of the circuit level and the device level of the switching converter so as to obtain an approximate analytical expression of the multi-scale state variable steady-state period solution of the switching converter.
In step S1, the simplified device mechanism model of the MOSFET includes a gate-level input resistor, a voltage-controlled current source, a first inter-electrode capacitor, a second inter-electrode capacitor, and an on-resistance; one end of the gate-level input resistor is connected with the grid of the MOSFET, and the other end of the gate-level input resistor is connected with one end of the second inter-pole capacitor; one end of the voltage-controlled current source is respectively connected with one end of the first interelectrode capacitor, one end of the on-resistance and the drain of the MOSFET; the other end of the voltage-controlled current source is respectively connected with the other end of the first interelectrode capacitor, the other end of the on-resistance and the source level of the MOSFET; the other end of the second inter-pole capacitor is connected with a source electrode of the MOSFET; wherein the expression of the voltage-controlled current source is iG＝gmuGSIn the formula, gmIs the forward transconductance of the MOSFET device, uGSIs the gate drive voltage signal of the MOSFET uGS＝s(1)uG，s(1)Non-linear control signal function uGIs the gate voltage magnitude of the MOSFET.
In step S1, the PN junction current equation of the diode is as follows:
Figure GDA0002830758540000031
in the formula iDIs the current of a diode, ISThe reverse saturation current of the diode can be found according to a parameter manual of the actually selected diode model; u shapeTIs thermodynamic voltage, U at normal temperatureT＝26mV；uDIs the forward voltage of the diode; will iDThe taylor expansion can be carried out to obtain the current-voltage characteristic of the equivalent nonlinear resistance of the diode as follows:
Figure GDA0002830758540000032
wherein n represents f (u)D) Order of Taylor expansion of Rn(uD) Is f (u)D) The remainder of the taylor expansion of (1).
In step S2, the nonlinear periodic function is:
Figure GDA0002830758540000033
in the formula, k represents the fitting degree to the switching function, d is the duty ratio of the switching control signal, and T is the period of the switching signal.
In step S3, the multi-scale unified non-linear mathematical model describing the switching converter is:
G0(p)x+G1f(1)(x)+G2f(2)(x)＝U (4)
wherein p represents a differential operator
Figure GDA0002830758540000041
x＝[iL uC0 uC1]TrRepresenting the multi-scale state variable vector of the switching converter system, the superscript Tr representing the transposition of the solving matrix, iLRepresenting instantaneous value of inductor current, uC0Representing instantaneous values of the output capacitor voltage, which are state variables describing the circuit characteristics of the switching converter and belong to the circuit level scale; u. ofC1First interelectrode capacitance C representing a model of a field effect transistor device mechanism1The voltage instantaneous value of (2) can describe the dynamic characteristics of the switching device and belongs to the state variable of the device level scale; g0(p)、G1、G2Coefficient matrixes related to the topological structure of the switching converter and circuit parameters are respectively provided; f. of(1)(x)＝(1-s(1))x＝s(2)x＝s(2)·[iL uC0 uC1]TrIs a non-linear vector function, where s(1)For the non-linear periodic function, S, established in step S2(2)＝(1-s(1))；f(2)(x)＝s(1)e 'is also a non-linear vector function, e' being a constant vector related to the drive signal; u is a vector related to the input voltage of the converter;
in step S4, a nonlinear periodic function S(1)Expansion into a Fourier series represented by equation (5):
Figure GDA0002830758540000042
where j is an imaginary unit, τ ═ ω t, ω ═ 2 π f,
Figure GDA0002830758540000043
Figure GDA0002830758540000044
f is the switching frequency of the switch and is,
Figure GDA0002830758540000045
is composed of
Figure GDA0002830758540000046
D is the duty ratio of the switching control signal, T is the period of the switching signal, and n is the order of function expansion;
non-linear periodic function s(2)Can be expanded to a fourier series represented by equation (6):
Figure GDA0002830758540000051
where j is an imaginary unit, τ ═ ω t, ω ═ 2 π f,
Figure GDA0002830758540000052
Figure GDA0002830758540000053
f is the switching frequency of the switch and is,
Figure GDA0002830758540000054
is composed of
Figure GDA0002830758540000055
The complex conjugate term of (a);
in step S5, the process of obtaining the approximate linear equivalent mathematical model of the switching converter is as follows:
according to the basic principle of the nonlinear system equivalent small parameter notation, the equation (4) is transformed to obtain an approximate linear equivalent mathematical model of the switching converter, wherein the approximate linear equivalent mathematical model is as follows:
Figure GDA0002830758540000056
in the formula, x0、x1、x2、……、xnThe main oscillation component, the first order correction quantity, the second order correction quantity, … … and the n order correction quantity of the state variable x are respectively;
Figure GDA0002830758540000057
are respectively a non-linear vector function f(1)(x) Neutral and x0、x1、x2、……、xnTerms having the same frequency content;
Figure GDA0002830758540000058
as a non-linear vector function f(2)(x) Neutral and x0Terms having the same frequency content;
Figure GDA0002830758540000059
are respectively a non-linear vector function f(1)(x) Neutral and x1、x2、……、xnTerms having the same frequency content;
Figure GDA00028307585400000510
Figure GDA00028307585400000511
are respectively a non-linear vector function f(2)(x) Neutral and x1、x2、……、xnTerms having the same frequency content;
the first equation in equation (7), called the principal oscillation equation, is used to determine the principal of the state variablesOscillation component x0(ii) a The remaining equations are called correction quantity equations for determining the correction quantity x of each order of the state variablei(ii) a The main oscillation equation can be a linear equation or a nonlinear equation and is related to the topological structure of the main circuit of the switching converter, and the correction equations are linear equations, so that the formula (7) is an approximate linear equivalent mathematical model;
in step S6, the specific steps of obtaining the converter state variable steady-state period solution are as follows:
s61, setting the main oscillation component of the switching converter as:
x0＝a00 (8)
in the formula, a00Is the direct current component of the state variable;
the first order correction is:
x1＝a11ejτ+c.c (9)
wherein c.c represents a complex conjugate term; a is11Is the 1 st harmonic amplitude of the first order correction;
the second order correction is:
x2＝a02+a22ej2τ+a32ej3τ+c.c (10)
in the formula, a02Is the direct component of the second order correction, a22Amplitude of 2 th harmonic of second order correction, a32Is the 3 rd harmonic amplitude magnitude of the second order correction;
s62, substituting the above expressions (8) to (10) into the nonlinear vector function f(1)(x) And f(2)(x) Obtaining:
Figure GDA0002830758540000061
Figure GDA0002830758540000062
Figure GDA0002830758540000063
Figure GDA0002830758540000071
s63, the above equations (8) to (16) are respectively substituted into the corresponding equation of the formula (7), to obtain:
Figure GDA0002830758540000072
in the formula, G0(0) Is the coefficient matrix G0(p) matrix obtained when p is 0, G0(j ω) is the coefficient matrix G0(p) matrix obtained when p is j ω, G0(j2 ω) is the coefficient matrix G0(p) matrix obtained by letting p j2 ω, G0(j3 ω) is the coefficient matrix G0(p) a matrix obtained by letting p be j3 ω;
s64, according to the result, obtaining an approximate expression of the steady state period solution of the multi-scale state variable x of the converter expressed by an exponential function or a trigonometric function form, wherein the approximate expression is as follows:
x≈x0+x1+x2
＝a00+a02+a11ejτ+a22ej2τ+a32ej3τ+c.c
＝a00+a02+2Re(a11)cosτ-2Im(a11)sinτ+2Re(a22)cos2τ
-2Im(a22)sin2τ+2Re(a32)cos3τ-2Im(a32)sin3τ (16)
in the formula, functions Re (·) and Im (·) represent the real part and imaginary part of the complex number, respectively.
Compared with the prior art, the invention has the following advantages and beneficial effects:
1. in the modeling process of the switching converter, a switching tube and a diode of the circuit are equivalent by using mechanism models thereof, a mathematical model which can describe state variables of a conversion device level and a circuit level simultaneously is established, the influence of the state variables of the device level on the circuit work can be reflected, and the unified modeling of the combination of the device level scale and the circuit level scale of the switching converter is realized.
2. Continuous and unified modeling of the converter is realized by adopting a continuous nonlinear periodic function to fit a traditional discrete switching function describing the on-off characteristics of the device.
Drawings
Fig. 1 is a circuit model of a Boost switching converter.
Fig. 2 is a simplified device mechanism model of a field effect transistor (MOSFET).
Fig. 3 is a multi-scale same model schematic diagram of a Boost switching converter considering a device model.
Fig. 4 is a functional image of a switch fitting function.
