Delay effects on the limit cycling behavior in an H-bridge resonant inverter with zero current switching control strategy

Delay effects on the limit cycling behavior in an H-bridge resonant inverter with zero current switching control strategy L. Benaderoa, F. Torresb, A. El Aroudic, C. Olallac, E. Ponce band L. Martinez-Salameroc aDepartament de F´ ısica, Universitat Polit` ecnica de Catalunya, Barcelona, Spain bDepartamento de Matem´ atica Aplicada, Escuela T´ ecnica Superior de Ingenier´ ıa, Universidad de Sevilla, Sevilla, Spain cDepartament d’Enginyeria Electr` onica, El` ectrica i Autom` atica, Universitat Rovira i Virgili, Tarragona, Spain [email protected], [email protected], [email protected], [email protected], [email protected], [email protected] Abstract—In this paper, bifurcations of limit cycles in a H-bridge LC resonant inverter under a zero current switching control strategy with delay in the switching action are analyzed. Mathematical analysis and numerical simulations show that the delay can degrade the quality of the oscillations and even inhibit them. 1. Introduction Resonant inverters are systems in which oscillations in an LC tank circuit are sustained by means of a switching network, thus converting a DC voltage into an AC one. We consider here self-sustained oscillations that are produced by switching the active branch of the bridge whenever the current in the LC tank becomes zero [1, 2, 3]. This zero current switching (ZCS) control strategy has the advantage of minimizing switching losses. A generalized model of a resonant DC-AC H-bridge inverter, which includes the parallel and series implementations, was introduced and its bifurcation scenery was analyzed in [4], without considering the delay. Here, the delay effect on the limit cycling behavior is analyzed from the switched model, and the results are confirmed by direct simulations. In Section 2, a normalized model of the inverter is introduced in the form of a delay differential system with only three parameters, and in Section 3, the bifurcation pattern is described highlighting the influence of the delay. 2. System description and mathematical modeling The circuit diagram of the system under study is depicted in Fig. 1. It consists of a generalized circuit including parasitic resistances in the energy storage elements, which can represent both series and parallel topologies. The following elements can be identified: the input voltage Vg; the output series resistance Ros; the inductor with inductance Land parasitic series resistor rls; the output parallel conductance Gop =1/Rop; the capacitor with capacitance C, parasitic parallel conductance gcp =1/rcp and parasitic series resistance rcs; and the switches S1, S2, S3and S4. The circuit operation is based on an automatically activated switching between two configurations. The switches are driven by the two signals δand its complementary iL S1 + − L vC + − S2 S3S4 + − Vg Crcp to S1and S4 to S2and S3 δ δ rcs rls Rop Ros iL Figure 1: Generalized schematic diagram of an LC resonant inverter. Note that if a resistor is missing, either its corresponding conductance or resistance vanishes. ¯ δ=1−δ. Note that δ=1 when the inductor current iL>0 and δ=0 if iL<0, so that a ZCS control is implemented in a natural way. In that diagram, δ=1 (0) forces the on (off) state in S1and S4and the off (on) state in S2and S3. Note that if the switching is inhibited, the capacitor voltage and the inductor current tend to be constant; specifically, if δ=1 (0), there is an equilibrium point with positive (negative) capacitor voltage and inductor current. Otherwise, a limit cycle is possible when the switching is active. This oscillating regime is built by a suitable aggregation of two orbits, each one being a part of the transient regime toward one of the equilibrium pair. However, this desired objective is only achieved for certain values of the parameters and some initial conditions, as it is shown later. First, a piecewise-linear model for the system shown in Fig. 1 will be obtained by ignoring the delay; for more details, see [4]. Let vCbe the capacitor voltage and iLthe inductor current. By applying KVL and KCL, one gets uVg=LdiL dt +iL(Ros +rls)+icsrcs +vC, iL=ics +(icsrcs +vC)Gop, where ics, that is the current through rcs, is ics =CdvC dt +gcpvC. The variable u=2δ−1 is determined by the control, such that u=1 (−1), that is δ=1 (0), if iL>0 (<0). After some algebra, the following model is obtained d dt vC iL!