Complex dynamics and nonlinear phenomenon, including low-frequency oscillation, ^{[1, 2]} period-doubling bifurcation, ^{[3, 4]} Hopf bifurcation, ^{[5, 6]} border collision bifurcation and chaos, ^{[7– 12]} exist extensively in switching DC-DC converters which can be regarded as nonlinear time-varying systems, and have a great influence on their performance. Therefore, it is of great significance in both theoretical research and engineering application to investigate the nonlinear phenomenon and the way to stabilize the converters.

The current-mode controlled converters have been widely studied and used due to their fast responses to input variation and easy-to-limit currents.^[13] Recently, the dynamical behaviors of current-mode controlled converters with resistive load have been reported by establishing discrete iterative map models and cycle-by-cycle simulation.^{[14, 15]} The period-doubling bifurcation and border collision in the second-order switching converter were revealed, which is meaningful for dynamics analysis.^[16] For the sake of making converters in stable operation, adding ramp compensation to the feedback control circuit is an efficient way which has received increasing attention. Based on the piece-wise smooth model of the switching converter, the equations of state transition were derived and the dynamics with ramp compensation were studied.^[17] The normalized mapping models of switching converters with ramp compensation were also established in order to illustrate the existence of three operation regions in parameter space.^{[18, 19]} It was reported that the ramp compensation has an influence on the stabilities and operation-mode shifts of converters.^[20] Currently, lots of researches have been carried out on converters with ramp compensation where the output voltage is considered as a constant voltage source, ^{[17– 19]} which can reduce the second-order system to the first-order system, however less attention is paid to the second-order converters with ramp compensation relatively and the influence of voltage feedback loop on their stabilities is ignored.

Switching DC-DC converters with constant current load (CCL) have a potential application in lighting emitting diode (LED), becoming an important research topic.^{[21, 22]} At present, there are a few researches of switching converters with CCL. The unique fast-slow scale instability under current-mode control without output voltage feedback is investigated in the converter with CCL^{[23, 24]} and verified theoretically by the eigenvalues of the Jacobian, ^[25] without discussing the stabilization or stable control of them.

In this paper, we take the peak current-mode controlled buck converter with CCL as a research object, adding ramp compensation and output voltage feedback to the converter, deducing the piece-wise smooth model and discrete iterative map model with two borders, exploring the complex dynamics of this converter and how the slope of ramp compensation and the proportional coefficient of outer voltage loop affect the stabilities.

A peak current-mode controlled buck converter with CCL is shown in Fig.  1, where the buck converter consists of an input voltage source E, a CCL I_o, a switch S_T, a diode S_D, an inductor L, and a capacitor C. The peak current-mode controller contains an inner current loop and an outer voltage loop, where the control pulse is obtained by comparing the inductor current i_L with reference current i_ref, which is composed of a constant current I_ref, a ramp compensation current, and the output voltage feedback. To maintain the order of the system, an error amplifier with the proportional component is considered in the output voltage feedback loop, where k is the proportional coefficient.

Fig.  1.

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	Fig.  1. Buck converter with CCL under peak current-mode control.

The operation principle of the system can be briefly described as follows. The flip– flop latch is set at the beginning of each switching cycle by the clock signal and the switch S_T is turned on. When the inductor current i_L increases linearly to the reference current i_ref, the switch S_T is turned off and the inductor current falls linearly until the end of the current switching cycle. If the inductor current i_L drops and remains zero during the off state of S_T, the converter operates in discontinuous conduction mode (DCM), otherwise in continuous conduction mode (CCM).

There are three switch operation states in the peak current-mode controlled buck converter with CCL. The dynamic equations can be obtained by defining the i_L and v_C as state variables.

In state 1, S_T is on and S_D is off. The dynamic equations are derived as follows:

In state 2, S_T is off and S_D is on. The equations are given as follows:

In state 3, S_T is off and S_D is off. The equations are given as follows:

It contains the first two states when the converter operates in CCM, while in DCM, states  1– 3 are involved and repeated periodically. From the dynamic equations of the converter we can see that the system is nonlinear and piecewise smooth, which makes it difficult to investigate the dynamics of the converter directly. By sampling synchronously for the inductor current and capacitor voltage at the beginning of each switching cycle, the discrete iterative map model can be derived.

The values of inductor current and capacitor voltage at the beginning of the n-th cycle and (n + 1)-th cycle are assumed to be i_n, v_n, i_{n+ 1}, and v_{n+ 1}, respectively. With the variation of circuit parameters, two borders I_b1 and I_b2 of the inductor current are defined in discrete state-space, as shown in Fig.  2. When i_n = I_b1 and i_{n+ 1} = i_ref, it is shown in Fig.  2(a) that the switch S_T is on during the whole period, indicating that the inductor current remains increasing. When i_n = I_b2 and i_{n+ 1} = 0, figure  2(b) reveals that the inductor current rises at first and falls to zero at the end of the period due to the off-time t_d.