FIG. 5a is a diagram of the inductor L current i according to the method and numerical calculation method of the present inventionLA waveform comparison graph of (c).
FIG. 5b shows a capacitance C of the present invention method and numerical calculation method0Voltage uC0A waveform comparison graph of (c).
FIG. 5C shows a capacitance C of the present invention method and numerical calculation method1Voltage uC1A waveform comparison graph of (c).
Detailed Description
To further illustrate the content and features of the present invention, the following detailed description of the embodiments of the present invention is provided with reference to the accompanying drawings, but the present invention is not limited thereto.
The nonlinear modeling method for simultaneously solving the multi-scale state variables of the switching converter provided by the embodiment comprises the following steps of:
s1, the switch device field effect transistor (MOSFET) is equivalent by using a simplified device mechanism model.
As shown in FIG. 2, the simplified device mechanism model of the field effect transistor (MOSFET) comprises a gate-level input resistor RGA pressure controlCurrent source iGFirst inter-electrode capacitance C1Second inter-pole capacitance C2And an on-resistance Rd. The gate-level input resistor RGOne end of the first electrode is connected with the grid of the MOSFET; the gate-level input resistor RGAnother terminal of (2) and a second inter-pole capacitance C2Is connected with one end of the connecting rod; the voltage-controlled current source iGOne end of each of the capacitors is connected to the first inter-electrode capacitor C1One terminal of (1), on-resistance RdOne end of the first and second electrodes is connected with the drain of the MOSFET; the voltage-controlled current source iGAnd the other end of the first and second capacitors are connected with the first inter-electrode capacitor C respectively1Another terminal of (1), on-resistance RdThe other end of the first diode is connected with the source electrode of the MOSFET; the second inter-electrode capacitor C2The other end of the first diode is connected with the source electrode of the MOSFET; wherein the voltage controlled current source iGIs expressed as iG＝gmuGSIn the formula, gmIs the forward transconductance of the MOSFET device, uGSIs the gate drive voltage signal of the MOSFET uGS＝s(1)uG，s(1)Non-linear control signal function uGIs the gate voltage magnitude of the MOSFET. (ii) a
And S2, equivalence is carried out on the diode device by using a nonlinear resistor, wherein the volt-ampere characteristic of the nonlinear resistor is described by a PN junction current equation of the diode.
The PN junction current equation for the diode is as follows:
Figure GDA0002830758540000091
in the formula iDIs the current of a diode, ISThe reverse saturation current of the diode can be obtained by looking up the value of the reverse saturation current according to a parameter manual of the actually selected diode model; u shapeTIs thermodynamic voltage, U at normal temperatureT＝26mV；uDIs the forward voltage of the diode. Will iDThe taylor expansion can obtain the current-voltage characteristic of the equivalent nonlinear resistance of the diode as follows:
Figure GDA0002830758540000092
wherein n represents f (u)D) Order of Taylor expansion of Rn(uD) Is f (u)D) The remainder of the taylor expansion of (1).
S3 fitting discrete switching function of switching device gate drive signal with continuous nonlinear periodic function
Figure GDA0002830758540000093
In the formula, k represents the fitting degree to the switching function, d is the duty ratio of the switching control signal, and T is the period of the switching signal.
S4, establishing a multi-scale unified non-linear mathematical model of the switching converter described by a differential equation
G0(p)x+G1f(1)(x)+G2f(2)(x)＝U (4)
Wherein p represents a differential operator
Figure GDA0002830758540000094
x＝[iL uC0 uC1]TrRepresenting the multi-scale state variable vector of the switching converter system, the superscript Tr representing the transposition of the solving matrix, iLRepresenting instantaneous value of inductor current, uC0Representing instantaneous values of the output capacitor voltage, which are state variables describing the circuit characteristics of the switching converter and belong to the circuit level scale; u. ofC1First interelectrode capacitance C representing a model of a field effect transistor device mechanism1The instantaneous value of the voltage can describe the dynamic characteristics of the switching device, and belongs to the state variable of the device level scale; g0(p)、G1、G2Coefficient matrixes related to the topological structure of the switching converter and circuit parameters are respectively provided; f. of(1)(x)＝(1-s(1))x＝s(2)x＝s(2)·[iL uC0 uC1]TrIs a non-linear vector function, where s(1)Is established in step S3Of a non-linear periodic function of s(2)＝(1-s(1))；f(2)(x)＝s(1)e 'is also a non-linear vector function, e' being a constant vector related to the drive signal; u is a vector related to the input voltage of the converter.
