An improved large signal model of full-bridge LLC converter

For the full-bridge LLC converter, it is necessary to establish a large signal model with low-complexity and high-precision when the current of filter inductor works under the continuous conduction mode (CCM) situation. In terms of low complexity, peak values of resonant state variables and average values of slow state variables are taken as the measured indexes between models and actual converter. Models based on the first harmonic approximation (FHA) are studied in depth because they have lower complexity than the existing models. From prospective of high precision, deficiencies of typical FHA model are explained in the light of harmonic generation mechanism of primary current and influence of Fourier series on the typical equivalent circuit. Therefore, transient process of each metallic oxide semiconductor field effect transistor (MOSFET) and its related fast recovery diode, transient process of Schottky rectifier diodes, piecewise equivalent circuit, appropriate selection of variables, and simplified modified dynamic equations are all considered. Furthermore, the unified proposed model is achieved according to the equivalent principle of indexes between aforementioned improved analyses and proposed model. Numerical results of typical FHA model and proposed model are given in accordance with the key indexes. Then, corresponding experimental results are also presented. Differences of indexes between the two models and the actual converter are compared respectively. Though complexity of proposed model is the same as that of typical FHA model, precision of proposed model is higher than that of typical FHA model.


Introduction
With the popularity of renewable energy generation, kinds of DC-DC converters play important roles in the power conversion system. In the photovoltaic (PV) system, DC-DC converter is connected to PV array and inverter can be taken as load. A large number of advanced maximum power point tracking (MPPT) algorithms have been successfully implemented in the DC-DC converter. These algorithms mainly include two-step algorithm for global MPPT [1], natural cubic-spline-guided Jaya algorithm (S-Jaya) [2], compensation power DC-DC converter for the distributed model-based MPPT (CPDC-DMBMPPT) [3], and model predictivebased controller (MPC) with a fixed step that is combined with the traditional incremental a1111111111 a1111111111 a1111111111 a1111111111 a1111111111 conductance (INC) algorithm [4]. In the electric vehicle (EV) system, DC-DC converter is intermediate link between different energy storages and DC bus. For example, in [5], high voltage battery-supercapacitor and DC motor are connected to the two ports of a kind of typical bidirectional DC-DC converter respectively. Proposed power-split strategy is developed to track real-time load profiles and determine cutoff frequency. In [6], dual battery energy source and DC-bus of different voltage levels are connected to the interfaces of proposed bidirectional DC-DC converter respectively. Furthermore, different modes of power transfer can be effectively carried out according to the hybrid model. In the modern electric ship medium voltage DC power system such as [7], DC-DC modular multilevel converter with a medium-frequency transformer is the promising transmission device. For the complicated converter, a novel fundamental period averaging (FPA) method is applied to obtain the steady states and dynamics. In the mining applications such as [8], DC-DC converter is necessary in the mobile mining equipment. In [8], the converter employs a passive LCL filter instead of high-frequency transformer so that core loss of transformer can be eliminated. Meanwhile, the proposed control strategy based on single phase pq theory is developed to ensure stable and robust operation.
Isolated full-bridge DC-DC converter is widely used in the aforementioned fields when electromagnetic isolation is required. Mastering the properties of full-bridge converter help researchers establish the appropriate mathematical model which mainly includes small signal model and large signal model. Furthermore, this kind of DC-DC converter can be effectively improved and designed. Small signal model can be the transfer functions of duty cycle to filter current, filter current to output voltage, duty cycle to output voltage, and input voltage to output voltage respectively [9][10]. All the transfer functions are formulated according to respective small signal equivalent circuit which decides characteristics of DC operating points and their neighborhood operating points. However, analysis on the characteristics of operating points is no longer appropriate when input voltage and load occur to a wide range of changes in the certain region. In this case, large signal model need to be founded and models in the following text are referred to it. Large signal model can be state equations on the average current of primary inductor and average voltage of parallel capacitor [11], the average output voltage and current [12], the average voltage of filter capacitor, average input current and output current [13], and so on. All the state equations are also derived from respective equivalent circuit which can reflect the properties of converter under the certain region. According to steady state and transient state of converter, large signal model can be obtained when input voltage and load work at the specific region.
Full-bridge LLC converter is one kind of isolated full-bridge DC-DC converters. Similarly, it is necessary to recognize the analyses bases related to steady state and transient state of converter before modeling and further design. In [14], proposed automatic resonant frequency tracking (ARFT) method is based on the model which contains phase and gain relationship of an exclusive variable pair in the LLC network. In addition, realistic ramp variation in resonant frequency is also reflected in the model. In [15], synchronous rectifier turn-on time is extended to improve control capability under light load condition. Investigated model cannot live without equivalent circuit which is made fully use of so that proposed Fourier series analysis can be developed. In [16], Lagrangian model is established according to the energy equivalency and equivalent circuit which includes magnetic circuit of proposed integrated transformer. State variables comply with the flux cancellation concept and operation rules of equivalent magnetic components. In summary, it is crucial to have knowledge of analyses on the steady state and transient state before establishing large signal model.
Analysis bases on the steady state properties of LLC converter are formulated in detail as follows. In [17][18][19], six operation modes are illustrated and they are classified by the relationship between switching frequency, resonant frequency, and conduction mode of output current. Equivalent circuits are normalized according to the FHA. Switching loss, conduction loss of switch devices, copper loss and core loss of inductive components are all considered in [18]. However, influence of transient processes related to switch devices on the resonance variables is neglected. In [20], the steady state characteristics are divided into three regions based on the FHA and corresponding equivalent circuit. Voltage conversion ratio, resonance frequency, characteristic impedance, quality factor, inductance ratio, normalized frequency, and equivalent load are all given. However, these relationships are obtained by the assumption that resonant network has well selectivity of sinusoidal signals. In [21] and [22], secondary leakage inductance is contained in the equivalent circuit. Developed analyses and equations are established according to the modified equivalent circuit. However, the equivalent converted load at primary side still comes from the typical FHA, which influences the precision accuracy of proposed circuit. In [23], equivalent isolated transformer circuit is improved by using the coupling coefficient to express inductances and turns ratio. Voltage gain and switching frequency can be predicted with high accuracy but equivalent