Abstract
As a new topology, Zsource converter can be potentially applied in emerging energy power fields. In contrast to traditional converters, Zsource converter under dualloop control exhibits particularity, but this property has been rarely investigated. In this study, the bifurcation and chaos phenomena are investigated in the Zsource converter under dualloop control. A dynamic model with a shootthrough state of the Zsource converter is initially derived. The stability of fixed points is then investigated. The parameter region of a steadystate operation is subsequently schemed. The evolution and mechanism of bifurcations and chaos are also analyzed in detail. Results show that the system is intermittently stable at relatively low ${V}_{ref}$, as ${V}_{ref}$ gradually increases, the system enters a bifurcation state and then exhibits chaos.
1. Introduction
As a new type of power converter, a Zsource converter, which can achieve singlestage buck/boost conversion, has been developed [1]; this converter can be potentially applied in the field of newgeneration energy because of its wide input voltage range [23]. For instance, the Zsource converter switches periodically under the influence of a control system, causing changes in system topology periodically; the voltage and current of a system then change between different stable points. Each kind of topology corresponds to a linear system, but this system is generally a piecewise nonlinear system. This characteristic results in a complex nonlinear dynamics behavior of physical quantities, such as bifurcation and chao of a system. These phenomena adverselyaffect the work performance of a system, particularly the critical state of sudden collapse, electromagnetic noise, unknown instability. Thesephenomena are external manifestations of the inherent nonlinear behavior of a Zsource converter [4]. A converter functions in a chaotic state and likely results in an unpredicted and uncontrolled system; as such, the performance of this converter is affected seriously. In some cases, a converter is completely unable to function. This condition causes great difficulty in the design and control of a system. Therefore, nonlinear dynamics theory can be applied to analyze and understand the nonlinear process and characteristics of a system. In this way, engineering practice can be effectively performed.
These complex nonlinear phenomena in power electronic systems have prompted many scholars to perform several studies. In the past 20 years, many studies were conducted on the theory of nonlinear dynamics, numerical simulation, and circuit experiment to analyze bifurcation and chaos in this kind of systems. These studies have observed quasiperiodic bifurcation, period doubling bifurcation, border collision bifurcation, intermittent bifurcation, chaos, and other nonlinear phenomena [58] in a DC/DC converter [911], DC/AC inverter [1213] and power factor correction converter (PFC) [14]. Another example is the Zsource converter that exhibits a unique shootthrough stateand achieves singlestate buckboost converting. The nonlinear behavior of this converter is more complex than that of traditional inverters. In anotherstudy, the bifurcation and chaos of a Zsource converter under peakcurrent control (singleloop) [15]. However, dualloop control can achieve multiobject control simulaneously; as such, dualloop control has been widely used. In the present study, a Zsource converter under dualloop control was used as our object, on the basis of the equation of state, the nonlinear behavior of the Zsource converter was modeled using a stroboscopic map method, simulated and analyzedto determine the bifurcation and chaos.
2. Principle of Zsource DC/DC converter
Fig. 1 illustrates the circuit diagram of a Zsource DC/DC converter based on currentmode dualloop control.
In Fig. 1, a twoport network consisting of the split inductors ${L}_{1}$ and ${L}_{2}$ and the capacitors ${C}_{1}$ and ${C}_{2}$ connected in an $X$ shape manner is used, to analyze this diagram, we assume that ${L}_{1}={L}_{2}=L$ and ${C}_{1}={C}_{2}=C$.
Fig. 1Schematic of the Zsource DC/DC converter based on currentmode dualloop control