S5, expanding the nonlinear periodic function of S3 into Fourier series
Figure GDA0002830758540000095
Where j is an imaginary unit, τ ═ ω t, ω ═ 2 π f,
Figure GDA0002830758540000101
Figure GDA0002830758540000102
f is the switching frequency of the switch and is,
Figure GDA0002830758540000103
is composed of
Figure GDA0002830758540000104
The complex conjugate term of (a);
similarly, said non-linear periodic function s(2)Can be expanded to a Fourier series shown in equation (6):
Figure GDA0002830758540000105
where j is an imaginary unit, τ ═ ω t, ω ═ 2 π f,
Figure GDA0002830758540000106
Figure GDA0002830758540000107
f is the switching frequency of the switch and is,
Figure GDA0002830758540000108
is composed of
Figure GDA0002830758540000109
The complex conjugate term of (a);
s6 obtaining approximate linear equivalent mathematical model of switching converter
Decomposing the nonlinear mathematical model of S4 by using a nonlinear analysis method to obtain an approximate linear equivalent mathematical model; the equivalent mathematical model is an equation set composed of a series of linear equations, and comprises an equation for solving the principal component of the state variable of the system and a plurality of equations for solving the correction quantity of each order of the state variable.
According to the basic principle of the nonlinear system equivalent small parameter notation, the formula (4) is transformed to obtain an approximate linear equivalent mathematical model of the switching converter as follows:
Figure GDA00028307585400001010
in the formula, x0、x1、x2、……、xnThe main oscillation component, the first order correction quantity, the second order correction quantity, … … and the n order correction quantity of the state variable x are respectively;
Figure GDA00028307585400001011
are respectively a non-linear vector function f(1)(x) Neutral and x0、x1、x2、……、xnTerms having the same frequency content;
Figure GDA00028307585400001012
as a non-linear vector function f(2)(x) Neutral and x0Terms having the same frequency content;
Figure GDA00028307585400001013
are respectively a non-linear vector function f(1)(x) Neutral and x1、x2、……、xnTerms having the same frequency content;
Figure GDA0002830758540000111
Figure GDA0002830758540000112
are respectively a non-linear vector function f(2)(x) Neutral and x1、x2、……、xnTerms having the same frequency content.
The first equation in equation (7), called the principal oscillation equation, is used to determine the principal oscillation component x of the state variable0(ii) a The remaining equations are called correction quantity equations for determining the correction quantity x of each order of the state variablei. The main oscillation equation can be a linear equation or a nonlinear equation and is related to the topological structure of the main circuit of the switching converter, and the correction equation is a linear equation, so that the equation (7) is an approximate linear equivalent mathematical model.
S7, obtaining approximate analytical expression of multi-scale state variable steady-state period solution of converter system
And (3) solving each equation in the equivalent model step by using a harmonic balance principle, and obtaining direct current components and correction quantities of each order of state variables of a circuit level and a device level of the switching converter system, so as to obtain an approximate analytical expression of the multi-scale state variable steady-state period solution of the converter system.
S71, setting the main oscillation component of the switching converter as:
x0＝a00 (8)
in the formula a00Is x0A direct current component of (a);
the first order correction is:
x1＝a11ejτ+c.c (9)
wherein c.c represents a complex conjugate term; a is11Is the 1 st harmonic amplitude of the first order correction;
the second order correction is:
x2＝a02+a22ej2τ+a32ej3τ+c.c (10)
in the formula a02Is the direct component of the second order correction, a22Amplitude of 2 th harmonic of second order correction, a32Is the 3 rd harmonic amplitude magnitude of the second order correction;
s72, substituting the above expressions (8) to (10) into the nonlinear function f(1)(x) And f(2)(x) Obtaining:
Figure GDA0002830758540000113
Figure GDA0002830758540000114
Figure GDA0002830758540000115
Figure GDA0002830758540000116
s73, the above equations (8) to (16) are respectively substituted into the corresponding equation of the formula (7), and the following can be obtained:
Figure GDA0002830758540000121
in the formula, G0(0) Is the coefficient matrix G0(p) matrix obtained when p is 0, G0(j ω) is the coefficient matrix G0(p) matrix obtained when p is j ω, G0(j2 ω) is the coefficient matrix G0(p) matrix obtained by letting p j2 ω, G0(j3 ω) is the coefficient matrix G0(p) a matrix obtained by letting p be j3 ω; s74, according to the result, the approximate expression of the steady state period solution of the multi-scale state variable x of the converter expressed by an exponential function or a trigonometric function form can be obtained as follows:
Figure GDA0002830758540000122
in the formula, functions Re (·) and Im (·) represent the real part and imaginary part of the complex number, respectively.