=A vC iL!+ub,(1) where A=−Gp C κ C −κ L−Rs L,b= 0 Vg L,(2) and the factor κ, the equivalent series resistance Rsand the equivalent parallel conductance Gpare κ=1 1+rcsGop ,Rs=Ros +rls +κrcs,Gp=gcp +κGop. Let det(A) and tr(A) be the determinant and trace of matrix Ain (2). Hence, the natural frequency ω0=√det(A) and the quality factor Q=−ω0/tr(A) of the LC tank are ω0=sRsGp+κ2 LC ,1 Q =Gp ω0C +Rs ω0L. Below, the model will be expressed in a canonical form with dimensionless parameters. Let βbe a first normalized parameter defined as β=QGp ω0C =1−RsQ ω0L =GpL GpL+CRs .(3) Note that 0 ≤β≤1. The case β=0 (1) arises when Gp=0 (Rs=0), that is the ideal series (parallel) inverter. Assuming Q>1/2, the eigenvalues of matrix Aare p±=−ω0 2Q±iω0s1−1 4Q2=ω0(σ±iν),(4) in which σ=−(2Q)−1and ν=p1−(2Q)−2. Let γbe a second normalized parameter, related to the relative energy losses of the LC tank, defined as γ=σ ν =−1 p4Q2−1 <0.(5) Now, by means of the following change of variables θ=νω0t,x= x1 x2!= κ Vg−LGp CVg 0νω0L Vg vC iL!,(6) system (1) can be reformulated as dx dθ =Ax +ub,(7) where matrix Aand vector bare redefined as A= 0 1 +γ2 −1 2γ!,b= 2βγ 1!.(8) Then, the new matrix Ahas the eigenvalues λ±=γ±i. Figure 2: Plots of βsn(γ), βcc(γ) and βhc(γ), in solid black, dashed red and dash-dotted blue, respectively. The line β= 1 marks the maximum value for this parameter and vertical lines give account of the corresponding minimum values of parameter γto avoid the homoclinic connection, critical crossing and fold bifurcation for any valid value of β. Considering an ideal switching forced by the condition iL=0, the switching function, in accordance to the change of variables (6), is given by the expression h(x)=x2. However, if a delayed switching action is considered, the control signal uis determined by a past state of the system. Let us consider, as an approximation to the problem, a fixed switching time delay Tddue to switches and driving circuitry. Then, the model of our system is the differential equation (7)-(8), together with the delayed switching function h(x,t)=h(x(t−τ)) =x2(t−τ),(9) where the normalized time delay τ=νω0Tdis a third parameter in addition to βand γ. Then, u=1 if h(x(t−τ)) > 0 and u=0 if h(x(t−τ)) <0. Note that system (7)-(9) with a constant value u=±1, has the following equilibria x±=(x1,x2)u=± 1−4βγ2 1+γ2,−2βγ 1+γ2!,(10) which may be constant solutions of the switched system, whenever β > 0, since x2>0. 3. Crossing limit cycle and its bifurcations Apart from the stable equilibria (10), the dynamics of system (7)-(9) can also be oscillatory. Without delay, the only possible stable oscillation is made up by two linked trajectories. However, with delay, more complex oscillations can occur. In the following, the conditions for existence of limit cycles without delay are revisited, and then, the case with delay is addressed. 3.1. Revisiting limit cycles existence without delay Solutions for any of the two linear configurations in (7), starting at x(0), can be expressed as x(θ)= Φ±(θ, x(0)) =φ(θ)x(0) −x±+x±,(11) where φ(θ) is the evolution operator given by φ(θ)=eγθ cos θ−γsin θ(1 +γ2) sin θ −sin θcos θ+γsin θ!. Note that without delay, the configuration changes whenever the orbit crosses the switching manifold Σ = {x=(x1,0),x1∈R}.