A small-signal modeling method for LLC resonant converter based on time-domain correction

Abstract

The present application relates to LLC resonant converter technical field, specifically to a kind of LLC resonant converter small signal modeling method based on time domain correction;Mainly include the following steps: establishing the equivalent model of LLC resonant converter each element and third-order average value equivalent circuit, LLC resonant converter small signal modeling method is reduced to second-order equivalent circuit by third-order average value equivalent circuit, add disturbance in second-order equivalent circuit, and separate large signal component and second-order small signal disturbance to obtain the analytical form of second-order average value small signal model;The LLC resonant converter small signal modeling method proposed in the scheme considers high precision and ease of use, improves the applicable frequency range, can guide LLC resonant converter controller design and stability analysis, the average value small signal model established by time domain analysis has higher precision, and analytical form transfer function is given, to facilitate practical engineering application.

Description

Technical Field

[0001] This invention relates to the field of LLC resonant converter technology, and more specifically to a small-signal modeling method for LLC resonant converters based on time-domain correction. Background Technology

[0002] DC-DC converters are power electronic devices that convert direct current into electrical energy. They are used to connect different DC buses and are applied in electric vehicles, smart grids, DC distribution networks, and battery energy storage. LLC resonant converters meet the requirements of high efficiency, high power density, and low electromagnetic interference of DC-DC converters, and therefore have been widely researched and applied. LLC resonant converters often employ pulse frequency modulation (PFM), which enables zero-voltage-switching (ZVS) turn-on of primary-side devices and zero-current-switching (ZCS) switching of secondary-side devices, effectively reducing switching losses. The aforementioned applications require strict regulation of the output voltage or current by the LLC converter, typically achieved through a closed-loop controller based on a small-signal model.

[0003] The most widely used method for small-signal modeling in LLC is the extended describing function (EDF) method. When applying EDF, the following assumptions are made: 1) The converter operates at the resonant frequency in steady state; 2) The amplitude of the disturbance signal is much smaller than that of the fundamental signal; 3) The frequency of the disturbance signal is lower than the switching frequency.

[0004] This results in the model being relatively accurate only at the resonant frequency, while in actual operation, the converter cannot always be at the resonant point and requires a wider frequency range, leading to a decrease in the accuracy of EDF small-signal modeling. Furthermore, the EDF method requires decomposing the LLC resonant current, resonant capacitor voltage, and magnetizing current into sinusoidal and cosine components, resulting in a total of seven state variables for the LLC converter's small-signal model, ultimately establishing a seventh-order system. Such high-order models are often quite complex and require order reduction in practical applications.

[0005] By modifying and simplifying the EDF method, a simple equivalent circuit model of the LLC resonant converter can be obtained, and an analytical expression for the third-order LLC small-signal model can be derived, which can assist in the design of the closed-loop feedback loop. The equivalent circuit model method can provide an analytical small-signal model, which is very convenient in practical applications and eliminates complex numerical calculations. However, it is slightly insufficient in terms of accuracy when operating away from the resonant point.

[0006] Discrete-time modeling methods can consider factors such as sampling, modulation effects, and delay effects of digital control loops, so they can establish accurate models of high-frequency or digital control converters. Using piecewise linear state-space equations, a difference equation of state variables at adjacent sampling instants is established. Then, through small-signal perturbation, linearization, and z-transform, a small-signal model of the LLC resonant converter in the discrete-time domain is directly obtained, which can be directly applied to the design of digital controllers. From the above analysis, it can be seen that the disadvantage of the discrete-time model is that the product of matrix exponents leads to complex calculations and long time consumption. This is especially true for LLC resonant converters because there are at least four states in each switching cycle. Moreover, the difference equation of the LLC converter is complex and only numerical solutions can be obtained. Therefore, the discrete-time model cannot obtain an analytical form of the transfer function, which is more complex in applications such as controller design.

[0007] In view of the following problems existing in the existing LLC small-signal modeling methods: 1) The accuracy of the analytical equivalent circuit model is low when deviating from the resonant frequency; 2) Although the time-domain or swept-frequency modeling has high accuracy, an analytical model cannot be obtained. Therefore, an analytical form LLC small-signal model that兼顾易用性和高精确度 is very important for guiding controller design and stability analysis. Summary of the Invention

[0008] Aiming at the deficiencies existing in the prior art, the purpose of the present invention is to provide a small-signal modeling method for LLC resonant converters based on time-domain correction that can兼顾准确度和易用性.

