CN108880225B - Nonlinear modeling method of flyback PFC converter - Google Patents

Nonlinear modeling method of flyback PFC converter Download PDF

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CN108880225B
CN108880225B CN201810743735.1A CN201810743735A CN108880225B CN 108880225 B CN108880225 B CN 108880225B CN 201810743735 A CN201810743735 A CN 201810743735A CN 108880225 B CN108880225 B CN 108880225B
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pfc converter
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常昌远
洪潮
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    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02M—APPARATUS FOR CONVERSION BETWEEN AC AND AC, BETWEEN AC AND DC, OR BETWEEN DC AND DC, AND FOR USE WITH MAINS OR SIMILAR POWER SUPPLY SYSTEMS; CONVERSION OF DC OR AC INPUT POWER INTO SURGE OUTPUT POWER; CONTROL OR REGULATION THEREOF
    • H02M1/00—Details of apparatus for conversion
    • H02M1/42—Circuits or arrangements for compensating for or adjusting power factor in converters or inverters
    • H02M1/4208—Arrangements for improving power factor of AC input
    • H02M1/4258—Arrangements for improving power factor of AC input using a single converter stage both for correction of AC input power factor and generation of a regulated and galvanically isolated DC output voltage
    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02M—APPARATUS FOR CONVERSION BETWEEN AC AND AC, BETWEEN AC AND DC, OR BETWEEN DC AND DC, AND FOR USE WITH MAINS OR SIMILAR POWER SUPPLY SYSTEMS; CONVERSION OF DC OR AC INPUT POWER INTO SURGE OUTPUT POWER; CONTROL OR REGULATION THEREOF
    • H02M1/00—Details of apparatus for conversion
    • H02M1/0003—Details of control, feedback or regulation circuits
    • H02M1/0012—Control circuits using digital or numerical techniques
    • Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
    • Y02—TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
    • Y02B—CLIMATE CHANGE MITIGATION TECHNOLOGIES RELATED TO BUILDINGS, e.g. HOUSING, HOUSE APPLIANCES OR RELATED END-USER APPLICATIONS
    • Y02B70/00—Technologies for an efficient end-user side electric power management and consumption
    • Y02B70/10—Technologies improving the efficiency by using switched-mode power supplies [SMPS], i.e. efficient power electronics conversion e.g. power factor correction or reduction of losses in power supplies or efficient standby modes

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Abstract

A non-linear modeling method of a Flyback PFC converter is characterized in that a peak current control Flyback PFC converter is modeled by a stroboscopic mapping method, and based on the model, a fast-scale instability phenomenon is numerically simulated, and the result shows that in a fast-scale instability region, not only is a period-doubled bifurcation phenomenon, but also boundary collision bifurcation and chaos occur. The invention mainly researches the influence of the value of the circuit parameter on the nonlinearity of the system, and specifically analyzes the position of the bifurcation point and the corresponding circuit parameter value by calculating the Jacobian matrix characteristic root, thereby providing a certain theoretical basis for the design of the flyback PFC circuit.

Description

Nonlinear modeling method of flyback PFC converter
Technical Field
The invention relates to a flyback PFC converter, in particular to a nonlinear modeling method of the flyback PFC converter, and belongs to the technical field of switching power supplies.
Background
In recent years, flyback PFC converters have been widely used in high-power electronic circuits as an effective means for suppressing harmonic pollution, and are used to solve the problem of harmonic pollution to the power grid which is becoming more serious. The flyback PFC converter is a strong nonlinear system due to the existence of nonlinear devices such as a switch and a multiplier. Therefore, the research on the working characteristics of the flyback PFC converter by using the nonlinear dynamics method has become a focus of attention in the academic and engineering fields, and the research on the nonlinear modeling has gained more and more attention.
