Guo-Dong Shi, Qiu-Juan Cao, Bo-Cheng Bao, and Zheng-Hua Ma

Dynamics and Operation-State Estimation of Current-Mode Controlled Flyback Converter

Guo-Dong Shi, Qiu-Juan Cao, Bo-Cheng Bao, and Zheng-Hua Ma

—By utilizing total magnetic fluxφof the primary and secondary windings of the flyback transformer as a state variable, the discrete-time model of current-mode controlled flyback converter is established, upon which the bifurcation behaviors of the converter are analyzed and two boundary classification equations of the orbit state shifting are obtained. The operation- state regions of the current-mode controlled flyback converter are well classified by two boundary classification equations. The theoretical analysis results are verified by power electronics simulator (PSIM). The estimation of operation-state regions for the flyback converter is useful for the design of circuit parameters, stability control of chaos, and chaos-based applications.

Index Terms—Bifurcation, chaos, discrete-time model, flyback converter.

1. Introduction

Bifurcation phenomena and border collision have been studied in buck, boost, and buck-boost converters[1]–[3]. In recent years, these phenomena have been also studied in peak-current-controlled superbuck converter[4], singleinductor dual-switching dc-dc converter[5], and boost PFC (power factor correction) converter[6]. However, few studies have been performed on isolated converter, such as flyback converter[7],[8]. In flyback converter, the secondary winding current of the transformer is zero when the switch is turned on, whereas the primary winding current of the transformer is zero when the switch is turned off. Thus the inductance currents of the primary and secondary windings of the transformer are discontinuous and can not be regarded as state variables to analyze the bifurcation behaviors. For this reason, total ampere-turns of the primary and secondary windings of the transformer have been presented as one of the variables to analyze the bifurcation behaviors on voltage-mode controlled flyback converter[8]. In this paper, we consider the total magnetic flux of the primary and secondary windings as a state variable to describe the dynamics of current-mode controlled flyback converter.

In current-mode controlled switching dc-dc converters, the operation-states may shift from continuous conduction mode (CCM) to discontinuous conduction mode (DCM), and three operation-state regions classified by stable period region, robust chaos region in CCM, and intermittent chaos region in DCM exist in the current-mode controlled switching dc-dc converters with the variations of circuit parameters, such as input voltage, output voltage, or reference current[4],[9]–[11]. Some methods have been developed to locate the boundary of the period-one zone of switching dc-dc converters[12]–[14], and an approach to locate two boundaries of three operation-state regions of current-mode controlled switching dc-dc converters has been proposed recently[11]. In this paper, the dynamics of a current-mode controlled flyback converter is analyzed through a discrete-time map model covering both CCM and DCM, and two boundary classification equations dividing three smooth operation-state regions over the parameter space of the converter are derived.

2. Modeling of Current-Mode Controlled Flyback Converter

2.1 Flyback Converter with Current-Mode Control

The schematic of current-mode controlled flyback converter is shown in Fig. 1, in which the main circuit topology is a second-order circuit consisting of an inductor L, a capacitor C, a switch S, a diode D, a load resistor R and a transformer.

Fig. 1. Current-mode controlled flyback converter.

The parameters of the transformer in Fig. 1 are the primary winding leakage inductanceL1, the secondary winding leakage inductanceL2, and the turn ratio of the primary and secondary windingsN1:N2. A timer generates a free-running clock which controls the operation of current-mode control loop. The converter is controlled by a feedback loop consisting of a comparator and a RS trigger.

Fig. 2 shows the waveforms of the currents and the corresponding total magnetic flux of the primary and secondary windings of the flyback transformer. The switch S is turned on at the beginning of each switching cycle, and during the time interval when the switch S is turned on, the primary winding currenti1rises linearly and the secondary winding currenti2equals to zero. The switch S is turned off wheni1increases to a reference currentIref, and during the time interval when the switch S is turned off,i1equals to zero andi2decreases. In CCM,i2is always non-zero, while in DCM,i2decreases to zero during the time interval when the switch S is turned off and remains at zero until the end of switching cycle. Because the switching frequency of flyback converter is usually much higher than the natural frequency of the converter, the dynamics of the outer voltage loop is much slower and can be ignored, and the output of the flyback converter can be represented by a constant voltage sourceVo[9],[11]. Under this assumption, the current-mode controlled flyback converter becomes onedimensional and the inductor current waveform becomes piecewise linear.

2.2 State Equations

Since two currentsi1andi2are discontinuous and can not be regarded as state variables[8], we consider the total magnetic fluxφof the primary and secondary windings as a state variable to describe the dynamics of the flyback converter. The expression ofφis

whereΨ1andΨ2are the magnetic flux linkages of the primary and secondary windings of the transformer, respectively.

