1562
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 61, NO. 3, MARCH 2014
Effects of Circuit Parameters on Dynamics of Current-Mode-Pulse-Train-Controlled Buck Converter Jin Sha, Jianping Xu, Member, IEEE, Bocheng Bao, and Tiesheng Yan
Abstract—Current-mode pulse train (CMPT) control technique, a novel discrete control technique for switching dc–dc converters, has completely different dynamics from the traditional pulsewidth modulation control technique. In this paper, a 1-D normal form of discrete-time map of CMPT-controlled buck converter operating in discontinuous conduction mode (DCM) is established, upon which the effects of the circuit parameters on the dynamics and the border collision bifurcation behaviors of CMPT-controlled DCM buck converter are observed and analyzed. According to discrete iterative maps of period-1, period-2, and period-3, the fixed point analyses of the corresponding periodicities are studied, and the mechanism of border collision bifurcation is revealed. Simultaneously, the parameter conditions of different periodicities are formulated, which are helpful to better understand the behaviors and to analyze the CMPT-controlled buck converter. Index Terms—Bifurcation, buck converter, current-mode pulse train (PT) (CMPT), discrete-time model.
I. I NTRODUCTION
P
ULSE TRAIN (PT) control technique, which realizes the output voltage regulation of switching dc–dc converters by applying high-power control pulses or low-power control pulses with discrete duty ratios, is a new kind of discrete control technique for switching dc–dc converters [1]–[6]. Without error amplifier and its corresponding compensation circuit, PT control technique benefits with fast transient response and is simple to design. PT control technique is thus suitable for various applications requiring fast transient response and high reliability. According to the way to generate high-power control pulse and low-power control pulse with discrete duty ratios, there are two kinds of PT controllers, voltage-mode PT (VMPT) controller and current-mode PT (CMPT) controller [4]. In VMPT controller, the duty ratios of high-power control pulse and low-power control pulse are preset, while in CMPT controller, the control currents are preset to generate high-power control Manuscript received August 15, 2012; revised November 22, 2012 and January 15, 2013; accepted March 20, 2013. Date of publication April 5, 2013; date of current version August 23, 2013. This work was supported by the National Natural Science Foundation of China under Grant 51177140 and by the Fundamental Research Funds for the Central Universities under Grant 2682013ZT20. J. Sha, J. Xu, and T. Yan are with the Key Laboratory of Magnetic Suspension Technology and Maglev Vehicle, Ministry of Education, School of Electrical Engineering, Southwest Jiaotong University, Chengdu 610031, China (e-mail:
[email protected];
[email protected];
[email protected]). B. Bao is with the School of Information Science and Engineering, Changzhou University, Changzhou 21300, China (e-mail: mervinbao@ 126.com). Digital Object Identifier 10.1109/TIE.2013.2257145
pulse and low-power control pulse. Thus, CMPT controller has inherent current control ability and is preferred. For PT-controlled switching dc–dc converters, the combination of control pulses (high-power control pulses and lowpower control pulses) in sequential switching cycles constitutes a control pulse repetition cycle. Once the circuit parameters are determined, the combination of the control pulses in the control pulse repetition cycle is fixed. The control performances of PTcontrolled switching dc–dc converters, such as output voltage ripple and output voltage precision, are controlled by the combination of control pulses in the control pulse repetition cycle. It should be noted that, for buck converter operating in discontinuous conduction mode (DCM), the power delivered from the input power source is completely consumed by the load. When the power delivered from the input power source is more than the rated load power, the output voltage will increase. Similarly, when the power delivered from the input power source is less than the rated load power, the output voltage will decrease. Thus, it is quite straightforward to control DCM buck converter by adjusting the combination of high-power control pulse PH and low-power control pulse PL in the control pulse repetition cycle, i.e., CMPT-controlled buck converter operates in period-n (n ≥ 2), not in period-1 as conventional pulsewidthmodulation (PWM)-controlled buck converter does. Therefore, in this paper, we will investigate the effects of circuit parameters on the dynamics of CMPT buck converter operating in DCM. Up to now, most of the modeling approaches of PT control techniques are mainly based on the energy balance principle [1], [3]–[5]. These modeling approaches are essentially a kind of approximate modeling by using averaging inductor current and output voltage, and they may not be accurate enough to reveal the dynamics of PT-controlled switching dc–dc converters. In addition, these approximate models only focus on the steadystate behavior of PT-controlled switching dc–dc converters under fixed circuit parameters, without considering the periodic transformation and the combination of control pulses under the variation of circuit parameters. In order to overcome the limitations of these existing modeling approaches, discrete-time modeling of the CMPT-controlled switching dc–dc converter is studied in this paper. Discrete-time modeling has been successfully used to investigate various nonlinear bifurcation phenomena in switching dc–dc converters [7], [9]–[16], [18]. Previous study results show that, except for some familiar bifurcations [7]–[10], such as period doubling bifurcation, saddle-node bifurcation, and Hopf bifurcation, border collision bifurcation is quite common
0278-0046 © 2013 IEEE
SHA et al.: EFFECTS OF CIRCUIT PARAMETERS ON DYNAMICS OF CMPT-CONTROLLED BUCK CONVERTER
in switching circuits [11]–[17]. Based on the fact that border collision bifurcation depends only on the local properties of the map in the neighborhood of the border [12], a normal form of the border collision bifurcation in the neighborhood of the border can be obtained by a coordinate transform. Similar to single-loop Σ−Δ modulator system [11], [16], a 1-D normal form of the discrete iterative map of CMPT-controlled buck converter operating in DCM is established in this paper. Based upon which, the dynamics of CMPT-controlled DCM buck converter, such as border collision bifurcations and fixed point analysis, are investigated. The investigation of the effects of circuit parameters (such as load resistance, input voltage, and preset control currents) on the dynamics of CMPT-controlled buck converter provides a way for engineers to understand the behavior of CMPT-controlled converters. The mechanism of periodic transformation under the variation of circuit parameters is investigated, and the parameter conditions of the periodicities are studied. The combination of control pulses of CMPT-controlled DCM buck converter in steady state is discussed. Based on which, engineers can design a CMPT-controlled switching converter to satisfy the requirement of practical application. This paper is organized as follows. A brief overview of CMPT control technique is presented in Section II. The 1-D normal form of the discrete-time map of CMPT-controlled buck converter operating in DCM is given in Section III. Upon which, the effects of circuit parameters on bifurcation behaviors are studied in Section IV. Section V analyzes the mechanism of border collision bifurcation through the fixed point analysis. The parameter condition of CMPT-controlled buck converter operating in DCM is summarized in Section VI. Section VII offers a conclusion. II. CMPT-C ONTROLLED B UCK C ONVERTER A. Review of CMPT Control Technique Fig. 1 shows the schematic diagram of CMPT-controlled buck converter, and Fig. 2 shows the time-domain waveforms of CMPT-controlled buck converter operating in DCM [5], [6]. The switch is turned on periodically at the beginning of each switching cycle and turned off when inductor current increases to the corresponding preset control current I. I = IH for highpower control pulse and I = IL for low-power control pulse, which can be unified as IH , if v ≤ Vref I= (1) IL , if v > Vref where v is output voltage and Vref is reference voltage. When v ≤ Vref , a high-power control pulse PH is required as active control pulse to increase the output voltage, and the switch will be turned off when inductor current increases to preset control current IH . Similarly, when v > Vref , a lowpower control pulse PL is required as active control pulse to decrease the output voltage, and the switch will be turned off when inductor current increases to preset control current IL . In steady state, the combination of high-power control pulse PH and low-power control pulse PL constitutes a control pulse repetition cycle. Let the numbers of high-power control
1563
Fig. 1. Schematic diagram of CMPT-controlled buck converter.
