Figure 2(a) shows a buck-boost converter, where v_g is the input voltage, v_o is the output voltage, R_C is the equivalent series resistance (ESR) of output capacitor C, v_C is the capacitor voltage, v_ESR is the voltage across the ESR, i_o, i_C, and i_L are load current, capacitor current, and inductor current, respectively. When the buck-boost converter operates in continuous conduction mode (CCM), its main steady-state waveforms are shown in Fig. 2(b) if the large ESR R_C is used.

Fig. 2.

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	Fig. 2. (a) Buck-boost converter and (b) its steady-state waveforms with large ESR.

In Fig. 2(a), the output voltage is the sum of v_C and v_ESR. If large ESR R_C is used, the output voltage ripple is dominated by the ripple voltage across the ESR and thus looks piece-wise linearly. As shown in Fig. 2(b), when the switch S₁ is ON, i_L increases with the slope of m₁ and C is discharged to supply the power to the load, which makes output voltage decrease. Due to the negative polarity of output voltage, output voltage in Fig. 2(b) exhibits negatively increasing. When the switch S₁ is OFF, i_L decreases linearly with increasing time t, C is charged, L and C supply the power to the load. In this duration, the output voltage also exhibits a negative increase due to a large ESR and the negative polarity of the output voltage. There exist two jump voltages in each switching cycle T_s because of the existence of ESR.

In the buck converter with peak voltage-mode control, the switch S₁ is turned on at the beginning of each switching cycle and turned off after the positive output voltage is increased to a control signal. For the buck-boost converter, however, after the switch S₁ is turned on, the negative output voltage increases, which means that the positive output voltage does not increase but decreases if the polarity of output voltage is inverted. Therefore, the peak voltage-mode control cannot be used for controlling the buck-boost converter. For the valley voltage-mode control, the switch S₁ is turned off at the beginning of each switching cycle and turned on after the positive output voltage decreases to a control signal. It is just the case for the buck-boost converter during the switch S₁ being in the OFF state (Fig. 2(b)) if the polarity of output voltage is inverted. Therefore, valley voltage-mode control can be applied to the buck-boost converter.

According to the polarity of the reference voltage, one valley voltage-mode controller for the buck-boost converter can be proposed. Figure 3(a) shows a valley voltage-mode controller with negative reference voltage, where the controller consists of an error amplifier, a comparator, a latch and a clock clk, K_v is the output voltage sensing coefficient (K_v > 0), and −V_ref is the reference voltage (V_ref > 0). It is noted that if the reference voltage is positive, i.e., V_ref < 0, the output voltage sensing coefficient must be negative, i.e., K_v < 0, and the two inputs of the comparator should be interchanged. The corresponding operation waveforms are shown in Fig. 3(b). It can be seen from Figs. 3(a) and 3(b) that the output voltage is used to generate the ramp signal v_s. The control objective of the valley voltage-mode controlled buck-boost converter is to make the negative peak value of v_s follow V_ref.

Fig. 3.

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	Fig. 3. (a) Valley voltage-mode controller for buck-boost converter and (b) corresponding operation waveforms.

At the beginning of every switching cycle, the clock resets the latch and makes the driver signal v_Q be low level, the switch S₁ is turned off, which causes i_L to decrease linearly from the initial value. The inductor current ripple Δi_L will flow through C when S₁ is OFF. Under the condition of large ESR, the voltage ripple of v_s is approximately equal to K_vΔi_LR_C. When v_s with negative polarity increases to V_ref, the output of the comparator sets the latch and makes the driver signal v_Q be high level, S₁ is turned on, which makes i_L increase and v_s (with negative polarity) increase till the end of the present switching cycle.