While the modeling method of the flyback PFC converter breaks through continuously, the research on the nonlinear phenomenon of the flyback PFC converter also obtains a series of results by virtue of the development of the model. In 2010, Orabi researches nonlinear behaviors of a flyback PFC converter under different load conditions, provides a load range of the flyback PFC converter under stable operation, and specifically analyzes subharmonic oscillation caused by multiple period bifurcation. In 2013, Tse further analyzes the cause of the nonlinear phenomenon of the switching power supply and gives the influence of different switching frequencies on the nonlinear phenomenon of the system, but the non-linear suppression method is mainly researched instead of further analyzing the connection between the switching frequency parameters and the nonlinear phenomenon. In 2016, AGosh further and deeply researches the nonlinear behavior of the flyback PFC converter, researches the bifurcation and chaotic state of a multiple period, and researches different nonlinear phenomena of the flyback PFC converter caused by different load capacitance value ranges by taking load capacitance as a bifurcation parameter. However, the calculation of the bifurcation point parameter is still a computer drawing method, and an accurate numerical value cannot be given.
Disclosure of Invention
The technical scheme adopted by the invention is as follows: a nonlinear modeling method of a flyback PFC converter is characterized in that: establishing a state equation of a main topological structure of the flyback PFC converter, establishing a discrete mapping model in a continuous sampling mode, and calculating specific ranges of different nonlinear states by adopting a calculation Jacobian matrix based on the discrete mapping model, wherein the method comprises the following steps:
1) establishing a master topological equation of state
The state equations of the power supply when the switch tube S1 is turned on and off are respectively listed, meanwhile, the related state equations are listed by considering that the switch tube S1 is always turned on in the whole switching period and the two states of the switch tube S1 are turned off after leading on in the period,
when the switching tube S1 is turned on, the state equation of the flyback PFC converter is:
Figure GDA0002228367440000021
Figure GDA0002228367440000022
wherein iLmIs primary side inductance current, LmIs a primary side inductance, VinFor input voltage, VCIs the output voltage, R is the load resistance, C is the load capacitance;
when the switching tube S1 is turned off, the state equation of the flyback PFC converter is:
Figure GDA0002228367440000023
Figure GDA0002228367440000024
the equation (2) gives the state space equation of the flyback PFC converter, and on the basis of the state space equation, discrete mapping is further utilized to constructThe modeling method gives an accurate model of the system, where N1And N2Respectively representing the number of winding turns of the primary side and the secondary side;
2) sampling state variables
The method comprises the steps of constructing a discrete mapping model of a peak current flyback PFC converter by adopting stroboscopic mapping, and when establishing the discrete mapping model of the flyback PFC converter controlled by the peak current, sampling a state variable and setting in＝iL(nT)，vn＝vC(nT)Respectively, the inductive current and the capacitor voltage are at nTSampled value of time, then in+1，vn+1Respectively, the sample values of the inductive current and the capacitor voltage at the moment (n +1) T, wherein when iL＝IrefSwitching occurs;
firstly, the conduction time of the switch tube is calculated, and if the loop adopts a peak current mode control mode, the reference current is
Iref＝K|sin(ωt)| (3)
Where K is the corresponding proportionality coefficient, ω is the switching tube angular frequency 50kHz and the line voltage frequency 50Hz, so that the reference current is equivalent to a fixed value during a switching cycle:
Iref＝K|sin(ωnT)| (4)
where n denotes the nth cycle in which the system is operating, and the line voltage in a switching cycle is also replaced by a fixed value, of
Vin＝VM|sinωt|＝VM|sinωnT| (5)
Wherein VMFor the input voltage amplitude, the conduction time of the switching tube is determined from equations (3), (4) and (5):
Figure GDA0002228367440000025
at the end time t of the conduction of the switching tubenThe method comprises the following steps:
Figure GDA0002228367440000031
wherein VnTo representThe output voltage at the end of the nth period adopts a mode of peak current mode control for the flyback PFC converter, namely the current value of the inductor at the moment when the conduction of the switching tube is finished is equal to the reference voltage value, so that the current of the inductor is equal to the reference voltage value
i(tn)＝Iref＝K|sin(ωt)| (8)
3) Considering the working state of the flyback PFC converter
According to the length of the opening time of the switching tube, dividing the working state of the flyback PFC converter into the following states, wherein T represents a switching period;