load is still derived from the FHA. In addition, the transient processes of switch devices are not considered. In [24], time domain analysis is introduced to improve the accuracy of voltage gain. Relationship between voltage gain, duty cycle, quality factor, and inductor ratio can be obtained by solving the numerical equations. However, Definition of quality factor still comes from the FHA. In [25], denominator of voltage gain is divided into resonant factor and load factor. They are related to the inductance ratio and quality factor respectively. The modified gain is obtained by linearizing and averaging proposed equivalent circuit and its theoretical curves. However, the transient characteristics of switches are neglected and it will cause deviations. In [26], proposed LLC converter is equivalent by applying the FHA. Input source and load are also derived from the FHA. Secondary leakage inductance is also neglected. However, it is necessary that the LLC network has well capacity of selecting fundamental wave so that boundary conduction mode can be strongly applied. In [27], frequency modulation and duty ratio modulation are synchronously implemented. Primary current and voltage of resonant capacitor are not sinusoidal because of the asymmetrical hybrid modulation. However, equivalent single output circuit is obtained by the typical FHA.
From what has been analyzed above, the following drawbacks can be concluded. (1) Dead time of the two MOSFETs at same arm of bridge is neglected. Furthermore, the analysis on the primary current is not exactly because influence of transient process is not considered. (2) Commutation process of two Schottky rectifier diodes is also neglected. It is important to take the process into account when filter inductor works under the CCM condition because not only output current but also primary current is affected by the process. (3) Resonant variables such as primary current, current of magnetizing inductance, and voltage of resonant capacitor are non-sinusoidal due to the transient processes of MOSFETs and Schottky rectifier diodes. Equivalent converted load at primary side is derived from the FHA and it needs to be modified under the CCM condition.
Analysis bases on the transient state properties of LLC converter are discussed in detail as follows. In [28], differential equations deduced from the corresponding equivalent circuit are transformed according to the typical averaging method. Averaged variables are investigated when load is sudden to change. Their trends of change are similar to the corresponding peak variables but not completely coincident. In [29], oscillation frequency and amplitude of output current are obtained by the FHA and extended describing functions (EDF) respectively. Method of transient current ripple reduction can be directly designed by the FHA and numerical solutions can be given by the EDF. In [30], feedback linearization is based on two-order dynamic equations derived from the equivalent FHA circuit. Characteristics of FHA are fully used of so that transient states can be further simplified. In [31], eigenvalues displacement for different operating conditions defined by switching frequency and displacement of dominant poles for different load are analyzed by equivalent resonant circuit deduced from the FHA. However, aforementioned analyses can be used when the LLC network has well sinusoidal. In [32], segmented equivalent circuits are obtained according to the different operation modes. Furthermore, state-plane composed of voltage of resonant capacitor and primary current is analyzed so that the optimal trajectory can be found during load transients. In [33], the main analysis method is similar to [32]. Voltage and average current of filter capacitor make up the two dimensional state-plane. In [34], normalized output voltage, primary current, and voltage of resonant capacitor are shown in the three dimensional state-plane. Furthermore, startup dynamic performance of voltage and average current of filter capacitor are analyzed in the time domain by the two dimensional state-plane. However, aforementioned variation tendencies of other unemployed variables are not clear due to the simplified state-planes in [32][33][34]. In order to get other something important, complex analytic variables need to be taken full use of. In [35] and [36], seventh-order state equations which contain real and imaginary parts are used. Average absolute primary current and output voltage are the output variables of converter. These dynamic equations are derived from the equivalent circuit composed of sine and cosine components. When the parameters suddenly change, transient process can be verified. However, it is benefit when the complexity of state variables is reduced. In [37], resonant variables and output voltage can be predicted by numerical equations during the startup process. Furthermore, the relationships between startup current, initial startup frequency, and duty cycle are presented. However, it is complicated that twelve modes need to be decomposed for the combination of different state variables.
From what has been discussed above, the following conclusions can be drawn. (1) Variables which are used to analyze transient state properties of converter need to be appropriate. A few numbers of variables result in the lack of something important and transient states are generally reflected by the approximate solutions of differential equations. On the contrary, a large numbers of variables lead to the high complexity of state equations. It is inconvenient to further design the converter by transient process. (2) Only the LLC network has well sinusoidal selectivity, can analytic equations based on the FHA be effectively used. If resonant variables are much more different from sinusoidal signal, the typical FHA cannot be applied directly and they need to be modified. Routine process of correction are much complex because large number of series are used.
In this paper, the condition that current of filter inductor works under CCM is investigated. In order to found a large signal model with low-complexity and high-precision, aforementioned five existing shortcomings have to be paid attention to. Firstly, three existing typical models are presented and respective complexities are considered according to the size of each table in the digital signal processor (DSP). Peak values of resonant state variables and average values of slow state variables are taken as the indexes between models and actual converter. Models based on the FHA are researched in the following text because they have the same lowest complexity. Secondly, equivalent circuit is obtained by the FHA. Differences between equivalent circuit and LLC converter are input source, simplified models of components, and equivalent load. Meanwhile, analyses on the steady state and transient state of model are given. Large signal model is established according to the aforementioned analyses. Variables of the nonlinear mathematical model are the instantaneous values of resonant variables and the average values of slow variables. Thirdly, reasons for the low accuracy of FHA under the studied condition are explained. There are two main viewpoints to formulate the reasons. They are harmonic generation mechanism of primary current and influence of Fourier series on the equivalent circuit respectively. In the meantime, work region of converter is given. Furthermore, combined with the five drawbacks mentioned above, improved equivalent circuit and corresponding improved analyses on the steady state and transient state are shown. Proposed mathematical model is established based on the equivalent principle of indexes. Correction coefficients need to be solved so that proposed model is certain. Lastly but very important, numerical operations and experiments need to be done. For the typical FHA model and the proposed model, steady state and transient state of mentioned indexes are calculated by Matlab. Correspondingly, results of actual converter are presented. What is more, accuracy of proposed model is verified by comparing differences of indexes between two models and actual converter.