Fig. 2. Time-domain waveforms of output voltage v, inductor current i, and control pulse voltage vp of CMPT-controlled buck converter operating in DCM.
pulse and low-power control pulse in the control pulse repetition cycle be denoted as μH and μL , respectively; then, the period of the control pulse repetition cycle is Ts = (μH + μL )T , and the converter operates in period-(μH + μL ). For CMPT-controlled DCM buck converter, the output voltage ripple is small enough to be ignored, i.e., the output voltage can be considered as a constant in a switching cycle. Thus, when high-power control pulse PH is applied, the average input current in a high-power control pulse cycle can be expressed as I¯in_H =
L I2 2(vin − v)T H
and the power delivered from the input power source to the load in high-power control pulse cycle is given by Po_H =
vin L I2 . 2(vin − v)T H
(2a)
In the same way, when low-power control pulse PL is applied, the average input current in a low-power control pulse cycle can be expressed as I¯in_L =
L I2 2(vin − v)T L
and the power delivered from the input power source to the load in low-power control pulse cycle is given by Po_L =
vin L I2 . 2(vin − v)T L
(2b)
1564
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 61, NO. 3, MARCH 2014
From (2), the circuit parameters of CMPT-controlled DCM buck converter should be designed such that, in a high-power control pulse cycle, more power than the rated load power is delivered from the input power source to the load to increase the output voltage, and in a low-power control pulse cycle, less power than the rated load power is delivered from the input power source to the load to decrease the output voltage [1], [2], [4], i.e., high-power control pulse PH and low-power control pulse PL determine the maximum and minimum rated load powers, respectively. Thus, high current threshold IH should be designed such that, at maximum rated load power, the converter should operate in DCM and not enter continuous conduction mode operation, but low current threshold IL can be designed such that the minimum rated load power can be as small as required, i.e., as τ1H = LIH /(vin − v), τ2H = LIH /v, and τ1H + τ2H ≤ T in DCM operation, and in (2), IH , and IL can be estimated as (vin − v)vT Lvin 2(vin − v)Po_L T . IL ≥ vin L
IH ≤
When the low current threshold of low-power control pulse is set to be zero, the CMPT control becomes current-mode pulse skip mode control, which provides a way to improve the light load efficiency [19], [20]. It should be noted also that, with the variation of circuit parameters, such as load resistance R, control current IH , IL , and so on, the CMPT-controlled DCM buck converter may operate from normal period-(μH + μL ), controlled by the combination of high-power control pulse PH and low-power control pulse PL , to period-1, controlled only by one high-power control pulse PH or one low-power control pulse PL . The period-1 operation controlled only by high-power control pulse PH and low-power control pulse PL determines the maximum and minimum rated load powers of CMPT-controlled DCM buck converter, respectively. When the load power is higher than the maximum rated load power, the high-power control pulse PH cannot increase the output voltage of the converter to higher than the desired voltage, and when the load power is lower than the minimum rated load power, the low-power control pulse PL cannot decrease the output voltage of the converter to lower than the desired voltage. From the aforementioned discussion, it is evident that different from traditional PWM controller which realizes the output voltage regulation of switching converter by adjusting the duty ratio cycle by cycle continuously, the CMPT controller realizes the output voltage regulation of switching converter by adjusting the control pulse combination of high-power control pulse and low-power control pulse in a control pulse repetition cycle. B. PSIM Simulation and Experimental Results Power simulation (PSIM) simulations for CMPT-controlled buck converter operating in DCM are performed for different load resistances and with the other circuit parameters as the fol-
Fig. 3. PSIM simulation results of CMPT-controlled buck converter under different load resistances R. (a) R = 1.8 Ω, period-1. (b) R = 7.5 Ω, period-1. (c) R = 2.98 Ω, period-2. (d) R = 3.75 Ω, period-3. (e) R = 3.21 Ω, period-9. (a1), (b1), (c1), (d1), and (e1) are time-domain waveforms of inductor current i and output voltage v, respectively. (a2), (b2), (c2), (d2), and (e2) are phase portraits in i−v plane, respectively.
lowing: vin = 12 V, Vref = 5 V, f = 50 kHz, L = 10 μH, C = 470 μF, IH = 5.6 A, and IL = 2.8 A. According to (2), the rated load power region can be given as [3.36 W, 13.44 W], and the corresponding load resistance region can be obtained as [1.86 Ω, 7.44 Ω]. Fig. 3 shows the simulation results of time-domain waveforms and phase portraits of inductor current versus output voltage. From Fig. 3, it can be seen that, for different load resistances R, the control pulse repetition cycle and the corresponding control pulse combination are different.
