Medium-Frequency Oscillation in Voltage-Controlled Full-Bridge Inverter with a Nonlinear Inductor

Abstract

An inverter, whose inductor current is periodic, is the key equipment for photovoltaic power generation, fuel cell power generation, etc. A nonlinear inductor, whose inductance will be changed by its current, can be used to improve the grid current quality of the inverter, although making the mathematical model of the inverter system more complex. In this paper, the voltage-controlled full-bridge inverter with a nonlinear inductor is investigated and its mathematical model is established. The influence of parameters on the dynamical behaviors of this system is analyzed by using the harmonic balance method and the Floquet theory. Consequently, the mechanism for the occurrence of medium-frequency oscillation is revealed. The hardware circuit for the voltage-controlled full-bridge inverter with a nonlinear inductor is designed and some experimental results are presented for confirmation. The research results show that the mechanism for the occurrence of medium-frequency oscillation is Hopf bifurcation and its oscillation frequency is between the line frequency and the switching frequency.

1. Introduction

Switching power converters, which include DC–DC converters, power factor correction (PFC) converters, DC–AC inverters, etc., mainly consist of semiconductors, inductors, capacitors, and loads, and have been widely employed in practical engineering systems such as photovoltaic (PV) power generation, fuel cell power generation, etc. [1]. For inductors in the switching power converters, as indicated in [2,3], two aspects need to be considered. (1) In order to reduce the THD of the system’s current and improve the whole system’s efficiency, the inductor needs to have a sufficiently high inductance under light-load current. (2) Under heavy-load current, it is hoped that the designed inductor would have low inductance to ensure that the inductor is in an unsaturated state. Obviously, it will be difficult for linear inductors to satisfy both these requirements simultaneously. In order to solve this problem, the nonlinear inductor has been proposed and designed [3,4,5,6,7,8]. For example, a ladder-gap nonlinear inductor whose inductance can be varied with the specified load was designed in [3]. Based on low-temperature co-fired ceramic (LTCC) technology, the multiwindow multipermeability nonlinear LTCC inductor was fabricated and applied to a DC–DC buck converter in [4]. In [5], the nonlinear inductor was employed in the interleaved power factor correction (PFC) converter, its Jiles–Atherton model was established, and the dynamic behaviors of the system were investigated. In [6], efficient nonlinear inductors were adopted in PV inverters and active PFC converters; the corresponding design method for nonlinear inductors was presented, where the research results showed its ability to perform at a size 32% smaller than linear inductors, with 20% less copper, and a total power loss that is 10% lower by using nonlinear inductors instead of linear inductors in a practical PV inverter. Furthermore, some researchers have applied nonlinear inductors to the three-phase grid-connected inverter [7], single-phase PV grid converters [8], DC–DC boost converters [9], and so on. Certainly, the research results above have greatly promoted the design of nonlinear inductors and their applications in switching power converters.

In addition, the extant research results in [10,11,12,13,14,15,16,17,18] show that the inverter is a type of strongly nonlinear system so that there are rich and complex nonlinear phenomena which will inevitably have an important influence on the performance of the inverter. For instance, in [10], bifurcation and chaos in the single-phase sinusoidal pulse width modulation (SPWM) full-bridge inverter with a linear inductor were analyzed by using the discrete model, and an adaptive carrier amplitude modulation control method was presented to control these discovered instabilities in order to ensure its stable operation. In [11], the exponential delay feedback control method used for controlling the chaotic phenomenon in the single-phase H-bridge photovoltaic inverter with a linear inductor was reported. In [12], the fast-scale instability and slow-scale instability in a single-phase parallel full-bridge inverter with a linear inductor was analyzed. Likewise, the complex nonlinear phenomena of the dual buck full-bridge inverter with a linear inductor were studied in [13]. The bifurcation phenomenon in parallel-connected voltage-source inverters with a linear inductor was analyzed by using the delay-feedback control method in [15]. The above research results provide good guidance for the parameter design of the inverter system in practical engineering to ensure its stable operation.

These achievements were all carried out on inverters with linear inductors; that is, the mathematical model of the inverter systems are piecewise linear models so that the analytical solution in each segment could be obtained, and the discrete model could be used to analyze such systems. However, with the introduction of the nonlinear inductor whose characteristics will make the dynamic behaviors of the inverter system more complicated, analyzing its strongly nonlinear system with the use of the discrete model will certainly prove to be even more difficult. Therefore, it is necessary to study the inverter with a nonlinear inductor that aims to provide a theoretical basis and design guidance for its engineering applications.