Existing models and complexity
Abbreviations in this section are defined in Table 1.
When analysis bases related to steady state and transient state of converter are done, establishing large signal model is natural step. There are several modeling methods to be referenced. In [38], prediction model with improved extreme learning machine (ELM) is proposed. Prediction accuracy is enhanced by considering the distribution of data through the use of L 2 norm. In [39], fine-grained activity recognition model is achieved by using improved ELM. The kernel risk-sensitive loss (KRSL) is incorporated into a novel multilayer neural network and identification accuracy can be further enhanced. In [40], generalized frequency-dependent averaged model is introduced by designing suitable jump mode which can approximate the transient states during switching time. The model can be treated as differential algebraic equations. In [41], approximate discrete-time model of nonisolated DC-DC converters is proposed and low-pass properties of converters are taken into account. Proposed model can capture the natural sampling characteristics and dimension of system matrices can be reduced. In [42], for the three-phase dual active bridge (3p-DAB) converter, generalized state space averaging (GSSA) model based on the dynamic phasor is developed. Proposed model is combined with state space averaging (SSA) method and precise for the stability analysis can be enhanced. In [43], generalized average models of dual active bridge (DAB) converters are proposed by using a triple phase shift modulation which includes single, dual and extended schemes. Modeling framework is a tradeoff between complexity and accuracy. Methods mentioned above have guiding significance for modeling. However, for the LLC converter, there are three typical models in the existing literatures and they are listed in detail as follows.
The studied full-bridge LLC converter is presented in Fig 1. Symbols in the figure are described in the nomenclature. In addition, the turn number ratio of primary side and secondary sides is N:1:1.
Þcos ot B Second step: Extended describing function.
i P ¼ ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi In conclusion, the final simplified large signal model can be expressed as follows: where X(t) is composed of i 1 , i 2 , i 3 , i 4 , u 1 , u 2 , and u C .

Model 2
Typical model 2 is deduced from the [37]. There are three steps to establish the model. Firstly, different states are analyzed from the view of input port and output port. The unified LC equivalent circuit can be obtained in each state. Furthermore, the equivalent impedance and angular frequency are defined and solutions of the equivalent circuit are given. Secondly, the equivalent parameters contained in the solutions of equivalent circuit under twelve modes are listed in detail. The modes are described as M n and the number n in each mode is fixed. Lastly, conduction modes and transition modes for the switches are defined as S in respectively. Similarly, the modes for the rectifier diodes are also expressed as S out . Relationships between the M n , S in , and S out are formulated and it is taken as the mode judger. Meanwhile, the voltage at secondary side of transformer at the current moment and the increment of output voltage can be obtained.
A First step.

Conduction modes for the switches
2. Transition modes for the switches 3. Modes for the rectifier diodes where X(n) is composed of i Le (n) and u Ce (n).

Model 3
Typical model 3 is derived from the [34]. There are three steps to establish the model. Firstly, the t k is defined as the instants when the structures of converter change and S is defined as the switching function. Secondly, the main differential equation and auxiliary equations during the two adjacent half cycles are formulated. Integral upper limit functions exist in these equations. Lastly, a homogeneous linear differential equation with constant coefficients is obtained by differentiating the integral upper limit functions and applying the elimination method. The new higher order differential equation is unified. Its characteristic equation needs to be solved by combining with piecewise differential equations which occur in the second step. Therefore, other variables can be obtained. All the variables can be described by simplified expressions. A First step: Definition of t k and S.