Fig.  2.

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	Fig.  2. Two boundaries of inductor current: (a) in = Ib1, in+ 1 = iref; (b) in = Ib2, in+ 1 = 0.

The two borders of the inductor current can be solved approximately from Fig.  2 and Eqs.  (1)– (6) as follows:

where T is the switching period and i_ref is given by

Here, m_c is the slope of the ramp compensation current and V_ref is the reference voltage.

If i_n ≤ I_b1, the switch S_T keeps the on-state throughout the switching cycle and the converter just operates in state 1. The stroboscopic map can be written as follows:

where

If I_b1 ≤ i_n ≤ I_b2, the converter operates in state 1 and state  2 within one switching cycle. After the on-time interval t_n of the switch S_T, the inductor current i_L rises to i_ref, and then S_T is turned off until the end of the switching cycle. The iterative map is described by

where t_n = ((i_ref − i_n)L)/(E − v_n), v_c (t_n) = (v_n − E)cos ω t_n + ω L(i_n − I_o)sin ω t_n + E, c₁ = i_ref − I_o, and c₂ = − v_c (t_n)/ω L.

If I_b2 ≤ i_n, the inductor current will rise at the beginning and drop to zero before the end of the switching cycle. The iterative map is given as

where

and t_d = Li_ref/v_c(t_n)

To verify the analysis results above, different values of k and m_c are selected for cycle-by-cycle simulation by Matlab software. Time-domain simulation waveforms and phase portraits of the converter are depicted by the Runge– Kutta algorithm via adopting a piecewise smooth switching model.

When E = 14  V and other parameters are the same as those in Fig.  7, let k = 0, k = 0.02 and k = 0.06 respectively, the inductor current i_L values and phase portraits (in the i_L– v_o plane) of the peak current-mode controlled buck converter with CCL are shown in Fig.  8.

Fig.  8.

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	Fig.  8. Time-domain waveforms of inductor current and phase portrait in iL– vo plane at k = 0 (a), 0.02 (b), and 0.06 (c).

Comparing Fig.  8(a) with Fig.  8(b), it can be seen that fast-scale and low-scale oscillation phenomenon are transferred to subharmonic oscillation by using relatively small proportional coefficient k. It can be seen clearly that the converter is stable with k = 0.06 while subharmonic oscillation exists with k = 0.02, which are consistent with the bifurcation diagram in Fig.  7.

Fixing E = 12  V and k = 0, the time-domain waveforms of inductor current i_L can be obtained as shown in Figs.  9(a)– 9(c) with different slopes of compensation ramp, i.e., m_c = 1000, m_c = 2000, and m_c = 6000, while figures  9(d)– 9(f) show the waveforms when k = 0.2 at the m_c values of 1000, 2000, and 6000, respectively.

Fig.  9.

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	Fig.  9. Time-domain waveforms of inductor current with (a) k = 0 and mc = 1000; (b) k = 0, and mc = 2000; (c) k = 0 and mc = 6000; (d) k = 0.2 and mc = 1000; (e) k = 0.2 and mc = 2000; (f) k = 0.2 and mc = 6000.

A comparison among Figs.  9(a)– 9(c) shows that the adoption of ramp compensation can eliminate both fast-scale accompanied by slow-scale oscillation and subharmonic oscillation. Moreover, the bigger the m_c, the better the results are. Comparing Fig.  9(a) with Fig.  9(d), it is shown that under the same m_c, the converter with a proper output voltage feedback seems to have a better performance due to its influence on slow-scale oscillation. The same conclusion is also inferred by comparing Fig.  9(b) with Fig.  9(e).

The experimental set-up is built with the same circuit parameters as those in the simulation analysis for verifying the simulation results of the stabilization in the peak current-mode controlled buck converter with CCL, by adding the ramp compensation and the output voltage feedback. The experimental results are shown in Fig.  10, in which Figs.  10(a), 10(c), and 10(e) show the waveforms of inductor current when k = 0 with different slopes of compensation ramp, i.e., m_c = 1000, 2000, and 6000, while Figs.  10(b), 10(d), and 10(f) display the waveforms when k = 0.2 at m_c = 1000, 2000, and 6000 respectively. It is concluded that the fast-scale oscillation accompanied by slow-scale oscillation and subharmonic oscillation can be eliminated gradually with increasing the value of m_c, while the adoption of a proper output voltage feedback can wipe off the slow-scale oscillation as indicated by the waveforms in Fig.  10, which agree well with the simulation results shown in Fig.  9, verifying the correctness of the theoretical analysis.