The schematic of a Zsource DC/DC converter based on dualloop control is shown in Fig. 1, where ${V}_{in}$ is the input voltage, ${V}_{ref}$ is the reference voltage, and ${P}_{e}$ is the error gain of the output voltage. This system mainly involves a clock signal that controls the turnon switch. This clock signal also sets the RS flipflop to 1 at regular intervals. At $Q=$0, the switch is turnoff, and the inductor current ${i}_{L}$ increases until the required ${I}_{ref}$ is reached. Afterward, the inductor inverts the output signal of the at $Q=$ 1. The current reference ${I}_{ref}$ is produced by amplifying the error between ${V}_{ref}$ and the actual output voltage. Currentmode dualloop control can be performed easily and limits the peak current to protect device; therefore, this system can be used in several applications.
On the basis of the shootthrough switch $Q$ states and the diode $D$ states, we can obtain the two working modes (Fig. 2).
Fig. 2DClink equivalent circuit of the Zsource converter
a) Equivalent circuit in shootthrough mode
b) Equivalent circuit in nonshootthrough mode
In working mode 1, the shootthrough state is represented as $Q$ on and $D$ off; the state equation is written as follows:
In working mode 2, the nonshootthrough state is represented as $D$ off and $Q$ on; the state equation is expressed as follows:
The state equation derived from Eqs. (1) and (2) is specified as Eq. (3) and Eq. (4), respectively:
where the state vector $\dot{x}$ is defined as $x=\left[\begin{array}{l}{i}_{L}\\ {u}_{c}\end{array}\right]$, ${i}_{L}$ is the inductor current of the Zsource network, ${u}_{c}$ is the capacitance voltage of the Zsource network, ${r}_{1}$ is resistor parasitics of the inductor, ${R}_{1}$ is the series resistance of capacitance, $R$ is the load resistance, and ${V}_{dc}$ is the output voltage:
${B}_{1}=0$${B}_{2}=\left[\begin{array}{c}1/L\\ 1/C/R\end{array}\right].$
3. Simulation of the nonlinear behavior of a Zsource DC/DC converter
The simulation results (Figs. 35) showed that ${U}_{c}$ is the capacitance voltage of the Zsource network and ${i}_{L}$ is the inductor current of the Zsource network. These figures show the typical waveforms of period1 state, period2 state, and chaotic state.
Fig. 3Typical waveforms of period1 state
a) Capacitance voltage
c) Phaseplane diagram
b) Inductor current
4. Dynamic modeling and analysis of a Zsource DC/DC converter
4.1. Stroboscopic mapping model
${d}_{n}$ is the duty cycle of the $n$th period $T$, where ${t}_{n}=nT$, ${x}_{n}=x\left(nT\right)$; other discrete parameters are defined by similar logic in which the state Eq. (3) and Eq. (4) as well as the discrete model Eq. (6) and Eq. (7) in the nth period are calculated. The discrete mapping equation can be expressed as follows:
$\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}={\varphi}_{1}\left({d}_{n}T\right)x\left({t}_{n}\right),\mathrm{}\mathrm{}\mathrm{}$$nT+{t}_{n}\le t<nT+{d}_{n}T),$
$\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}={\varphi}_{2}\left({\stackrel{}{d}}_{n}T\right)x\left({t}_{n}+{d}_{n}T\right)+{\int}_{0}^{{\stackrel{}{d}}_{n}T}{\varphi}_{2}\left({\stackrel{}{d}}_{n}T\tau \right){B}_{2}{V}_{in}d\mathrm{\tau},\mathrm{}\mathrm{}\mathrm{}\left(nT+{d}_{n}T\le t<\left(n+1\right)T\right),$
where ${\varphi}_{1}\left({d}_{n}T\right)={e}^{{A}_{1}{d}_{n}T}$, ${\varphi}_{2}\left({\stackrel{}{d}}_{n}T\right)={e}^{{A}_{2}{\stackrel{}{d}}_{n}T}$, ${\stackrel{}{d}}_{n}=1{d}_{n}$.
Fig. 4Typical waveforms of period2 state
a) Capacitance voltage
c) Phaseplane diagram
b) Inductor current
Fig. 5Typical waveforms of choatic state
a) Capacitance voltage
c) Phaseplane diagram
b) Inductor current
${A}_{1}$ and ${A}_{2}$ are both invertible matrixes; as a result the integral terms of these equations can be simplified and expressed as follows:
Eq. (8) can be substituted in Eq. (9) and expressed as the discrete mapping Eq. (10):
The switching function can be derived as follows:
$\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}={k}_{1}{\varphi}_{1}\left({d}_{n}T\right){x}_{n}\left[{V}_{ref}{k}_{2}{\varphi}_{1}\left({d}_{n}T\right){x}_{n}\right]\cdot Pe,$
