Period-doubling bifurcation in two-stage power factor correction converters using the method of incremental harmonic balance and Floquet theory

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1 Perioddoubling bifurcation in twostage power factor correction converters using the method of incremental harmonic balance and Floquet theory Wang FaQiang( ), Zhang Hao( ), and Ma XiKui( ) State Key Laboratory of Electrical Insulation and Power Equipment, School of Electrical Engineering, Xi an Jiaotong University, Xi an 70049, China (Received 8 July 20; revised manuscript received 5 September 20) In this paper, perioddoubling bifurcation in a twostage power factor correction converter is analyzed by using the method of incremental harmonic balance (IHB) and Floquet theory. A twostage power factor correction converter typically employs a cascade configuration of a preregulator boost power factor correction converter with average current mode control to achieve a near unity power factor and a tightly regulated postregulator DC DC Buck converter with voltage feedback control to regulate the output voltage. Based on the assumption that the tightly regulated postregulator DC DC Buck converter is represented as a constant power sink and some other assumptions, the simplified model of the twostage power factor correction converter is derived and its approximate periodic solution is calculated by the method of IHB. And then, the stability of the system is investigated by using Floquet theory and the stable boundaries are presented on the selected parameter spaces. Finally, some experimental results are given to confirm the effectiveness of the theoretical analysis. Keywords: twostage power factor correction converter, incremental harmonic balance, Floquet theory, perioddoubling bifurcation PACS: a, Jc, 45.0.Hj DOI: 0.088/674056/2/2/ Introduction In engineering applications, the power factor correction (PFC) converter is one of the basic requirements of power systems since it can force the input current to follow the input voltage to obtain a near unity power factor. Therefore, many topologies of PFC converter and various control methods have been proposed in Refs. [] [7]. In general, by considering tight output regulation, there are two types of typical active PFC converter, [5 7] i.e., singlestage PFC converter, and twostage PFC converter. For the singlestage PFC converter, the preregulator PFC converter and the postregulator DC DC converter share the same control circuit to reduce the cost and complexity. However, the low efficiency and the variation of the bulk capacitor voltage lead to its limitation in practice. For the twostage PFC converter, the preregulator PFC converter and the postregulator DC DC converter has its own control circuit, and accordingly it can achieve both power factor correction and output regulation, and this leads to many advantages, such as low total harmonic distortion (THD), high power factor, and constant bulk capacitor voltage. Thus, the modeling, dynamical behaviours, and control methods of the twostage PFC converter have received a great deal of attention. Especially, since the complex behaviours in PFC converter will have an adverse effect on its performance, [8 7] the complex behaviours in the twostage PFC converter have also been more and more concerned in recent years. [8 2] For example, Orabi et al. pointed out that the perioddoubling bifurcation, which occurs at the line frequency, could be observed in the twostage PFC converter through computer aided simulations and circuit experiments. [8] Chu et al. proposed the doubleaveraging method to analyse this perioddoubling bifurcation theoretically and gave the critical condition of the stable operation for practical design. [20] However, if the accuracy of the results is highly improved Project supported by the National Natural Science Foundation of China (Grant No ), the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No ), the Fundamental Research Funds for the Central Universities of China, and the State Key Laboratory of Electrical Insulation and Power Equipment of China (Grant No. EIPE0303). Corresponding author. faqwang@mail.xjtu.edu.cn c 202 Chinese Physical Society and IOP Publishing Ltd