(12) Furthermore, in the subset of the switching manifold Σs= {x=(x1,0),−1≤x1≤1}, the escaping sliding conditions are satisfied, i.e., the vector field points outward both sides of Σs. Due to this property, the sliding subset Σsplays a relevant role in the existence of unstable sliding cycles. These cycles are boundaries between the region of attraction of the oscillatory dynamics and that of different equilibria. A complete classification of limit cycle configurations for system (7-9), without delay and with parameter β > 0, appears in Theorem 1 in [4]. In such theorem, limit cycle conditions are summarized. For the sake of completeness, such conditions are reproduced below. (a) If 0 < β < βhc(γ) then there exist one stable crossing limit cycle and two unstable sliding limit cycles. (b) If β=βhc(γ) then there exist one stable crossing limit cycle and two homoclinic connections to the origin. (c) If βhc(γ)< β < βcc(γ) then there exist one stable crossing limit cycle and one unstable sliding limit cycle. (d) If β=βcc(γ) then there exist one stable crossing limit cycle and one unstable critical crossing limit cycle. (e) If βsn(γ)<β<βcc(γ) then there exist two crossing limit cycles having opposite stability. (f) If β=βsn(γ) then there is one crossing limit cycle which is semi-stable. (g) If β>βsn(γ) then there are no crossing limit cycles. Consequently, four regions are found in the parameter plane (β, γ) defined by the three functions βsn(γ), βcc(γ) and βhc(γ), which are codimension-one lines corresponding to a smooth fold, also called saddle-node, bifurcation of cycles, to a critical crossing-sliding cycle and to a double homoclinic saddle connection, respectively. These functions are depicted in Fig. 2, and some cases for limit sets are shown in Fig. 3. 3.2. Limit cycles bifurcations under delay action Although the switching is theoretically induced at time instants such that the orbit crosses Σ, actually the transition between the two configurations is delayed due to the non ideal features of the switches. In Fig. 4(a), the stable limit cycle has been computed for three values of the delay, resulting in smaller cycles when the delay is increased, until this closed orbit collides with Σfor the highest value of τ. (a) β<βhc(γ) (b) β=βhc(γ) (c) βhc(γ)< β < βcc(γ) (d) β=βcc(γ) (e) βcc(γ)< β < βsn(γ) (f) β=βsn(γ) Figure 3: Limit sets for parameters τ=0, γfixed and β given in the caption. The equilibrium points and the outer stable limit cycle are depicted in blue color and the unstable cycles in red color. The black cycle in (f) is the non hyperbolic limit cycle at the fold bifurcation. The red straight line corresponds to the sliding set Σs. (a) (x1,x2) plane (b) xc 1,−xs 1(τ) (c) xs 2(τ) Figure 4: (a) Limit cycles for τ∈ {0,1,2.252586...}. (b) Crossing and switching values of x1and (c) switching values of x2, versus τ. Fixed parameters β=1, γ =−0.15. Dots in (b-c) are from simulations in the steady state. Let xs=(xs 1,xs 2) be the point of the orbit at the actual switching instant, xc=(xc 1,0) be the point where the orbit crosses Σ, and θsbe the half-period of the cycle. Taking into account the vector field symmetry, limit cycles with delay can be obtained by solving the equation set Φ+(θs,xs)=−xs,Φ+(θs−τ, xs)=xc.(13) Note that the four algebraic equations in (13) must be solved for the set of four unknowns {θs,xs 1,xs 2,xc 1}. In Fig. 4(b-c), xc 1,−xs 1and xs 2have been represented versus the delay τ. The curves have been computed from (13), and the dots have been obtained by long time running simulations. The fixed parameters βand γused in the quoted figure are such that, when increasing τ, the standard limit cycle is annihilated by a border collision with the switching manifold Σ, as it can be appreciated in Fig. 4(a). This critical value of τcan be computed by forcing xs 2=0, taking τas the forth unknown in (13). Notice that, in the border collision condition, xs∈Σs. There is, however, another possible bifurcation for the stable limit cycle, when delay is increased, which can be understood as an evolution of the smooth saddle-node bifurcation of cycles mentioned in the above section. In this case, for some critical values of