To investigate the dynamics of a valley voltage-mode controlled buck-boost converter, an especial sampled-data modeling is performed by assuming that the increasing and decreasing slopes of the inductor current are constant during each switching cycle. This condition is reasonable in almost all practical designs because the output voltage ripple has a trivial influence on the slope of the inductor current. The capacitor voltage v_C and inductor current i_L at the beginning of the nth switching cycle are denoted as v_n and i_n. When the switch S₁ is turned on, inductor current i_n+k can be obtained easily from Fig. 3(b) as follows: 1 where t_off is the off time duration of switch S₁ in the nth switching cycle and m₂ = −v_o/L is the decreasing slope of i_L. From Fig. 2, during the off time duration of switch S₁, the capacitor current i_C is equal to −(i_L + i_o). Therefore, the capacitor voltage v_n+k can be derived as 2 Similarly, i_L and v_C at the end of the nth switching cycle can be expressed as 3 4 where m₁ = v_g/L is the increasing slope of i_L.

From Fig. 3(b), the sensed output voltage v_o(n + k) at the turn-on instant of the switch S₁ is equal to the reference voltage V_ref, i.e., K_vv_o(n + k) = V_ref. Therefore, the control constraint can be expressed as follows: 5

Significantly, it is found that instability exists in the valley voltage-mode controlled buck-boost converter when duty ratio D is less than 0.5 (D < 0.5), which is the same as that in the valley V² controlled boost converter.^[6] Therefore, the dynamical analysis in this paper mainly focuses on the case of D > 0.5. Equations (1)–(5) constitute a complete two-dimensional sampled-data model of the valley voltage-mode controlled buck-boost converter, based on which, when input voltage v_g = 4 V, V_ref = 0.602 V, K_v = 0.1, V_c = −0.6 V, T_s = 20 μs, L = 20 μH, C = 940 μF, and R = 3 Ω keep constant while R_C is varied from 20 mΩ to 10 mΩ, bifurcation diagrams of inductor current and output voltage of the valley voltage-mode controlled buck-boost converter are obtained, as shown in Fig. 4.

Fig. 4.

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	Fig. 4. (color online) Bifurcation diagrams of (a) iL versus RC and (b) vo versus RC.

From Fig. 4, it is shown that the valley voltage-mode controlled buck-boost converter has complex dynamical behaviors. With the decrease of R_C, the orbit of the converter is transformed from steady period-1 state to subharmonic oscillation (period-2 state), owing to the first period-doubling bifurcation at point A where R_C = 19.2 mΩ. As R_C gradually decreases, the converter goes from the period-2 state to the period-4 state due to the second period-doubling bifurcation at point B where R_C = 12.25 mΩ. From Fig. 4, it can also be known that a period-8 state region exists in the bifurcation diagram during R_C = 11.69 mΩ ∼ 11.6 mΩ, owing to the third period-doubling bifurcation taking place at point C where R_C = 11.69 mΩ. As R_C decreases further, the period-8 state disappears and a chaotic orbit appears at R_C = 11.6 mΩ. When R_C decreases to 10.65 mΩ finally, the converter with valley voltage-mode control is invalid.

The following is analyzed in a similar way when C is taken as a bifurcation parameter. By fixing R_C = 19 mΩ, the inductor current bifurcation diagram can be obtained and shown in Fig. 5(a) with C changing from 500 μF to 1000 μF, as well as its output voltage bifurcation diagram drawn in Fig. 5(b).

Fig. 5.

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	Fig. 5. (color online) Bifurcation diagrams with the variation of C: (a) iL versus C; (b) vo versus C.

It is found from Fig. 5 that as the output capacitor C decreases from 1000 μF to 573.9 μF, the orbit of the converter has a transition from steady period-1 state to multi-period state due to period-doubling bifurcation. With C decreasing to 573.9 μF, the orbit enters into a chaotic state. When C ≤ 526.7 μF, the chaotic state disappears, and the voltage-mode controlled buck-boost converter is invalid. In the ranges of C = 950.2 μF–605.2 μF, 605.2 μF–578.3 μF, and 578.3 μF–573.9 μF, the system operates in period-2, period-4, and period-8 due to the first, the second, and the third period-doubling bifurcation, respectively, at the points D, E, and F shown in Fig. 5, where C = 950.2 μF, 605.2 μF, and 578.3 μF.