In this paper, the mathematical model of the voltage-controlled full-bridge inverter with a nonlinear inductor is established, the influence of system’s parameters on its dynamic behaviors is analyzed, and the mechanism of medium frequency oscillation in the system is revealed. Additionally, a hardware circuit for the system is designed, and some experimental results used to verify the correctness of the theoretical analyses and the existence of the medium-frequency oscillation phenomenon are presented.

2. Nonlinear Inductor’s Model and Its Experimental Verification

For the nonlinear inductor, its inductance can be described by its static inductance and dynamic inductance [3]. The mathematical model for the nonlinear inductor’s static inductance, provided in the system-level power electronics simulation software and named Piecewise Linear Electrical Circuit Simulation (PLECS) developed by the Swiss company Plexim GmbH, is given as [19]:

$$L_E={\mathrm{A}}_1+{\mathrm{A}}_2\frac{\arctan ({\mathrm{A}}_3{i}_N)}{i_N}$$

(1)

where A₁, A₂, and A₃ are the parameters of the nonlinear inductor, arctan is the arctangent function, i_N is the current through the nonlinear inductor, and L_E is the static inductance of the nonlinear inductor.

Note that the expression for the magnetic flux linkage of the nonlinear inductor is φ; = L_Ei_N, i.e., φ = A₁i_N + A₂ arctan(A₃i_N). Subsequently, the voltage across this nonlinear inductor is v_LN = dφ/dt, that is, the relationship between the current i_N through this nonlinear inductor and its voltage v_LN satisfies the following formula:

$$v_{LN}=\frac{{\mathrm{A}}_1{+\mathrm{A}}_2{\mathrm{A}}_3{+\mathrm{A}}_1{\mathrm{A}}_3{}^2{i}_N{}^2}{1+{\mathrm{A}}_3{}^2{i}_N{}^2}\frac{d{i}_N}{dt}$$

(2)

Additionally, based on the definition of dynamic inductance L_N, its expression is the following:

$$L_N=\frac{d\phi }{d{i}_N}=\frac{{\mathrm{A}}_1{+\mathrm{A}}_2{\mathrm{A}}_3{+\mathrm{A}}_1{\mathrm{A}}_3{}^2{i}_N{}^2}{1+{\mathrm{A}}_3{}^2{i}_N{}^2}$$

(3)

The test circuit consisting of an AC voltage source v_S(t), linear resistor R, nonlinear inductor L_N, and linear inductor L is shown in Figure 1a, where R = 1 Ω, L = 110 μH, and v_S(t) = 18sin(2000πt)V. Notably, L_N is the dynamic inductance of the nonlinear inductor. According to Figure 1a, the experimental test data including the voltage across the linear inductor v_L, the voltage across the nonlinear inductor v_LN, and the voltage across the linear resistor v_R can be obtained. Additionally, since the linear resistor is R = 1 Ω, the current i_N through the nonlinear inductor can be represented by the voltage v_R. Thus, the dynamic inductance of the nonlinear inductor can be calculated as follows:

$$L_N=L\frac{v_{LN}}{v_L}$$

(4)

By making the dynamic inductance L_N described in (3) as the goal of the fitting function and substituting the test data obtained from the experiments in (3), A₁, A₂, and A₃ shown in (3) could be obtained by using the nlinfit function in the Matlab software; the fitting results are A₁ = 1.5845 × 10⁻⁵, A₂ = 0.0013, and A₃ = 0.3932. Figure 1b shows the comparisons between the fitted nonlinear inductance L_N with the current i_N and the corresponding experimental test results. One can see that the results from these two sides are basically consistent with each other, indicating the effectiveness of the nonlinear inductor model and the correctness of the fitting technique.

3. Mathematical Model and Approximate Calculation

By replacing the linear inductor in the full-bridge inverter with the above-mentioned nonlinear inductor L_N, the circuit schematic diagram of the voltage-controlled full-bridge inverter with nonlinear inductor L_N could be obtained and is shown in Figure 2. The power circuit of this system consists of an input DC voltage v_in, four power switches (Q₁, Q₂, Q₃, and Q₄), a nonlinear inductor L_N (whose inductor current is i_N), a linear capacitor C (whose voltage is v₀), and a load resistor R_L. Note that four power switches (Q₁, Q₂, Q₃, and Q₄) form the inverter bridge, and they are turned on or off by their drive signals. Specifically, when the drive signal is at a high level, the corresponding power switches are turned on, otherwise they are turned off. Moreover, the driving signals of Q₁ and Q₄ are the same, the driving signals of Q₂ and Q₃ are the same, and Q₁ and Q₂ are turned on complementarily.