the parameter set {β, γ, τ}, a nonhyperbolic limit cycle exists, such that when decreasing τ, two crossing limit cycles, the outer stable and the inner unstable, are given. Simulations show that this bifurcation, including the switching delay, has the same qualitative features than the ideal case (without delay) analyzed in [4]. In Fig. 5(a-b), the two bifurcations for crossing limit cycles have been represented in the plane (γ, τ), with β= 1. The red (blue) line corresponds to the smooth (bordercollision) case. Note the existence of a codimension-two bifurcation point for the intersection of those codimensionone lines. The critical values γ1=−0.26239683 and τ1= 0.48856227 have been determined for this point for β=1. Note also that if γ > γ1(γ < γ1), the border collision takes place for the stable (unstable) crossing cycle. Figure 5(c) is a diagram similar to that in Fig. 4(b), but with a more negative value of γ, in order to have a saddlenode bifurcation of cycles. In this diagram, blue (red) lines stand for stable (unstable) cycles. Here, the red line ends at a border collision of the unstable crossing cycle, similar the one explained above for the stable one. 3.3. Some estimation of a safe value for delay The delay action introduces an important degree of complexity so that the determination of the boundary between the regions of attraction of the desired stable limit cycle and the equilibrium points is a difficult task. Although a formal analysis of this subject is out of the scope of this paper, we approach the problem by determining a critical value of the delay such that an orbit starting at the origin, x0=(0,0), reaches the sliding subset of the switching manifold, Σs. This value is determined by the set of equations, with unknowns {ˆ θ, ˆτ, ˆxs 1,ˆxc 1}, Φ+(ˆ θ, x0)=ˆ xs,Φ+(ˆ θ−ˆτ, x0)=ˆ xc, where ˆτis the critical value of τ,ˆ θis the flight time from the origin to a point ˆ xs∈Σs, with 0 <ˆxs 1<1, and ˆ xc∈Σ, is the crossing point of the orbit, with ˆxc 1>1. The dashed green line in Fig. 5(a) corresponds to ˆτ(γ) with β=1. Note that ˆτβ−1 hc (1)=0. From this approach, we conclude that it is advisable a parameter set of the system below the dashed green line in Fig. 5(a) diagram, in order to guarantee the oscillatory dynamics under starting conditions near the origin of the phase state. 4. Conclusions A smooth fold and a border collision bifurcations of cycles have been found for a H-bridge self resonant inverter if (a) τ(γ) (b) τ(γ) (c) −xc 1,−xs 1(τ) Figure 5: (a-b) Saddle-node in red color and border collision in blue color bifurcations of crossing limit cycles in the parameter plane (γ, τ) with fixed parameter β=1, and (b) is a zoom. (c) Crossing and switching values of x1versus parameter τin its valid interval, with fixed parameters β=1, γ =−0.27; the values for the stable and for the unstable cycles are in blue and red colors respectively. the switching delay is considered. The amplitude of the oscillation and some curves in the parameter space for these bifurcations have been computed. Acknowledgments This work has been sponsored by the Spanish Agencia Estatal de Investigaci´ on (AEI) and the Fondo Europeo de Desarrollo Regional (FEDER) under grant DPI201784572-C2-1-R, by the Spanish Ministerio de Ciencia e Innovaci´ on under grant MTM2015-65608-P, and by the Consejer´ ıa de Econom´ ıa y Conocimiento de la Junta de Andaluc´ ıa under grant P12-FQM-1658. References [1] C. S. Tang, Y. Sun, Y. G. Su, S. K. Nguang and A. P. Hu,“Determining multiple steady-state ZCS operating points of a switch-mode contactless power transfer system,” IEEE Transactions on Power Electronics, vol. 24, no. 2, pp. 416–425, 2009. [2] R. Bonache-Samaniego, C. Olalla and L. Mart´ ınezSalamero, “Design of self-oscillating resonant converters based on a variable structure systems approach,” IET Power Electronics, vol. 57, no. 1, pp. 111–119, 2015. [3] R. Bonache-Samaniego, C. Olalla and L. 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