CN106484962A - A kind of symbolic analysis method of the resonance type wireless transmission system based on E class inversion - Google Patents

A kind of symbolic analysis method of the resonance type wireless transmission system based on E class inversion Download PDF

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CN106484962A
CN106484962A CN201610839092.1A CN201610839092A CN106484962A CN 106484962 A CN106484962 A CN 106484962A CN 201610839092 A CN201610839092 A CN 201610839092A CN 106484962 A CN106484962 A CN 106484962A
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陈艳峰
吴美玉
张波
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South China University of Technology SCUT
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    • GPHYSICS
    • G06COMPUTING; CALCULATING OR COUNTING
    • G06FELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00Computer-aided design [CAD]
    • G06F30/30Circuit design
    • G06F30/36Circuit design at the analogue level
    • G06F30/367Design verification, e.g. using simulation, simulation program with integrated circuit emphasis [SPICE], direct methods or relaxation methods
    • GPHYSICS
    • G06COMPUTING; CALCULATING OR COUNTING
    • G06FELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00Computer-aided design [CAD]
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Abstract

The invention discloses a kind of symbolic analysis method of the resonance type wireless transmission system based on E class inversion, the method combines the principle of harmonic balance, the complicated solving process meeting the multi-state variable circuit of ZVS condition is the process that matrix operationss and linear equation (group) solve, compare more existing modeling and analysis methods, the inventive method is except being capable of analytically analytic transformation device state variable ripple peak-to-peak value size, the impact to changer working condition for the energy-storage travelling wave tube order change, in the case of the order not increasing state equation, also rapidly obtain analytic solutions steady-state period of state variable, and can be used for analyzing the harmonic componentss of state variable.

Description

A kind of symbolic analysis method of the resonance type wireless transmission system based on E class inversion
Technical field
The present invention relates to the modeling of resonance type wireless transmission system and analysis field, refer in particular to one kind and meet ZVS (Zero Voltage Switching) condition the resonance type wireless transmission system of E class inversion symbolic analysis method, specifically, profit With the ultimate principle feature of equivalent small parameter method, the method seeking the periodic solution expression formula of the state variable meeting ZVS condition circuit.
Background technology
Past is directed to E class inverter circuit and its application circuit meeting ZVS (Zero Voltage Switching) condition Conventional modeling and analysis method have:Model based on generalized mean method, discrete iteration mapping model, the segmentation of simplification circuit method Linear model, Singular-Perturbation Method model, multifrequency averaging method.
Generalized mean method is based on the thought that signal is carried out with frequency domain decomposition, is to become by increasing state variable harmonic constant Amount;(list of references 1 " R.E.ect, " Small-signal modeling of a DC-DC Class-E piezoconverter based on generalized averaging method,"2011 IEEE I SoIE,2011, pp.396-401.).Discrete iteration mapping model is made by calculating method circuit being iterated solve, list of references 2 (P.C.K.ect,"State-Space Modeling of a Class E2Converter for Inductive Links,"in IEEE T o P E,pp.3242-3251,June 2015.).Simplifying circuit method is using equivalent current source Method, carries out Equivalent Modeling to circuit, draws the equivalence value based on input current for the circuit state variable, list of references 3 (Chen Wen, The symbolic analysis [J] of mound aquatic .E class A amplifier A. South China Science & Engineering University's journal (natural science edition), 1997,08:89-93.).Very Different method of perturbation is to be analyzed according to the subsystem of different time scales, list of references 4 (P.D, K.W.E.C.ect. " Singular perturbation modeling technique and analysis for class-E DC-DC converter using piezoelectric transformer,"in IET Power Electronics,pp.518- 526,Dec 2008.).Multifrequency averaging method is all to divide a circuit into the different subsystem of frequency to be modeled analyzing, with reference to literary composition Offer 5 (C.B, E.O.ect, " Dynamic Model of Class-E Inverter With Multi-frequency Averaged Analysis,"in IEEE T I E,pp.3737-3744,Oct.2012.).Above-mentioned existing E class circuit and There is the periodic solution expression formula that coefficient matrix is excessive, cannot draw circuit, input current value, not in the modeling method of its application circuit The shortcomings of size of consideration choke induction.