the first mode is as follows: t is tn≥T
The switch tube is always conducted in the whole switching period, and then
Figure GDA0002228367440000032
Figure GDA0002228367440000033
And a second mode: t is tn< T, solving the formula (5) to obtain the characteristic root of the equation:
Figure GDA0002228367440000034
Figure GDA0002228367440000035
wherein N is N2/N1；
And a third mode: t is tnT and the system enters DCM operation mode, i.e. the current in the inductor is already 0 at the end of the period, then
Figure GDA0002228367440000036
Wherein, tcRepresenting a pending coefficient associated with a particular control mode of the control circuit;
by the derivation, a voltage-current relation between an n period and an n +1 period is established, the state of the circuit at the end of any period can be calculated through multiple iterations, and a discrete mapping model of the system is preliminarily established;
4) jacobian matrix and bifurcation analysis
From in+1＝in＝iQ，vn+1＝vn＝vQFinding the stationary point iQ，vQIn practical application, it is desirable that the flyback PFC converter can stably operate in a cycle one state, at this time, the system only switches between a mode one and a mode two, but does not go through a mode three, so that only the jacobian matrix of the mapping equation (12) needs to be analyzed to obtain the jacobian matrix as the equation (13), which is the obtained nonlinear model:
Figure GDA0002228367440000041
the eigen equation of the jacobian matrix at the stationary point can be expressed as:
det(λI-J)＝0 (14)
wherein I represents a unit vector, and lambda is a characteristic root to be solved; calculating the system stationary point i by a recursion formulaQ,vQThus, a Jacobian matrix in the field of the stationary points is obtained, the stationary points are determined by a nonlinear equation system (13), and the unstable boundary of the system can be determined by analyzing the characteristic root track obtained by the formula (14);
5) and (4) realizing the nonlinear modeling process of the flyback PFC converter according to the Jacobian matrix obtained in the step (4), and analyzing the nonlinear state of the circuit through a formula (14).
The invention has the advantages and obvious effects that: according to the invention, a stroboscopic mapping method is adopted to model the peak current control flyback PFC converter, a group of piecewise smooth mapping equations is obtained, and based on the model, the fast-scale instability phenomenon is numerically simulated, and the result shows that in the fast-scale instability region, not only is a period-doubled bifurcation phenomenon, but also boundary collision bifurcation and chaos occur. The invention mainly researches the influence of the value of the circuit parameter on the nonlinearity of the system, and specifically analyzes the position of the bifurcation point and the corresponding circuit parameter value through the calculation of the Jacobian matrix characteristic root, thereby providing a theoretical basis for the design of the flyback PFC circuit.
Drawings
Fig. 1 is a circuit configuration of a typical flyback PFC converter;
FIG. 2 is a discrete iterative mapping model;
FIG. 3 shows the inductor current i after steady stateLmA waveform diagram of (a);
FIG. 4 shows the voltage phase 0 < theta1The waveform of the time inductor;
FIG. 5 shows the voltage phase θ1＜θ＜θ2The waveform of the time inductor;
FIG. 6 shows the voltage phase θ2The waveform of the inductor when theta is less than pi;
FIG. 7 is a graph of the on duty cycle of the switch during one cycle;
fig. 8 shows an inductor current waveform of 100 Ω;
fig. 9 shows an inductor current waveform of 150 Ω.
Detailed Description
Fig. 1 shows a typical structure of a flyback PFC converter, which is based on a control system including a rectifier bridge, a flyback topology, and a control IC connected in sequence. The power factor characterizes the utilization efficiency of the system to the power grid, wherein the power factor is defined as follows, and the power factor is defined as the ratio of active power P to apparent power S, namely:
Figure GDA0002228367440000051
wherein, URFor effective value of grid voltage, IRFor an effective value of the input current, UIFor the effective value of the fundamental wave of the input voltage, IIFor the effective value of the fundamental wave of the input current,
Figure GDA0002228367440000052
the displacement factor reflects the magnitude of the phase between the fundamental current and the voltage. PF is therefore reduced to the following formula:
Figure GDA0002228367440000053
equation (16) shows that the PF of the system reaches the maximum value when the input voltage and the input current are equal in phase under the condition of a certain distortion degree of the current waveform, so that a reference current I can be assumed for the flyback PFC converterref＝K sinωt (17)
Where K is a constant, and the reference current shown in equation (17) is used as the peak value of the input current, it can be ensured that the phase of the overall envelope of the input current is similar to the phase of the input current, thus ensuring a high PF.