The control circuit of this system is composed of the measurement circuit for the output floating voltage v₀, the bleeder circuit consisting of R_vi and R_vd, the PI compensator, the SPWM generator, and the driving circuit. The output voltage of the PI compensator is v_vf and v_ref = V_ref sin(ω₀t). Assuming that all components are ideal, i.e., their parasitic parameters are not considered since their values are very small, taking the voltage v₀, the current i_N, and the voltage v_vf as the circuit variables, and based on the KCL and KVL law, the mathematical model of the system described by differential equations can be obtained as follows:

$$\{\begin{array}{l}\frac{d{i}_N}{dt}=\frac{1+{\mathrm{A}}_3{}^2{i}_N{}^2}{{\mathrm{A}}_1{+\mathrm{A}}_2{\mathrm{A}}_3{+\mathrm{A}}_1{\mathrm{A}}_3{}^2{i}_N{}^2}((2S-1)v_E-v_0)\\ \frac{d{v}_0}{dt}=\frac{i_N}{C}-\frac{v_0}{R_{L}C}\\ \frac{d{v}_{vf}}{dt}=\frac{k_M{R}_{vf}{v}_0}{R_{vi}C{R}_L}-\frac{k_M{v}_0}{R_{vi}{C}_{vf}}-\frac{k_M{R}_{vf}{i}_N}{R_{vi}C}+k_1\sin ({\omega }_{0}t)+k_2\cos ({\omega }_{0}t)\end{array}$$

(5)

where

$$\{\begin{array}{l}{k}_1=(\frac{1}{R_{vi}}+\frac{1}{R_{vd}})\frac{V_{ref}}{C_{vf}}\\ k_2=(\frac{R_{vf}}{R_{vi}}+\frac{R_{vf}}{R_{vd}}+1)V_{ref}{\omega }_0\end{array}$$

(6)

Notably, S = 1 means that Q₁ and Q₄ are turned on, while Q₂ and Q₃ are turned off. S = 0 means that Q₂ and Q₃ are turned on, while Q₁ and Q₄ are turned off. Clearly different from the full-bridge inverter with a linear inductor, Equation (5) is always nonlinear for both S = 1 and S = 0 so that it will be difficult to obtain its analytical solution, and the discrete model cannot be adopted to analyze the stability of the voltage-controlled full- bridge inverter with a nonlinear inductor.

As for the bipolar SPWM generator, it consists of a comparator COM, a triangular carrier v_tri, and a power switch driver. Note that the triangle carrier v_tri is expressed as follows:

$$v_{tri}=\{\begin{array}{l}-({\omega }_{sw}t-2\pi k-2\pi )\frac{2V_{tri}}{\pi }-V_{tri},2\pi k+\pi \le {\omega }_{sw}t\le 2\pi k+2\pi \\ ({\omega }_{sw}t-2\pi k)\frac{2V_{tri}}{\pi }-V_{tri},2\pi k\le {\omega }_{sw}t\le 2\pi k+\pi \end{array}$$

(7)

where ω_sw and V_tri are the angular frequency and the amplitude of the triangular carrier, respectively. The triangle carrier v_tri is compared with the voltage v_vf to generate the SPWM signal. If v_tri < v_vf, the comparator COM output is at a high level, while if v_tri > v_vf, the comparator COM outputs a low level.

For the full-bridge inverter with a nonlinear inductor, the angular frequency ω_sw of the triangular carrier v_tri is much larger than the angular frequency ω₀ of the modulating wave v_ref; that is, the switching frequency f_sw(f_sw = ω_sw/2π) is much larger than the power frequency f₀(f₀ = ω₀/2π). Therefore, the average modeling method proposed in [20] can be used to establish the average model of the system, and the duty ratio 〈d〉 can be computed as follows:

$$d=\frac{1}{2}(1+\frac{\langle v_{vf}\rangle }{V_{tri}})$$

(8)

Therefore, the average model of the voltage-controlled full-bridge inverter with a nonlinear inductor can be given as:

$$\{\begin{array}{l}\frac{d\langle i_N\rangle }{dt}=\frac{1+{\mathrm{A}}_3{}^2{\langle i_N\rangle }^2}{{\mathrm{A}}_1{+\mathrm{A}}_2{\mathrm{A}}_3{+\mathrm{A}}_1{\mathrm{A}}_3{}^2{\langle i_N\rangle }^2}(\frac{v_E}{V_{tri}}\langle v_{vf}\rangle -\langle v_0\rangle )\\ \frac{d\langle v_0\rangle }{dt}=\frac{\langle i_N\rangle }{C}-\frac{\langle v_0\rangle }{R_{L}C}\\ \frac{d\langle v_{vf}\rangle }{dt}=\frac{k_M{R}_{vf}\langle v_0\rangle }{R_{vi}C{R}_L}-\frac{k_M\langle v_0\rangle }{R_{vi}{C}_{vf}}-\frac{k_M{R}_{vf}\langle i_N\rangle }{R_{vi}C}+k_1\sin ({\omega }_{0}t)+k_2\cos ({\omega }_{0}t)\end{array}$$