2 or the original system is very complex, the doubleaveraging method may be too difficult to identify the type of instability. Most importantly, the twostage PFC converter is a periodic structural vibration and thus the solution of the system is also periodic. Accordingly, if we use a more general method to directly analyse the stability of the periodic orbit of the system, the results will be more straightforward and easier to understand. Note that only the perioddoubling bifurcation at the line frequency, which is generally called slowscale instability, [8 20] is concerned with in this paper. The incremental harmonic balance (IHB), which was originally developed by Lau et al. [22] for treating periodic structural vibrations and widely applied to tackle different kinds of dynamic problems, [23 27] is a semianalytical method and is remarkably effective in computer implementation for obtaining the periodic solutions. For example, Ge et al. utilised the method of IHB and Floquet theory to investigate the bifurcation and chaotic responses in a singleaxis rate gyro mounted on a space vehicle. [23] Raghothama et al. used the IHB to investigate the periodic motions of nonlinearly geared rotorbearing system. [24] The method was also applied by Shen et al. to investigate the bifurcation and the routetochaos of Mathieu Duffing oscillator. [27] In this paper, we apply the method of IHB and Floquet theory to investigate the perioddoubling bifurcation in the twostage PFC converter, i.e., the approximate periodic solution of the simplified model is calculated by the IHB, and the stability of the system is identified by using the Floquet theory. The rest of the paper is organized as follows. In the next section, the simplified model of the twostage PFC converter under some assumptions is derived. In Section 3, the approximate periodic solution of the simplified model is calculated by the IHB and the stability of the system is identified by Floquet theory. In Section 4, the perioddoubling bifurcation in the circuit system is analysed and some typical figures are given. In Section 5, some experimental data are given to confirm the effectiveness of the method of IHB and Floquet theory. Finally, some concluding remarks and comments are given in Section Twostage PFC converter and its simplified model A twostage PFC converter typically consists of a preregulator Boost PFC converter with average current mode (ACM) control and a postregulator DC DC Buck converter with voltage feedback control, as depicted in Fig.. Obviously, this system is a rather complex nonlinear one for theoretical analysis. Fortunately, if the DC DC Buck converter is well controlled, the tightly regulated DC DC Buck converter under a fixed load condition can be replaced by a constant power sink, [8,20] and its detail is shown in Fig. 2. Note that the ACM control is usually realized by IC UC3854. [28] PFC Boost converter average current control DCDC Buck converter voltage feedback control load tightly regulated DCDC Buck converter Fig.. Schematic of the twostage PFC converter. The constant power sink can be described as follows. i PFC = P 0, () ηv PFC where v PFC is the output voltage of the preregulator boost PFC stage, P 0 is the constant power, i PFC is the current through the constant power sink, and η is the efficiency of the DC DC Buck converter. One can see that the constant power sink has a negative impedance characteristic