The method of the present embodiment will be described in detail with reference to fig. 1, 2, 3, 4, 5a, 5b, and 5 c.
FIG. 1 is a schematic diagram of a Boost switching converter, where STDenotes a field effect transistor (MOSFET) switching device, SDDenotes a diode device, VSRepresenting a DC source, R representing a load resistance, C0Representing capacitance, L represents inductance;
FIG. 2 is a simplified model of a field effect transistor device, wherein RGRepresenting the gate resistance, C1Denotes a first inter-electrode capacitance, C2Representing the second inter-pole capacitance, iGRepresenting a voltage-controlled current source, RdRepresents the on-resistance;
FIG. 3 is a schematic diagram of a multi-scale same model of Boost switching converter with device model taken into consideration, wherein circuit parameters include switching frequency fs of 25kHz, inductance L of 330 μ H, and capacitance C 010 muF, input DC voltage V S10V, 20 Ω, forward transconductance g of the MOSFETm1S, a first interelectrode capacitance C of the MOSFET11000pF, MOSFET on-resistance Rd0.15 Ω, drive signal u of MOSFETGThe duty ratio d of the switching signal is equal to 0.5;
fig. 4 is a functional image of the continuous switch fitting function, where k is 1000;
according to the method, the main oscillation component and the first-order and second-order correction quantities of the Boost switching converter are calculated, and finally the main oscillation component and the first-order and second-order correction quantities are added to obtain an expression of a multi-scale state variable steady-state period analytical solution of the converter, wherein the expression comprises the following steps:
Figure GDA0002830758540000131
the ripple waveforms of the state variables calculated in the steady state by the method of the present invention and the numerical calculation method based on the longgutta method are compared, and the obtained waveform results are respectively shown in fig. 5a, 5b, and 5c, and the numerical algorithm parameters are consistent with the parameters used in the calculation by the symbolic analysis method used in the method of the present invention. In the figure, the dotted line shows the waveform of the calculation result of the proposed method, and the solid line shows the waveform of the result of the numerical algorithm simulation calculation.
It can be seen from the figure that the degree of fitting between the two wave curves is very high, which indicates that the method proposed by the present invention is effective. Continuous unified modeling of the converter is realized by adopting a continuous switch fitting function to replace a piecewise switch function, the change of the variable of different scales of the converter can be clearly seen through a steady-state period analytic solution expression of the multi-scale state variable of the converter, and the influence of the variable change of the device scale on the converter can be seen.
The above-described embodiments are only preferred embodiments of the present invention, and not intended to limit the scope of the present invention, 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 they are included in the scope of the present invention.

Claims (5)

1. The nonlinear modeling method for simultaneously solving the multi-scale state variables of the switching converter is characterized by comprising the following steps of: in the method, a switching tube and a diode are equivalent by using mechanism models thereof in a modeling process, and a continuous nonlinear periodic function is adopted to fit a discrete switching function for describing the on-off characteristics of a device, so as to obtain a uniform nonlinear continuous mathematical model for describing state variables of a switching converter circuit level and the device level; solving by using a nonlinear analysis method, an approximate analytical expression of the steady state period solution of the state variables of the switching converter circuit level and the device level can be obtained at the same time, namely the steady state period analytical solution of the multi-scale state variables of the switching converter can be obtained; which comprises the following steps:
s1, the MOSFET of the switching device is equivalent by using a simplified device mechanism model; equating the diode device with a nonlinear resistor, wherein the volt-ampere characteristic of the nonlinear resistor is described by a PN junction current equation of the diode;
s2, fitting a discrete switching function of the gate drive signal of the switching device by using a continuous nonlinear periodic function;
s3, establishing a multi-scale unified nonlinear mathematical model of the switching converter described by a differential equation;
s4, expanding the nonlinear periodic function of the step S2 into Fourier series;
s5, acquiring an approximate linear equivalent mathematical model of the switching converter;
decomposing the nonlinear mathematical model established in the step S3 by using a nonlinear analysis method to obtain an approximate linear equivalent mathematical model of the nonlinear mathematical model, wherein the equivalent mathematical model is an equation set consisting of a series of linear equations and comprises an equation for solving the main component of the state variable of the system and a plurality of equations for solving the correction quantity of each order of the state variable;
s6, obtaining an approximate analytical expression of a multi-scale state variable steady-state period solution of the switching converter;
and (4) solving each equation in the equivalent mathematical model in the step (S5) step by using a harmonic balance principle, and obtaining direct current components and correction quantities of each order of state variables of the circuit level and the device level of the switching converter so as to obtain an approximate analytical expression of the multi-scale state variable steady-state period solution of the switching converter.
