Slope Compensation Design for a Peak Current-Mode Controlled Boost-Flyback Converter

Juan-Guillermo Muñoz 1, Guillermo Gallo 2, Fabiola Angulo 1,* and Gustavo Osorio 1 1 Departamento de Ingeniería Eléctrica, Electrónica y Computación, Percepción y Control Inteligente, Facultad de Ingeniería y Arquitectura, Universidad Nacional de Colombia—Sede Manizales, Bloque Q, Campus La Nubia, Manizales 170003, Colombia; jgmunozc@unal.edu.co (J.-G.M.); gaosoriol@unal.edu.co (G.O.) 2 Departamento de Ingeniería Electrónica y Telecomunicaciones, Automática, Electrónica y Ciencias Computacionales (AE&CC), Instituto Tecnológico Metropolitano, Medellín 050013, Colombia; ggalloh@unal.edu.co * Correspondence: fangulog@unal.edu.co


Introduction
The main purpose of power converters is to change the level voltage. Currently, this task is achieved by controlling a converter through pulse width modulation (PWM) such that the system is described by a set of dynamic equations. Power converters can be modeled as a piecewise linear dynamic system [1][2][3], and all exhibit a plethora of nonlinear phenomena depending on the parameter values used. Such behaviors, which are currently being examined at length [4][5][6][7], include period-doubling bifurcations, subharmonics and chaos [2,8,9].
The main goal of a converter is usually to contribute a load with a desired voltage; in this sense, it is important to compute and analyze the stability of the Period-1 orbit and to study its complex dynamics (a complete revision of stability analysis methods applied to power converters can be found in [10]). The behaviors of a power converter are often determined by plotting bifurcation diagrams [4,5,11,12] using the Poincaré map [3]. In these diagrams, as a parameter value changes, the Poincaré map of the steady state is plotted. The stability of the Period-1 orbit is also analyzed by presenting the Poincaré map as a monodromy matrix such that by analyzing eigenvalues of the monodromy matrix (Floquet multipliers), it is possible to determine the stability of a Period-1 orbit [13]. Several studies also combine bifurcations and monodromy matrix analyses [6,14].
High step-up power converters are some of the main devices used in photovoltaic applications [15][16][17][18][19] due to the low output voltage of solar panels. With such applications, efficiency is a key issue, so single-stage and the coexistence of Period-1, Period-2 and chaotic orbits with varying coupling coefficients has been proven via bifurcation analysis [52]. However, neither analytical expressions for computing the value of the compensation ramp of a peak-current controlled BF converter, nor experimental data that confirm the corresponding analytical expression have been even published.
An analytical expression is first obtained by simplifying the problem. Corresponding results are compared to those derived from three sources. (a) Results can first be derived from analytical expressions computed with a complete model and using the monodromy matrix [13]. In this case, the monodromy matrix is computed analytically, and its eigenvalues are calculated as the parameter varies. The largest absolute value of its eigenvalues is called the LAVE. (b) Numerical results can also be obtained from bifurcation diagrams computed by brute force and (c) from experiments carried out in a lab prototype of 100 W. All results show good agreement.
The rest of this paper is organized as follows. In Section 2, the operation mode of the boost-flyback converter is described, as well as that of peak-current mode control. In Section 3, computations for obtaining the mathematical expression for slope compensation are presented. In Section 4, numerical and experimental results are shown and compared. Numerical results are obtained using the derived formula for a particular converter using parameters similar to those used in the experimental setup, including the nonideal model (internal resistance for certain components). The experimental results are obtained from a 100-watt lab prototype and they are presented and compared to numerical ones. Finally, Section 5 concludes.

