LLC resonant converter analysis in PO mode

The PO mode under time-domain analysis is a kind of discontinuous conduction mode (DCM) which is beneficial to reduce the switching loss of the converter, but the calculation of the gain and frequency characteristics of the converter in this mode is cumbersome and difficult to be solved. Based on the analysis method of time domain approximation, the PO operating mode characteristics of the LLC resonant converter are analyzed, and a simple and accurate voltage gain calculation formula is theoretically deduced. Simple analytical expressions for the PO mode power boundary conditions are given based on the voltage variations of the magnetizing inductance and capacitance. A simulation model of the studied converter is established, and the simulation can prove that the proposed voltage gain formula and PO mode boundary conditions are correct.


Introduction
The fundamental harmonic analysis (FHA) takes the resonant cavity voltage and current approximately equivalent to sinusoidal quantity, which has more simplification, simple analysis, and high practicality.However, the LLC analysis of intermittent conduction mode has large errors and low precision [1][2].The time-domain analysis method has a complex analysis degree, and the voltage gain expression is difficult to express.Although the analysis accuracy is high, it has to be solved by mathematical software, and the practicality is low [3][4].Both have their advantages.
The PO mode occurs in the under-resonant state, in which the voltage gain of the converter decreases monotonically with increasing frequency [5].The switching tube of the inverter circuit can realize ZVS and the diode of the rectifier circuit does not need to reverse the recovery process, and the stability of the control closed loop can be ensured.Therefore, this is the most beneficial operating mode to reduce losses [6].PON and PN cannot guarantee that the switch tube achieves ZVS.OPO mode will produce high circulation conduction loss [7].A detailed analysis of each mode was carried out in [8], indicating the connection between each mode and the boundary of PO is OPO and PON.In [9], the PO mode is studied using time domain analysis, and the converter is designed according to the performance curve.In [10], the converter uses a varying inductor, which cleverly avoids the effect of switching frequency on the mode of operation and reduces the variables.In [11], the power boundary conditions of PO mode were given based on the subinterval analysis method.Still, only the power boundary conditions between PON and PO were studied, and the boundary conditions between OPO and PO were not studied.
This paper uses time domain approximation analysis to avoid tedious time domain analysis and make up for the inaccuracy of FHA.The voltage gain equation is derived by analyzing the converter's PO mode of operation.At the same time, the corresponding boundary conditions are deduced from the voltage variations of the resonant capacitance and magnetizing inductance in the PO mode boundary.The corresponding simulations are carried out to verify that.

Time domain approximation analysis
A full-bridge LLC resonant converter is the research object.The converter uses complementary communication signals, as shown in Figure 1.

Figure 1. Full bridge LLC converter.
There are six basic modes in the time domain waveform of the LLC resonant converter.If time domain analysis is performed, transcendent equations will occur, and analytical expressions for most modes of operation cannot be obtained.It is difficult to perform quantitative analysis, which is detrimental to analyzing the design.Since the PO mode appears within the under-resonant state, this paper focuses on fs<fr.Figure 2 shows its main operating waveforms.Not only the dead time is neglected, but also the common resonant waveforms of the three resonant elements are simplified.The following quantities are defined for ease of analysis: In the period t0~t1, the waveform characteristics can be listed: In the period of t1~t2, since Lm is usually much larger than Lr, and the period is short, the magnetizing inductor current can be approximated to a constant value Im.

∋ ( ∋( ∋( ∋ (
According to the symmetry, there are: It follows from Equation ( 4) and the continuity of iLr and iLm at time t1: The average value of the current flowing through the diode in half a cycle is the output current, that is: During this period, the voltage of the resonant capacitor is also limited by the following: ∋( From Equation ( 5) and ( 9), the voltage gain can be obtained as:

Boundary conditions for the PO mode
Based on the modal analysis, the PO mode is bounded by PON and OPO, and it is not rigorous to use the voltage of the magnetizing inductor as the boundary condition because when the input voltage of the resonator jumps, the voltage of the magnetizing inductor will jump as well.It is not feasible to judge the boundary condition directly by the inductor voltage but indirectly by the resonant capacitor voltage.PON and PO two modes of physical process is the difference between the former resonant capacitor voltage stress is high, that is, uCr(t) amplitude is large, resulting in not until u1(t) next jump, O state is over, rectifier circuit commutation, N state appears; The latter until the next jump of u1(t), uCr(t) is also not enough to commutation the rectifier circuit and the N state cannot appear.Therefore, the operating mode boundary conditions of PON and PO are as follows: The simultaneous Equations ( 1), (2), and (5) are obtained: The difference between PO and OPO is that the voltage on the magnetizing inductor does not reach nV0 due to uCr(t), so the boundary conditions of PO and OPO are as follows: The simultaneous Equations ( 1) and ( 5) are obtained: From Equations ( 8), ( 10), (11), and (12), the boundary conditions for the PO mode can be obtained as: Figure 3 shows the effect of k on the PO mode boundary at different frequencies when the load is constant.It is clear that the closer fn is to 1, the larger the PO mode boundary range is; that is, the easier it is to realize PO mode, and the smaller fn is, the more difficult it is to realize the PO mode.For the converter to realize PO mode at full switching frequency, the condition that must be met is that the converter can operate in PO mode at the lowest switching frequency.Lm/Lr will gradually decrease to a constant value at the same switching frequency.The PO mode boundary conditions are less affected by the value of k.

