Switching DC–DC converters are strongly nonlinear systems with complex nonlinear phenomena, such as low-frequency oscillation,^[1,2] period-doubling bifurcation, border collision bifurcation, chaos, and so on.^[3–11] With the variation of circuit parameters, the converter systems may fall into chaotic state via various bifurcations, which have negative effects on their performance. Hence, it is of great importance to study nonlinear phenomena of the converters both in theory and application.

Nonlinear phenomena of DC–DC converters have been studied for many years. Previous literature mainly focus on peak current mode Boost converter^[12–14] and current-mode Buck–Boost converter,^[15–17] while research on peak current mode Buck converter started relatively late, and most of them employ an approximate one-dimensional discrete model, which fails to reflect the actual working situations of the converters.^[3,18] Although there are some research studies that use a two-dimensional discrete model, their physical meanings are ambiguous.^[19]

With the blooming development of science and technology, switching mode power supply develops rapidly in the direction of high frequency, miniaturization and light weight. Meanwhile, high-frequency switching power converters are widely applied in all kinds of portable electronic devices. Higher switching frequency means lower ripple of output voltage, thus reducing the required values of inductor and capacitor. Consequently, the sizes of passive devices can be reduced, making switching power converters smaller. However, most existing research is confined to low-frequency conditions (hundreds of kilohertz or lower).^[20,21] Obviously, it is of great necessity to study those nonlinear phenomena in high-frequency switching power converters (switching frequency up to megahertz region), and to discover the relationship between switching frequency and system stability.

In this paper, the authors will study the nonlinear phenomena in high-frequency peak current mode Buck converter working in CCM. A precise two-dimensional discrete model is established in Section 2. In Section 3, the bifurcation diagrams of the system are plotted with the aid of MATLAB, meanwhile, circuit simulations are carried out with PSIM and the corresponding time domain waveforms as well as phase portraits in i_L–v_C plane are plotted with MATLAB. In addition, the critical reference current at the period-doubling point and the border of inductor current are derived in this part. In the following section, we derive the Jacobian matrix and analyze the stability of the system based on the roots of characteristic equations. Then in Section 5, an experimental circuit is built and tested to verify the validity of theoretical analysis. Section 6 gives the conclusion.

The stability of the system can be determined by computing the eigenvalues at the fixed point. Choosing circuit parameter φ (e.g., I_ref or f) as the bifurcation parameter, and the discrete iterative mapping model of the converter in CCM (i.e., equations (5) and (6)) can be rewritten in the following equation:

Let i_n+1 = i_n = i_Q, v_n+1 = v_n = v_Q, then the fixed point v_Q, i_Q can be obtained according to Eq. (11). The Jacobian matrix in the neighbourhood of the fixed point can be described as

The stability of the system is determined by the eigenvalues λ₁ and λ₂ at the fixed point. The corresponding φ when one of the eigenvalues equals to −1 is the parameter value at the period-doubling bifurcation point.^[3,4,18] Using I_ref as bifurcation parameter while keeping the other parameters of the system unchanged, the obtained eigenvalues and system state are shown in Table 1.

Table 1.

Table 1.

Table 1. Eigenvalues and system state at different Iref with f = 1 MHz. . Iref/A Eigenvalue λ1 Eigenvalue λ2 State 0.8 −0.6436 0.9439 period-1 orbit 1.0 −0.9929 0.9461 period-1 orbit 1.002 −0.9972 0.9461 period-1 orbit 1.0033 −1.0000 0.9461 period-doubling bifurcation 1.01 −1.0148 0.9462 period-2 orbit

Table 1. Eigenvalues and system state at different Iref with f = 1 MHz. .

As can be seen from Table 1, when I_ref increases to 1.0033 A, the period-doubling bifurcation occurs and the system switches from period-1 orbit to period-2 orbit. The analysis results and the bifurcation diagram in Fig. 2 are basically consistent.

Next, by serving f as bifurcation diagram, the obtained eigenvalues and system state as I_ref equals to 0.8 A, 1 A, and 1.2 A are shown in Tables 2–4, respectively.

Table 2.

Table 2.

Table 2. Eigenvalues and system state at different f with Iref = 0.8 A. . f/MHz Eigenvalue λ1 Eigenvalue λ2 State 1.0 −0.6436 0.9439 period-1 orbit 1.6 −0.6886 0.9652 period-1 orbit 2.25 −0.7119 0.9753 period-1 orbit 3.0 −0.7268 0.9815 period-1 orbit

Table 2. Eigenvalues and system state at different f with Iref = 0.8 A. .

Table 3.

Table 3.

Table 3. Eigenvalues and system state at different f with Iref = 1 A. . f/MHz Eigenvalue λ1 Eigenvalue λ2 State 0.5 −0.8293 0.8923 period-1 orbit 1.0 −0.9929 0.9461 period-1 orbit 1.02 −0.9965 0.9471 period-1 orbit 1.05 −1.0017 0.9486 period-doubling bifurcation

Table 3. Eigenvalues and system state at different f with Iref = 1 A. .