From the analysis above, it is shown that the valley voltage-mode controlled buck-boost converter with the variations of output-capacitor time-constant R_C and C has the same routes from stable period-1 to period-2, period-4, and period-8 via the first, the second, and the third period-doubling bifurcation, entering into chaos and ultimately being an invalid state.

By investigating the eigenvalues of the Jacobian around the fixed point, the stability of the equilibrium state can be determined. Furthermore, by studying the movements of the eigenvalues of the Jacobian under circuit parameter variations, the stability status such as the occurrence of bifurcations and boundaries of operating regimes can be identified.

The Newton–Raphson method or other numerical algorithms^[20,21] can be used to obtain the eigenvalues of Jacobian at the fixed point X_Q = [i_L v_o]^T by putting x_n+1 = x_n = X_Q. The Jacobian of the sampled-data model around the fixed point X_Q can be derived as 6 where J₁₁ = ∂i_n+1/∂i_n, J₁₂ = ∂i_n+1/∂v_n, J₂₁ = ∂v_n+1/∂i_n, and J₂₂ = ∂v_n+1/∂v_n.

From Eqs. (1)–(5), J_ij in Eq. (6) are given as where Δ = i_n + i_o − m₂(CR_C + t_off).

The eigenvalues can be obtained by solving the characteristic equation in λ 7

Based on the sampled-data model described by Eqs. (1)–(5) and the Jacobian given by Eq. (6), the bifurcation and stability characteristics of the valley voltage-mode controlled buck-boost converter can be obtained by examining the movements of the eigenvalues under the variations of the specific parameters. The movements of the eigenvalues reveal the bifurcation phenomena and the way parameter variations affect the operation of the buck-boost converter with valley voltage-mode control. If all eigenvalues of the Jacobian evaluated at the fixed point are inside a unit circle, the buck-boost converter is stable. Any movements of the eigenvalues crossing from the interior of the unit circle to the exterior indicate a loss of stability of the crossing point.

Here, special attention should be paid to the loci under R_C and C variation. Listed in Tables 1 and 2 are the eigenvalues for different values of R_C and C respectively. Figure 6 illustrates the loci movements of the eigenvalues when (a) C = 940 μF and R_C varies from 19.5 mΩ to 18.9 mΩ and (b) R_C = 19 mΩ, and C varies from 960 μF to 930 μF. The arrows of the locus movements indicate the directions of movements of the eigenvalues with decreasing R_C and C.

Table 1.

Table 1.

Table 1. Eigenvalues for different ESRs. . RC/mΩ Eigenvalues Remarks 19.17 –0.9971, 0.4483 CCM period-1 19.18 –0.9979, 0.4480 CCM period-1 19.19 –0.9989, 0.4478 CCM period-1 19.20 –1.0001, 0.4476 period-doubling bifurcation 19.21 –1.0015, 0.4474 CCM period-2 19.22 –1.0034, 0.4473 CCM period-2

Table 1. Eigenvalues for different ESRs. .

Table 2.

Table 2.

Table 2. Eigenvalues for different output capacitors. . C/μF Eigenvalues Remarks 950.95 –0.9986, 0.4480 CCM period-1 950.70 –0.9992, 0.4479 CCM period-1 950.45 –0.9998, 0.4477 CCM period-1 950.20 –1.0005, 0.4476 period-doubling bifurcation 949.50 –1.0012, 0.4475 CCM period-2 947.00 –1.0021, 0.4474 CCM period-2

Table 2. Eigenvalues for different output capacitors. .

Fig. 6.

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	Fig. 6. (color online) Movements of eigenvalues for valley voltage-mode controlled buck-boost converter: (a) loci of eigenvalues when C = 940 μF and RC varies from 19.5 mΩ to 18.9 mΩ; (b) loci of eigenvalues when RC = 19 mΩ and C varies from 960 μF to 930 μF. Arrows indicate the directions of movements of the eigenvalues with decreasing RC and C.