FIG. 2 shows a process for establishing a discrete mapping model, in which a strobe mapping is used to construct the discrete mapping model of the peak current flyback PFC converter, as shown in FIG. 2, i is setn＝iL(nT)，vn＝vC(nT) are respectively the sampling values of the inductive current and the capacitance voltage at nT moment, in+1，vn+1Respectively, the sample values of the inductive current and the capacitor voltage at the moment (n +1) T, wherein when iL＝IrefSwitching occurs.
FIG. 3 shows the inductor current i after the system enters a steady stateLmThe waveform diagram shows that the inductive current passes through three different states in a power frequency period, and when the phase of the input voltage is 0 < theta1In time, the inductive current shows an unstable phenomenon (multiple period bifurcation and chaos); when the phase of the input voltage is theta1＜θ＜θ2When the current is in a stable first-period state, the inductive current enters a stable first-period state; when the phase of the input voltage is theta2When theta is less than pi, the inductive current enters an unstable state again (multiple period bifurcation and chaos). This phenomenon is related to the DC-DC converter, the inputThe period division phenomenon occurring when the voltage is too low is similar, but the input voltage changes periodically for the flyback PFC converter, so the non-linear phenomenon also shows a certain intermittence. As analyzed above, the flyback PFC converter may intermittently (input voltage is too low) generate instability during every 1/2 input voltage cycles.
To specifically analyze the non-linear behavior of the inductor current in each cycle, the waveforms of the inductor current in the three states are amplified as shown in fig. 4, 5, and 6, respectively, where fig. 4 shows the voltage phase 0 < θ1The waveform of the time-dependent inductance, theta is given in FIG. 51＜θ＜θ2The waveform of the time-dependent inductance is given by θ in FIG. 62The waveform where < theta < pi.
As shown in FIG. 7, given a distribution of the on duty of the switch over a period, it can be found that θ is analyzed as described above1And theta2The duty cycle likewise enters the multiple cycle bifurcation state. And at t-1, the duty cycle of the switch conduction reaches 1. The above analysis shows that in an unstable region, the circuit state is frequently switched among the mode 1, the mode 2 and the mode 3, and there are not only the accumulation of the multiple periods but also the occurrence of boundary collisions, so that it can be seen that the method has very important theoretical significance for the research on the nonlinear behavior of the flyback PFC converter.
Fig. 8 shows the waveform of the inductor current when the load resistance R is 100 Ω, and θ can be obtained from the graph1＝21.6°，θ2At 156.8 °, the stable phase range is θ2-θ1135.2. The resistance value R was changed to 100 by changing the load resistance, and the inductor current waveform at this time was as shown in fig. 8, from which θ was obtained1＝58，θ2At 149, the stable phase range at this time is: theta2-θ191. Further increasing the load resistance R, fig. 9 shows the waveform of the inductor current when R is 150 Ω, as can be seen from fig. 9, and the system exhibits non-linearity throughout the switching period. The above analysis shows that as the load resistance R increases, the stability region of the flyback PFC converter in one cycle decreases continuously.