where ${k}_{1}=\left[\text{1,}\text{}\text{0}\right]\text{,}$${k}_{2}=\left[\text{0,}\text{}\text{1}\right]\text{,}$ and $Pe=\text{100}$ at $\sigma ({x}_{n},{d}_{n})=\text{0;}$ as a result, the state of the converter is altered.
Eq. (10) and Eq. (11) can be simultaneously calculated as a nonlinear discrete mapping equation of a Zsource converter.
4.2. Stability analysis based on Jacobian matrix method
As a result of changes in the circuit parameter of a system and the effect of external factors, the system likely operates from a stable working mode to unstable working mode. Jacobian matrix method is used to analyze the local bifurcation in the nonlinear dynamics system. We can then establish the Jacobian matrix based on a fixed point.
Assuming that the controlled system is stabilized at onecycle state, and make the input voltage and reference voltage keep constant, let ${x}_{n}=\widehat{x}+{X}_{Q}$, ${d}_{n}=\widehat{d}+D$, ${X}_{Q}$ and $D$ are the stationary solutions.
The perturbation and linearization of Eq. (5) are then expressed as follows:
At $\sigma ({x}_{n},{d}_{n})=\text{0}$, Eq. (13) is obtained:
Therefore, we can derive Eq. (14) expressed as follows:
Eq. (14) can be substituted to Eq. (12) toobtain Eq. (15):
We can then obtain the Jacobian matrix:
where:
We can set the eigenvalue of Jacobian matrix to $\lambda $. According to the system stability, the system is stable when $\lambda $ is located in the unit circle of a complex plane. Stability is observed when the Jacobian matrix corresponds to the disturbance ratio of two successive iterations (the latter term to the former term). At ratio > 1, the disturbance of the system increases; thus, the system is likely unstable by increasing the iteration number. At ratio < 1, the system is stable. Therefore, the system is likely unstable as the reference voltage is altered.
The periodic solutions ${X}_{Q}$ and $D$ are calculated before the eigenvalue of the Jacobian matrix is obtained. The following solutions are used:
Let ${x}_{n+1}={x}_{n}={X}_{Q}$, $x(nT+{d}_{n}T)={X}_{D}$, and ${d}_{n}=D$ in Eq. (8) and Eq. (9) to obtain the following equations:
Eq. (19) and Eq. (20) are then derived:
The switching function satisfies Eq. (21):
Eq. (19) and Eq. (21) can be simultaneously calculated to obtain the stable periodic solutions ${X}_{Q}$ and $D$; the eigenvalue of the Jacobian matrix can then be obtained to determine the stability of the system.
5. Numerical simulation and result analysis
Our simulation is based on the state equations derived in Section 4. We investigated the dynamics of the state variable by using ${V}_{ref}$ as a bifurcation parameter. The circuit parameters used in our simulation are listed as follows: ${C}_{1}={C}_{2}=C=\text{1,000}\text{}\text{\mu F}$, ${L}_{1}={L}_{2}=L=\text{1}\text{}\text{mH}$, ${V}_{in}=\text{60}\text{}\text{V}$, $R=\text{6}\text{}\text{\Omega}$, ${R}_{1}=\text{0.03}\text{}\text{\Omega}$, ${r}_{1}=\text{0.5}\text{}\text{\Omega}$, and $f=$10 kHz. We obtained a bifurcation diagram (Fig. 6). Fig. 6 summarizes ${u}_{c}$ at the beginning of each switching period as ${V}_{ref}$ increases.
Fig. 6Bifurcation diagram of uc
Fig. 6 also shows that the system exhibits a period1 orbit at relatively low ${V}_{ref}$. This orbit indicates that the system is stable. This orbit should also be stabilized at a widened range. In the Zsource converter under dualloop control, the period1 orbit is intermittent which observed from the partially enlarged detail. As ${V}_{ref}$ increases, the period2 orbit is observed, and the branch point is approximately ${V}_{ref}=\text{30}\text{}\text{V}$. As ${V}_{ref}$ increases, chaos occurs. Therefore, Fig. 6 illustrates the path from stability to chaos.
Fig. 7 shows the effect of variations in ${V}_{ref}$ on the stability of the Zsource converter. The system is initially stable, and the twoJacobian matrixeigenvalues are mapped in the unit circle. As ${V}_{ref}$ is gradually increased, one of the eigenvalues leaves the unit circle. This result indicates a period2 bifurcation, rapidly results in instability and chaos. The eigenvalues are listed in Table 1.
Fig. 7Movement of eigenvalues for a dualloop controlled Zsource converter