Simulation validation
The relevant simulation model was established in MATLAB to verify the correctness of the theoretical analysis.Table 1 is the LLC's partial parameter; other parameters will be determined later.Different values of k are chosen, and the proposed PO mode gain equation is verified using parameters R ranging from light to heavy loads to make the simulation results convincing.At the same time, the corresponding FHA is added, and the theoretical analysis results of this paper, FHA analysis results, and simulation results are compared.
The relationship between M and fn is studied under different R and k in PO mode.Firstly, the converter is made to work in PO mode, calculated from equation 10: Lr/Cr is chosen as 1944.8, and k is chosen as 4.12 for simulation.Figure 4 shows the relationship between M and fn under different R. Figure 4 shows that R has little effect on the gain, which corresponds to Equation (10) and is in line with the derivation in this paper.At fn=0.6, the simulation error of the proposed calculation method is the largest, 4.4%, and the simulation gain error between different resistance values is 7.3%, indicating that R has little influence on the gain in PO mode.The FHA performance, however, is somewhat worse, with a minimum error of 10.64% and a maximum error of 24.7% at fn=0.6.Higher accuracy is only found close to the resonant frequency.
Keeping Lr/Cr and R=3 Ω unchanged, the relationship between fn and M under different k is studied.As shown in Figure 5, the maximum error is still at fn=0.6.When k=4, the error between the proposed calculation method and the simulation is 4.4%; when k=8, the error between the proposed calculation method and the simulation is 1.4%.FHA still lags behind the computational method of this paper, and its errors are 10.6% and 4.8% in the same cases, respectively.It can also be noticed that the error at k=4 is larger than the error at k=8 because a large value of Lm will make the magnitude of iLr(t) much larger than im(t) so that the approximate invariance within the O mode brings about a smaller error (Figure 2), which more closely follows the simplified condition.In PO mode, the overall simulation results are closer to the calculated results of Equation (10) in this paper than in FHA mode, with higher accuracy even far from the resonant frequency.The calculation method in this paper has high precision, does not make rough simplifications like FHA, avoids complex region analysis, and analyzes the LLC resonant converter operating mode.

Conclusions
In this paper, based on the analysis method of time domain approximation, the steady-state characteristics of the LLC resonant converter operating in PO mode are analyzed, and the voltage gain expression in PO mode is derived.Unlike FHA, the formula has high precision.Boundary conditions in PO mode are derived from the voltage variation relationship between the magnetizing inductor and capacitor in the converter.Finally, simulation tests verify the voltage gain equation and the PO mode's boundary conditions.At the same time, it has some guiding significance for designing an LLC resonant converter working in PO mode.

Figure 2 .
Figure 2. PO mode steady state waveform of LLC converter.

Figure 3 .
Figure 3.Effect of k on PO mode boundaries at different frequencies.

Figure 4 .
Figure 4. Relationship between fn and M with different values of R.

Figure 5 .
Figure 5. Relationship between fn and M with different values of k.Finally, three points, A(3,2), B(4,10), and C(4,35), as shown in Figure 6, are selected for the verification of the PO mode boundary conditions of the LLC resonant converter.As can be seen from the figure, the converter at point A is in OPO mode, the converter at point B is in PO mode, and the converter at point C is in PON mode.Figures 7, 8, and 9 show the three-point simulation figures.Two O modes and one P mode appear in the half cycle of Figure 7, so the simulation result at point A is OPO mode; one P mode and one O mode appear in the half cycle of Figure 8.The simulation result at point B is PO mode, and one P mode, one O mode, and one N mode appear in the half cycle of Figure 9; the simulation result at point C is PON mode, which corresponds to Figure 6 and verifies the boundary conditions of the proposed PO modes.
2.2 The gain formula for the PO modeAt time t0, iLr and iLm are equal:

Table 1 .
The parameters of the LLC resonant converter.