Claims (1)

1. A nonlinear modeling method of a flyback PFC converter is characterized in that: establishing a state equation of a main topological structure of the flyback PFC converter, establishing a discrete mapping model in a continuous sampling mode, and calculating specific ranges of different nonlinear states by adopting a Jacobian matrix based on the discrete mapping model, wherein the method comprises the following steps:
1) establishing a master topological equation of state
The state equations of the power supply when the switch tube S1 is turned on and turned off are respectively listed, meanwhile, the related state equations are listed by considering that the switch tube S1 is always turned on in the whole switching period and the two states of the switch tube S1 are turned off after leading on in the period,
when the switching tube S1 is turned on, the state equation of the flyback PFC converter is:
Figure FDA0002261061280000011
Figure FDA0002261061280000012
wherein iLmIs primary side inductance current, LmIs a primary side inductance, VinIs an input voltage vCThe voltage is the capacitor voltage, namely the output voltage, R is the load resistor, and C is the load capacitor;
when the switching tube S1 is turned off, the state equation of the flyback PFC converter is:
Figure FDA0002261061280000013
and (3) a state space equation of the flyback PFC converter is given by formula (2), and on the basis of the state space equation, an accurate model of the system is further given by using a discrete mapping modeling method, wherein N is1And N2Respectively representing the number of winding turns of the primary side and the secondary side;
2) sampling state variables
A discrete mapping model of the peak current flyback PFC converter is constructed using strobe mapping,when a discrete mapping model of a flyback PFC converter controlled by peak current is established, a state variable needs to be sampled and i is setn＝iL(nT)，vn＝vC(nT) are respectively the sampling values of the inductive current and the capacitance voltage at nT moment, in+1，vn+1Respectively, the sampled values of the inductor current and the capacitor voltage at the time of (n +1) T, wherein T represents the switching period when iL＝IrefSwitching occurs;
firstly, the conduction time of the switch tube is calculated, and if the loop adopts a peak current mode control mode, the reference current is
Iref＝K|sin(ωt)| (3)
Where K is the corresponding proportionality coefficient, ω is the switching tube angular frequency 50kHz and the line voltage frequency 50Hz, so that the reference current is equivalent to a fixed value during a switching cycle:
Iref＝K|sin(ωnT)| (4)
wherein n represents the nth switching cycle during which the system is operating, and the line voltage during a switching cycle is also replaced by a fixed value
Vin＝VM|sinωt|＝VM|sinωnT| (5)
Wherein VMFor the input voltage amplitude, the conduction time of the switching tube is determined from equations (3), (4) and (5):
Figure FDA0002261061280000021
at the end time t of the conduction of the switching tubenThe capacitor voltage is:
Figure FDA0002261061280000022
the flyback PFC converter adopts a peak current mode control mode, namely the current value of the inductor at the moment when the conduction of the switching tube is finished is equal to the reference voltage value, so that the current i (t) of the inductorn)＝Iref＝K|sin(ωt)| (8)
3) Considering the working state of the flyback PFC converter
According to the length of the opening time of the switching tube, dividing the working state of the flyback PFC converter into the following states;
the first mode is as follows: t is tn≥T
The switch tube is always conducted in the whole switching period, and then
Figure FDA0002261061280000023
Figure FDA0002261061280000024
And a second mode: t is tn<And T, solving the formula (5) when the system does not enter a DCM working mode to obtain an equation characteristic root:
Figure FDA0002261061280000025
Figure FDA0002261061280000026
wherein N is N2/N1；
And a third mode: t is tn<T and the system enters DCM working mode, i.e. the current in the inductor is already 0 at the end of the period, then
Figure FDA0002261061280000027
Wherein, tcRepresenting a pending coefficient associated with a particular control mode of the control circuit;
by the derivation, a voltage-current relation between an n period and an n +1 period is established, the state of the circuit at the end of any period can be calculated through multiple iterations, and a discrete mapping model of the system is preliminarily established;
4) jacobian matrix and bifurcation analysis
From in+1＝in＝iQ，vn+1＝vn＝vQFinding the stationary point iQ，vQIn practical application, it is desirable that the flyback PFC converter can stably operate in a cycle one state, at this time, the system only switches between a mode one and a mode two, but does not go through a mode three, so that only the jacobian matrix of the mapping equation (12) needs to be analyzed to obtain the jacobian matrix as the equation (13), which is the obtained nonlinear model:
Figure FDA0002261061280000031
the eigen equation of the jacobian matrix at the stationary point can be expressed as:
det(λI-J)＝0 (14)
wherein I represents a unit vector, and lambda is a characteristic root to be solved; calculating the system stationary point i by a recursion formulaQ,vQThus, a Jacobian matrix in the field of the stationary points is obtained, the stationary points are determined by a nonlinear equation system (13), and the unstable boundary of the system can be determined by analyzing the characteristic root track obtained by the formula (14);
5) and (4) realizing the nonlinear modeling process of the flyback PFC converter according to the Jacobian matrix obtained in the step (4), and analyzing the nonlinear state of the circuit through a formula (14).
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