SHA et al.: EFFECTS OF CIRCUIT PARAMETERS ON DYNAMICS OF CMPT-CONTROLLED BUCK CONVERTER
When R = 1.8 Ω, the load is too heavy to be controlled by the CMPT controller, the output voltage v is always lower than the reference voltage Vref , high-power control pulse PH is thus continuously applied as active control pulse as shown in Fig. 3(a1), and the limit cycle with period-1 as shown in Fig. 3(a2) is generated. In this case, the control pulse repetition cycle consists of only high-power control pulse PH . When R = 7.5 Ω, the load is too light to be controlled by the CMPT controller, v is always higher than Vref , low-power control pulse PL is thus continuously applied as active control pulse as shown in Fig. 3(b1), and the limit cycle with period-1 as shown in Fig. 3(b2) is generated. In this case, the control pulse repetition cycle consists of only low-power control pulse PL . Fig. 3(c1) shows the case when R = 2.98 Ω, one high-power control pulse and one low-power control pulse are applied alternately, and the limit cycle with period-2 as shown in Fig. 3(c2) is generated. In this case, the control pulse repetition cycle consists of one high-power control pulse and one lowpower control pulse, and the control pulse combination can be denoted as 1PH − 1PL . With the increase of R, the ratio between the number of low-power control pulse μL and the number of high-power control pulse μH increases. Fig. 3(d1) shows the case when R = 3.75 Ω, one high-power control pulse followed by two low-power control pulses are applied as active control pulses, and the limit cycle with period-3 as shown in Fig. 3(d2) is generated. In this case, the control pulse repetition cycle consists of one high-power control pulse followed by two lowpower control pulses, and the control pulse combination can be denoted as 1PH − 2PL . With the variation of R, more complicated control pulse combinations in a control pulse repetition cycle may occur. Fig. 3(e1) shows the case when R = 3.21 Ω and nine switching cycles, including four high-power control pulse cycles and five low-power control pulse cycles, make up a control pulse repetition cycle. In this case, the control pulse repetition cycle consists of one subrepetition cycle of 1PH − 2PL followed by three subrepetition cycles of 1PH − 1PL , i.e., the control pulse combination can be denoted as 1(1PH − 2PL ) − 3(1PH − 1PL ), and limit cycle of period-9 as shown in Fig. 3(e2) is generated. A setup of CMPT-controlled buck converter with the aforementioned circuit parameters has been implemented. Fig. 4(a)–(e) shows the corresponding experimental results for R’s of 1.8, 7.5, 2.98, 2.48, and 3.21 Ω, respectively. These results are in good agreement with the simulation results. III. M ODELING OF CMPT-C ONTROLLED B UCK C ONVERTER O PERATING IN DCM A. State Equations For CMPT-controlled buck converter operating in DCM, there are three switch states in each switch cycle: 1) switch S on and diode D off; 2) switch S off and diode D on; and 3) switch S off and diode D off. The time durations of these three switch states in high-power control pulse cycle and lowpower control pulse cycle are defined as τ1H , τ2H , τ3H , and
1565
Fig. 4. Experimental results of CMPT-controlled buck converter under different load resistances R. (a) R = 1.8 Ω, period-1. (b) R = 7.5 Ω, period-1. (c) R = 2.98 Ω, period-2. (d) R = 3.75 Ω, period-3. (e) R = 3.21 Ω, period-9. (a1), (b1), (c1), (d1), and (e1) are time-domain waveforms of inductor current i and output voltage v, respectively. (a2), (b2), (c2), (d2), and (e2) are phase portraits in i−v plane, respectively.
τ1L , τ2L , τ3L , respectively, as shown in Fig. 2. For convenience, these time durations are unified as τjH , if vn ≤ Vref τj = (3) τjL , if vn > Vref where j = 1, 2, 3 denotes different switch states.
1566
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 61, NO. 3, MARCH 2014
The state equations corresponding to these three switch states can be given as x˙ = Aj x(t) + Bj vin ,
tj−1 ≤ t < tj j = 1, 2, 3
(4)
where x is state vector defined as x = [i, v]T ; i is inductor current; v is capacitor voltage; vin is input voltage; [tj−1 , tj ] is the time interval of the jth switch state; and τj = tj − tj−1 , Aj , and Bj are state matrix and input matrix of the jth switch state, which are given by 0 − L1 0 0 A1 = A 2 = 1 = A 3 1 1 0 − RC − RC C 1 0 . B1 = L B 2 = B3 = 0 0
B. One-Dimensional Discrete-Time Model Let the inductor current and capacitor voltage at the beginning of the nth and the (n + 1)th switching cycle be in and vn and in+1 and vn+1 , respectively. Based on (4), the corresponding solutions in these three switch states can be derived. Switch State 1 [t0 , t1 ]: The state equations in switch state 1 are nonautonomous ordinary differential equations, and the solutions of the state variables in switch state 1, i(t) and v(t), are given by ⎧
⎨ i(t) = e−α(t−t0 ) −vRin cos ω(t−t0 )+k1 sin ω(t−t0 ) + vRin , −α(t−t0 ) [(vn −vin ) cos ω(t−t0 )+k2 sin ω(t−t0 )] ⎩ v(t) = e + vin where α = 1/2RC, ω = (1/LC) − α2 , k1 = ((R − αL)vin /ωRL) − (vn /ωL), and k2 = −(α/ω)(vn + vin ). The inductor current and capacitor voltage at the end of switch state 1 are i(t1 ) and v(t1 ), respectively. As i(t1 ) = I, the time duration of switch state 1 can be appreciatively obtained as (vn − vin ) + (vn − vin )2 + 4αLIvn τ1 = . 2αvn Switch State 2 [t1 , t2 ]: The state equations in switch state 2 are autonomous differential equations. The initial conditions of switch state 2 are i(t1 ) = I and v(t1 ), and the timedomain solutions of the state variables in switch state 2, i(t) and v(t), are obtained as i(t) = e−α(t−t1 ) [I cos ω(t−t1 )+k3 sin ω(t−t1 )] (6) v(t) = e−α(t−t1 ) [v(t1 ) cos ω(t−t1 )+k4 sin ω(t−t1 )] where k3 = (αI/ω) − (v(t1 )/ωL1 ) and k4 = I/ωC − αv(t1 )/ω. The inductor current at the end of switch state 2 is i(t2 ) = 0, and the time duration of switch state 2 can be given by τ2 =
1 arctan(−I/k3 ). ω
Switch State 3 [t2 , t3 ]: The initial conditions of switch state 3 are i(t2 ) = 0 and v(t2 ), and the time-domain solutions of the state variables in switch state 3, i(t) and v(t), can be obtained as i(t) = 0 (t−t2 ) (7) v(t) = v(t2 )e− RC . The time duration of switch state 3 is τ3 = T − τ , where τ = τ1 + τ2 . From (5)–(7), the 2-D discrete-time model of CMPTcontrolled DCM buck converter can be described as in+1 = 0 vn+1 = f (vn , I) = e−α(2T −τ ) [(vn − vin ) cos ωτ + k2 sin ωτ ]
α + e−α(2T −τ1 −τ ) vin cos ωτ2 + sin ωτ2 . (8) ω It is remarked that the discrete-time model (8) is a 2-D piecewise-smooth discrete system with the control parameter I switching between IH and IL . Since the inductor current at the end of each switching cycle is zero, the 2-D discrete-time model (8) can be simplified as a 1-D model vn+1 = f (vn , I).