Content of the invention
It is an object of the invention to overcoming the shortcoming and defect of prior art, one kind is provided to meet ZVS (Zero Voltage Switching) the symbolic analysis method of the resonance type wireless transmission system of E class inversion of condition is it is considered to choke induction, inductance are posted Raw parameter, conducting resistance, and can quickly obtain analytic solutions steady-state period of free position variable.
For achieving the above object, technical scheme provided by the present invention is:A kind of resonance type wireless based on E class inversion is defeated The symbolic analysis method of electric system, comprises the following steps:
1) set up the mathematical model of resonance type wireless transmission system under the conditions of ZVS;
1.1) according to circuit theory, row write the piecewise linearity differential equation of wireless power transmission systems:
G1(p)+G2(p) f (x)=u (1)
Electric capacity on same branch road, inductance carries out series connection and calculates, x=[i in above formula1i2i3v1v2v3]TExpression system State variable, i1Represent inductance L1On electric current, i2Represent inductance L2On electric current, i3Represent inductance L4On electric current, v1Table Show electric capacity C1On voltage, v2Represent electric capacity C2With C3Voltage after series connection, v3Represent electric capacity C4On voltage;Differential operator in formula P=d/dt, G1(p)、G2P () is coefficient matrix;F (x)=s (t) x is non-linear phasor function;
Switch function s (t) characterizes the state of switch S, and it is defined as:
Wherein D is dutycycle, equal to the ON time of switch and the ratio in cycle;
1.2) state variable x and switch function s (t) are all expanded into the form of principal part and a small amount of remainder sum:
Formula (3), (4) are substituted into f (x)=s (t) x, merges identical order remainder in a small amount, obtain:
Items in formula (5) are expressed as fi=fim+εRi+1(i=0,1,2 ... .), wherein fimComprise fiIn all and xi There is the item of same frequency spectrum composition, and and xiThe Xiang Ze with different spectral composition belongs to Ri+1, for determining xi+1Frequency become Point;A small amount of labelling ε shows Ri+1It is fimSingle order in a small amount, i.e. Ri+1<fim
Wherein:
1.3) according to principle of harmonic balance, by principal part in the expansion (7) of described state variable x and switch function s (t) and It is as follows that i-th rank remainder a small amount of does Fourier expansion:
Wherein akiRepresent the amplitude of the k subharmonic composition of the i-th rank correction, bmIt isConjugate complex number, described switch letter Number s (t) expansion coefficient expressions are as follows:
In formula (8)
2) according to principle of harmonic balance, coefficient expansion (8) is substituted into Fourier expansion formula (7), solving state becomes successively The main oscillations component of amount and each rank correction;
Coefficient matrix G1(p), G2P () is changed into G1(jk ω), G2(jkω),k∈EirRepresent humorous in current i-th rank correction Ripple number of times k, same after the definition of i, k.
2.1) seek main oscillations component
Resonance inversion circuit, contains DC quantity and first harmonic amount, i=0 in main oscillations, is set to:
Work as k=0, when 1, by x0、s0In substitution formula (6), and by fomIn (1) formula of substitution, obtain:
Try to achieve the main oscillations component of transducer status variable by formula (10):
2.2) seek each rank correction
According to main oscillations component remainder R1In the harmonic componentss that contain, i=1, if the first-order correction form of state variable As follows:
Wherein c.c represents conjugation item, same afterwards;From the harmonic componentss in the first-order correction of state variable, k=2, generation Enter f in formula (6)1, obtain first-order correction expression formula:
It is obtained in that with regard to harmonic amplitude a according to formula (13)01And ak1System of linear equations;
Parameter is substituted into the expression formula of gained current order correction, if each harmonic amplitude phase of current order correction In comparison, first-order correction is less than an order of magnitude, then be not required to do the correction of higher order, conversely, continue according to said process continuing Seek the correction of higher order time;
3) main oscillations component is added with each rank correction, obtains analytic solutions expression steady-state period with regard to state variable Formula.