(9)

where 〈d〉, 〈i_N〉, 〈v₀〉, and 〈v_vf〉 are the average values of d, i_N, v₀, and v_vf in one switching period, respectively.

One can see that the voltage-controlled full-bridge inverter with a nonlinear inductor described by (9) is a complex nonlinear and non-autonomous circuit system. Generally speaking, for this inverter system, it is expected that the average value of its output voltage v₀ should be approximately a sine wave, and the harmonic is expected to be as low as possible; therefore, it can be assumed as:

$$\langle v_0\rangle =a_0+{\displaystyle \sum _{n=1}^N\left[a_n\cos (n\tau )+b_n\sin (n\tau )\right]}$$

(10)

where τ = ω₀t and

$$\{\begin{array}{l}\langle {v^{\prime }}_0\rangle =\frac{d\langle v_0\rangle }{dt}\\ \langle {v^{″}}_0\rangle =\frac{d^2\langle v_0\rangle }{d{t}^2}\\ {\langle v_0\rangle }^{(-1)}={\displaystyle \int \langle v_0\rangle dt}\end{array}$$

(11)

hence,

$$\{\begin{array}{l}\langle i_N\rangle =C\langle v{\prime }_0\rangle +\langle v_0\rangle /R_L\\ \langle v_{vf}\rangle =\frac{V_{tri}}{v_E}(\langle i{\prime }_N\rangle \frac{({\mathrm{A}}_1{+\mathrm{A}}_2{\mathrm{A}}_3){+\mathrm{A}}_1{\mathrm{A}}_3{}^2{\langle i_N\rangle }^2}{1+{\mathrm{A}}_3{}^2{\langle i_N\rangle }^2}+\langle v_0\rangle )\\ \langle i{\prime }_N\rangle =C\langle v\prime {\prime }_0\rangle +\langle v{\prime }_0\rangle /R_L\end{array}$$

(12)

By substituting the first expression and the third expression of (12) with (9), we can get the following:

$$\begin{array}{l}f(a_0,a_1,\dots ,a_N,b_1,\dots ,b_N)=({\mathrm{A}}_1{+\mathrm{A}}_2{\mathrm{A}}_3{+\mathrm{A}}_1{\mathrm{A}}_3{}^2{(C\langle {v^{\prime }}_0\rangle +\frac{\langle v_0\rangle }{R_L})}^2)\\ (C\langle {v^{″}}_0\rangle +\frac{\langle {v^{\prime }}_0\rangle }{R_L})-(1+{\mathrm{A}}_3{}^2{(C\langle {v^{\prime }}_0\rangle +\frac{\langle v_0\rangle }{R_L})}^2)(\frac{v_E}{V_{tri}}(\frac{k_M{R}_{vf}{\langle v_0\rangle }^{(-1)}}{R_{vi}C{R}_L}\\ -\frac{k_M{\langle v_0\rangle }^{(-1)}}{R_{vi}{C}_{vf}}-\frac{k_M{R}_{vf}(C\langle v_0\rangle +{\langle v_0\rangle }^{(-1)}/R_L)}{R_{vi}C}-\frac{k_1}{{\omega }_0}\cos (\tau )+\frac{k_2}{{\omega }_0}\sin (\tau ))-\langle v_0\rangle )\end{array}$$

(13)

Notably, if the bigger number N is selected, more accurate results can be obtained. However, by considering that the amplitudes of high order harmonics are very small when compared to that of the fundamental, choosing a small value for N is enough. Then, substituting (10) and (11) with (13) and taking N = 4, and based on the principle of harmonic balance, we can get the following:

$$\{\begin{array}{l}{\displaystyle \underset{0}{\overset{2\pi }{\int }}f(a_0,a_1,a_2,a_3,a_4,b_1,b_2,b_3,b_4)d\tau }=0\\ {\displaystyle \underset{0}{\overset{2\pi }{\int }}f(a_0,a_1,a_2,a_3,a_4,b_1,b_2,b_3,b_4)\cos (\tau )d\tau }=0,{\displaystyle \underset{0}{\overset{2\pi }{\int }}f(a_0,a_1,a_2,a_3,a_4,b_1,b_2,b_3,b_4)\cos (2\tau )d\tau }=0\\ {\displaystyle \underset{0}{\overset{2\pi }{\int }}f(a_0,a_1,a_2,a_3,a_4,b_1,b_2,b_3,b_4)\cos (3\tau )d\tau }=0,{\displaystyle \underset{0}{\overset{2\pi }{\int }}f(a_0,a_1,a_2,a_3,a_4,b_1,b_2,b_3,b_4)\cos (4\tau )d\tau }=0\\ {\displaystyle \underset{0}{\overset{2\pi }{\int }}f(a_0,a_1,a_2,a_3,a_4,b_1,b_2,b_3,b_4)\sin (\tau )d\tau }=0,{\displaystyle \underset{0}{\overset{2\pi }{\int }}f(a_0,a_1,a_2,a_3,a_4,b_1,b_2,b_3,b_4)\sin (2\tau )d\tau }=0\\ {\displaystyle \underset{0}{\overset{2\pi }{\int }}f(a_0,a_1,a_2,a_3,a_4,b_1,b_2,b_3,b_4)\sin (3\tau )d\tau }=0,{\displaystyle \underset{0}{\overset{2\pi }{\int }}f(a_0,a_1,a_2,a_3,a_4,b_1,b_2,b_3,b_4)\sin (4\tau )d\tau }=0\end{array}$$

(14)

By solving (14), the coefficients a₀, a₁, a₂, a₃, a₄, b₁, b₂, b₃, and b₄ can be calculated by using the improved differential evolution method, and the expression for the average output voltage 〈v₀〉 of the voltage-controlled full-bridge inverter with a nonlinear inductor can be obtained.

The circuit parameters of the voltage-controlled full-bridge inverter with a nonlinear inductor are chosen as v_E = 12 V, C = 20 μF, R_L = 5 Ω, V_tri = 2 V, V_ref = 1 V, f_sw = 25 kHz, f₀ = 50 Hz, C_vf = 102 nF, R_vf = 1 kΩ, R_vi = 36 kΩ, R_vd = 36 kΩ, and k_M = 0.25. The approximate solution of the voltage 〈v₀〉 is calculated as:

$$\begin{array}{ll}\langle v_0\rangle =& -0.187207-1.263136\cos (\tau )+7.158382\sin (\tau )\\ & -0.100880\cos (2\tau )-0.178769\sin (2\tau )-0.057748\cos (3\tau )\\ & -0.137595\sin (3\tau )-0.018889\cos (4\tau )-0.109892\sin (4\tau )\end{array}$$

(15)

Likewise, the current 〈i_N〉 and the voltage 〈v_vf〉 from (12) can be obtained.

Figure 3 shows the comparisons between the numerical simulations obtained from (9) and the approximate calculation results obtained from the harmonic balance (HB) method. It can be seen that the results from these two sides are consistent with each other, which indicates that the harmonic balance method is effective for calculating the approximate solution of the voltage-controlled full-bridge inverter with a nonlinear inductor.

4. Theoretical Analysis on Oscillation Phenomena

From (9), it can be seen that the voltage-controlled full-bridge inverter with a nonlinear inductor is a type of periodic time-varying nonlinear non-autonomous system, and its solution is the periodic solution as shown in (15). Therefore, the Floquet theory can be used to judge the stability of the periodic solution descried in (15). Supposing that this system has a small perturbation near its periodic solution, the following equation can be used:

$$\{\begin{array}{l}{i}_N=\langle i_N\rangle +\mathsf{\Delta }{i}_N\\ v_0=\langle v_0\rangle +\mathsf{\Delta }{v}_0\\ v_{vf}=\langle v_{vf}\rangle +\mathsf{\Delta }{v}_{vf}\end{array}$$

(16)

where

$$\{\begin{array}{l}\langle i_N\rangle \gg \mathsf{\Delta }{i}_N\\ \langle v_0\rangle \gg \mathsf{\Delta }{v}_0\\ \langle v_{vf}\rangle \gg \mathsf{\Delta }{v}_{vf}\end{array}$$

Then, linearize (9) at its periodic solution (〈i_N〉, 〈v₀〉, 〈v_vf〉), the linearization matrix Y(τ) can be obtained as follows:

$$Y(\tau )=\left[\begin{array}{ccc}\frac{a_{11}}{{\omega }_0}& \frac{a_{12}}{{\omega }_0}& \frac{a_{13}}{{\omega }_0}\\ \frac{1}{{\omega }_{0}C}& -\frac{1}{{\omega }_0{R}_{L}C}& 0\\ -\frac{k_M{R}_{vf}}{{\omega }_0{R}_{vi}C}& \frac{k_M{R}_{vf}}{{\omega }_0{R}_{vi}C{R}_L}-\frac{k_M}{{\omega }_0{R}_{vi}{C}_{vf}}& 0\end{array}\right]$$