2. The method of claim 1, wherein the method comprises: in step S1, the simplified device mechanism model of the MOSFET includes a gate-level input resistor RGA voltage-controlled current source iGA first interelectrode capacitance C1A second inter-pole capacitor C2And an on-resistance Rd(ii) a The gate-level input resistor RGOne end of the first electrode is connected with the grid of the MOSFET, and the other end of the first electrode is connected with the second inter-electrode capacitor C2Is connected with one end of the connecting rod; the voltage-controlled current source iGOne end of each of the capacitors is connected to the first inter-electrode capacitor C1One terminal of (1), on-resistance RdOne terminal of (1) and drain connection of MOSFETConnecting; the voltage-controlled current source iGAnd the other end of the first and second capacitors are connected with the first inter-electrode capacitor C respectively1Another terminal of (1), on-resistance RdThe other end of the first diode is connected with the source electrode of the MOSFET; the second inter-electrode capacitor C2The other end of the first diode is connected with the source electrode of the MOSFET; wherein the voltage-controlled current source iGIs expressed as iG＝gmuGSIn the formula, gmIs the forward transconductance of the MOSFET, uGSIs the gate drive voltage signal of the MOSFET uGS＝s(1)uG，s(1)A non-linear control signal function uGIs the gate voltage magnitude of the MOSFET.
3. The method of claim 1, wherein the method comprises: in step S1, the PN junction current equation of the diode is as follows:
Figure FDA0002830758530000021
in the formula iDIs the current of a diode, ISThe reverse saturation current of the diode can be found according to a parameter manual of the actually selected diode model; u shapeTIs thermodynamic voltage, U at normal temperatureT＝26mV；uDIs the forward voltage of the diode; will iDThe taylor expansion can be carried out to obtain the current-voltage characteristic of the equivalent nonlinear resistance of the diode as follows:
Figure FDA0002830758530000022
wherein n represents f (u)D) Order of Taylor expansion of Rn(uD) Is f (u)D) The remainder of the taylor expansion of (1).
4. The method of claim 1, wherein the method comprises: in step S2, the nonlinear periodic function is:
Figure FDA0002830758530000031
in the formula, k represents the fitting degree to the switching function, d is the duty ratio of the switching control signal, and T is the period of the switching signal.
5. The method of claim 1, wherein the method comprises: in step S3, the multi-scale unified non-linear mathematical model describing the switching converter is:
G0(p)x+G1f(1)(x)+G2f(2)(x)＝U (4)
wherein p represents a differential operator
Figure FDA0002830758530000032
x＝[iL uC0 uC1]TrRepresenting the multi-scale state variable vector of the switching converter system, the superscript Tr representing the transposition of the solving matrix, iLRepresenting instantaneous value of inductor current, uC0Representing instantaneous values of the output capacitor voltage, which are state variables describing the circuit characteristics of the switching converter and belong to the circuit level scale; u. ofC1First interelectrode capacitance C representing a model of a field effect transistor device mechanism1The voltage instantaneous value of (2) can describe the dynamic characteristics of the switching device and belongs to the state variable of the device level scale; g0(p)、G1、G2Coefficient matrixes related to the topological structure of the switching converter and circuit parameters are respectively provided; f. of(1)(x)＝(1-s(1))x＝s(2)x＝s(2)·[iL uC0 uC1]TrIs a non-linear vector function, where s(1)For the non-linear periodic function, S, established in step S2(2)＝(1-s(1))；f(2)(x)＝s(1)e 'is also a non-linear vector function, e' being a constant vector related to the drive signal; u is a vector related to the input voltage of the converter;
in step S4, a nonlinear periodic function S(1)Expansion into a Fourier series represented by equation (5):
Figure FDA0002830758530000033
where j is an imaginary unit, τ ═ ω t, ω ═ 2 π f,
Figure FDA0002830758530000041
Figure FDA0002830758530000042
f is the switching frequency of the switch and is,
Figure FDA0002830758530000043
is composed of
Figure FDA0002830758530000044