The present invention compared with prior art, has the advantage that and beneficial effect:
Be can be seen that by the solution formula of institute of the present invention extracting method and ask the E class inversion meeting ZVS condition using this method Analytic solutions steady-state period of resonance type wireless transmission system circuit state variable, according to matrix operationss with ask the linear equation (group) can To draw the analytic solutions expression formula of all state variables of circuit rather than equivalent analytical expression, as long as being set up according to circuit theory As the state equation of formula (1) matrix form, then coefficient expressions are substituted into each rank correction formula, by simple matrix Multiplication and division plus and minus calculation and the unit that disappears can be obtained by with regard to circuit state variable stable state solution's expression.Compare over and increase order Or the method for solving of interative computation, the solution procedure of institute of the present invention extracting method combines the feature of equivalent small parameter method, avoids For thoroughly discussing of single status variable content, the steady state solution of gained has obvious physical significance, according to adopting the present invention The form of the steady state solution that institute's extracting method obtains, can be clearly seen that the harmonic componentss that each state variable is comprised, is conducive to The resonance type wireless transmission system circuit of E class inversion is launched deeper into analysis.
Brief description
Fig. 1 is the resonance type wireless transmission system circuit model of E class inversion.The parameter declaration of in figure is as follows:
Fig. 2 is voltage v under this paper circuit parameter1Waveform with switch ends electric current.Wherein, abscissa be emulation when Between, vertical coordinate represents amplitude.
Fig. 3 is v under side circuit parameter in Fig. 11Spectrogram.Wherein, abscissa is frequency, and vertical coordinate represents amplitude.
Fig. 4 a is presently disclosed method and electric current i in simulation software1Comparison diagram.Wherein, abscissa is emulation Time, vertical coordinate represents current value.
Fig. 4 b is presently disclosed method and electric current i in simulation software2Comparison diagram.Wherein, abscissa is emulation Time, vertical coordinate represents amplitude.
Fig. 4 c is presently disclosed method and electric current i in simulation software3Comparison diagram.Wherein, abscissa is emulation Time, vertical coordinate represents amplitude.
Fig. 4 d is presently disclosed method and voltage v in simulation software1Comparison diagram.Wherein, abscissa is emulation Time, vertical coordinate represents amplitude.
Specific embodiment
With reference to specific embodiment, the invention will be further described.
The symbolic analysis method of the resonance type wireless transmission system based on E class inversion described in the present embodiment, specifically profit With the ultimate principle feature of equivalent small parameter method, the method seeking the periodic solution expression formula of the state variable meeting ZVS condition circuit, Comprise the following steps:
1) set up the resonance type wireless transmission system mathematical model of the E class inversion meeting ZVS condition;
1.1) according to the circuit theory in Fig. 1, row write the piecewise linearity differential equation of wireless power transmission systems:
G1(p)+G2(p) f (x)=u (1)
Electric capacity on same branch road, inductance carries out series connection and calculates, x=[i in above formula1i2i3v1v2v3]TExpression system State variable, i1Represent inductance L1On electric current, i2Represent inductance L2On electric current, i3Represent inductance L4On electric current, v1Table Show electric capacity C1On voltage, v2Represent electric capacity C2With C3Voltage after series connection, v3Represent electric capacity C4On voltage.Differential operator in formula P=d/dt, G1(p)、G2P () is coefficient matrix.F (x)=s (t) x is non-linear phasor function.
Switch function s (t) characterizes the state of switch S, and it is defined as:
Wherein D is dutycycle, equal to the ON time of switch and the ratio in cycle.