Half cycle kth (t k < t < t k+1 )
Main equation: Auxiliary equations: Main equation: Auxiliary equations: Model is applied in the DSP and look-up table method is widely used. Therefore, complexity of model can be defined as the size of each table in this paper. According to the (1), (3), and (4), it can be concluded that state variables at next moment are determined by state variables, input voltage, load, switching angular frequency corresponding to different modes at present moment no matter what the functions F are when the tables are used. State variables, input voltage, and load at present moment are deterministic and optional modes of switching angular frequency at present moment are determined by the number of internal functions. The table means state variables at next moment under the certain state variables, input voltage, and load at present moment. In other words, size of each table is product of the dimension of X and the number of internal functions related to ω. The value is namely the complexity of corresponding model. Complexity for the different existing models can be seen in Table 2.
For the LLC converter, resonant state variables are u Ca , i Lp , and i Lm . Slow state variables are i Lf and u C . Most important indexes between models and actual converter are peak values of resonant state variables and average values of slow state variables because they can reflect basic characteristics of LLC converter and they are direct references of further designing converter. Analyses on the FHA provide concise relationships between indexes and it is convenient to establish models. Variables in the FHA models are instantaneous values of resonant state and average values of slow state, but only the indexes are available in the DSP. Namely, dimension of state variables in the FHA models is five. In addition, mode of switching angular frequency in the FHA models is deterministic because the switching angular frequency is not compound function. So the number of internal functions related to switching angular frequency is one. Size of each table is five and it represents the complexity of FHA models. Given the lowest complexity, models based on the FHA analyses are formulated in detail as follows.

Typical FHA model
Abbreviations in this section are defined in Table 3

Analysis on the steady state
According to the [20], U AB(eq) , R eq(FHA) , f, f r , Z, Q, l, f n , M, and U o can be listed as follows: ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ð1 þ l À l When the converter works under the steady state, Fig 2 is fully utilized. Indexes U Ca_peak , I Lp_peak , I Lm_peak , I Lf_av , and U C_av are the basic elements of large signal model and the following  Large signal model of converter expressions of them are shown:

Analysis on the transient state
When the input source or the load happens to sudden change, converter works under the transient state. The R eq(FHA) can be expressed as follows: Imagine that initial time of transient state is t 0 and final time of transient state is t 1 . Average value of R eq(FHA) is applied in Fig 2 and it can be indicated in the following expression: where the two differential terms are estimated according to the corresponding old and new steady state. Solving process of (11) can be seen in the Appendix.
where initial values of states are the peak values under the old steady state.

Large signal model
In summary, a unified model needs to be founded so that the steady state and transient state of indexes can be well formulated. Definition of variables mentioned above continues to be used. Following nonlinear mathematical equations are taken as the large signal model based on the typical FHA: where U C is actual average voltage of C in the theoretical analysis. Differential terms of I Lf and U C(eq) are zero under the steady state. Solving process of aforementioned model can be seen in the Appendix.

Insufficiency analysis on the typical FHA
Abbreviations in this section are defined in Table 4 except that those have been appeared in aforementioned sections. When the current of filter inductor works under the CCM condition, the typical FHA is not precise to analyze the properties of converter because of the common drawbacks mentioned in the introduction. Furthermore, both harmonic generation mechanism of primary current and influence of Fourier series on the equivalent circuit can explain the reason that why aforementioned deficiencies about typical FHA cannot be neglected. The situation of transient state is similar to that of steady state in every switching period. So the situation of steady state is analyzed in this section.

Harmonic generation mechanism of primary current
In Fig 1, primary current is also the current of L P . In fact, voltage between A and B is the square wave and it is described as u AB . The cycle of u AB is as same as the switching cycle T. According to the Fig 2 and definitions in the tables of nomenclature, following equation is easily obtained: The u P and i P are the quasi square waves which are synchronously changing with u AB due to the interaction of rectifier diodes, large filter inductor, and filter capacitor. i P is almost constant in the time t on . Meanwhile, i Lm monotonously and linearly changes in this time. According to the (14), changing situation of i Lp is the same as that of i Lm in the time t on . Furthermore, following equations can be obtained: It is inferred that u P and u Ca both monotonously change in the time t on . In other words, i Lp is always nonnegative or nonpositive in this time.
Each first half of switching period is defined from moment 0 to moment T/2, and each second half of switching period is defined from moment T/2 to moment T. When the commutation process of rectifier diodes and the nearly constant current of filter inductor are taken into consideration, the load R needs to satisfy the following inequalities so that primary current i Lp is nonnegative at the moment T/2 and i Lp is nonpositive at the moment T: where the (18) represents the work region of converter. u AB and i Lp at first half of switching period are taken as the example. When the load R meets the (18), i Lp at the moment T/2 satisfies the following inequality: In addition, following inequality can be achieved according to the monotonicity of u Ca : Based on (19) and (20), the relationship of i Lp between the moment 0 and moment T/2 can be listed as follows: From what has been analyzed above, the diagram of u AB and i Lp at first half of switching period is shown in Fig 3. Similarly, they are reverse at second half of switching period. It is the reason for harmonic generation in the converter.

Influence of Fourier series on the equivalent circuit
According to the Fig 3, it can be known that u AB can be taken as square wave. The Fourier series of u AB is presented as follows: In the light of definition of t D , following equality can be shown: What is more, the following inequality can be obtained: The aforementioned relationship (24) indicates that the transition process of u AB cannot be simply depicted by its fundamental component.
In accordance with Fourier series of square wave and definition of effective value, variables u S , U S , i S , I S , U o , and I o can be described as follows: Furthermore, in terms of definition of resistance, R S(eq) can be shown as follows: The aforementioned relationship (30) states that R S(eq) cannot be replaced directly by the equivalent load derived from the typical FHA.
According to the (24) and (30), the conclusion can be drawn that Fourier series of variables in the converter need to be applied with several more items when the output of LLC network has not well sinusoidal selectivity. It can be observed from Fig 3 that the primary current is composed of many harmonic components and it cannot be equivalent to the single fundamental component effectively. Therefore, for the studied situation, analyses on the steady state and transient state need to be corrected. Furthermore, the modified large signal model is obtained in the following section.