Table 1Eigenvalues of different values of the reference voltage Vref for dualloop control
${V}_{ref}$  $D$  ${\lambda}_{1}$  ${\lambda}_{2}$  System state  ${V}_{ref}$  $D$  ${\lambda}_{1}$  ${\lambda}_{2}$  System state 
23  0.4792  0.8084  0.9604  stable  38  0.4979  0.9291  1.0323  unstable 
24  0.4809  0.8184  0.9663  stable  42  0.5008  0.9493  1.0447  unstable 
25  0.4825  0.8284  0.9722  stable  46  0.5037  0.9696  1.0573  unstable 
26  0.4841  0.8384  0.9781  stable  48  0.5052  0.9798  1.0637  unstable 
27  0.4857  0.8485  0.9841  stable  50  0.5066  0.9899  1.0701  unstable 
28  0.4873  0.8585  0.9900  stable  51  0.5081  1.0020  0.2879  unstable 
29  0.4888  0.8686  .9960  stable  52  0.5095  1.0409  0.2602  unstable 
30  0.4904  0.8786  1.0020  bifurcation  53  0.5109  1.0707  0.2344  unstable 
31  0.4919  0.8887  1.0080  unstable  54  0.5124  1.1028  0.2099  unstable 
32  0.4934  0.8988  1.0140  unstable  55  0.5138  1.1466  0.1865  unstable 
34  0.4949  0.9089  1.0201  unstable  57  0.5166  1.1804  0.1422  unstable 
Table 1 shows that $\left{\lambda}_{1}\right$ is approximately equal to 1 at ${V}_{ref}=$30 V. On the basis of numberical calculations, we can obtain the following: at values exceeding ${V}_{ref}=$30 V, ${\lambda}_{1}=\text{0.8786,}$ and ${\lambda}_{2}=\text{1.0020}$, where ${V}_{ref}$ is the critical value of the unstable system, the system experiences a bifurcated and chaotic state. At ${V}_{ref}<$30 V, eigenvalues < 1; as ${V}_{ref}$ increases, $\left{\mathrm{\lambda}}_{1}\right$ moves toward –1. This result indicates the occurrence of period2 bifurcation; as ${V}_{ref}$ continuously increases, $\left{\lambda}_{2}\right$ moves toward 1 at ${V}_{ref}=$ 50 V, which happens jump. The two eigenvalues change along the real axis, and $\left{\mathrm{\lambda}}_{1}\right$ changes more rapidly than $\left{\lambda}_{2}\right$. If any of these eigenvalues is > 1, the system becomes unstable.
As the system is controlled in period –1, the boundaries of a stable region or an unstable region are presented in this section. ${V}_{in}$ and ${V}_{ref}$ are selected as the variations. The operation boundaries (Fig. 8) are derived from the analytical solutions and cyclebycycle simulation. The two results complement each other. The operation boundary provides essential designoriented information in which system parameters are selected systematically.
Fig. 8Stable region boundary with altered parameters
6. Experiment verifications
To verify the theoretical analysis result, we established an experimental circuit prototype of the Zsource converter under dualloop control (Fig. 9). The main circuit of the Zsource converter is shown in Fig. 9(a) and the control signal is generated using RTlab equipment [Fig. 9(b)]. All of the parameters are similar to those in Section 4. The experimental waveforms in Fig. 10 are period1, period2, and chaos respectively. Fig. 10(a) shows stable period1 at ${V}_{ref}=$25 V, Fig. 10(a) shows the stable period1 at ${V}_{ref}=$25 V. Fig. 10(b) shows the stable period2 at ${V}_{ref}=$32 V. Fig. 10(c) shows the chaos phenomenon. The experimental results are consistent with the theoretical analysis. These results also confirm the correctness of the theoretical analysis. In Fig. 10, the oscilloscope channel1 displays the Zsource capacitor voltage ${u}_{c}$, and channel2 shows the Zsource inductance current ${i}_{L}$.
Fig. 9Experimental platform
a) Main circuit
b) RTlab controller
Fig. 10Experimental waveform
a) Period1, ${V}_{ref}=$25 V
b) Period2, ${V}_{ref}=$32 V
c) Chaos, ${V}_{ref}=$67 V
7. Conclusions
The Zsource converter can be potentially applied in emerging energy technologies and distributed generation. In this study, the Zsource DC/DC converter is used as the object under dualloop control. The stroboscopic mapping model is established to analyze the nonlinear behavior of the converter. Simulations and experimental results are presented for verification purposes. The researchers of this study expanded the nonlinear research fields involving converters; the system provides an important theoretical basis and is of practical significance in engineering.
References