since the current i PFC decreases when the voltage v PFC increases, and vice versa. This is different from a preregulator PFC converter with a resistive load, so that we cannot simply extend the stability conditions of a preregulator PFC converter with a resistive load to a practical twostage PFC converter. [9] In other words, the stability of the twostage PFC converter should be reconsidered completely. The system in Fig. 2. is also a high order piecewise one with low frequency excitation signal even if the postregulator DC DC Buck converter is replaced by a constant power sink. Thus, without losing the main dynamical behaviours, the following two assumptions should be made for simplicity. (i) All the components in the system are regarded as ideal and the voltage across the sensing resistor R s is ignored. Also, the efficiency of the DC DC Buck converter is assumed to be 00%, i.e., η. Additionally, the output of the secondorder filter v ff is regarded as a constant value since its ripple is very

3 i in D L v in Q C v PEC P 0 v PFC R s i ref v p R i v z C p V ramp R ffl R mo C z R z v con C ffl C ff2 v ffl R ff2 v ff R ff3 R ac IM IM=AB/C 2 B C A v vf R vf C vf V ref R vi R vd Fig. 2. Boost PFC converter terminated by a constant power sink. small. Its value is [28] V ff = 0.9R ff3 R ff R ff2 R ff3 V in, (2) where V in is the rootmeansquare (rms) value of the input voltage. (ii) The system only operates in continuous conduction mode (CCM). Also, if only the perioddoubling bifurcation occurring at the line frequency is considered and the current compensator is well designed, the current i in can be considered as following the current programming signal i ref linearly. [20] Under the above two assumptions, the simplified model can be obtained as dv vf = v ( vf R vi R ) vd V ref dt C vf R vf C vf R vf R vi R vd C vf v PFC, R vi C vf dv PFC dt = P 0 C v PFC R movin 2( cos(2ω lt)) R s Vff 2R (v vf ), acc v PFC (3) where ω l is the angular frequency, v vf is the output voltage of the voltage compensator of the ACM control, and V ref is the desired reference voltage of the preregulator boost PFC stage. Obviously, the obtained simplified model is a nonlinear nonautonomous system. 3. Theoretical derivations First, the periodic solution of the simplified model of the twostage PFC converter is derived by using the IHB method. And then, the stability of the circuit system is identified by using Floquet theory. 3.. Formulation for IHB Let ω 0 = 2ω l and τ = ω 0 t. the simplified model (3) of the twostage PFC converter can be rewritten as follows: v vf = α ω 0 v vf F ω 0 β ω 0 v PFC, v PFC = F 2 ω 0 v PFC γ ω 0 v PFC (v vf )( cos(τ)), (4) where v vf = dv vf /dτ, v PFC = dv PFC/dτ, α = /C vf R vf, β = /R vi C vf, γ = R mo Vin 2/C R s Vff 2R ac, ( F = R vi R ) vd V ref, C vf R vf R vi R vd C vf and F 2 = P 0 /C. From the first equation of Eq. (4), the solution for v PFC can be derived as v PFC = ω 0 β v vf α β v vf F β. (5) Substituting Eq. (5) into the second equation of Eq. (4) yields

4 ω 3 0v vf v vf αω 2 0(v vf ) 2 αω 2 0v vf v vf α 2 ω 0 v vf v vf F ω 2 0v vf F αω 0 v vf = β 2 F 2 β 2 γ(v vf )( cos(τ)), (6) where v vf = d 2 v vf /dτ 2. Let v vf = V vf v vf, (7) where V vf is the guessed solution and v vf is the incremental part. By substituting Eq. (7) into Eq. (6) and ignoring the higherorder terms of v vf since their values are very small, the following incremental equation is derived where K v vf K 2 v vf K 3 v vf = M M 2, (8) K = (ω 3 0V vf αω 2 0V vf F ω 2 0), (9) K 2 = ω 3 0V vf αω 2 0V vf αω 2 0V vf α 2 ω 0 V vf F αω 0, (0) K 3 = αω 2 0V vf α 2 ω 0 V vf γβ 2 ( cos(τ)), () M = ω0v 3 vf V vf αω0v 2 vf V vf F ω0v 2 vf αω0v 2 vf V vf α 2 ω 0 V vf V vf F αω 0 V vf, (2) M 2 = β 2 F 2 β 2 γ( cos(τ))(v vf ). (3) Let the voltage V vf and v vf be expressed in the following harmonic series of