Proposed model
Abbreviations in this section are defined in Table 5  R P includes primary AC resistance of isolated transformer, equivalent series resistance (ESR) of C a , and current sampling resistor. R S1 and R S2 are the same secondary AC resistances of

C eq
Equivalent drain-source capacitance k 1 -k 5 Correction coefficients transformer respectively. R S represents the general term of R S1 and R S2 . R Lf includes AC resistance of L f and current sampling resistor. When improved equivalent circuit and large signal model are proposed, the insufficiencies mentioned in the introduction are taken into account at the same time. Correspondingly, analyses on the steady state and transient state of converter are formulated in detail as follows. For every MOSFET and corresponding fast recovery diode, the transient process of midpoint voltage U AB is mainly determined by the junction capacitances C DS and C R in each

Analysis on the steady state
where the two functions can be fitted in accordance with the related datasheet. C DS and C R can fully charge and discharge when work region satisfy the (14). Furthermore, the common equivalent C eq is proposed so that the transient process can be effectively simplified. Based on the u AB in Fig 3, C eq can be expressed as follows: According to the (33), it can be known that C eq is varied with U in . The transient process can be formulated by the following equation:  The t on and T/2 have been defined in the aforementioned section. U S(av) and I S(av) are the average voltage and current of secondary winding from moment 0 to moment t on respectively. Average voltage and current of secondary winding, average voltage of L M from moment t on to moment T/2 are U S(av1) , I S(av1) , U Lm1 respectively. Relationships of U S(av1) , I S(av1) , and U Lm1 are expressed as follows:  condition. When the converter operates under the steady state, the corresponding resistance R loss is used to equate the aforementioned loss approximately. R loss is constant under the certain steady state but changes under the different steady states.
According to the Fig 4, it can be known that other conduction loss is caused by the R P , R S (R S1 = R S2 ), R Lf , and R D . The R D is equivalent resistance in the forward conduction mode and it can be inferred as follows: where k and b are the fitting coefficients according to the forward volt-ampere characteristics. u D and i D can be seen from the Fig 7. I D is the average value of i D at the certain time interval. The first half of switching period from moment 0 to moment T/2 is investigated and it is opposite to second half of switching period from moment T/2 to moment T. According to the Figs 3, 4, 7, 8, and mentioned above, the equivalent circuit from moment 0 to moment T/2 is proposed in Figs 9 and 10. In Fig 9, the influence of L f and C is neglected because voltage and current at secondary side are almost constant. Equivalent resistance R eq(pro) can reflect the volt-ampere characteristic of secondary side. In Fig 10,  Characteristic of pulse source U AB can be seen in Fig 3. C a , L P , R P , and L M are the same as that in Fig 4. Average current of i s1 is shown in the (36). Based on the definition of resistance, R eq(pro) is expressed as follows: The T h is used to represent the first half of switching period. The instantaneous voltage of C a , instantaneous current of L P and L M at the moment 0 are u Ca (0), i Lp (0), and i Lm (0) respectively. Similarly at the moment t on , they are u Ca (t on ), i Lp (t on ), and i Lm (t on ) respectively. According to the aforementioned analyses and the equivalent circuit shown in Figs 9 and 10, the following algebraic equations are listed by piecewise averaging and linearization: In the aforementioned equations, i Lp (0) is zero and nine unknown variables are t on , R loss , R eq(pro) , R eq(proposed) , u Ca (0), i Lm (0), u Ca (t on ), i Lp (t on ), and i Lm (t on ) respectively. The steady states can be reflected by applying the nine equations. In addition, it is -u Ca (0), i Lp (t on ), and -i Lm (0) that are U Ca_peak , I Lp_peak , and I Lm_peak respectively. Meanwhile, I Lf_av and U C_av can be shown as follows: The definitions of variables are in accordance with that in the typical FHA model. Suppose that the correction coefficients from k 1 to k 5 are corresponding to the u Ca , i Lp , i Lm , I Lf , and U C respectively. Proposed equivalent circuit can be seen in Fig 11 and it is originated from the typical FHA. Considered the invariance of zero-state response after step change of parameters, modified dynamic equations are obtained according to the principle that the indexes are approximate to the transient process which is only influenced by the characteristic equation of proposed equivalent circuit. Solving method of k 1 to k 5 can be seen in the next subsection.
The equivalent primary load in Fig 11 comes from R eq(proposed) which can be expressed as follows in the transient process: where R o(eq) is the equivalent load of converter under the new steady state. R o(eq) is solved in accordance with the (40a) to (40i) and it is equal to R eq(proposed) /N 2 when the converter works under the steady state. Imagine that the initial and final moments of transient state are t 0 and t 2 respectively. Average value of R eq(proposed) in Fig 11 is presented as follows: where the two differential terms can be estimated in the light of relevant old and new steady state. Solving process of (42) is the same as that of (11). Large signal model of converter Transient process of indexes can be approximately expressed as the positive envelopes of following state equations which the structure is similar to the (12): where the initial values of states are the indexes under the old steady state.