(9)
C. One-Dimensional Normal Form Since I is a control variable determined by (1), there is a border vn = Vref in the 1-D discrete-time model (9). It is known that the border collision bifurcation only depends on the behavior of the map in the neighborhood of the border. The linearization of the 1-D discrete-time model (9) around the border gives ∂f (vn , I) (vn −Vref ) (10) vn+1 = f (vn , I) |vn=Vref + ∂vn vn=Vref
where
∂f (vn , I) ∂vn
= e−α(2T −τ ) [δ1 cos ωτ − δ2 sin ωτ ] vn =Vref
+e−α(2T −τ1 −τ ) [(1 − δ1 ) cos ωτ2 + δ3 sin ωτ2 ]
and δ1 , δ2 , and δ3 are defined as ∂τ δ1 = 1 − 2αvin ∂vn 2 ∂τ 1 Vref ∂τ 2 δ2 = − ω − α vin α+ ω LC ∂vn ∂vn 2 vin 2 ∂τ1 2 ∂τ2 δ3 = − ω −α 2α ω ∂vn ∂vn ∂τ ∂τ1 ∂τ2 = + ∂vn ∂vn ∂vn ∂τ1 −v 2 − 2αLIVref vin in = + 2 2 ∂vn 2αVref 2αVref (Vref − vin )2 + 4αLIVref ∂τ1 2αVref ∂vn − 1 −ατ ∂τ2 e 1 I cos ωτ1 = 2 2 2 ∂vn ω L 3 + I ) (k ∂τ1 vin ∂τ1 ω 2 − α2 Vref ∂v − LC ∂vn + α n + 2 ω 3 L (k3 + I 2 ) × e−ατ1 I sin ωτ1 .
SHA et al.: EFFECTS OF CIRCUIT PARAMETERS ON DYNAMICS OF CMPT-CONTROLLED BUCK CONVERTER
Fig. 5. Bifurcation diagrams with the increase of R. (a) Bifurcation diagram based on the 1-D discrete-time model (9). (b) Bifurcation diagram based on the 1-D normal form (11). (c) Period-2 window; zoom-in view of (b). (d) Illustration of border collision bifurcation; zoom-in view of (b).
By the coordinate transformation xn = vn − Vref , the border is now given by xn = 0, and the 1-D normal form is given by FH (xn ) = aH xn +bH , if xn ≤ 0 xn+1 = F (xn , I) = (11) FL (xn ) = aL xn +bL , if xn > 0 where FH (·) and FL (·) stand for the 1-D normal forms when high-power control pulse and low-power control pulse are applied, respectively, and the parameters aH , bH , aL , and bL are given by ∂f (vn , IH ) aH = ∂vn vn =Vref bH = f (vn , IH )|vn=Vref − Vref ∂f (vn , IL ) aL = ∂vn vn =Vref bL = f (vn , IL )|vn =Vref − Vref . Based on 1-D normal form (11), the dynamical analysis of CMPT-controlled buck converter operating in DCM can be performed. IV. B ORDER C OLLISION B IFURCATION IN CMPT-C ONTROLLED B UCK C ONVERTER O PERATING IN DCM In this section, the effects of load resistance R, input voltage vin , and preset control current I on the dynamics of output voltage are investigated, from which the dependence of bifurcation behaviors of CMPT-controlled buck converter on its circuit parameters can be obtained. A. Effect of Load Resistance on Bifurcation Behavior Fig. 5 shows the bifurcation diagrams of CMPT-controlled DCM buck converter with load resistance R as bifurcation parameter. In order to ensure DCM operation of buck converter, with the other circuit parameters being the same as given earlier, the load resistance R should be in the region [1.5 Ω, 8 Ω].
1567
Fig. 5(a) is obtained from the 1-D discrete-time model (9), and Fig. 5(b)–(d) is obtained from the 1-D normal form (11). From Fig. 5(a) and (b), it can be known that the 1-D discretetime model (9) can be well approximated by the corresponding 1-D normal form (11). From the bifurcation diagram as shown in Fig. 5(a) and (b), it is known that, with the increase of R, the converter goes from period-1 to multiperiodicities and, finally, to period-1. Define the orbit over the border xn = 0 as “low-power orbit” and the orbit below the border xn = 0 as “high-power orbit;” then, when R < 1.870 Ω, only high-power orbit exists, and, when R > 7.463 Ω, only low-power orbit exists. It should be noted that the load resistance region [1.860 Ω, 7.440 Ω] corresponding to high-power and low-power control pulses, derived from (2), is under the assumption that the output voltage of the converter is constant. While from 1-D discrete-time model (9) and 1-D normal form (11), the load resistance region corresponding to high-power and low-power control pulses is [1.870 Ω, 7.463 Ω], very close to the load resistance region [1.860 Ω, 7.440 Ω]. The high-power orbit collides with the border at R = 1.870 Ω, which results in border collision bifurcation. The border collision bifurcation evolves from period1 to multiperiodicity with the increase of load resistance R. Similarly, the low-power orbit collides with the border at R = 7.463 Ω, resulting in border collision bifurcation. The border collision bifurcation evolves from period-1 to multiperiodicity with the decrease of load resistance R. From Fig. 5(b) and (c), it can be known that the periodicity regions are not continuous, e.g., period-1 with control pulse combination of 1PH appears in the region [1.5 Ω, 1.870 Ω], period-2 with control pulse combination of 1PH − 1PL appears in the region [2.978 Ω, 3.012 Ω] [as shown in Fig. 5(c)], and period-3 with control pulse combination of 2PH − 1PL and 1PH − 2PL appears in the regions [2.476 Ω, 2.496 Ω] and [3.741 Ω, 3.768 Ω], respectively [as shown in the zoom-in bifurcation diagrams in Fig. 5(b)]. It is found that lower periodicity has larger region, e.g., the region of period-1 is larger than that of period-2, and the region of period-2 is larger than that of period-3. In the higher periodicity region, there exist some lower periodic regions defined as “low periodic windows.” Fig. 5(d) shows the zoom-in bifurcation diagram of Fig. 5(b), which serves to reveal the high periodicities and the border collision bifurcation in detail. With the increase of R, the orbit A, nearest to the border in the multiperiodicity, collides with the border at R = 3.20127 Ω and turns into some new orbits as shown in the region E, in which a low-power orbit must be included. Simultaneously, other multiperiodic orbits, such as orbits B, C, and D, also turn into some new orbits, respectively. As a result, a new multiperiodicity is born. From 1-D normal form (11), the eigenvalue can be derived as dx n+1 = aH , if xn ≤ 0 n (12) λ = dxdxn+1 if xn > 0. dxn = aL , With the circuit parameters given earlier, the eigenvalue is real, and its absolute value is less than one, as shown in Fig. 6(a), which indicates that the converter works in stable periodic state. Fig. 6(b) shows that the maximal Lyapunov
1568
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 61, NO. 3, MARCH 2014
goes from multiperiodicities and, finally, to period-1. The lowpower orbit collides with the border at IL /IH = 0.79, resulting in border collision bifurcation. The border collision bifurcation evolves from period-1 to multiperiodicity with the decrease of IL /IH . From the zoom-in view in Fig. 7(c), the period-2 window appears in the region [0.49, 0.504]. The corresponding maximal Lyapunov exponent is shown in Fig. 7(d), which is less than one, meaning that the system is stable for IL /IH . Fig. 6. Eigenvalues and maximal Lyapunov exponent with the variation of R. (a) Eigenvalues based on the 1-D normal form (11). (b) Maximal Lyapunov exponent.
C. Dynamics of CMPT Control Technique From aforementioned discussion, it can be known that the dynamics of CMPT control technique is quite different from that of traditional PWM control technique, which can be concluded as follows.