τ V vf = a 0 N [a n cos(nτ) b n sin(nτ)], (4) n= v vf = a 0 N [ a n cos(nτ) b n sin(nτ)], (5) n= where N is a natural number and is to be determined in the required accuracy, a n and b n are the increments of a n and b n, respectively. Define Thus E = [ cos(0), cos(τ),..., cos(nτ), sin(τ),..., sin(nτ) ], (6) A = [a 0, a,..., a N, b,..., b N ], (7) A = [ a 0, a,..., a N, b,..., b N ]. (8) E = [0, sin(τ),..., N sin(nτ), cos(τ),..., N cos(nτ)], (9) E = [0, cos(τ),..., N 2 cos(nτ), sin(τ),..., N 2 sin(nτ)]. (20) Hence V vf = EA T, V vf = E A T, V vf = E A T, (2) v vf = E A T, v vf = E A T, v vf = E A T. (22) Substituting Eqs. (2) and (22) into Eqs. (8) (3) respectively, the following equations can be obtained K E A T K 2 E A T K 3 E A T = M M 2, (23) K = (ω 3 0E A T αω 2 0EA T F ω 2 0), (24) K 2 = ω 3 0E A T 2αω 2 0E A T α 2 ω 0 EA T F αω 0, (25) K 3 = αω 2 0E A T α 2 ω 0 E A T γβ 2 ( cos(τ)), (26) M = ω 3 0E A T E A T αω 2 0E A T EA T F ω 2 0E A T αω 2 0E A T E A T α 2 ω 0 E A T EA T F αω 0 E A T, (27) M 2 = β 2 F 2 β 2 γ( cos(τ))(ea T ). (28) With the implementation of the Galerkin procedure, [27] the differential equation of the system becomes 2π 0 2π = 0 E T (K E K 2 E K 3 E) A T dτ E T (M M 2 )dτ. (29) Since the term A T has no relation to τ, the above expression can be changed into the following form K A T = M, (30) where K = M = 2π 0 2π 0 E T (K E K 2 E K 3 E)dτ, (3) E T (M M 2 )dτ. (32) Also, since the matrix K is full rank, the values for A T can be derived by multiplying both sides of the Eq. (30) by K, i.e., A T = K M. (33) First of all, an initial guess of the coefficients in A T is centred upon Eq. (33) and the solution to A T is found. The solution to A T are added to A T such that A T h = A T h A T h. (34) The process is ended when the value of M is sufficiently small. In this way, the solution to A T can be calculated

5 3.2. Stability of the periodic solution After obtaining the solution of A T, the steady state of the simplified model is also obtained, which is in periodic form, and its stability can be investigated by introducing a small perturbation v PFC to V PFC and v vf to V vf, i.e., v PFC = V PFC v PFC, (35) v vf = V vf v vf, (36) where V vf has been given in Eq. (4) and V PFC is given as follows: V PFC = N [( nan ω 0 αb n β β n= ( ω0 nb n αa n β β ) sin(nτ) ) cos(nτ) β F α β a 0. (37) Thus, the following equations can be obtained by linearization of Eq. (4) at (V vf, V PFC ). V = Y (τ)v, V = ( v vf, v PFC ) T R 2, (38) where α β Y (τ) = ω 0 ω 0 γ( cos(τ)) F 2 γ(v vf )( cos(τ)). ω 0 v PFC ω 0 vpfc 2 (39) Obviously, Y (τ) is a periodic matrix with the same period as (V vf, V PFC ), i.e., Y (τ T ) = Y (τ). If Φ(τ) = [Φ (τ), Φ 2 (τ)] is the fundamental solution to Eq. (38), which satisfies the initial condition Φ(0) = I, Φ(τ T ) is also the fundamental solution. Therefore, the relation between Φ(τ T ) and Φ(τ) can be expressed as [27] Φ(τ T ) = T M Φ(τ), (40) ] where T M is the transition matrix. By taking τ = 0 and using Φ(0) = I, the transition matrix T M is written as T M = Φ(T ). (4) From the calculated eigenvalues of the transition matrix T M, the stability of the system and the type of bifurcation can be determined. If all the eigenvalues are within the unit circle with its centre at the origin of the complex plane, the periodic solution is stable. Otherwise, the periodic solution is unstable. That is, a bifurcation occurs. [29] For example, the perioddoubling bifurcation will occur if there is a real eigenvalue moving out of the unit circle in the negative direction (cross at ) with remaining eigenvalues staying inside the unit circle. The Hopf bifurcation will occur if there is a pair of conjugate eigenvalues moving out of the unit circle with remaining eigenvalues staying inside it. First, the period T (= 2π) is divided into N k subintervals, and the