Large signal model
where the k 1 to k 5 are obtained by combining the (5) to (9)  From what has been analyzed above, it can be concluded that the precision of large signal model based on the typical FHA and the proposed method is the accuracy of indexes under the steady state and transient state in the two models.

Design and selection of parameters
Abbreviations in this section are defined in Table 6 except that those have been appeared in aforementioned sections.
Parameters of actual converter are designed and selected according to the Fig 12. It mainly contains driving circuit and main circuit. There are four same driving circuits except the driving sources. Every driving circuit is connected to the gate-source of corresponding MOSFET in parallel. Main circuit is the core of converter and it can be divided into six parts in the following description. Symbols of main circuit are the same as that in Fig 1.

Driving circuit
Driving circuit of Q1 is taken as an example and it is shown in Fig 13. The driving source U G1 generates pulse signal and it comes from the TMS320F2812, optocoupler separation circuit, and amplifying circuit. R 11 is the driving resistor which has effect on the switching action of Q1. V Z1 is the zener diode whose reverse voltage is the same as the amplitude of U G1 . Generally, the value is fifteen volts. R 12 is the discharge resistor for the gate of Q1 and it is about ten thousand ohms. The design and selection of R 11 are formulated as follows. R 11 is related to the gate-source capacitance C GS and gate-drain capacitance C GD . So C GS and C GD need to be fitted according to the datasheets. Both of them are the nonlinear functions of u DS and the functions are symbolically shown as follows: When the maximum value of u DS is certain, the average values of C GS and C GD will be known and they are expressed as C GS(av) and C GD(av) respectively.
In order to obtain the enough damping, following inequality is listed so that the oscillation of driving current can be suppressed: ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi L G C GSðavÞ s : ð46aÞ In order to prevent the malfunction of Q1 when it turns off, the inequality can be listed as follows: Based on the aforementioned two inequalities and the actual type of chip resistor, R 11 can be obtained effectively. Large signal model of converter

Main circuit
Parameters of the main circuit are designed and selected according to the following six parts. They are isolated transformer, resonant capacitor, filter inductor, filter capacitor, MOSFETs and corresponding fast recovery diodes, and rectifier diodes respectively.
1 Isolated transformer. In this paper, the bobbin of isolated transformer is type of EE. Parameters of transformer are designed and selected in the following order.

Turn ratio N
In order to make the output voltage can reach U o(max) when the input voltage is U in(max) , following relationship can be obtained: where N needs to be rounded.
4. Wire of secondary side Given the skin effect, relationships of them are similar to that of primary side and they are presented as follows:

Verification of window area
The windings must meet the following inequality. If it does not meet the expression, the wire has to be twined again.
L wireðPÞ D wireðPÞ þ 2L wireðSÞ D wireðSÞ < W Tr 6. AC resistance R (P) and R (S) The AC resistances of wire at primary and secondary side need to be redefined according to where the units of ρ and D wire(P) are O.mm 2 /m and mm respectively.

Leakage inductance L P and excitation inductance L M
A way of designing the LLC converter is proposed in [45] and [46]. However, the leakage inductance and excitation inductance of transformer are hardly regulated. So in this paper, the L P and L M need to be obtained by short-circuit test and open-circuit test which are shown in the following Figs 14 and 15. The U p(source) is a high frequency sinusoidal source. The I p(short) and U s(open) are taken as ideal ammeter and voltmeter respectively.
Furthermore, the following equations are deduced from the aforementioned schematic diagrams. Values of L P and L M can be estimated according to the following equations and a large Large signal model of converter number of tests: 2 Resonant capacitor C a . In order to make the output current always lower than the I o (max) when the switching frequency is f max and voltage C a reaches half of U in(max) , the C a is selected according to the following equation: where C a is obtained in the way of worst case analysis.
3 Filter inductor L f . The bobbin of filter inductor is also type of EE. Parameters of inductor are designed and selected in the following order.

Inductance
Value of L f is calculated according to the following equation so that the current ripple is not more than ten percent of I o(max) when the switching frequency, input voltage, and output voltage are f min(n) , U in(max) , and U o(n) respectively: where L f is obtained in the way of worst case analysis.

The turn number
Based on the definition of inductance with air gap, following relationship is shown: where the N Lf needs to be rounded.

Verification of B S(Lf)
The theoretical magnetic flux density must meet the following equation so that L f is not magnetic saturation. If it does not meet the expression, δ Lf needs to be modified again.