(17)

where

$$\{\begin{array}{l}{a}_{11}=(\frac{v_E}{V_{tri}}\langle v_{vf}\rangle -\langle v_0\rangle )(\frac{2{\mathrm{A}}_3{}^2\langle i_N\rangle }{{\mathrm{A}}_1{+\mathrm{A}}_2{\mathrm{A}}_3{+\mathrm{A}}_1{\mathrm{A}}_3{}^2{\langle i_N\rangle }^2}-\frac{2{\mathrm{A}}_1{\mathrm{A}}_3{}^2\langle i_N\rangle (1+{\mathrm{A}}_3{}^2{\langle i_N\rangle }^2)}{{({\mathrm{A}}_1{+\mathrm{A}}_2{\mathrm{A}}_3{+\mathrm{A}}_1{\mathrm{A}}_3{}^2{\langle i_N\rangle }^2)}^2})\\ a_{12}=-\frac{1+{\mathrm{A}}_3{}^2{\langle i_N\rangle }^2}{{\mathrm{A}}_1{+\mathrm{A}}_2{\mathrm{A}}_3{+\mathrm{A}}_1{\mathrm{A}}_3{}^2{\langle i_N\rangle }^2}\\ a_{13}=\frac{1+{\mathrm{A}}_3{}^2{\langle i_N\rangle }^2}{{\mathrm{A}}_1{+\mathrm{A}}_2{\mathrm{A}}_3{+\mathrm{A}}_1{\mathrm{A}}_3{}^2{\langle i_N\rangle }^2}\frac{v_E}{V_{tri}}\end{array}$$

(18)

The period of the linearization matrix Y(τ) is consistent with the period of 〈v₀〉. Therefore, the linearized system can be expressed as follows:

$$\dot{x}=Y(\tau )x,\begin{array}{c}\end{array}x=(\mathsf{\Delta }{i}_N;\mathsf{\Delta }{v}_0;\mathsf{\Delta }{v}_{vf})\in R^3$$

(19)

If Φ(τ) is the periodic solution of (19), then Φ(τ + T) is also the periodic solution of (19), and there is the following:

$$\mathsf{\Phi }(\tau +T)=M\mathsf{\Phi }(\tau )$$

(20)

where M is the transition matrix. Let τ = 0 and Φ(0) = I, and there is the following:

$$M=\mathsf{\Phi }(T)$$

(21)

According to the Floquet theory, if the Floquet multipliers of the transition matrix M are all located in the unit circle, the periodic time-varying system is stable [21]. Otherwise,

• If one of the Floquet multipliers of the transition matrix M passes through the unit circle from the negative real axis and the rest remains in the unit circle, the period-doubling bifurcation occurs in the periodic time-varying system;
• If one of the Floquet multipliers of the transition matrix M passes through the unit circle from the positive real axis and the rest remains in the unit circle, the pitch-fork bifurcation occurs in the periodic time-varying system;
• If a pair of conjugate Floquet multipliers of the transition matrix M goes out of the unit circle and the rest remains in the unit circle, then Hopf bifurcation occurs in the periodic time-varying system.

Since the period of τ is 2π, that is T = 2π. Divide T into equal subparts N_k, then t_k = T_k/N_k. This is defined as follows:

$$Y_k=\frac{1}{\mathsf{\Delta }k}{\displaystyle {\int }_{t_k{}_{-1}}^{t_k}Y(\tau )}d\tau$$

(22)

Hence, the transition matrix M can be given as follows:

$$\begin{array}{ll}M& ={\displaystyle \prod _{i=N_k}^1\left[\exp (Y_i{\mathsf{\Delta }}_i)\right]}\\ & =\exp (Y_{N_k}{\mathsf{\Delta }}_{N_k})\exp (Y_{N_k-1}{\mathsf{\Delta }}_{N_k-1})\dots \exp (Y_1{\mathsf{\Delta }}_1)\\ & ={\displaystyle \prod _{i=N_k}^1\left[I+{\displaystyle \sum _{j=1}^{N_j}\frac{{(Y_i{\mathsf{\Delta }}_i)}^j}{j!}}\right]}\end{array}$$