The converter can be regarded as a system with a variable structure that toggles its topologies according to the states of the switches as shown in Fig. 2. Typically, when the converter operates in DCM, three switch states can be identified as follow:

State 1: switch S on and diode D off.

State 2: switch S off and diode D on.

State 3: switch S off and diode D off.

Fig. 2. Waveforms of the inductance currents and the magnetic flux.

When the converter operates in CCM, only two switch states are identified by State 1 and State 2, i.e., the state 3 does not appear in CCM. Thus, the system operating in both CCM and DCM can be described by the following state equations.

State 1: When S is on, D is off,i2= 0,φ=L1i1/N1anddφ/dt= (L1/N1)di1/dt. The state equation is

State 2: When S is off, D is on,i1= 0, soφ=L2i2/N2anddφ/dt=(L2/N2)di2/dt. The relationship between the primary inductorL1and the secondary inductorL2is. Thus the state equation is

State 3: When the converter operates in DCM, both S and D are off,i1=i2= 0. The state equation is

2.3 Two Borders

For current-mode controlled flyback converter operating in DCM, there are two borders in the discrete state-space. The borderφb1is defined as the total magnetic flux of the primary and secondary windings at the beginning of switching cycle when the currenti1reachesIrefjust at the end of the switching cycle. The borderφb2is defined as the total magnetic flux of the primary and secondary windings at the beginning of switching cycle which decreases to zero just at the end of the switching cycle. Fig. 3 shows the total magnetic flux waveforms of current-mode controlled flyback converter at these borders. Fig. 3 (a) shows the evolution of total magnetic flux ifφn=φb1, where the currenti1reachesIrefat the end of the switching cycle, and the switch S remains on throughout this switching cycle. Fig. 3 (b) shows the evolution of total magnetic flux ifφn=φb2, where the currenti2decreases to zero at the end of the switching cycle.

Based on the definitions of these two borders, we can easily obtain two borders of the total magnetic flux,φb1andφb2, as below:

Fig. 3. Total magnetic flux waveforms of current-mode controlled flyback converter with two borders: (a)φn=φb1,φn+1=L1Iref/N1and (b) φn=φb2, φn+1= 0.

2.4 Discrete-Time Model

With these two borders given by (5) and (6), there are three types of orbits between consecutive clock instants.

1)φn≤φb1. The switch remains on throughout the switching cycle, and the map is easily derived from (2) and given by

2)φb1＜φn＜φb2. The magnetic flux of the primary winding increases toL1Iref/N1and then the magnetic flux of the secondary winding decreases until the end of the switching cycle, and the map is derived from (3) and given by

3)φn≥φb2. The inductance currenti2of the secondary winding decreases to zero during thenth switching cycle, i.e., the converter enters DCM. Thus, at the end of thenth switching cycle, we have

Thus, the discrete-time model of current-mode controlled flyback converter can then be written in the form:

Based on three piecewise linear equations containing two borders, the discrete map model of current-mode controlled flyback converter is obtained and the corresponding dynamical analysis can be performed.

3. Dynamics and Operation-State Estimation

3.1 Dynamics of Current-Mode Controlled Flyback Converter

By utilizing the discrete-time model (10), the bifurcation to chaotic behavior of current-mode controlled flyback converter with circuit parameters variation can be effectively exhibited. The numerical simulations are performed by using MATLAB software platform with following parameters:Iref= 1.2 A,L1= 1 mH,T= 100 μs, andN1:N2= 3:2. If we fixE= 7 V and takeVoas bifurcation parameter, we can obtain the magnetic flux bifurcation diagram with the increasing ofVoas shown in Fig. 4 (a). If we fixVo= 9 V and takeEas bifurcation parameter, then we can obtain the magnetic flux bifurcation diagram with the increasing ofEas shown in Fig. 4 (b). In Fig. 4, two bordersφb1andφb2are plotted by using the dashed line and dash-dot line, respectively.

From Fig. 4, it is found that the current-mode controlled flyback converter has complex dynamical behaviors. First consideringVoas variable, whenVoincreases gradually, a period-doubling bifurcation occurs atVo= 4.67 V. After the occurrence of period-doubling bifurcation, the unstable periodic orbit with period-two collides with the borderφb1, resulting in a border collision bifurcation at the same parameter value, and the operation-state of the converter goes into the robust chaos with CCM from the stable period. WhenVoincreases further, the chaotic orbit collides with the borderφb2atVo= 8 V, resulting in another border collision bifurcation, and the operation-state of the converter shifts into the intermittent chaos with DCM from the robust chaos with CCM. Then consideringEas variable, whenEdecreases gradually, a reverse period-doubling bifurcation occurs atE= 13.5 V. After the occurrence of period-doubling bifurcation, the converter operates in DCM, i.e., the currenti2remains at zero in the durations of some switching cycles, the unstable periodic orbit collides with the borderφb2, resulting in a border collision bifurcation at the same parameter, and the operation-state of the converter directly enters into sub-harmonics with DCM. WhenEdecreases further, the unstable periodic orbit collides with the borderφb1atE=L1Iref/T= 12 V, resulting in the nonzero periodic orbit to be folded. The folded non-zero periodic orbit collides with the borderφb2atE= 10.8 V, and a period-four orbit emerges from the period-two orbit. The operation-state of the converter jumps into the intermittent chaos with DCM from the period-sixteen orbit atE=8.17 V.