Fig. 7. Bifurcation diagrams. (a) Bifurcation diagram with the increase of vin . (b) Maximal Lyapunov exponent with the increase of vin . (c) Bifurcation diagram with the increase of IL /IH . (d) Maximal Lyapunov exponent with the increase of IL /IH .
exponent is less than zero for all values of R, which illustrates that there is no chaotic state and the system is stable. B. Effects of Input Voltage and Preset Control Currents on Bifurcation Behaviors From the 1-D normal form (11), the bifurcation diagrams of CMPT-controlled DCM buck converter with input voltage vin and IL /IH as bifurcation parameters can be obtained, as shown in Fig. 7. Fig. 7(a) shows the bifurcation diagrams with input voltage vin as bifurcation parameter when load resistance R = 3 Ω and the other circuit parameters are the same as given earlier. In order to make buck converter operate in DCM, input voltage vin ≥ 9 V is required. From Fig. 7(a), it is known that, with the increase of vin , the converter goes from multiperiodicities and, finally, to period-1. The high-power orbit collides with the border at vin = 74.8 V, resulting in border collision bifurcation. The border collision bifurcation evolves from period-1 to multiperiodicity with the decrease of vin . From the zoom-in view in Fig. 7(a), the period-2 window appears in the region [11.96 V, 12.13 V]. The corresponding maximal Lyapunov exponent is shown in Fig. 7(b), which is less than one, meaning that the system is stable. Fig. 7(c) shows the bifurcation diagrams with IL /IH as bifurcation parameter when R = 3 Ω, vin = 12 V, IH = 5.6 A, and the other circuit parameters are the same as given earlier. It can be known that, with the increase of IL /IH , the converter
1) CMPT-controlled buck converter operates in multiperiodicity, and its dynamics is caused by border collision bifurcation. Typical bifurcations, such as period doubling bifurcation, saddle-node bifurcation, and Hopf bifurcation, do not exist in CMPT control buck converter. 2) Low periodic window exists in the high-periodic region. Different from the periodic window in smooth map which is caused by saddle-node bifurcation and thought to be a common feature of chaos, low periodic window in CMPT-controlled buck converter is caused by the border collision bifurcation. 3) In CMPT-controlled buck converter, there are period-3 orbits, while in continuous system, period-3 means chaos. 4) Although the chaotic orbit does not exist, but when load power is beyond the region of rated load power, as shown in Figs. 3(a) and (b) and 4(a) and (b), the output voltage will be lower or higher than the reference voltage; in this case, the converter operates in period-1. 5) The effects of load resistance R and preset control current I on the dynamics of CMPT-controlled buck converter are similar. With the increase of load resistance R and preset control current I, both low-power orbits and high-power orbits move upward. The periodic transformation occurs when high-power orbit collides with the border, resulting in border collision bifurcation. Different from the effects of R and I on the dynamics, with the increase of input voltage vin , both low-power orbits and high-power orbits move downward. The periodic transformation occurs when low-power orbit collides with the border, resulting in border collision bifurcation. However, the variations of load resistance R, preset control current I, and input voltage vin do not affect the stability of CMPT-controlled DCM buck converter. V. M ECHANISM OF B ORDER C OLLISION B IFURCATION AND F IXED P OINT A NALYSIS According to the 1-D normal form, the discrete iterative maps can be obtained, and the fixed points in different periodic states can be studied. The mechanism of border collision bifurcation can also be revealed. In this section, we only focus on the analysis with the variation of load resistance R; similarly conclusions can also obtained for the case with the variations of
SHA et al.: EFFECTS OF CIRCUIT PARAMETERS ON DYNAMICS OF CMPT-CONTROLLED BUCK CONVERTER
Fig. 8.
1569
Parameters bH and bL in model (10) with the variation of R.
Fig. 9. Three cases of first iterative map. (a) bH ≤ 0 and bL < 0. (b) bH > 0 and bL < 0. (c) bH > 0 and bL ≥ 0.
other parameters, such as input voltage vin and preset control current I. A. Period-1 Orbit Fig. 9 shows three cases of first iterative map of 1-D normal form, marked with the solid line. From (11), border xn = 0 divides the first iterative map into two regions, RH with xn ∈ (−∞, 0) and RL with xn ∈ (0, +∞), and F (xn , I) is smooth in each of the regions RH and RL . Furthermore, aH and aL are the slopes of first iterative map in the regions RH and RL , respectively. From Fig. 6(a), it can be known that, for all load resistance R, 0 < aH < 1 and 0 < aL < 1. Thus, if the intersection points xH and xL between the diagonal and first iterative map in the regions RH and RL exist, xH and xL must be stable fixed points. From Fig. 8, it can be known that bL < bH , bH , and bL increase from negative to positive with the increase of R. According to the values of bH and bL , there are three different first iterative maps. Case 1) bH ≤ 0 and bL < 0: When R is in the region [1.500 Ω, 1.870 Ω], as shown in Fig. 8, bH ≤ 0 and bL < 0. Fig. 9(a) shows the corresponding first iterative map; there is only one stable fixed point in the region RH , which indicates that the converter operates in stable period-1 with control pulse combination of 1PH . From (11), the stable fixed point can be expressed as xH = bH /(1−aH ). With the increase of R, bH increases and xH also increases along the diagonal direction. When R = 1.870 Ω, bH = 0 and xH overlaps with the origin and transforms from a stable fixed point into an unstable fixed point, implying the occurrence of border collision bifurcation. Case 2) bH > 0 and bL < 0: When R is in the region (1.870 Ω, 7.486 Ω), as shown in Fig. 8, bH > 0 and bL < 0. Fig. 9(b) shows the corresponding first iterative map, and there is no fixed point. Thus, the period-1 orbit does not exist in this case.
Fig. 10. Period-2 orbit. (a) Second iterative map with aL bH + bL < 0 and aH bL + bH > 0. (b) Two boundaries: Solid line indicates the boundary case of aL bH + bL = 0 and aH bL + bH > 0 and dashed line indicates the boundary of aL bH + bL < 0 and aH bL + bH = 0. (c) aL bH + bL and aH bL + bH with the variation of R.