Table 4.

Table 4.

Table 4. Eigenvalues and system state at different f with Iref = 1.2 A. . f/MHz Eigenvalue λ1 Eigenvalue λ2 State 0.2 −0.7713 0.7422 period-1 orbit 0.28 −0.9632 0.8235 period-1 orbit 0.296 −0.9958 0.8336 period-1 orbit 0.3 −1.0037 0.8359 period-doubling bifurcation

Table 4. Eigenvalues and system state at different f with Iref = 1.2 A. .

It can be seen from Table 2 that when I_ref is relatively small (I_ref = 0.8 A), the system is stable even when f reaches 3 MHz. According to Tables 3 and 4, as I_ref increases to 1 A and 1.2 A, the corresponding switching frequencies when the period doubling bifurcation occurs are about 1.05 MHz and 0.3 MHz respectively, which are consistent with the results presented in Fig. 4.

In order to verify the correctness of previous theoretical analysis, an experimental circuit is built to test the nonlinear phenomena of the peak current mode Buck converter. The full schematic diagram and the photograph of the experimental circuit are shown in Figs. 6 and 7 respectively.

Fig. 6.

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	Fig. 6. Schematic diagram of the peak current mode Buck converter.

Fig. 7.

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	Fig. 7. Photograph of the experimental circuit.

In the main topology, the power tube is μPA2791 and the driver chip is UCC27524. The feedback path is mainly composed of current sampling circuit, an amplifier (AD620), a comparator (LM311) and an R-S flip-flop (CD4043BC). In this paper, we adopt DCR current sampling method^[23] to obtain a voltage that is proportional to inductor current, and then the voltage is amplified by AD620, hence, the signal input into the inverting input port of the comparator is a reference voltage V_ref that is numerically equal to the reference current I_ref.

The waveforms of inductor current and output voltage as well as phase portraits in i_L–V_o plane at 1 MHz switching frequency when reference voltage V_ref is 0.95 V, 1.05 V, and 1.2 V are shown in Fig. 8.

Fig. 8.

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	Fig. 8. Experimental waveforms of inductor current and output voltage ((a)–(c)) as well as phase portraits in iL–Vo plane ((d)–(f)) at Vref = 0.95 V ((a) and (d)), 1.05 V ((b) and (e)), 1.2 V ((c) and (f)) with f = 1 MHz.

As can be seen from Fig. 8, with the increase of V_ref, the converter experiences the stable period-1 orbit, period-2 orbit and chaotic state successively, indicating the converter system tends to be unstable with the augment of V_ref at a certain switching frequency. The experimental results are consistent with previous analysis and simulation results.

Then we will verify the influence of switching frequency variation on system state. According to Figs. 8(c) and 8(f), when f = 1 MHz and V_ref = 1.2 V, the peak current mode Buck converter is working in chaotic state. Keeping the reference voltage V_ref = 1.2 V unchanged, the waveforms of inductor current and output voltage as well as phase portraits in i_L–V_o plane with f equals to 600 kHz and 250 kHz are shown in Fig. 9.

Fig. 9.

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	Fig. 9. Experimental waveforms of inductor current and output voltage as well as phase portraits in iL–Vo plane at f = 600 kHz ((a) and (c)), 250 kHz ((b) and (d)) with Vref = 1.2 V.

Combining Figs. 8(c) and 8(f) with Fig. 9, it can be found that with the decrease of f, the converter experiences the chaotic state, period-2 orbit and stable period-1 orbit in sequence, namely the converter system is tending towards stability with the decrease of f at a certain reference voltage, which proved the experimental results are basically consistent with previous analysis and simulation results.

In this paper, the focus of our research is to study nonlinear phenomena in high frequency peak current mode Buck converter and to discover the relationship between switching frequency and system stability. First, we establish the two-dimensional discrete mapping model of the converter. After that, we derive the reference current at the period-doubling point and the border of inductor current. Next, the bifurcation diagrams are obtained through numerical simulation and plotted with the aid of MATLAB. Meanwhile, time domain waveforms and phase portrait in i_L–v_C plane are obtained through PSIM simulation, and are plotted with MATLAB on the basis of simulation data. Then, the Jacobian matrix is derived and the stability of the system is analyzed based on the eigenvalues of characteristic equations. Finally, to testify the validity of theoretical analysis, the experiment is carried out on a test circuit. Both the simulation and experimental results show that the converter tends to be unstable with the increase of reference current I_ref, as the corresponding switching frequency f is approaching to low-frequency stage continuously when the period-doubling bifurcation occurs. Moreover, with the increase of f, the stable working range of the system becomes smaller for the corresponding I_ref decreases when the period-doubling bifurcation occurs.