kth interval is k = t k t k, where t k = T k/n k. And then, the value of the timevarying Y (τ) can be approximated to the following constant matrix in the kth interval when N k is chosen to be sufficiently large since Y (τ) is continuous with respect to τ. So, let Y k = k tk the transition matrix T M is t k Y (τ)dτ; (42) T M = [exp(y i i )] i=n k N j (Y i i ) = I j, (43) j! i=n k j= where N j denotes the number of terms in the approximation of the constant matrix Y k exponential and Y k = k tk t k α β ω 0 ω [ ] 0 γ( cos(τ)) tk dτ ω 0 v PFC k t k F 2 γ(v vf )( cos(τ)) ω 0 vpfc 2 dτ. (44) From the above derivations, one can see that the stability of the nonlinear nonautonomous system is directly identified by detecting the stability of the obtained approximate periodic solution. 4. Bifurcation analysis It is necessary to analyse the perioddoubling bifurcation in detail so as to understand the nonlinear phenomenon and to determine the boundaries about the stable operation. Note that, since the objective of the preregulator PFC converter is to achieve a near

6 unity power factor, the higherorder harmonic contents can be ignored, and here we take N = 3. Additionally, we take V in = 75 V, f l = 60 Hz, T s = 0.9 µs, L = 3 mh, R s = 0.5 Ω, R mo = 3.9 kω, R i = 3.9 kω, R ac = 620 kω, R vi = 50 kω, C vf = 44 nf, R z = 30 kω, C p = 220 pf, C z = nf, R ff = 90 kω, R ff2 = 9 kω, R ff3 = 22 kω, C = 07 µf, R vd = 33 kω, and P 0 = 20 W. First, the perioddoubling bifurcation is generally investigated as the variation of the resistance R vf. Let R vf = 50 kω, the approximate periodic solution of the simplified model is calculated by using the IHB, the results are shown as follows: V vf = cos(τ) cos(2τ) cos(3τ) sin(τ) sin(2τ) sin(3τ), (45) V PFC = cos(τ) cos(2τ) cos(3τ) sin(τ) sin(2τ) sin(3τ). (46) Meanwhile, by comparing the results from the IHB on the simplified model with the results from the timeintegration on the complete model, as plotted in Fig. 3, it is found that it is reasonable and effective to use the simplified model to replace the system terminated by a constant power sink, when the current compensator is well designed, and the IHB is used effectively to calculate the approximate periodic solution of the simplified model. Also, it is reasonable to choose N = 3 to describe the approximate periodic solution and to assume that the system only operates in CCM operation. The eigenvalues of the transition matrix T M can be obtained: λ = , λ 2 = Obviously, the two eigenvalues are staying inside the unit circle, and this leads to stable operation of the system. But, when the value of the resistance R vf is varied, the eigenvalues of the transition matrix T M may change, as shown in Table and Fig. 4(a). From Table and Fig. 4(a), it is clear that there is a real eigenvalue moving out of the unit circle in the negative direction (cross by ) with the remaining eigenvalue staying inside the unit circle for increasing Table. The eigenvalues for different R vf. R vf /kω λ λ 2 State stable stable stable stable stable stable perioddoubling bifurcation unstable.8.7 (a) time integration IHB (b) time integration IHB.6 V vf V.5 v PFC /V t/s t/s Fig. 3. The solutions of the system by using timeintegration for the complete model (solid line) and the IHB for the simplified model (dashed line): (a) the voltage v vf and (b) the voltage v PFC