Verification of window area
The windings must satisfy the following inequality. If it does not meet the relationship, the wire has to be twined again.
6. AC resistance R (Lf) The AC resistance of wire is redefined in the light of Δ Lf and DC resistance. So it can be formulated as follows: where the units of ρ and D wire(Lf) are O.mm 2 /m and mm respectively.
4 Filter capacitor C. Value of C is obtained in accordance with the following equation so that the voltage ripple is not more than the standard ripple 0.2V when the switching frequency, input voltage, and output voltage are f min(n) , U in(max) , and U o(n) respectively: where C is obtained in the way of worst case analysis. 5 MOSFETs Q1~Q4 and fast recovery diodes V D1~VD4 . U Q , I Q , U VDR , and I VD come from datasheets based on the specific type of MOSFET and fast recovery diode respectively. In order to set aside enough margins, the following inequalities need to be satisfied: 6 Rectifier diodes D1 and D2. U DR and I D come from datasheets based on the specific type of Schottky rectifier diode respectively. They need to meet the following inequalities so that enough margins can be set aside: In this paper, aforementioned parameters are listed in Tables 7 and 8 to 13. Furthermore, all the relationships mentioned above are applied so that the specific components can be obtained. Main specifications of transformer and filter inductor are same and it is listed in Table 7 individually. Values of other components are shown in Tables 8 to 13.
According to aforementioned design and selection of parameters, the actual converter can be obtained. Main components and values are summed up in Table 14. Following numerical verifications are also presented in accordance with this table. The unit of f min(n) is kHz. Other physical quantities are all used the international system of units.
https://doi.org/10.1371/journal.pone.0205904.t008 Table 9. Values of components.  The units of f max , δ Lf , Δ Lf , and D wire(Lf) are kHz, mm, mm, and mm respectively. Other physical quantities are all used the international system of units.

Symbol
https://doi.org/10.1371/journal.pone.0205904.t010 Table 11. Values of components. The units of ΔU o(min) , C GS(av) , and C GD(av) are mV, pF, and pF respectively. Other physical quantities are all used the international system of units.

Numerical results
According to the Fig 4, main parameters of actual converter are listed in Table 14. It includes the main body of converter except for the switching frequency, input voltage, and load. The later three parameters will be given in Table 15 individually so that the different working conditions described from A1 to A8 can be well presented. In addition, the work region satisfies the (18).  Table 16. They are also the solutions of large signal model from (13a) to (13f) under the steady state. Given that I Lm_peak cannot be measured directly and it has to be replaced by I Lp_peak −I Lf_av /N in the actual converter, so I Lm_peak are not necessary to be listed in the following sections. represents the R eq(FHA) under old steady state and R eq1(FHA) represents the R eq(FHA) under new steady state. When working conditions are sudden to change from A1 to A3, from A2 to A4, from A5 to A7, from A6 to A8 respectively, following relationships derived from the (11) can be concluded according to the old steady state and new steady state shown in Table 16: u Ca peak : À 2:9588e À 8300:4978t þ 17:5532 ð66aÞ i Lp peak : À 0:0267e À 8300:4978t þ 0:1584 ð66bÞ State (c).

Proposed model
In order to obtain the typical drain-source capacitance C DS of IRFP 460, relationships between C oss , C rss , and u DS are shown in Fig 16 according to the corresponding datasheet.
Following function is obtained by fitting the two curves in Fig 17: The C eq in (33) can be operated by combining (69) and (70). When the input voltage is 100V, the C eq is 277.55pF. When the input voltage is 120V, the C eq is 268.41pF.
Forward volt-ampere characteristic of V50100PW is shown in Fig 18 according  conditions are sudden to change from A1 to A3, from A2 to A4, from A5 to A7, from A6 to A8 respectively, following relationships derived from the (42) can be generalized according to the old steady state and new steady state presented in Table 19:  The transition time is 0.8456 milliseconds. State (c).
These expressions can be taken as the approximate positive envelopes of large signal model from (44a) to (44f) under the transient state.

Experimental results
Actual full-bridge LLC converter has been designed and the main parameters are shown in Table 14. Working conditions verified by the experiment are the same as that in Table 15. According to the theoretical analyses and previous numerical results, the results of experimental verification are corresponding to the steady state and transient state respectively.

Results of steady state
In terms of input voltage and load at the same switching frequency, the working conditions A1 and A4 can be taken as the boundaries of A2 and A3 respectively. Similarly, the working conditions A5 and A8 can be also treated as the boundaries of A6 and A7 respectively. So the experimental results of key variables related to A1, A4, A5, and A8 are given under the steady state.
A1. Under the A1 condition, the u AB and i Lp can be seen in Fig 19. The u Ca and i Lp can be seen in Fig 20. The U C and I Lf can be seen in Fig 21. The i Lp is measured by 1O sampling resistor and the I Lf is measured by 0.1O sampling resistor.

A4.
Under the A4 condition, the u AB and i Lp can be seen in Fig 22. The u Ca and i Lp can be seen in Fig 23. The U C and I Lf can be seen in Fig 24. The   Tables 20 and 21 respectively.

Results of transient state
The state (a), state (b), state (c), and state (d) mentioned in the numerical results continue to be used. In order to verify the approximate expressions of u Ca_peak and i Lp_peak obtained in the 2. The u Ca and i Lp always satisfy the differential relation in the actual converter. Therefore, the variation tendency of u Ca_peak and i Lp_peak is always coincident in the transient process.
Therefore, the transition time t tr determines extent of approximation about the indexes under the transient state. The lower fractional errors of t tr between models and converter mean the higher extent of approximation.

Interpretation of results
1 Observing the key indexes under steady state and transient state in the following indicated tables, equations, and figures, we can draw a conclusion as follows.
Proposed model: Therefore, the maximum fractional errors of U Ca_peak , I Lp_peak , U C_av , and I Lf_av are 9.96%, 5.34%, 2.44%, and 7.78% respectively under the eight working conditions. In addition, the Therefore, the maximum fractional error of t on is 2.74% in the eight working conditions.