(23)

where

$$Y_k=\left[\begin{array}{ccc}\frac{1}{{\omega }_0}\frac{1}{\mathsf{\Delta }k}{\displaystyle {\int }_{t_k{}_{-1}}^{t_k}{a}_{11}}d\tau & \frac{1}{{\omega }_0}\frac{1}{\mathsf{\Delta }k}{\displaystyle {\int }_{t_k{}_{-1}}^{t_k}{a}_{12}}d\tau & \frac{1}{{\omega }_0}\frac{1}{\mathsf{\Delta }k}{\displaystyle {\int }_{t_k{}_{-1}}^{t_k}{a}_{13}}d\tau \\ \frac{1}{{\omega }_{0}C}& -\frac{1}{R_L{\omega }_{0}C}& 0\\ -\frac{k_M{R}_{vf}}{{\omega }_0{R}_{vi}C}& \frac{k_M{R}_{vf}}{{\omega }_0{R}_{vi}C{R}_L}-\frac{k_M}{{\omega }_0{R}_{vi}{C}_{vf}}& 0\end{array}\right]$$

(24)

According to (23), the Floquet multiplier of the transition matrix M can be calculated and used to determine the stability and bifurcation types of the periodic time-varying system. With the selected parameters in the previous section and the periodic solution obtained in (15), three Floquet multiples of the transition matrix M are calculated as λ₁ = 2.9667 × 10⁻⁴, λ₂ = 2.6986 × 10⁻²¹, and λ₃ = −2.0678 × 10⁻²⁰, respectively. Evidently, these three Floquet multiples remain in the unit circle, indicating that the voltage-controlled full-bridge inverter with a nonlinear inductor is in stable operation.

The variations of three Floquet multipliers of the transition matrix M with respect to the different capacitor C_vf in the PI compensator are shown in

Table 1. Floquet multipliers of the transition matrix M with respect to the different values of C_vf.

Cvf	λ1	λ2,3	|λ2,3|	State
120.0nF	0.0010	−5.7784 × 10−20, −2.8184 × 10−20	Less than 0.01	Stable
100.0nF	0.0003	−4.1838 × 10−21, 7.4738 × 10−21	Less than 0.01	Stable
80.0nF	2.8828 × 10−5	−3.9112 × 10−22 ± 3.3437 × 10−22i	Less than 0.01	Stable
60.0nF	7.2088 × 10−7	−2.2764 × 10−23, 3.3481 × 10−23	Less than 0.01	Stable
40.0nF	3.2698 × 10−10	−1.1158 × 10−26 ± 5.6199 × 10−26i	Less than 0.01	Stable
20.0nF	2.5304 × 10−21	−1.5602 × 10−35 ± 7.3916 × 10−34i	Less than 0.01	Stable
10.0nF	−1.5264 × 10−37	−4.8659 × 10−22 ± 1.8747 × 10−22i	Less than 0.01	Stable
4.0nF	−1.3293 × 10−16	−0.9869 ± 0.1610i	0.9999	Stable
3.9nF	−4.9487 × 10−16	−1.7504 ± 3.5006i	3.9139	Unstable

. Additionally, two Floquet multiples (λ₂ andλ₃) with different capacitors C_vf are plotted in Figure 4. From Table 1 and Figure 4, when C_vf = 4 nF, three Floquet multiples (a real number and a pair of complex conjugates) of the transition matrix M of the voltage-controlled full-bridge inverter with a nonlinear inductor are all located in the unit circle, indicating that this system is stable. However, when C_vf = 3.9 nF, a pair of conjugate complex Floquet multiples of the transition matrix M are located outside the unit circle, indicating that the voltage-controlled full-bridge inverter with a nonlinear inductor is in unstable operation. Therefore, when the capacitor C_vf gradually decreases from 4 nF to 3.9 nF, there is a pair of complex conjugate Floquet multipliers penetrating from inside the unit circle and moving outwards, that is, Hopf bifurcation is occurring in this system. Therefore, when C_vf ≥ 4 nF, the voltage-controlled full-bridge inverter with a nonlinear inductor is stable, while when C_vf is less than or equal to 3.9 nF, the system is unstable.

Therefore, when C_vf = 102 nF and its value is greater than 4 nF, the voltage-controlled full-bridge inverter with a nonlinear inductor should be stable and there is no oscillation in the current i_N and the voltage v₀, as shown in Figure 3. However, when C_vf = 2 nF and its value is less than 3.9 nF, the voltage-controlled full-bridge inverter with a nonlinear inductor is unstable. The calculated three Floquet multipliers of the corresponding transition matrix M under C_vf = 2 nF are λ₁ = 1.1686 and λ_2,3 = 3.2608 × 10¹⁶ ± 7.2821 × 10¹⁵i, respectively. It can be seen that these three Floquet multipliers are located outside of the unit circle, which further proves that the voltage-controlled full-bridge inverter with a nonlinear inductor under C_vf = 2 nF is indeed unstable.