7 resistance R vf. This denotes that the perioddoubling bifurcation occurs and the bifurcation point is R vf = kω. Therefore, the system will operate in stable operation at R vf < kω. Otherwise, it will be in unstable operation. Also, the above conclusion can be obtained by the bifurcation diagram, as shown in Fig. 4(b). Furthermore, we record the critical value, at which perioddoubling bifurcation occurs, when the other parameters varies at R vf = 200 kω/250 kω, as shown in Figs. 5(a), 5(b), and 5(c). One can observe that the stable region increases as R vf decreases, while the perioddoubling bifurcation develops as C decreases (as R vd increases, R vi decreases, the R ff3 decreases, the output power P 0 increases, and the C vf decreases). Additionally, for the same circuit parameters and R vf = 250 kω, the comparison about the boundaries in the parameter space of P 0 versus C vf between the preregulator PFC converter with a resistive load and the twostage PFC converter are shown in Fig. 5(d). It is found that the perioddoubling bifurcation in the twostage PFC converter develops as the output power P 0 increases and its stable region is A, but the one in the preregulator PFC converter with a resistive load develops as the output power P 0 decreases and its stable region is A A 2. Therefore, indeed we cannot simply extend the stability conditions of a preregulator PFC converter with a resistive load to a practical twostage PFC converter. Imaginary part 0 λ 2 λ 0 Real part (a) v PFC /V (b) R vf /kw Fig. 4. The dynamical behaviours with different R vf : (a) the corresponding Table and (b) the bifurcation diagram. 30 (a) 530 (b) 20 stable R vf =250 kw 50 R vf =250 kw stable C /uf 0 00 R vf =200 kw unstable R vi kw 490 unstable R vf =200 kw R vd kw R ff3 kw (c) stable R vf =250 kw (d) stable A C vf /nf R vf =200 kw unstable C vf /nf pre regulator PFC converter with a resistive load A 2 A 3 two stage PFC converter unstable P 0 /W P 0 /W Fig. 5. The theoretical results: (a) the boundaries of the twostage PFC converter in the parameter space of R vd versus C ; (b) the boundaries of the twostage PFC converter in the parameter space of R ff3 versus R vi ; (c) the boundaris of the twostage PFC converter in the parameter space of P 0 versus C vf ; (d) the comparison results between the twostage PFC converter and the preregulator PFC converter with a resistive load of R vf = 250 kω

8 5. Experimental verification To verify the theoretical analysis, an experimental circuit prototype of a twostage PFC converter is constructed. The preregulator boost PFC converter under ACM control is accomplished by IC UC3854, and the postregulator DC DC Buck converter is controlled by a typical PI controller. The full schematic diagram with detailed specifications indicated is shown in Fig. 6. In our experiment, an Agilent 0074C voltage probe is used to detect the voltage v PFC, the voltage v 0, and the voltage v vf. A Tektronix A622 current probe is used to detect the inductor current i in. An Agilent DSO604A digital oscilloscope is used to display the signals from the probes. Firstly, according to the dictation in Section 4, the Chin. Phys. B Vol. 2, No. 2 (202) perioddoubling bifurcation will occur with increasing R vf. In depth, as described in Table, the system will operate in period operation with R vf < kω. Otherwise, it will operate in period2 operation. Let R vf = 50 kω, which is smaller than the bifurcation point, the timedomain waveforms of the inductor current i in, the voltage v vf, the voltage v PFC, and the voltage v 0 are shown in Fig. 7(a). It is easy to see that the system is in stable operation. Furthermore, for convenience to compare the results from the circuit experiment with the results from the IHB on the simplified model and timeintegration on the complete model (Fig. 3), the value of per vertical division about the voltage v vf and the voltage v PFC are reduced, as v in 75 V(rms) 60 Hz KBU808 i in L D v PFC Q 2 L 2 3 mh 2 mh MUR560 IRFP250 v in Q C R IRFP250 MUR560 D C mf 00 W v 0 R s 0.5 W R ffl 90 kw C ffl 0. mf 9 kw C ff mf R i 3.9 kw C R z ac.8 kw R mo 620 kw 3.9 kw nf R z i C p 30 kw 00 pf ref 220 pf 0 kw v ffl mf 2 6 R ff2 R 9 vf 50 kw UC v 6 ff C vf R R C T ff3 T 6.2 kw 22 kw V nf 2.2 nf 00 mf cc8 V TLP250 R vi 50 kw R vd V ramp2 LM3 C v R v 