Boost-Flyback Converter: Modeling and Control
A peak-current controlled boost-flyback converter is depicted in Figure 1. The boost-flyback components are denoted with black lines while the controller is denoted with gray lines. The boost-flyback consists of two coupled inductors (L p , L s ), two capacitors (C 1 , C 2 ), one MOSFET (S) and two diodes (D 1 , D 2 ). The MOSFET is controlled while the diodes commutate depending on their degree of polarization. As the name implies, the union of a boost and flyback converter achieves high gains and high levels of efficiency, while the stress voltage of semiconductor devices decreases relative to that of a standard flyback [23,53]. Figure 1. Boost-flyback converter with peak current-mode control.
As three semiconductor devices are used, there are eight possible switch configurations or states: E 1 ... E 8 . However, it has been shown that only six states have physical meaning [54], and it has been also proven that the controlled system exhibits a Period-1 orbit switching between four states, as described in Table 1 [51]. A schematic diagram of the steady state current behavior of a Period-1 solution is presented in Figure 2. States E 1 and E 2 are present when the MOSFET is turned on, and states E 3 and E 4 are present when the MOSFET is turned off. Table 1. States of the Period-1 orbit. From E 1 , the system evolves as follows: A change from E 1 to E 2 occurs when i s = 0 at t = t 1 ; the system switches from E 2 to E 3 when the switching condition is satisfied at t = DT, which is referred to as the duty cycle and which corresponds to the ratio between the time at which the MOSFET is turned on and the period T, i.e.,: D = t u=1 /T; E 3 changes to E 4 when i p = 0 (at t = t 2 ), and finally, when t = T, the system returns to E 1 . The set of differential equations describing the Period-1 orbit is shown below: State 2: E 2 , t ∈ (kT + t 1 kT + DT]: where V in is the input voltage, i p and i s are the primary and secondary currents, respectively, V C 1 and V C 2 are the voltages across capacitors C 1 and C 2 and M =k L p L s is the mutual inductance, withk as the coupling coefficient and Peak current-mode control is widely used for the control of power converters [41,47,51]. A general schematic diagram of the boost-flyback converter with the proposed controller is depicted in Figure 1. When peak current-mode control is used, a fixed switching frequency is obtained, and current behaviors are very similar to those shown in Figure 2. At the start of the period, the MOSFET is active, the current i p increases and the current i s declines to i s = 0; at time t 1 , dynamic equations describing the system change while the MOSFET continues on until i p is equal to the reference current I * c at t = DT. At t = DT, switches stop until the next cycle begins. Signal I * c is composed of two parts: the first (denoted as I c ) is provided by a PI controller applied to the output voltage error e = V re f − V out . The second part corresponds to the signal supplied by the compensation ramp V r = A r T mod(t/T). Thus, the reference current can be expressed as: where k p and k i [A/(V.t)] are parameters associated with the PI controller and A r [A]corresponds to the amplitude of the compensation ramp. As a result of the controller, there is only one switching cycle per period. At the start of the period, the switch turns on, and it remains on until switching condition i p = I * c is achieved (the corresponding duty cycle). When i p = I * c , the switch opens and remains open until the next period starts. As sliding is not possible (i.e., there is only one round of commutation per cycle), the switching condition can be expressed as: where D ∈ [0, 1] is the duty cycle.

Slope Compensation Design
To our knowledge, the related literature has not reported on a means of determining the slope of the compensation ramp of a boost-flyback converter, which can be used to stabilize the Period-1 orbit. The objective of this section is to analyze the slopes of currents flowing through inductors to find an analytical expression to determine the slope of a compensation ramp and thus to guarantee the stability of the Period-1 orbit. Figure 3 presents the behavior of currents flowing through primary and secondary coils when the system operates within the Period-1 orbit described by states E 1 , E 2 , E 3 and E 4 . Slopes are clearly marked in the figure.

Assumptions
In the analysis, the following approximations are considered. (i) For all elements and devices, the internal resistances are zero. (ii) The steady-state output of the PI-controller (I c ) is constant, and hence, its derivative is zero. However, as can be seen in the procedure, the constant value is not needed to compute the final expression. (iii) Voltages V C 1 and V C 2 are constant and can be computed as a function of the duty cycle D; V C 1 is the output of the boost component, and V C 2 is the output of the flyback component, taking into account the coupling factor k < 1 (see Appendix A for a complete derivation of the formula. (iv) Finally, all currents can be mathematically expressed as straight lines such that slopes associated with i p include m 1 , m 2 and m 3 , while slopes associated with i s includem 1 ,m 3 andm 4 (see Figure 3). These slopes can be computed from Equations (2)-(5) as follows: In a similar way as the slope compensation in a boost power converter is designed considering the stability of the Period-1 orbit [33], in this paper, we present an analysis of the stability of the Period-1 orbit based on information on current slopes and on conditions that should be met to guarantee the stability of the controlled system. To analyze the stability of the Period-1 orbit, small perturbations are applied at the beginning of the cycle, and its corresponding value is measured at the end of period T. When the magnitude of perturbation increases, the Period-1 orbit is unstable; by contrast, when the magnitude of perturbation decreases, the orbit is stable.