Advantages of proposed model
Based on the Table 2  There are a number of harmonics exist in the resonant variables under studied CCM condition. It has been fully considered in the improved analyses which are the foundation of proposed model. Maximum fractional errors of key indexes, t on under the steady state, and maximum fractional error of transition time are all acceptable and reasonable. On the other hand, precision of the two models and deficiencies of FHA are considered. It can be draw a conclusion that proposed model is more applicable than typical FHA model when nonstandard sinusoidal signals occur in the LLC network.
In summary, complexity of proposed model are the same that of typical FHA model but precision of proposed model is higher than that of typical FHA model when current of filter inductor works under CCM situation. Facing with the condition, proposed model is more applicable than typical FHA model.

Summarization of observations
According to the whole theoretical analyses, numerical results, and experimental results, key observations of typical FHA model and proposed model can be summarized in terms of complexity, precision, and effectiveness respectively.
1. Complexity of models can be defined as the size of each table in the DSP. Given the application of models and the universality of look-up table method, complexity of models is defined according to the size of each table which means state variables at next moment under the certain state variables, input voltage, and load at present moment.
Therefore, complexity of models is determined by the dimension of state variables and the number of internal functions related to switching angular frequency.
2. Peak values of resonant state variables and average values of slow state variables can be taken as indexes between models and actual converter. For the LLC converter, peak values of the three resonant state variables and average values of the two slow state variables are basic elements which can reflect properties of converter.
In addition, they can be taken as the direct references in further design. Therefore, they are the indexes which are used to measure different models and actual converter.
1. Deficiencies of typical FHA can be explained by harmonic generation mechanism of primary current and influence of Fourier series on the equivalent circuit. Typical FHA analyses are based on the well sinusoidal selectivity of LLC network. When the current of filter inductor works under CCM, resonant variables are distorted because the phenomenon results from the transient process of MOSFETs and rectifier diodes. Therefore, insufficiencies of typical FHA can be analyzed on the harmonic generation mechanism of primary current and influence of Fourier series on the equivalent circuit.
2. For the proposed model, it is important to take fully use of the improved analyses on the steady state and transient state.
Proposed model is originated from the improved analyses on the converter. It is similar to the typical FHA analyses that the steady state and transient state are further analyzed in the improved method. The transient process of MOSFETs and related fast recovery diodes, transient process of Schottky rectifier diodes, piecewise equivalent circuit, appropriate selection of variables, and simplified modified dynamic equations are all considered in the improved analyses. It is these considerations that lay the foundation for proposed model.

Effectiveness.
It is an effective way that the equivalent principle of indexes between improved analyses and proposed model is utilized.
Proposed model is established according to the equivalent principle of indexes between improved analyses and proposed model. It can ensure the proposed model has low complexity and high precision. Given the importance and reference value of indexes under the steady state and transient state, it is appropriate that improved analyses and proposed model is equivalent in meaning of indexes.
In summary, the improved analyses and the equivalent principle of indexes are the core of proposed model when converter works under CCM. For the proposed model, complexity and precision are guaranteed by the core.

Conclusion
This paper proposes an improved FHA model based on the improved analyses and the equivalent principle of indexes in the certain work region so that large signal model has low complexity and high precision when the current of filter inductor works under CCM. Firstly, complexity of models is defined as the size of each table in the DSP. Peak values of resonant state variables and average values of slow state variables are regarded as the indexes between models and actual converter. Complexity of models based on the FHA is lowest in the existing literatures and this paper. Secondly, equivalent circuit based on the typical FHA is derived. Steady state and transient state are briefly analyzed. Therefore, corresponding large signal model called typical FHA model is obtained. Thirdly, insufficiency of typical FHA under this CCM condition is formulated in detail. The two main points are harmonic generation mechanism of primary current and influence of Fourier series on the equivalent circuit. Meanwhile, the work region is given. Furthermore, transient process of switches at the arms of full-bridge, transient process of Schottky rectifier diodes, and piecewise equivalent circuit under the steady state are taken into account. Appropriate selection of variables and simplified modified dynamic equations under the transient state are also considered. Equivalent circuit and improved FHA model based on the equivalent principle of indexes are obtained. Corresponding large signal model is called proposed model. Lastly, numerical results and experimental results are achieved. Rationality of analyses on the proposed model can be verified by comparing numerical results of proposed model with related experimental results. Complexity of proposed model is the same as that of typical FHA model. Meanwhile, it can be further generalized that proposed model is closer to actual converter in meaning of indexes by comparing the differences between typical FHA model, proposed model, and actual converter.

Solving process of (11)
In this paper, the derivative variables in (11) represent the change rate of I Lf and U C(eq) from old steady state to new steady state when the converter happens to step change. Under this situation, I Lf and U C(eq) vary monotonously because of the over-critical damping peculiarity in the LCR network composed of filter inductor L f , filter capacitor C, and equivalent load.
I Lf and U C(eq) are described as I Lf_o and U C(eq)_o under the old steady state respectively. Correspondingly, they are described as I Lf_n and U C(eq)_n under the new steady state respectively. All the values are known by the means of steady-state analysis. The time interval between t 0 and t 1 in (11) is described as Δt. Derivative variables in (11) can be estimated according to the following relationships: In summary, the complete solving process of (13a) to (13f) has been generalized. It can be regarded as the applying process of steady state analysis and transient state analysis on the converter.
Supporting information S1 File. Parameters of actual converter. (PDF)