D is the duty ratio of the switching control signal, T is the period of the switching signal, and n is the order of function expansion;
non-linear periodic function s(2)Can be expanded to a fourier series represented by equation (6):
Figure FDA0002830758530000045
where j is an imaginary unit, τ ═ ω t, ω ═ 2 π f,
Figure FDA0002830758530000046
Figure FDA0002830758530000047
f is onThe frequency of the switch-off is,
Figure FDA0002830758530000048
is composed of
Figure FDA0002830758530000049
The complex conjugate term of (a);
in step S5, the process of obtaining the approximate linear equivalent mathematical model of the switching converter is as follows:
according to the basic principle of the nonlinear system equivalent small parameter notation, the equation (4) is transformed to obtain an approximate linear equivalent mathematical model of the switching converter, wherein the approximate linear equivalent mathematical model is as follows:
Figure FDA00028307585300000410
in the formula, x0、x1、x2、……、xnThe main oscillation component, the first order correction quantity, the second order correction quantity, … … and the n order correction quantity of the state variable x are respectively;
Figure FDA0002830758530000051
are respectively a non-linear vector function f(1)(x) Neutral and x0、x1、x2、……、xnTerms having the same frequency content;
Figure FDA0002830758530000052
as a non-linear vector function f(2)(x) Neutral and x0Terms having the same frequency content;
Figure FDA0002830758530000053
are respectively a non-linear vector function f(1)(x) Neutral and x1、x2、……、xnTerms having the same frequency content;
Figure FDA0002830758530000054
Figure FDA0002830758530000055
are respectively a non-linear vector function f(2)(x) Neutral and x1、x2、……、xnTerms having the same frequency content;
the first equation in equation (7), called the principal oscillation equation, is used to determine the principal oscillation component x of the state variable0(ii) a The remaining equations are called correction quantity equations for determining the correction quantity x of each order of the state variablei(ii) a The main oscillation equation can be a linear equation or a nonlinear equation and is related to the topological structure of the main circuit of the switching converter, and the correction equations are linear equations, so that the formula (7) is an approximate linear equivalent mathematical model;
in step S6, the specific steps of obtaining the converter state variable steady-state period solution are as follows:
s61, setting the main oscillation component of the switching converter as:
x0＝a00 (8)
in the formula, a00Is the direct current component of the state variable;
the first order correction is:
x1＝a11ejτ+c.c (9)
wherein c.c represents a complex conjugate term; a is11Is the 1 st harmonic amplitude of the first order correction;
the second order correction is:
x2＝a02+a22ej2τ+a32ej3τ+c.c (10)
in the formula, a02Is the direct component of the second order correction, a22Amplitude of 2 th harmonic of second order correction, a32Is the 3 rd harmonic amplitude magnitude of the second order correction;
s62, substituting the above expressions (8) to (10) into the nonlinear vector function f(1)(x) And f(2)(x) Obtaining:
Figure FDA0002830758530000061
Figure FDA0002830758530000062
Figure FDA0002830758530000063
Figure FDA0002830758530000064
s63, the above equations (8) to (16) are respectively substituted into the corresponding equation of the formula (7), to obtain:
Figure FDA0002830758530000065
in the formula, G0(0) Is the coefficient matrix G0(p) matrix obtained when p is 0, G0(j ω) is the coefficient matrix G0(p) matrix obtained when p is j ω, G0(j2 ω) is the coefficient matrix G0(p) matrix obtained by letting p j2 ω, G0(j3 ω) is the coefficient matrix G0(p) a matrix obtained by letting p be j3 ω;
s64, according to the result, obtaining an approximate expression of the steady state period solution of the multi-scale state variable x of the converter expressed by an exponential function or a trigonometric function form, wherein the approximate expression is as follows:
x≈x0+x1+x2
＝a00+a02+a11ejτ+a22ej2τ+a32ej3τ+c.c
＝a00+a02+2Re(a11)cosτ-2Im(a11)sinτ+2Re(a22)cos2τ-2Im(a22)sin2τ+2Re(a32)cos3τ-2Im(a32)sin3τ (16)
in the formula, functions Re (·) and Im (·) represent the real part and imaginary part of the complex number, respectively.
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