Analysis of Currents in the Primary Coil
At switching time t = DT, a pair of equations is fulfilled in Figure 3: one to its left and the other to its right. When defining the slope of the compensation ramp as m sc = Ar T , at the switching time, the following equation is satisfied: Based on perturbation observed in the initial condition, the last equation can be expressed as follows: By subtracting Equation (10) from (9), we obtain the following: From (11),t In a similar way, the analysis illustrated to the right of the switching time leads to the following equation.
Taking into account the perturbation, this equation is given by: Subtracting (14) from (13),

Analysis of Currents in the Secondary Coil
Now, the expressions for the current i s and its perturbationĩ s (0) are computed. At t = t 1 , they are as follows: and: Subtracting (18) from (17), it is obtained: Replacing (12) in (19), we have:ĩ From this equation,dT can be expressed as: Now, at t = t 2 , the following equation is fulfilled, At the same time t = t 2 , the perturbed equation is: Now, subtracting (22) from (23), we have: Replacing (16) in (24), we obtain: Finally, by replacing Equation (21) in (25), we find an expression that relates to the secondary coil current at the beginning of the cycle with its value shown at the end. This expression is given by: where:

Stability Condition
Then, the stability of the Period-1 orbit is given by the absolute value of α. When |α| > 1, the periodic orbit is unstable; when |α| < 1, it is asymptotically stable; and when |α| = 1, it corresponds to the limit of stability. To guarantee that the system operates within a Period-1 orbit, the slope of the compensation ramp must satisfy the following expression:

Results
Parameters associated with the converter and experiment are presented in Table 2. The first two columns are needed to simulate the system, and the other parameters describe the electronic circuit.

Numerical Results
To compare the results obtained using Equation (28) with the analytical results, we determined the stability of the Period-1 orbit from the saltation matrix associated with switching times [13,55,56]. A complete analysis of the stability and computation of the saltation matrix for this system can be found in [51]. Parameter values used for the simulations and experiments are given in Table 2. Voltages V C 1 and V C 2 are computed from Equation (7); the slopes of straight lines are calculated from Equation (8); and the output voltage V out corresponds to the desired output voltage V re f and |α| = 1. With these data, the desired output voltage varies, and the limit value of slope compensation m sc is obtained. Figure 4a shows results obtained from the proposed approach (see (28)) and V re f ∈ (90, 130) V. Figure 4b presents the exact computation using the saltation matrix. Values of A r exceeding the stability limit guarantee the stability of a Period-1 orbit. In addition, for V re f = 100 V, the limit value of the compensation ramp is close to A r = 1.94 A, and for V re f = 120 V, it is close to A r = 3.25 A (see Figure 4). Figure 4c compares the analytical approach proposed in this paper with the exact value obtained from the saltation matrix; the result is expressed as a percentage. As is shown, the lower the reference voltage, the higher the error value is. In fact, for gain factors greater than six, the approach generates better results. This is due to the assumptions in (7): as the gain factor decreases, the gains of boost and flyback parts cannot be separated. Even more, for gains close to two, only the boost converter works, and the flyback part is voided.  To verify these results, we consider a more complete model of the converter that uses internal resistances of the primary inductor, secondary inductor and MOSFET, as well as the shunt resistance to measure the current (see Table 2). In Figure 5a,c, bifurcation diagrams varying the slope of the ramp are computed while the desired output voltage remains fixed (see Figure 5a for V re f = 100 V and Figure 5c for V re f = 120 V). In Figure 5b,d, the behavior of the largest absolute value of eigenvalues (LAVE) is shown for the same reference voltage values. In these cases, limit values of the compensation ramp are A r = 2.035 A and A r = 3.21 A. These results complement those computed from Equation (27) and Figure 4. Slight displacement is observed between numerical values obtained from equations and from the LAVE due to the assumptions applied.     Using the previous limit values, we can find the ramp slopes to stabilize the Period-1 orbit. To prove the robustness of the controller and to compare the system's behaviors for two values of A r , changes in the load are induced first by applying V re f = 100 V and A r = 1.8 A. When the load is changed, it can be observed that for the full range of load resistance values, the Period-1 orbit is unstable (see Figure 6a). However, when we fix the slope of the ramp at A r = 2.2 A (the limit of stability is close to two), the Period-1 orbit is stable for the full range of load resistance values (see Figure 6b). In the second case, we establish V re f = 120 V and A r = 3 A. When the load is changed, it can be observed that for the full range of load resistance values, the Period-1 orbit is unstable (see Figure 6c). However, when we fix the slope of the ramp at A r = 3.4 A (the limit of stability is close to 3.2), the Period-1 orbit is stable for the full load range, as is shown in Figure 6d.