The two Lyapunov exponents of the two-dimensional system are represented as^[22,23] 8 and the maximal Lyapunov exponent is given by 9 Here, eig(J_n J_n–1 ⋯ J₁) is the function to obtain the eigenvalues of (J_n J_n–1 ⋯ J₁) and max(λ_m1, λ_m2) is the function to acquire the maximal value between λ_m1 and λ_m2.

By using Eqs. (6), (8), and (9), the maximal Lyapunov exponents corresponding to Figs. 4 and 5 are shown in Figs. 7(a) and 7(b), respectively. From Fig. 7(a), the maximal Lyapunov exponent vanishes as R_C ≤ 10.65 mΩ, the valley voltage-mode controlled buck-boost converter is invalid. As R_C increases, the maximal Lyapunov exponent becomes positive from zero when 10.65 mΩ < R_C ≤ 11.6 mΩ, the converter is in a chaotic state. Compared with the scenario in the bifurcation diagram, the maximal Lyapunov exponent has dramatic changes at some points where chaotic orbits jump to the periodic orbits, which indicates that multiple periodic windows exist in the chaotic region. At the points A, B, and C shown in Fig. 7(a), where R_C = 19.2 mΩ, 12.25 mΩ, and 11.69 mΩ, respectively, the maximal Lyapunov exponent reaches zero from the negative, which means that the converter undergoes the period-doubling bifurcation, corresponding to the points A, B, and C of the bifurcation diagram shown in Fig. 4.

Fig. 7.

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	Fig. 7. (color online) Maximal Lyapunov exponents for valley voltage-mode controlled buck-boost converter when (a) RC varies from 10 mΩ to 20 mΩ and (b) C varies from 500 μF to 1000 μF.

Similarly, in Fig. 7(b), when C > 950.2 μF, 605.2 μF < C ≤ 950.2 μF and 578.3 μF < C ≤ 605.2 μF, the converter is in a periodic state with CCM operation, the maximal Lyapunov exponent becomes negative. When C = 950.2 μF, C = 605.2 μF, and C = 578.3 μF, the maximal Lyapunov exponent equals zero, which means that the converter exhibits a period-doubling bifurcation, corresponding to the points D, E, and F of the bifurcation diagram shown in Fig. 5. In the range of 526.7 μF–573.9 μF, the maximal Lyapunov exponent changes from negative to positive, which indicates the converter is in a chaotic state. When C ≤ 526.7 μF, the maximal Lyapunov exponent equals zero, the corresponding dynamical behavior of the buck-boost converter with valley voltage-mode control is invalid as shown in Fig. 4.

According to the above discussion of bifurcation behavior in Subsection 3.1, it is found that the valley voltage-mode controlled buck-boost converter with the variations of R_C and C presents the same dynamical behavior, including the transition from period-doubling bifurcation to chaos, and to invalid control finally.

In Subsection 3.2, it is mentioned that the variations of eigenvalues can evaluate the parameter range of the converter working in a stable state. When the eigenvalue of first iteration map x_n+1 = f(x_n) is equal to −1, the first period-doubling bifurcation occurs and drives the system entering into period-2 from period-1.

From Eq. (7), the eigenvalues λ_1,2 of the characteristic equation for the sampled-data modeling can be expressed as 10 Hence, by putting min(λ₁, λ₂) = −1 and substituting Eq. (6) into Eq. (10), the first period-doubling bifurcation boundary as well as the stability boundary τ₁ can be yielded as 11 Equation (11) indicates that the first period-doubling bifurcation of valley voltage-mode controlled buck-boost converter is influenced by output-capacitor time-constant (the product of R_C and C), input voltage v_g, output voltage v_o, inductance L, load resistance R, and the switching period T_s. The system will operate in the stable period-1 region when the left part of Eq. (11) is greater than the right one, i.e., the stable region is located above the boundary τ₁.

While the converter is transferred from steady period-1 state to period-2 state, the second iteration map x_n+2 = f(f(x_n)) = f⁽²⁾ (x_n) is existent. Similarly, making x_n+2 = x_n = X_Q2, the corresponding fixed point X_Q2, the Jacobian and the eigenvalues are obtained. The second period-doubling bifurcation boundary τ₂ can be obtained by letting , equal −1. Using the same method can obtain the third period-doubling bifurcation boundary τ₃.