Fig. 4. Dynamics of current-mode controlled flyback converter: (a) Bifurcation diagram withVoincreasing and (b) Bifurcation diagram withEincreasing.

The above analysis results show that there exist three operation-state regions, i.e., stable period region, robust chaos region with CCM, and intermittent chaos region with DCM, in current-mode controlled flyback converter. Especially, the super-stable periodic orbits exist in DCM due to the occurrence of zero eigenvalue[9]. The converter shows weak chaos and strong intermittency, which means that the chaotic behavior becomes weak in DCM[10].

3.2 Boundary Classification Equations for Operation-State

From (10), the eigenvalueλof the characteristic equation for current-mode controlled flyback converter is given byλ= –N1Vo/N2E. To ensure stable operation,λmust fall between –1 and 1[15]. The first period-doubling occurs whenλ= –1. Hence, by puttingλ= –1, the first boundary classification equationσ1for the operation mode shifting from the stable period-one to subharmonics and chaos will be

which implies that if the circuit parameters satisfyσ1＞ 0, i.e.Vo＜N2E/N1, the converter operates with periodic oscillation, otherwise with subharmonics or chaotic oscillation.

It is clear that the maximum total magnetic flux at the end of thenth switching cycle isφn+1,max=L1Iref/N1. When the total magnetic flux of the flyback converter reaches the borderφb2, the border collision bifurcation and operation-state shift occur. Under this condition, there existsφb2=φn+1,max, i.e. Therefore, the second boundary classification equationσ2for the operation-state region shifting from CCM to DCM will be

Fig. 5. Estimation of operation-state regions for current-mode controlled flyback converter: (a) parameter space map ofVoandEand (b) corresponding division of operation-state regions.

If the circuit parameters satisfyσ2＜ 0, i.e.,Vo＞N2L1Iref/N1T, the converter will operate in DCM, otherwise in CCM.

It is remarkable thatσ1only depends on the input voltageE, the output voltageVo, and the turns ratio of the primary and secondary windings of the transformer, whileσ2depends on all of circuit parameters of current-mode controlled flyback converter except for the input voltageE.

3.3 Estimation of Operation-State Regions

Considering the circuit parameters with rangesE= 2～18 V andVo= 2 ～ 12 V, and letingIref= 1.2 A,L1= 1 mH,T= 100 μs, andN1:N2= 3:2, we can obtain the parameter space map as shown in Fig. 5 (a). The higher periodicities are depicted with deeper gray levels, the darker shade areas imply chaos, while the white and shallower gray areas mean low period.

The three smooth operation-state regions of currentmode controlled flyback converter over the parameter space can be divided by above boundary classification equationsσ1andσ2. Fig. 5 (b) shows the regions of the orbit operation-states corresponding to Fig. 5 (a). From (11) and (12), the two boundaries areσ1:Vo=N2E/N1andσ2:Vo=N2L1Iref/N1Trespectively. The boundaryσ1is called as the first period-doubling bifurcation borderline and the boundaryσ2is called as the operation mode shifting borderline. The regions of stable period-one, robust chaos in CCM, and intermittent chaos in DCM are shown in Fig. 5 (b). From the parameter space map, the different operation-state regions of the converter can be demonstrated clearly.

It is visible that the 1-D bifurcation diagrams ofφnversusVoandEin Fig. 4 (a) and Fig. 4 (b) can be obtained along the paths from point A to point B and from point C to point D in Fig. 5 (a), respectively. It should also be noted that the boundary classification equationσ2will lose its physical significance when current-mode controlled flyback converter is located in stable operation-state region.

3.4 PSIM Simulation Results

In order to verify theoretical analysis, the PSIM (power electronics simulator) simulations of the current-mode controlled flyback converter are performed with the parameters as mentioned above, by first fixingE= 7 V and thenVo= 9 V. With the increasing ofVoorE, the waveforms of the primary winding currenti1and the secondary winding currenti2of the flyback transformer are obtained in Fig. 6 and Fig. 7. Fig. 6 depicts that with the variation of output voltageVo: 4 V, 4.68 V, 7 V and 10 V, the period-1 with CCM, period-2 with CCM, robust chaos with CCM, and intermittent chaos with DCM occur respectively. Fig. 7 describes that with the variation of inputvoltageE: 5 V, 9.5 V, 12 V, and 14 V, the intermittent chaos with DCM, period-4 with DCM, period-2 with DCM, and period-1 with CCM occur respectively. These simulation results are consistent with those results shown in Fig. 4.