Experimental Results
To validate the numerical results, an experimental lab prototype that can deliver 100 watts to the load was designed and implemented.The complete design of the circuit is shown in Figure 7. A ferrite core type E is used to design coupled inductors, and the number of turns is calculated using the approach proposed in [57]. Values of the different circuit elements are given in Table 2. The current in the primary coil is measured with non-inductive shunt-resistance r shunt (LTO050FR0100FTE3) and then with an instrumentation amplifier IC 1 ; the output voltage is measured through a voltage divider consisting of R a and R b . The signal generated from the voltage divider feeds amplifier IC 1 . The MOSFET is an IRFP260N with low internal resistance. Finally, two ultrafast diodes RHRP30120 (D 1 and D 2 ) are used.
The controller is applied using operational amplifiers (IC 2 ). The compensation ramp and clock signals are generated using an LM555(IC 4 ). The amplitude of the compensation ramp is adjusted with a span resistor R span , and V B compensates for the offset. Constants k p and k i are associated with the PI controller and are obtained from R 2 , R 3 , R 4 and C 3 . The measured signals are scaled to 0.196 from voltage gains (A g 1 and A g 2 ). Constant G v is given by voltage divider R b /(R a + R b ).
Four experiments employed to validate the results shown in the previous section were carried out. All figures of the experimental results show reference current I * c , primary coil current i p , secondary coil current i s and output voltage V out . Therefore, the output voltage and current in the secondary coil are scaled by a factor of 10. The reference current and the current in the primary coil are scaled by a factor of 0.196, as mentioned above.
Vcc Dis Thr Ctl Gnd Trg Out Rst For V re f = 100 V (the load resistance is fixed at R = 200 Ω; see Table 2), two values of slope compensation are tuned: A r = 1.8 A and A r = 2.2 A. When A r = 1.8 A, the limit set is a Period-2 orbit, as is shown in Figure 8a, but when the ramp compensation increases to A r = 2.2 A, it converts to a Period-1 orbit (Figure 8b). For the second experiment, V re f = 120 V. In a similar way, two values of slope compensation are tuned: A r = 3 A and A r = 3.4 A. The behaviors of I * c , i p , i s and V out are shown in Figure 9a,b. For A r = 3 A, a high-period orbit appears, and for A r = 3.4 A, the Period-1 orbit is stable. These results complement information provided by Equation (28), and the formula can be used to tune the slope of the compensation ramp.

Conclusions
This paper enhances the knowledge of the controller design for a boost-flyback converter, which is currently a topic of study.
To achieve high gains with a stable Period-1 orbit when a boost-flyback converter is used, it is necessary to add a compensation ramp to the design. In this work, an analytical expression for computing the value of the compensation ramp slope is presented and mathematically proven. For gains of greater than six, the approach given in this paper has an error of less than 5%.
In general, the results of the equation derived from our computations agree with those of experiments, with minor discrepancies in exact solutions observed for gains of less than six, mainly because certain assumptions are too strong to apply to the real system, and these are not included in the model for the sake of simplicity. This difference is negligible for high step-up gains, for which our approach offers the major benefit of using a formula that guarantees stability while preventing overcompensation and the use of very complex computations. Acknowledgments: The authors would like to thank Ángel Cid Pastor and Abdelali el Aroudi from GAEIResearch Center, Universitat Rovira i Virgili, Spain, for their assistance in obtaining experimental results.

Conflicts of Interest:
The authors declare no conflict of interest. The founding sponsors had no role in the design of the study; in the collection, analyses or interpretation of data; in the writing of the manuscript; nor in the decision to publish the results.

Appendix A
In this Appendix, the procedure to find the ratio between input and output voltages for a flyback converter when coupling factork is different from zero is presented: The flyback converter operates in two topologies named State 1 and State 2, which are depicted in Figure A1. Voltage equations in primary and secondary coils are given in the general form as: Depending on the state, voltages and currents can be approximated as: such that the average values can be calculated as: Taking into account k < 1, i.e., M Lp = Ls M , we have: to finally find Doing k = 1, it is easy to prove that this ratio is the same as that reported for a non-magnetically coupled flyback converter.