1.2) state variable x and switch function s (t) are all expanded into the form of principal part and a small amount of remainder sum:
Formula (3), (4) are substituted into f (x)=s (t) x, merges identical order remainder in a small amount, obtain:
Items in formula (5) are expressed as fi=fim+εRi+1(i=0,1,2 ... .), wherein fimComprise fiIn all and xi There is the item of same frequency spectrum composition, and and xiThe Xiang Ze with different spectral composition belongs to Ri+1, for determining xi+1Frequency become Point.A small amount of labelling ε shows Ri+1It is fimSingle order in a small amount, i.e. Ri+1<fim.
Wherein:
1.3) according to principle of harmonic balance, by principal part in the expansion (7) of described state variable x and switch function s (t) and It is as follows that i-th rank remainder a small amount of does Fourier expansion:
Wherein akiRepresent the amplitude of the k subharmonic composition of the i-th rank correction.bmIt isConjugate complex number, described switch letter Number s (t) expansion coefficient expressions are as follows:
In formula (8)
2) according to principle of harmonic balance, coefficient expansion (8) is substituted into Fourier expansion formula (7), solving state becomes successively The main oscillations component of amount and each rank correction;
Coefficient matrix G1(p)、G2P () is changed into G1(jkω)、G2(jkω),k∈EirRepresent humorous in current i-th rank correction Ripple number of times k, same after the definition of i, k;
2.1) seek main oscillations component
Resonance inversion circuit, contains DC quantity and first harmonic amount, i=0 in main oscillations, is set to:
Work as k=0, when 1, by x0、s0In substitution formula (6), and by fomIn (1) formula of substitution, obtain:
Tried to achieve the main oscillations component of transducer status variable by formula (10) in formula (10):
2.2) seek each rank correction
According to main oscillations component remainder R1In the harmonic componentss that contain, i=1, if the first-order correction form of state variable As follows:
Wherein c.c represents conjugation item, same afterwards;From the harmonic componentss in the first-order correction of state variable, k=2, generation Enter f in formula (6)1, obtain first-order correction expression formula:
It is obtained in that with regard to harmonic amplitude a according to formula (13)01And ak1System of linear equations;
Parameter is substituted into the expression formula of gained current order correction, if each harmonic amplitude phase of current order correction In comparison, first-order correction is less than an order of magnitude, then be not required to do the correction of higher order, conversely, continue according to said process continuing Seek the correction of higher order time;
3) main oscillations component is added with each rank correction, obtains analytic solutions expression steady-state period with regard to state variable Formula.
Symbolic analysis method below for the resonance type wireless transmission system using above-mentioned E class inversion for the instantiation is carried out Computing, for its state variable x=[i of circuit in literary composition1i2i3v1v2v3]T, state equation is as follows:
Form described by corresponding (1) it is known that
Circuit parameter is with reference to list of references (Z.W., X.W.and B.Zh.A magnetic coupled resonance WPT system design method of double-end impedance converter networks with Class-E amplifier[J].I ES,2015,:The parameter of ZVS condition is met in 003093-003098.), as follows:
Circuit parameter Parameter value Circuit parameter Parameter value
L1(r1) 66μH(0.03Ω) C1 6nF
L2 8μH C2 3.6nF
L3(r3) 36.09μH(0.69Ω) C3 0.7018nF
L4(r4) 36.3μH(0.69Ω) C4 0.6798nF
M 1.96μH Ron 0.02Ω
Vin 10V RL 34.5Ω
fs 1MHz D 0.5
Foregoing circuit parameter is emulated, the voltage v1 in Fig. 2 and the current waveform of switch can be obtained, have above-mentioned known to figure Parameter meets ZVS condition, according to voltage v in Fig. 31Spectrogram can be seen that the amplitude of four-time harmonic is just very little, a demand Solution is to four-time harmonic.
Ask main oscillations component, first-order correction and the second order correction of circuit according to step above, now due to second order In correction, the Amplitude Ration main oscillations component of each harmonic is much smaller, therefore does not continue to seek higher order correction, and circuit is through two ranks Revised steady-state period, analytic solutions form was as follows:
In formula, Re (aik) represent take aikReal part, Im (aik) represent take aikImaginary part, xdcAnd xacRepresent that state becomes respectively The direct current component of amount and of ac.akiExpression formula as follows:
Parameter substitution formula (15) and (16) can be obtained analytic solutions steady-state period is:Notice that we do not list v2、v3Expression Formula, is due to generally and need to be not concerned with v2、v3.