Considering the other four sets of circuit parameters locating in the different operation-state regions divided by two boundary classification equationsσ1andσ2in Fig. 5 (b), we can further obtain the simulation results as shown in Fig. 8. Fig. 8 (a) shows that when the circuit parameters are located in CCM robust chaos region in Fig. 5 (b), the converter is in chaotic state with CCM. Fig. 8 (b) depicts that when the circuit parameters are just located at the boundaryσ2in Fig. 5 (b), the secondary winding currenti2decreases to zero at the end of some switching cycle, and the operation-state shifts between CCM and DCM. The circuit parameters for Fig. 8 (c) are located in DCM intermittent chaos region in Fig. 5 (b), thus the converter is in chaotic state with intermittency. While the circuit parameters locating in DCM intermittent chaos region in Fig. 5 (b) are selected, the converter operates at period-2 with DCM, as shown in Fig. 8 (d).

Fig. 6. Simulation waveforms whileVois increased along the path from point A to point B in Fig. 5 (a): (a) CCM, period-1 forVo= 4 V, (b) CCM, period-2 forVo= 4.68 V, (c) CCM, robust chaos forVo= 7 V, and (d) DCM, intermittent chaos forVo= 10 V.

Fig. 7. Simulation waveforms whileEis increased along the path from point C to point D in Fig. 5 (a): (a) DCM, intermittent chaos forE= 5 V, (b) DCM, period-4 forE= 9.5 V, (c) DCM, period-2 forE= 12 V, and (d) CCM, period-1 forE= 14 V.

Fig. 8. Simulation waveforms whileEandVoare arbitrarily selected in different operation-state regions of Fig. 5(b): (a) CCM, robust chaos forE= 3 V andVo= 6 V, (b) shifting between CCM and DCM, critical robust chaos forE= 6 V andVo= 8 V, (c) DCM, intermittent chaos forE= 5 V andVo= 10 V, and (d) DCM, period-2 forE= 12 V andVo= 11 V.

4. Conclusions

With the variations of circuit parameters such as input voltage and output voltage, the current-mode controlled flyback converter exhibits two borders and its operationstates can shift between stable period-1, robust chaos in CCM, and intermittent chaos in DCM via period-doubling bifurcation and border collision bifurcation. Two boundary classification equations of operation-state regions can characterize the constitutive relations that the current-mode controlled flyback converter shifts among different operation-states. Utilizing total magnetic fluxφof the primary and secondary windings as a state variable, we establish the discrete-time model with two borders, analyze the bifurcation behaviors, and obtain two boundary classification equations. The estimated operation-states by two boundary classification equations are verified by PSIM simulation results of the current-mode controlled flyback converter.

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Guo-Dong Shiwas born in Jiangsu, China in 1956. He is currently a professor with the School of Information Science and Engineering, Changzhou University, Changzhou, China. His research interests include electrical automation and applications, and artificial intelligence.

Qiu-Juan Caowas born in Jiangsu, China in 1988. She is currently pursuing the M.S. degree with School of Information Science and Engineering, Changzhou University. Her research interests include the control technology of switching power converter.

Zheng-Hua Mawas born in Jiangsu, China in 1962. He is currently a professor with the School of Information Science and Engineering, Changzhou University. His research interests include power electronic technology and embedded development with applications.

Bao

the B.S. and M.S. degrees in electronic engineering from the University of Electronics Science and Technology of China, Chengdu, China in 1986 and 1989, respectively, and the Ph.D. degree from the Department of Electronic Engineering, Nanjing University of Science and Technology, Nanjing, China in 2010. Dr. Bao is currently a professor with the School of Information Science and Engineering, Changzhou University. His research interests include bifurcation and chaos, analysis and simulation in power electronic circuits, and nonlinear circuits and systems.

Manuscript received December 6, 2012; revised February 10, 2013. This work was supported by the National Natural Science Foundation of China under Grant No. 51277017 and the Natural Science Foundation of Changzhou, Jiangsu Province, China under Grant No. CJ20120004.

B.-C. Bao is with the School of Information Science and Engineering, Changzhou University, Changzhou 213164, China (Corresponding author e-mail: mervinbao@126.com).

G.-D. Shi, Q.-J. Cao, and Z.-H. Ma are with the School of Information Science and Engineering, Changzhou University, Changzhou 213164, China (e-mail: sgd@cczu.edu.cn; jsjma@126.com; qiujuan@126.com).

Digital Object Identifier: 10.3969/j.issn.1674-862X.2013.03.013