Case 3) bH > 0 and bL ≥ 0: When R is in the region [7.486 Ω, 8 Ω], as shown in Fig. 8, bH > 0 and bL ≥ 0. Fig. 9(c) shows the corresponding first iterative map; there is only one stable fixed point xL = bL /(1−aL ) in the region RL , which indicates that the converter operates in stable period-1 with control pulse combination of 1PL . With the decrease of R, bL decreases and xL also decreases along the diagonal direction. When R = 7.486 Ω, bL = 0 and xL overlaps with the origin and transforms from a stable fixed point into an unstable fixed point, implying the occurrence of border collision bifurcation. B. Period-2 Orbit The second iteration map can be obtained from (11) as ⎧ FH (FH (xn )) = a2H xn ⎪ ⎪ ⎪ +a b +b , ⎪ if xn ≤ − abH H H H ⎪ H ⎪ ⎪ ⎪ FL (FH (xn )) = aH aL xn ⎪ ⎪ ⎨ + a b +b , if − abH < xn ≤ 0 L H L H xn+2 = F (F (x )) = a a x (13) H L n H L n ⎪ ⎪ ⎪ + a b +b , ⎪ if 0 < xn ≤ − abLL H L H ⎪ ⎪ ⎪ 2 ⎪ F (F (x )) = aL xn ⎪ ⎪ ⎩ L L n + aL bL +bL , if xn > − abLL . The solid line in Fig. 10(a) represents second iterative map of (13). The second iterative map can be divided into four regions RH1 , RH2 , RL1 , and RL2 with three borders −bH /aH , 0, and −bL /aL as shown in Fig. 10(a). As the period-2 orbit of CMPT-controlled buck converter must consist of one lowpower orbit and one high-power orbit, there are two stable fixed points in the regions RH1 and RL1 , i.e., [−bH /aH , −bL /aL ]. From (13), for FL (FH (xn )) = xn and FH (FL (xn )) = xn , two fixed points are given by xH = (aL bH + bL )/(1 − aH aL )
1570
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 61, NO. 3, MARCH 2014
and xL = (aH bL + bH )/(1 − aH aL ), which locate in the regions RH1 and RL1 , respectively. The two fixed points satisfy the conditions of xH = (aL bH + bL )/(1 − aH aL ) < 0 and xL = (aH bL + bH )/(1 − aH aL ) > 0. As 1 − aH aL > 0, the parameter of period-2 satisfies following conditions: a L bH + b L < 0
aH bL + bH > 0.
(14)
Thus, the two boundaries of period-2 are aL bH + bL = 0 and aH bL + bH = 0, as shown in Fig. 10(b), in which solid line indicates the border of aL bH + bL = 0 and the dashed line indicates the border of aH bL + bH = 0. As aL bH + bL increases from negative to positive, the stable fixed point xH increases to zero and then changes to an unstable fixed point, and period-2 disappears, which means that, when aL bH + bL > 0, the period-2 does not exist. Similarly, as aH bL + bH decreases from positive to negative, the stable fixed point xL decreases to zero and then changes to an unstable fixed point and period-2 disappears, which means that, when aH bL + bH < 0, the period-2 does not exist. Fig. 10(c) shows the parameters of aL bH + bL and aH bL + bH with the increase of R. When R = 2.978 Ω and R = 3.012 Ω, there exist aH bL + bH = 0 and aL bH + bL = 0, respectively; thus, the period-2 orbit is located in the region [2.978 Ω, 3.012 Ω], which satisfies the conditions of (14) and is consistent with the result in Fig. 5(c). C. Period-3 Orbit From (11), the third iteration map can also be obtained as (15), shown at the bottom of the page. It is known that only when the parameter R is located in period-2 region, both aL bH + bL ≥ 0 and aH bL + bH < 0 observed from Fig. 10(c) exist; otherwise, these two parameter conditions cannot be satisfied simultaneously. Thus, 0 < aH aL < 1, −(aL bH + bL )/aH aL < xn ≤ 0, and 0 < xn ≤ −(aH bL + bH )/aH aL in (15) cannot be satisfied simultaneously. Consequently, there are two period-3 orbits with two different control pulse combinations. 1) When −(aL bH + bL )/aH aL < xn ≤ 0, the converter operates in period-3 with control pulse combination of 1PH − 2PL . 2) When 0 < xn ≤ −(aH bL + bH )/aH aL , the converter operates in period-3 with control pulse combination of 2PH − 1PL .
xn+3
Fig. 11. Period-3 orbit. (a) Third iterative map for the control pulse combination with 1PH − 2PL . (b) Parameter region of period-3 with control pulse combination of 1PH − 2PL . (c) Parameter region of period-3 with control pulse combination of 2PH − 1PL .
Fig. 11(a) shows the third iterative map of (15) with the control pulse combination of 1PH − 2PL , where the third iterative map can be divided into seven regions RH1 , RH2 , RH3 , RH4 , RL1 , RL2 , and RL3 with six borders −(aH bH + bH )/a2H , −bH /aH , −(aL bH + bL )/aH aL , 0, −bL /aL , and −(aL bL + bL )/a2L (the border of −(aH bL + bH )/aH aL does not exist). Three stable fixed points are xH = a2L bH + aL bL + bL / 1 − aH a2L xL1 = (aH aL bL + aL bH + bL )/ 1 − aH a2L xL2 = (aH aL bH + aH bL + bH )/ 1 − aH a2L which are distributed within the regions of RH1 , RL1 , and RL2 , i.e., [−(aH bH + bH )/a2H , 0], [0, −bL /aL ], and [−bL /aL , −(aL bH + bL )/aH aL ], respectively. For xH < 0, xL1 > 0, and 1 − aH a2L > 0, the parameter of period-3 with control pulse combination of 1PH − 2PL satisfies the following conditions [13]: a2L bH + aL bL + bL < 0 aH aL bL + aL bH + bL > 0.
⎧ FH (FH (FH (xn ))) = a3H xn + a2H bH + aH bH + bH , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ FL (FH (FH (xn ))) = a2H aL xn + aH aL bH + aL bH + bL , ⎪ ⎪ ⎪ ⎪ ⎪ FH (FL (FH (xn ))) = a2H aL xn + aH aL bH + aH bL + bH , ⎪ ⎪ ⎪ ⎨ F (F (F (x ))) = a a2 x + a2 b + a b + b , L L H n H L n L L L L H = ⎪ FH (FH (FL (xn ))) = a2H aL xn + a2H bL + aH bH + bH , ⎪ ⎪ ⎪ ⎪ ⎪ FL (FH (FL (xn ))) = aH a2L xn + aH aL bL + aL bH + bL , ⎪ ⎪ ⎪ ⎪ FH (FL (FL (xn ))) = aH a2L xn + aH aL bL + aH bL + bH , ⎪ ⎪ ⎪ ⎪ ⎩ F (F (F (x ))) = a3 x + a2 b + a b + b , L
L
L
n
L n
L L
L L
L
(16)
if xn ≤ − aH baH2+bH H
if − aH baH2+bH < xn ≤ − abH H H
L if − abH < xn ≤ − aLabHHa+b H L aL bH +bL if − aH aL < xn ≤ 0 H if 0 < xn ≤ − aHabHLa+b L aH bL +bH if − aH aL < xn ≤ − abLL if − abLL < xn ≤ − aL baL2+bL
if xn > − aL baL2+bL L
L
(15)
SHA et al.: EFFECTS OF CIRCUIT PARAMETERS ON DYNAMICS OF CMPT-CONTROLLED BUCK CONVERTER
1571
TABLE I C ONTROL P ULSE C OMBINATIONS IN S OME T YPICAL P OWER P ULSE R EPETITION C YCLES
Fig. 11(b) shows the parameters of a2L bH + aL bL + bL and aH aL bL + aL bH + bL with the increase of R. When R is in the region [3.741 Ω, 3.768 Ω], period-3 with control pulse combination of 1PH − 2PL exists, and these parameters satisfy the conditions of (16), which is the same as the result in Fig. 5(b). A similar conclusion can be yielded for period-3 with the control pulse combination of 2PH − 1PL . Three stable fixed points can be easily obtained as xH1 = a2H bL + aH bH + bH / 1 − aH a2L aL xH2 = (aH aL bL + aH bL + bH )/ 1 − a2H aL xL2 = (aH aL bL + aL bH + bL )/ 1 − aH a2H aL which are distributed within the regions of [−(aH bH + bH )/a2H , −bH /aH ], [−bH /aH , 0], and [0, −(aH bL + bH )/aH aL ], respectively. For xH2 < 0, xL > 0, and 1 − a2H aL > 0, the corresponding conditions of parameter are given by a H a L bL + a H bL + b H < 0 a2H bL + aH bH + bH > 0.