Fig.  10.

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	Fig.  10. Experimental waveforms of inductor current under (a) k = 0 and mc = 1000; (b) k = 0.2 and mc = 1000; (c) k = 0 and mc = 2000; (d) k = 0.2 and mc = 2000; (e) k = 0 and mc = 6000; (f) k = 0.2 and mc = 6000.

Assuming x_n = [i_n, v_n] and x_{n+ 1} = [i_{n+ 1}, v_{n+ 1}] to be the sampling values of state valuables at nT and (n + 1)T respectively, the fixed point X_Q = [I    V]^T can be solved by making x_{n+ 1} = x_n = X_Q while the Jacobi matrix around the X_Q can be deduced under the discrete iterative model and shown as follows:

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.

If i_n ≤ I_b1, the elements of the Jacobi matrix are expressed as follows: J₁₁ = cos ω T, J₁₂ = − sinω T/Lω , J₂₁ = ω Lsinω T, and J₂₂ = cosω T.

If I_b1 ≤ i_n ≤ I_b2, the elements of the Jacobi matrix are expressed as follows:

where

and

If I_b2 ≤ i_n, the elements of the Jacobi matrix are expressed as follows:

where

and

The eigenvalues can be obtained by solving the following characteristic equation:

According to the bifurcation diagrams in Figs.  3 and 4, the case where i_n ≤ I_b1 will not happen in the area of the period-1 state and its stabilities will be influenced when I_b1 ≤ i_n ≤ I_b2 in the peak current-mode controlled buck converter with CCL. The period-doubling bifurcation will take place when the eigenvalue equals − 1 and the characteristic equation can be expressed as

Based on Eq.  (18), the equation of the stability boundary containing m_c and k can be further obtained.

The orbit of the peak current-mode controlled buck converter with CCL is transferred from subharmonic oscillation (period-2 state) to CCM robust chaos when the inductor current reaches the border current I_b1 at the moment when the inductor current is a minimum value. While the inductor current is a maximum value, it reaches the border current I_b2, causing the operation of the converter to shift from CCM to DCM. The two formulas can be obtained as follows:

where

and i_{n+ 1, max} = i_ref

To make it more clearly, two formulas can be deduced further from Eqs.  (7)– (9), which give an expression for the two boundary curves. One is the boundary of period-2 state and CCM robust chaos and is expressed as

the other is the boundary of mode transferring from CCM to DCM, and given by

The two formulas both contain m_c and k, based on which the boundary curves can be described in parameter maps.

For the engineering applications, it is imperative to illustrate the boundaries of these dynamical behaviors mentioned previously, by knowing all of the operation regions of employing ramp compensation or the output voltage feedback. By exploring the relationships between k and E, k and I_o, m_c and E, m_c and I_o, the operation regions can be obtained in the parameter space with the variations of k and m_c respectively, owing to the formulas expressed in Eqs.  (18), (21), and (22).

As it can be seen from Fig.  11, there are four ways to stabilize the converter: they are to increase E, I_o, and m_c, and to choose an appropriate value of k. Three operating boundaries including period-doubling, I_b1, and I_b2 divide the work place into CCM period-1 region, CCM period-2 region, CCM robust chaos region and DCM weak chaos region in terms of selected circuit parameters, which provide design-oriented information for ensuring the stable operation of peak current-mode controlled buck converters with CCL.

Fig.  11.

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	Fig.  11. Operation regions varying with k [(a) k– E and (b) k– Io] and with mc [(c) mc– E and (d) mc– Io].

In this paper, we investigate the dynamical behaviors and stabilizations of peak current-mode controlled buck converters with CCL. Discrete iterative map models with two inductor current borders are established to study the complex dynamics of the converter by depicting its bifurcation diagram and Lyapunov exponent spectrum. The influences of the slope m_c of the compensation ramp and proportional coefficient k of the output voltage feedback loop on the stability of the converter are explored based on the iterative map model. To testify the validities of simulation and theoretical analysis, the experiment is carried out. The operation regions within variation of k and m_c are obtained by analyzing the loss of stabilities and border collision.

Research results in this paper point out that there exist four operation regions in the peak current-mode controlled buck converter with CCL, i.e., CCM period-1, CCM period-2, CCM robust chaos, and DCM weak chaos, which are divided by the period-doubling bifurcation and two boundaries of border collision for the changing of k and m_c. It is also concluded that the ramp compensation can eliminate both fast-scale oscillation accompanied by slow-scale oscillation and subharmonic oscillation, broaden the stable area of the converter, and transfer the operation mode. When the output voltage feedback loop is introduced, the fast-scale and slow-scale subharmonic oscillation can be gradually eliminated and the stable region can be extended slowly.