Figure 5 shows the stability boundaries in the parameter space of the capacitor C_vf and the capacitor C. One can see that with the decrease of capacitance C_vf or the increase of capacitance C, the voltage-controlled full-bridge inverter with a nonlinear inductor is more likely to be unstable. Likewise, in Figure 5, for the load resistance R_L = 5 Ω, the stable region is B₁ + B₂. However, for the load resistance R_L = 10 Ω, the stable region is B₁. Hence, as the load resistance R_L increases, the voltage-controlled full-bridge inverter with a nonlinear inductor has a smaller stable region; that is, the system is more likely to become unstable.

5. Experimental Results

According to the circuit structure given in Figure 2, the designed hardware experimental circuit for the voltage-controlled full-bridge inverter with a nonlinear inductor is shown in Figure 6. In the experiments, the current probe Tek A622 and the voltage differential probe P5200A were used to measure the current and the voltage in the voltage-controlled full-bridge inverter with a nonlinear inductor, and the Tek MDO-34 oscilloscope was used to capture the waveforms in probes. Four power switches are realized by using IRFP460, and the driver circuit used for driving these four power switches is realized by IR2130, the operational amplifier is LF356, the comparator is LM311, and the phase inverter is MAX628. In addition, the voltage sensor LV25-P (i_P: i_M = 1000: 2500) is used to measure the floating voltage v₀, and its magnification k_M is calculated as follows:

$$k_M=\frac{v_M}{v_0}=\frac{i_M{R}_M}{i_P{R}_P}=\frac{2500\times 200}{1000\times 2000}=\frac{1}{4}$$

(25)

The other parameters remain unchanged, except when choosing a different capacitance, i.e., C_vf = 102 nF and C_vf = 2 nF, the experimental results of the voltage-controlled full-bridge inverter with a nonlinear inductor can thus be observed. Figure 7a shows the experimental results of the current i_N and the voltage v₀ under C_vf = 102 nF. Thus, when C_vf = 102 nF, the system is in stable operation. Moreover, the maximum value of the voltage v₀ from the experiment is 7.267 V, and the maximum value of the current i_N in the experiment is 1.522 A. According to (12) and (15), the approximate calculated results for the voltage v₀ and the current i_N are 7.482 V and 1.497 A, respectively. Moreover, by comparing Figure 7a with Figure 3a,b, it can be seen that the experimental results are consistent with the theoretical calculation results, which verifies the correctness of the established model and the effectiveness of the approximate calculations.

In addition, with the aid of the frequency spectrum analysis of the PLECS and choosing 50 Hz as the fundamental frequency, the experimental curves of the current i_N and the voltage v₀ can be obtained, as shown in Figure 7c,d, respectively. One can see that in the range of 0~1 kHz, the harmonics of both the current i_N and the voltage v₀ have smaller amplitudes than the fundamental wave. In the range of 1–2 kHz, the amplitude of each harmonic can be approximately zero.

Figure 8a illustrates the experimental curves of the current i_N and the voltage v₀ under C_vf = 2 nF, while Figure 8b is the close-up view of Figure 8a. Figure 8c,d are the harmonic spectrums of the current i_N and the voltage v₀, respectively. It can be seen that when C_vf = 2 nF, the voltage-controlled full-bridge inverter with a nonlinear inductor is in unstable operation and there is oscillation; the oscillation frequency is between the power frequency of 50 Hz and the switching frequency of 25 kHz. This oscillation phenomenon can be called medium-frequency oscillation. In the region of the oscillation frequency, both the current i_N and the voltage v₀ have larger harmonics. Therefore, the occurrence of medium-frequency oscillation will increase the difficulty of the system filter design; likewise, will increase the voltage stress and the current stress of the power switches in the system.

6. Conclusions

In this paper, the results from theoretical analysis, approximate calculation, stability determination, and circuit experiment show that in the voltage-controlled full-bridge inverter with a nonlinear inductor, the medium-frequency oscillation whose oscillation frequency is between the power frequency and the switching frequency causes the occurrence of Hopf bifurcation, making the filter design for the system more difficult since this oscillation should be filtered. Hence, the medium-frequency oscillation occurring in the voltage-controlled full-bridge inverter with a nonlinear inductor should be avoided in practical engineering.