As shown in Fig. 3(b), when the valley voltage-mode controlled buck-boost converter operates in OFF state, the variation of v_s is positive for the duration of nT_s ∼ (nT_s + t_off). If the variation of v_s is negative for the duration of nT_s ∼ (nT_s + t_off), the comparator cannot set the latch and the driver signal v_Q is always at a low level, S₁ is turned off in the whole switching period. In this condition, the convertor cannot work and the control technology is invalid. According to v_s = K_vv_o = K_v(v_C + v_ESR), we have 12

Supposing that the peak inductor current is I_max during the n-th switching cycle, the inductor current at t₁ can be found to be 13

The output capacitor current i_C = −(i_L + i_o) when S₁ is turned off, therefore the variations of v_C and v_ESR during nT_s ∼ (nT_s + t₁) are 14 15

For the buck-boost converter in CCM, the peak inductor current can be expressed as 16 Combining Eqs. (12) and (14)–(16), equation (12) can be rewritten as 17 Because K_v is positive and v_o is negative, by considering t₁ = t_off in the whole OFF state, equation (17) can be simplified into 18 From Eq. (18), we know if the valley voltage-mode controlled buck-boost converter can work normally during one switching cycle, the output-capacitor time-constant must satisfy Eq. (18). Otherwise, the valley voltage-mode control of the buck-boost converter will be invalid. Thus, we can obtain the invalidated boundary τ₄ 19 The valley voltage-mode control for the buck-boost converter is invalid when the left part of Eq. (19) is less than the right one. The region located below the boundary τ₄ which should be avoided in practical design, is called a forbidden region.

Considering the output-capacitor time-constant parameters with R_C = 10–20 mΩ and C = 500–1000 μF, and the other circuit parameters are chosen to be the same as those in Fig. 4, the operating-state regions of R_C and C can be obtained from Fig. 8. Figure 8(a) shows the operation spaces divided by boundary equations. The two-parameter dynamic behavior distribution divided by the sampled-data model as shown in Fig. 8(b), where the higher periodicities are depicted in grey color, the white area means low period, and the darkest shade refers to chaos.

Fig. 8.

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	Fig. 8. (color online) Operation spaces varying with output-capacitor time-constant CRC: (a) regions divided by boundary equations; (b) regions divided by sampled-data model.

Comparing Fig. 8(a) with Fig. 8(b), it is easily noted that the operation spaces divided by boundary equations are consistent with the regions divided by the sample data model. There exist five operation state regions, i.e., CCM period-1, CCM period-2, CCM period-4, CCM period-8 and chaos, and the forbidden region, of the valley voltage-mode controlled buck-boost converter over the parameter space which can be divided by four stability boundaries τ₁–τ₄. The CCM period-8 and chaotic region contains a CCM period-8 region which is so weak that the change of grey level is obscure.

By choosing a horizontal line with C = 940 μF in Fig. 8(b), the bifurcation process will be obtained when the R_C varies among 10 mΩ–20 mΩ, which is shown in Fig. 4. In the same way, the bifurcation process while the C varies among 500 μF–1000 μF will be obtained via a vertical line with R_C = 19 mΩ in Fig. 8(b), the corresponding bifurcation diagrams are observed from Fig. 5. The period bifurcation points A–C in Fig. 4 and D–F in Fig. 5 are in complete agreement with the results in Fig. 8, which verifies the correctness and feasibility of operating spaces.

As shown in Fig. 8, the different operation state regions of the converter can be demonstrated clearly. It is found that the system is stable when the output-capacitor time-constant CR_C is relatively large, unstable and even invalid when CR_C is small. Using such operation spaces the designer can place the nominal operating point away from behaviour boundaries, especially from the forbidden region.