Peng Fangzheng Zsource inverter. IEEE Transactions on Industry Applications, Vol. 39, Issue 2, 2003, p. 504510.

Baoming Ge, AbuRub H., Peng Fangzheng, et al. An energystored quasiZSource inverter for application to photovoltaic power system. IEEE Transactions on Industrial Electronics, Vol. 60, Issue 10, 2013, p. 44684481.

Dehghan S. M., Mohamadian M., Varjani A. Y. A new variablespeed wind energy conversion system using permanentmagnet synchronous generator and ZSource Inverter. IEEE Transactions on Energy Conversion, Vol. 24, Issue 3, 2009, p. 714724.

Zhang Bo Study of nonlinear chaotic phenomena of power converters and their applications. Transactions of China Electrotechnical Society, Vol. 20, Issue 12, 2005, p. 16.

Zhusubaliyev Zhanybai T., Soukhoterin E. A., Mosekilde E. Quasiperiodicity and bordercollision bifurcations in a DCDC converter with pulsewidth modulation. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, Vol. 50, Issue 8, 2003, p. 10471057.

Dai Dong, Ma Xikui, Li Xiaofeng Border collisionbifurcations and chaos in a class of piecewise smoothsystems with two boundaries. Acta Physica Sinica, Vol. 52, Issue 11, 2003, p. 27292736.

Zhang Hao, Ma Xikui, XueBianling, Liu Weizeng Study of intermittent bifurcations and chaos in boost. Chaos, Solitons and Fractals, Vol. 23, 2005, p. 431444.

Xie Fan, Yang Ru, Zhang Bo Bifurcation and border collision analysis of voltagemodecontrolled flyback converter based on total ampereturns. IEEE Transactions on Circuits and Systems I: Regular Papers, Vol. 58, Issue 9, 2011, p. 22692280.

Premalatha L., Vanaja Ranjan P. Spectral analysis of DCDC buck converter with chaotic dynamics. Annual IEEE INDICON, 2005, p. 605608.

Basak B., Parui S. Exploration of bifurcation and chaos in buck converter supplied from a rectifier. IEEE Transactions on Power Electronics, Vol. 25, Issue 6, 2010, p. 15561564.

Deivasundari P., Uma G., Poovizhi R. Analysis and experimental verification of Hopf bifurcation in a solar photovoltaic powered hysteresis currentcontrolled cascadedboost converter. IET Power Electronics, Vol. 6, Issue 4, 2013, p. 763773.

FeiHu Hsieh, HenKung Wang, PoLun Chang, HsuanChiang Wu Nonlinear dynamic behaviors in voltagemode controlled singlephase halfbridge inverters via varying proportional gain. International Conference on Machine Learning and Cybernetics, 2011, p. 12741278.

Iu H. H. C., Robert B. Control of chaos in a PWM currentmode Hbridge inverter using timedelayed feedback. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, Vol. 50, Issue 8, 2003, p. 11251129.

Meng Huang, Tse C. K., SiuChung Wong, Cheng Wan, Xinbo Ruan Lowfrequency Hopf bifurcation and its effects on stability margin in threephase PFC power supplies connected to nonideal power grid. IEEE Transactions on Circuits and Systems I: Regular Papers, Vol. 60, Issue 12, 2013, p. 33283340.

Chen Yan, Zheng Yong Nonlinear behavior analysis of Zsource DC/DC converter based on current control. Journal of Vibroengineering, Vol. 15, Issue 3, 2013, p. 15761584.
About this article
This study is supported by the Basic and Frontier Research Program of Chongqing Municipality (Grant No. cstc2013jcyjA0531) and the Science and Technology Special Foundation of Banan District of Chongqing (Grant No. 2012Q120).