4.75 mf 5. kw v f LF356 V ref V R 33 kw R 2 2 kw Fig. 6. Full schematic diagram of the experimental circuit..00 V 2.00 V 50.0 V 50.0 V ms 0.00 ms stop 2.70 V 500 mv 5.00 V ms 0.00 ms stop 50 mv (a) (b) acquire menu Acq mode averaging Avgs Realtime Serial decode acquire menu Acq mode averaging Avgs Realtime Serial decode Fig. 7. Measured waveforms from the circuit experiment for R vf = 50 kω; (a) the inductor current i in (the first channel: V/div), the voltage v vf (the second channel: 2 V/div), the voltage v PFC (the third channel: 50 V/div), and the voltage v 0 (the fourth channel: 50 V/div), time scale: 0 ms/div; (b) the voltage v PFC (upper trace: 5 V/div) and the voltage v vf (lower trace: 500 mv/div), time scale 0 ms/div

9 .00 V 2.00 V 50.0 V 50.0 V ms 0.00 ms stop 2.73 V 500 mv 5.00 V ms 0.00 ms stop 00 mv acquire menu Acq mode averaging Avgs Realtime Serial decode acquire menu Acq mode averaging Avgs Realtime Serial decode Fig. 8. Measured waveforms from the circuit experiment for R vf = 250 kω: (a) the inductor current i in (the first channel: V/div), the voltage v vf (the second channel: 2 V/div), the voltage v PFC (the third channel: 50 V/div), and the voltage v 0 (the fourth channel: 50 V/div), time scale 0 ms/div; (b) the voltage v PFC (upper trace: 5 V/div) and the voltage v vf (lower trace: 500 mv/div), time scale 0 ms/div. shown in Fig. 7(b). Obviously, the results from those three aspects are in good agreement with one other. But, when R vf = 250 kω, which is bigger than the bifurcation point, the experimental results are shown in Fig. 8. It can be observed that the system is losing stability and operates in period2 operation. Secondly, in order to obtain the boundaries of the stable region, we locate the critical points where perioddoubling bifurcation occurs. Taking R vf = 200 kω, we determine the values of C at the bifurcation point for different values of R vd. From the results shown in Fig. 9, we see clearly that the experimental data are also in good agreement with the theoretical results. Thus, it is confirmed that the method of IHB along with Floquet theory is an effective semianalytical one to analyse the perioddoubling bifurcation in a twostage PFC converter. C \mf stable R vd kw theoretical result experiment unstable Fig. 9. Stable boundaries in the parameter space of R vd versus C from the theoretical result (solid line) and circuit experiment ( ). 6. Conclusion The approximate periodic solution of the simplified model, which has been derived for a twostage PFC converter, has been calculated by using the IHB method. Then, both the perioddoubling bifurcation and the bifurcation point are identified by Floquet theory. The results of timedomain voltage waveform from the timeintegration, the IHB and the circuit experiment are in good agreement with one other. The results of the stable boundaries of the system from the method of IHB and Floquet theory and circuit experiment are also in good agreement with each other. In addition, by comparing with the doubleaveraging method, it is found that the method of IHB and Floquet theory is simpler, straightforward, easier to understand and generally applied. Therefore, the method of IHB along with Floquet theory is an effective semianalytical method to analyse the perioddoubling bifurcation in a twostage PFC converter. References [] Tse C K 2003 Int. J. Circ. Theor. Appl [2] Ki S K, Cheng D K W and Lu D D C 2008 IET Power Electron. 72 [3] Ren H P, Liu D 2005 Chin. Phys [4] Wang F Q, Zhang H, Ma X K and Li X M 2009 Acta Phys. Sin (In Chinese) [5] Rustom K and Batarseh I 2003 IEEE Int. Conf. Industr. Technol. December 0 2, 2003, Maribor, Slovenia, Vol. 2, pp [6] Lee K Y and Lai Y S 2009 IET Power Electron [7] Neba Y, Ishizaka K and Itoh R 200 IET Power Electron

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