Time-domain waveforms and phase portraits are usually used to visualize nonlinear phenomena, which can also be used to validate the correctness of the sampled-data model. According to Figs. 2 and 3(a), a circuit simulation model is built by using PSIM software. The circuit parameters are chosen to be the same as those in Fig. 4. Figure 9 shows simulation results of the time-domain waveforms (steady-state output voltage, sensed output voltage, and inductor current) and the phase portraits (inductor current versus output voltage) of the valley voltage-mode controlled buck-boost converter with different ESRs.

Fig. 9.

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	Fig. 9. (color online) Simulation results of valley voltage-mode controlled buck-boost converter with different ESRs: (a) RC = 20 mΩ; (b) RC = 16 mΩ; (c) RC = 11 mΩ.

By substituting the circuit simulation parameters into Eqs. (11) and (19), two critical output capacitor time-constants of the converter state transiting can be gained. The converter will move from steady period-1 state to period-2 state at R_C = 19.15 mΩ calculated from Eq. (11). The valley voltage-mode controlled buck-boost converter is invalid at R_C = 10.64 mΩ calculated from Eq. (19). When R_C = 20 mΩ > 19.15 mΩ, which means that the output capacitor time constant is greater than the right part of Eq. (11), the valley voltage-mode controlled buck-boost converter operates in CCM and stable period-1 state as shown in Fig. 9(a1), the output voltage of the buck-boost converter is negative and looks piece-wise linear. From Figs. 9(b1) and 9(c1), it can be seen clearly that the converter is in period-2 state with R_C = 16 mΩ, and in chaos with R_C = 11 mΩ separately. Their corresponding phase portraits are shown in Figs. 9(a2), 9(b2), and 9(c2). It can be seen that the stable state is transformed into an unstable state by using a relatively small output capacitor ESR, which is consistent with the bifurcation diagram in Fig. 4 and the operation space distribution in Fig. 8.

To validate the invalid region mentioned in bifurcation diagrams, valley voltage-mode controlled buck-boost converters with different ESRs, which operate in CCM period-1 and invalid state, are further investigated. The time-domain simulation waveforms of output voltage and inductor current under ESR variation are shown in Fig. 10. The system operates in CCM period-1 with R_C = 20 mΩ for the durations of 20 ms–30 ms. At time t = 30 ms, through changing R_C = 10.5 mΩ (which is smaller than 10.65 mΩ in Fig. 4, and also located in the forbidden region in Fig. 8(a)), the inductor current descends to zero quickly while the output voltage is gradually raised to zero, which means that the valley voltage-mode controlled buck-boost converter is invalid and thus the converter does not work.

Fig. 10.

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	Fig. 10. Simulation waveforms of output voltage and inductor current under ESR variation.

To verify the theoretical analysis and simulation results given earlier and to illustrate the effect of output capacitor time-constant, the experimental studies of the voltage-mode controlled buck-boost converter are performed with the same circuit parameters: v_g = 4 V, V_ref = 0.602 V, K_v = 0.1, L = 40 μH, C = 940 μF, R = 6 Ω, and T_s = 20 μs. A photograph of the experimental set-up is shown in Fig. 11, where switch S₁ is the N-channel power MOSFET (IRF740), switch S₂ is the diode (ES4J), the comparator is high-speed LM319, the flip-flop latch consists of 74LS02, and the control signals are generated by the driver A3120, respectively.

Fig. 11.

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	Fig. 11. (color online) Photograph of the experimental set-up.

Figure 12(a) shows that for R_C = 20 mΩ, the converter operates in the period-1 state. For R_C = 16 mΩ and R_C = 11 mΩ, the converter operates in the period-2 state and chaos, as shown in Figs. 12(b) and 12(c). These experimental results are consistent with the theoretical analysis and simulation results. In hardware experiment, at the moment when the switch turns on or off, there are some spikes existing in the output voltage due to many parasitic parameters in circuit components.

Fig. 12.

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	Fig. 12. (color online) Experimental results of valley voltage-mode controlled buck-boost converter with different ESRs: (a) RC = 20 mΩ; (b) RC = 16 mΩ; (c) RC = 11 mΩ.