pulse combinations in a control pulse repetition cycle are symmetric. In Table I, it can be observed that the control pulse combinations of the periodic states are symmetric about the diagonal of the table. Note that, for μH = μL , all the control pulse combinations on the diagonal is the combination of 1PH − 1PL of period-2; thus, the control pulse combinations of the periodic states are symmetric about the period-2, e.g., the possible control pulse combinations of period-7 are 1(1PH − 2PL ) − 2(1PH − 1PL ) and 1(2PL − 1PH ) − 2(1PL − 1PH ). 2) If the number of high-power control pulse σ × μH and the number of low-power control pulse σ × μL have common factor σ(σ > 1), then the converter will operate in period-k, with k = μH + μL , instead of in period(σ × k). 3) If the control pulse combination of period-k (k = μH + μL ) is known, the corresponding iteration map can be obtained, in which the two equations around the border can be expressed as
(17)
Fig. 11(c) shows the parameters aH aL bL + aH bL + bH and a2H bL + aH bH + bH with the increase of R. Period-3 with the control pulse combination of 2PH − 1PL exists, when R is in the region [2.476 Ω, 2.496 Ω], and these parameters satisfy the conditions of (17), which are the same as the result in Fig. 5(b). VI. PARAMETER C ONDITION OF P ERIODIC S TATE The control pulse combinations and parameter conditions of period-1, period-2, and period-3 are presented through the fixed point analysis in the aforementioned sections. Similarly, the parameter conditions corresponding to other higher periodic states can be analyzed in the same way. Some conclusions of the control pulse combinations with part of them given in Table I and the parameter conditions corresponding to different periodic states are summarized as follows. 1) As aL bH + bL ≥ 0 and aH bL + bH < 0 can only be satisfied simultaneously in period-2, the periodicities with the same number of control pulses but different control
xn+k =
F (· · · (FH (FL (xn )))) = aμHH aμLL xn + ξ1 F (· · · (FL (FH (xn )))) = aμHH aμLL xn + ξ2
if xn ≤ 0 if xn > 0 (18)
where ξ1 = aμHH −1 aμLL bH + aμHH −1 aμLL −1 bL + · · · ξ2 = aμHH aμLL −1 bL + aμHH −1 aμLL −1 bH + · · · and, in (18), F (·) = FH (·) for high-power control pulse and F (·) = FL (·) for low-power control pulse. There are k items in ξ1 and ξ2 ; the first two items of ξ1 and ξ2 are independent of the control pulse combination, and the other items of ξ1 and ξ2 correspond to control pulses in the control pulse combination. Based on (18), two stable fixed points around the border can be expressed as follows: xH = respectively.
ξ1 1 − aμHH aμLL
xL =
ξ2 1 − aμHH aμLL
(19)
1572
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 61, NO. 3, MARCH 2014
Fig. 12. Parameter condition of period-9. (a) Estimating parameter condition by (20). (b) Local bifurcation diagram of period-9 with four high-power orbits and five low-power orbits.
As xH ≤ 0, xL > 0, and 0 < 1 − aμHH aμLL < 1, the parameter condition of period-k with control pulse combination can be expressed as ξ1 ≤ 0
ξ2 > 0.
(20)
Let us take the converter operating in period-9, with four high-power control pulses and five low-power control pulses, as an example. According to (20), the parameter condition of period-9 with control pulse combination of 1(1PH − 2PL ) − 3(1PH − 1PL ) can be evaluated as ξ1 = a3H a5L bH +a3H a4L bL +a3H a3L bL +a2H a3L bH +a2H a2L bL +aH a2L bH +aH aL bL +aL bH +bL < 0 (21a) ξ2 = a4H a4L bL +a3H a4L bH +a3H a3L bL +a2H a3L bH +a2H a2L bL +aH a2L bH +aH aL bL +aL bH +bL > 0. (21b) From aforementioned parameter condition, the parameter region of R can be depicted in Fig. 12(a), which locates in [3.207 Ω, 3.215 Ω]. The corresponding bifurcation diagram around the period-9 window is plotted in Fig. 12(b). Note that two parameter regions are consistent. It should be noted that the parameter regions of periodicities are very narrow, even for the largest one, period-2. Because of the error in the actual circuits, it is hard to control the converter operating in one expected periodicity. VII. C ONCLUSION In this paper, a 1-D normal form of CMPT-controlled buck converter operating in DCM has been established. Based on which, border collision bifurcation behaviors and the effects of circuit parameters on the bifurcation are studied. According to the 1-D normal form, the iterative maps of different periodic states are constructed, and the mechanism of border collision bifurcation is revealed through the fixed point analysis. Finally, parameter condition corresponding to the periodicity with control pulse combination is summarized. As a discrete control strategy, CMPT control technique has completely different dynamics from the traditional PWM control technique. The effects of circuit parameters on the dynamics of CMPT-controlled DCM buck converter are analyzed. With the variation of circuit parameters, the maximal Lyapunov exponent of CMPT-controlled DCM buck converter is always less than zero, which means that CMPT-controlled
DCM buck converter does not have chaotic state. The converter operates in different periodicities and has a transition from one periodicity to other periodicity caused by the border collision bifurcation. The CMPT control technique provides better stability and reliability than the conventional PWM control technique does, which makes it easy to design a stable and reliable switching converter in industrial electronics. Finally, the parameter conditions corresponding to different periodicity are deducted, which provide a way to get the parameter regions and to the analysis of CMPT-controlled buck converter. The analysis of parameter conditions gives a considerable insight to fulfill the design requirements. R EFERENCES [1] M. Telefus, A. Shteynberg, and M. Ferdowsi, “Pulse train control technique for flyback converter,” IEEE Trans. Power Electron., vol. 19, no. 3, pp. 757–764, May 2004. [2] M. Ferdowsi, A. Emadi, M. Telefus, and A. Shteynberq, “Suitability of pulse train control technique for BIFRED converter,” IEEE Trans. Aerosp. Electron Syst., vol. 41, no. 1, pp. 181–189, Jan. 2005. [3] M. Ferdowsi, A. Emadi, M. Telefus, and