By Symbolic Analysis Method of the present invention and PSIM software state variable waveform comparison in stable state, such as Fig. 4 a, Shown in 4b, 4c, 4d, analogous diagram adopts parameter in list of references.
It can be seen that electric current i1Direct current component difference 0.02A belongs in range of error, other electric currents, the contrast of voltage Curve matching obtains very well, illustrates that method proposed by the invention is effective.Be can be seen that by parsing solution formula and adopt this method Seeking analytic solutions steady-state period of circuit state variable, as long as setting up the state equation as formula (1) form, then coefficient being expressed Formula substitutes into each rank correction formula, and the resonance type wireless that can be obtained by E class inversion by simple matrix operationss and the unit that disappears is defeated The state variable stable state solution's expression of electric system, can clearly be seen that by this expression formula the harmonic wave in state variable becomes Point, by the expression formula of harmonic amplitude coefficient it can be seen that the impact to transducer status variable for the element order.
Embodiment described above is only the preferred embodiments of the invention, not limits the practical range of the present invention with this, therefore The change that all shapes according to the present invention, principle are made, all should cover within the scope of the present invention.

Claims (1)

1. a kind of symbolic analysis method of the resonance type wireless transmission system based on E class inversion is it is characterised in that include following walking Suddenly:
1) set up the mathematical model of resonance type wireless transmission system under the conditions of ZVS;
1.1) according to circuit theory, row write the piecewise linearity differential equation of wireless power transmission systems:
G1(p)+G2(p) f (x)=u (1)
Electric capacity on same branch road, inductance carries out series connection and calculates, x=[i in above formula1i2i3v1v2v3]TThe shape of expression system State variable, i1Represent inductance L1On electric current, i2Represent inductance L2On electric current, i3Represent inductance L4On electric current, v1Represent electricity Hold C1On voltage, v2Represent electric capacity C2With C3Voltage after series connection, v3Represent electric capacity C4On voltage;Differential operator p=in formula D/dt, G1(p)、G2P () is coefficient matrix;F (x)=s (t) x is non-linear phasor function;
Switch function s (t) characterizes the state of switch S, and it is defined as:
s ( t ) = 1 , o n : 0 &le; t &le; D T 0 , o f f : D T &le; t &le; T - - - ( 2 )
Wherein D is dutycycle, equal to the ON time of switch and the ratio in cycle;
1.2) state variable x and switch function s (t) are all expanded into the form of principal part and a small amount of remainder sum:
s ( t ) = s 0 + &Sigma; i = 1 &infin; &epsiv; i s i - - - ( 3 )
x = x 0 + &Sigma; i = 1 &infin; &epsiv; i x i - - - ( 4 )
Formula (3), (4) are substituted into f (x)=s (t) x, merges identical order remainder in a small amount, obtain:
f ( x ) = f 0 + &Sigma; i = 1 &infin; &epsiv; i f i - - - ( 5 )
Items in formula (5) are expressed as fi=fim+εRi+1(i=0,1,2 ... .), wherein fimComprise fiIn all and xiThere is phase With the item of spectrum component, and and xiThe Xiang Ze with different spectral composition belongs to Ri+1, for determining xi+1Frequency content;In a small amount Labelling ε shows Ri+1It is fimSingle order in a small amount, i.e. Ri+1<fim
f ( x ) = f 0 m + &epsiv;R 1 + &Sigma; i = 1 &infin; ( &epsiv; i f i m + &epsiv; i + 1 R i + 1 ) - - - ( 6 )
Wherein:
f 0 = s 0 x 0 f 1 = s 0 x 1 + s 1 x 0 f 2 = s 0 x 2 + s 1 x 1 + s 2 x 0 f 3 = s 0 x 3 + s 2 x 1 + s 1 x 2 + s 3 x 0 ...... - - - ( 7 )
1.3) according to principle of harmonic balance, by principal part and i-th in the expansion (7) with switch function s (t) for described state variable x It is as follows that rank remainder a small amount of does Fourier expansion:
x i = a 0 i + &Sigma; k &Element; E i r &infin; ( a k i e j k &tau; + b &OverBar; k i e - j k &tau; ) - - - ( 7. a )
s ( t ) = b 0 + &Sigma; m = 1 &infin; ( b m e j m &tau; + b &OverBar; m e - j m &tau; ) - - - ( 7. b )
Wherein akiRepresent the amplitude of the k subharmonic composition of the i-th rank correction, bmIt isConjugate complex number, described switch function s T () expansion coefficient expressions are as follows:
b 0 = D b m = 0.5 ( &alpha; m - i&beta; m ) - - - ( 8 )
In formula (8)
2) according to principle of harmonic balance, coefficient expansion (8) is substituted into Fourier expansion formula (7), successively solving state variable Main oscillations component and each rank correction;
Coefficient matrix G1(p)、G2P () is changed into G1(jkω)、G2(jkω),k∈EirRepresent harmonic wave in current i-th rank correction Number k, same after the definition of i, k;
2.1) seek main oscillations component
Resonance inversion circuit, contains DC quantity and first harmonic amount, i=0 in main oscillations, is set to:
x0=a00+a10e+c.c (9)
Work as k=0, when 1, by x0、s0In substitution formula (6), and by fomIn (1) formula of substitution, obtain:
G 1 ( 0 ) a 00 + G 2 ( 0 ) ( b 0 a 00 + b 1 a &OverBar; 10 + b &OverBar; 1 a 10 ) = u G 1 ( j &omega; ) a 10 + G 2 ( j &omega; ) b 0 a 10 + G 2 ( j &omega; ) b 1 a 00 = 0 - - - ( 10 )
Try to achieve the main oscillations component of transducer status variable by formula (10):
a 00 = ( G 1 ( 0 ) + G 2 ( 0 ) ( b 0 + b 1 B &OverBar; + b &OverBar; 1 B ) ) - 1 u a 10 = - G 2 ( j &omega; ) - 1 ( G 1 ( j &omega; ) + G 2 ( j &omega; ) b 0 ) b 1 a 00 - - - ( 11 )
2.2) seek each rank correction
According to main oscillations component remainder R1In the harmonic componentss that contain, i=1, if the first-order correction form of state variable is as follows:
x 1 = a 01 + &Sigma; k &Element; E 1 r ( a k 1 e j k &tau; + c . c ) - - - ( 12 )
Wherein c.c represents conjugation item, same afterwards;From the harmonic componentss in the first-order correction of state variable, k=2, substitute into formula (6) f in1, obtain first-order correction expression formula:
G 1 ( j 2 &omega; ) a 21 + G 2 ( j 2 &omega; ) ( b 0 a 21 + b 2 a 00 + b 3 a &OverBar; 10 ) = - G 2 ( j 2 &omega; ) b 1 a 10 - - - ( 13 )
It is obtained in that with regard to harmonic amplitude a according to formula (13)01And ak1System of linear equations;
Parameter is substituted into the expression formula of gained current order correction, if each harmonic amplitude of current order correction compares Upper first-order correction is less than an order of magnitude, then be not required to do the correction of higher order, conversely, continue according to said process to continue to ask more The correction of high order;
3) main oscillations component is added with each rank correction, obtains analytic solutions expression formula steady-state period with regard to state variable.
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CN111030313A (en) * 2019-12-30 2020-04-17 华南理工大学 Method for designing ZVS (zero voltage switching) working parameters of E-type inverter of wireless power transmission system
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CN112953280B (en) * 2021-03-16 2023-09-19 西安理工大学 Design method of E-type high-frequency inverter circuit parameters
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