A. Shteynberq, “Pulse regulation control technique for flyback converter,” IEEE Trans. Power Electron., vol. 20, no. 4, pp. 798–805, Jul. 2005. [4] M. Qin, J. P. Xu, G. H. Zhou, and Q. B. Mu, “Analysis and comparison of voltage-mode and current-mode pulse train control buck converter,” in Proc. IEEE 4th ICIEA, Xi’an, China, May 2009, pp. 2924–2928. [5] M. Qin and J. P. Xu, “Multiduty ratio modulation technique for switching DC–DC converters operating in discontinuous conduction mode,” IEEE Trans. Ind. Electron., vol. 57, no. 10, pp. 3497–3507, Oct. 2010. [6] M. Qin and J. P. Xu, “Improved pulse regulation control technique for switching DC–DC converters operating in DCM,” IEEE Trans. Ind. Electron., vol. 60, no. 5, pp. 1819–1830, May 2013. [7] J. H. Chen, K. T. Chau, and C. C. Chan, “Analysis of chaos in currentmode-controlled DC drive systems,” IEEE Trans. Ind. Electron., vol. 47, no. 1, pp. 67–76, Feb. 2000. [8] C. K. Tse and M. Di Bernardo, “Complex behavior in switching power converters,” Proc. IEEE, vol. 90, no. 5, pp. 768–781, May 2002. [9] J. P. Wang, B. C. Bao, J. P. Xu, G. H. Zhou, and W. Hu, “Dynamical effects of equivalent series resistance of output capacitor in constant ontime controlled buck converter,” IEEE Trans. Ind. Electron., vol. 60, no. 5, pp. 1759–1768, May 2013. [10] B. C. Bao, G. H. Zhou, J. P. Xu, and Z. Liu, “Unified classification of operation state regions for switching converters with ramp compensation,” IEEE Trans. Power Electron., vol. 26, no. 7, pp. 1968–1975, Jul. 2011. [11] O. Feely and L. O. Chua, “The effect of integrator leak in Σ−Δ modulator,” IEEE Trans. Circuits Syst., vol. 38, no. 11, pp. 1293–1305, Nov. 1991. [12] G. H. Yuan, “Shipboard crane control, simulated data generation, and border-collision bifurcations,” Ph.D. dissertation, Univ. Maryland, College Park, MD, USA, 1997. [13] S. Banerjee, P. Ranjan, and C. Grebogi, “Bifurcation in two-dimensional piecewise smooth maps: Theory and applications in switching circuits,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 47, no. 5, pp. 633–643, May 2000. [14] P. Jain and S. Banerjee, “Border-collision bifurcations in one-dimensional discontinuous maps,” Int. J. Bifur. Chaos, vol. 13, no. 11, pp. 3341–3351, Nov. 2003. [15] S. Kapat, S. Banerjee, and A. Patra, “Discontinuous map analysis of a DC–DC converter governed by pulse skipping modulation,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 57, no. 7, pp. 1793–1801, Jul. 2010. [16] O. Feely, “Discontinuous piecewise-linear discrete-time dynamics: Maps with gaps in electronic systems,” COMPEL, Int. J. Comput. Math. Elect. Electron. Eng., vol. 30, no. 4, pp. 1296–1306, Nov. 2011. [17] Z. T. Zhusubaliyew, E. Mosekilde, and O. O. Yanochkina, “Torusbifurcation mechanisms in a DC/DC converter with pulse widthmodulated control,” IEEE Trans. Power Electron., vol. 26, no. 4, pp. 1270–1279, Apr. 2011. [18] F. Xie, R. Yang, and B. Zhang, “Bifurcation and border collision analysis of voltage-mode-controlled flyback converter based on total ampereturns,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 58, no. 9, pp. 2269– 2280, Sep. 2011.
SHA et al.: EFFECTS OF CIRCUIT PARAMETERS ON DYNAMICS OF CMPT-CONTROLLED BUCK CONVERTER
[19] W. R. Liou, M. L. Yeh, and Y. L. Kuo, “A high efficiency dual-mode buck converter IC for portable applications,” IEEE Trans. Power Electron., vol. 23, no. 2, pp. 667–677, Mar. 2008. [20] S. Kapat, S. Banerjee, and A. Patra, “Voltage controlled pulse skipping modulation: Extension towards the ultra light load,” in Proc. IEEE ISCAS, Taipei, Taiwan, May 2009, pp. 2649–2652.
Jin Sha was born in Shandong, China, in 1987. She received the B.S. degree in electrical engineering and automation from Southwest Jiaotong University, Chengdu, China, in 2009, where she is currently working toward the Ph.D. degree in the School of Electrical Engineering. Her research interests include control technique of switching-mode power supplies and modeling, simulation, and analysis of dynamical behavior of switching dc–dc converters.
Jianping Xu (M’10) received the B.S. and Ph.D. degrees in electronic engineering from the University of Electronic Science and Technology of China, Chengdu, China, in 1984 and 1989, respectively. Since 1989, he has been with the School of Electrical Engineering, Southwest Jiaotong University, Chengdu, China, where he has been a Professor since 1995. From November 1991 to February 1993, he was with the Department of Electrical Engineering, University of Federal Defense Munich, Munich, Germany, as a Visiting Research Fellow. From February 1993 to July 1994, he was with the Department of Electrical Engineering and Computer Science, University of Illinois, Chicago, IL, USA, as a Visiting Scholar. His research interests include modeling, analysis, and control of power electronic systems.
1573
Bocheng Bao received the B.S. and M.S. degrees in electronic engineering from the University of Electronic 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. He has more than 20 years experience in industry and has served several enterprises as Senior Engineer and General Manager. From June 2008 to January 2011, he was a Professor with the School of Electrical and Information Engineering, Jiangsu Teachers University of Technology, Changzhou, China. He is currently a Professor in the School of Information Science and Engineering, Changzhou University, Changzhou. His research interests include bifurcation and chaos, analysis and simulation in power electronic circuits, and nonlinear circuits and systems.
Tiesheng Yan was born in Shanxi, China, in 1981. He received the B.S. degree in mechatronic engineering from Lanzhou Railway University, Lanzhou, China, in 2002 and the M.S. degree in power electronics and electrical transmission from Southwest Jiaotong University, Chengdu, China, in 2005, where he is currently working toward the Ph.D. degree in the School of Electrical Engineering. His research interests include control technique of switching-mode power supplies, modeling and simulation of switching ac–dc converters, power factor correction converters, and the development of new converter topologies.