[0009] To solve the above technical problems, the technical solution provided by the present invention is: The small-signal modeling method for LLC resonant converters based on time-domain correction includes the following steps:

[0010] (1) Establish an equivalent model of each component of the LLC resonant converter and a third-order average equivalent circuit;

[0011] (2) Reduce the third-order average equivalent circuit obtained in step (1) to a second-order equivalent circuit;

[0012] (3) Add a perturbation quantity to the second-order equivalent circuit obtained in step (2), and separate the large-signal component and the second-order small-signal perturbation quantity to obtain an analytical form of the second-order average small-signal model.

[0013] Further, in step (1), an equivalent model of each component of the LLC resonant converter and a third-order average equivalent circuit are established through simplified time-domain analysis, and the following assumptions are made for the simplified time-domain analysis:

[0014] The time of the LC resonance stage is half of the resonance period, 0.5T r , where T rIt represents the resonant period. Within one period, the LLC includes four operating modes;

[0015] Analyze the first two modes:

[0016] In Mode I (t0 - t1), the resonant inductor and the resonant capacitor resonate with each other. The time-domain equation for this mode is expressed as:

[0017]

[0018]

[0019]

[0020] In the above equation, i Lr (t), v Cr (t) and i Lm (t) represent the resonant current, the resonant capacitor voltage, and the magnetizing inductor current respectively. I p is the peak value of the resonant current; Lr and Cr represent the values of the resonant inductor and the resonant capacitor respectively; ω r is the resonant angular frequency, and is the initial phase angle of the resonant current; Z r is the resonant impedance, and According to the time-domain equation of Mode I (t0 - t1), the values of the resonant current, the resonant capacitor voltage, and the magnetizing inductor current at t0 and t1 can be obtained as follows:

[0021]

[0022]

[0023]

[0024] In the above equation, n is the transformer turns ratio; V i is the input voltage; V o is the output voltage; I L0 and I L1 are the resonant currents at t0 and t1 respectively; V C0 and V C1 are the resonant capacitor voltages at t0 and t1 respectively; Lm represents the value of the magnetizing capacitor;

[0025] In Mode II (t1 - t2), the resonant inductor, the resonant capacitor, and the magnetizing inductor resonate together. The resonant inductor current and the magnetizing inductor current are equal and approximately linearly varying; the resonant capacitor voltage for this mode is expressed as:

[0026]

[0027] In the above equation, IL1 The resonant current at time t2. Based on the resonant capacitor voltage in mode II (t1 - t2), the value of the resonant capacitor voltage at time t2 can be obtained as follows:

[0028]

[0029] In the above formula, V C2 is the resonant capacitor voltage at time t2. According to the half - cycle symmetry of the waveform, the values of the resonant current, resonant capacitor voltage, and exciting inductor current at time t2 are respectively:

[0030] i Lr (t2) = I L2 = -i Lr (t0) = -I L0 ;

[0031] v Cr (t2) = V C2 = -v Cr (t0) = -V C0 ;

[0032] i Lm (t2) = I L2 = -i Lm (t0) = -I0;

[0033] By solving the equations simultaneously for the values of the resonant current, resonant capacitor voltage, and exciting inductor current at t0, t1, and t2, we get:

[0034]

[0035]

[0036] In the above formula, k is the ratio of the exciting inductor to the resonant inductor, k = L m / L r ;

[0037] Dividing both sides of the above equation by nV o , we can obtain the simplified gain expression as:

[0038] We have

[0039] In the above formula, M represents the voltage gain, f n is the normalized switching frequency, f n = f s / f r ; f s is the switching frequency, f s = 1 / T s , f r is the resonant frequency;

[0040] According to the above simplified time-domain analysis, for the resonant inductor L r and the resonant capacitor C r perform average value equivalent modeling, and the process is as follows:

[0041] Equivalent the rectifier current i rec to an equivalent rectifier current i rec_aver in the form of a continuous sine half-wave. To ensure the correctness of the equivalence, it is necessary to ensure that the average value of the equivalent rectifier current within half a switching period is equal to the sum of the current flowing through the output filter capacitor and the load current, that is, to satisfy:

[0042]

[0043] In the above formula, I rec_averm is the peak value of the equivalent rectifier current i rec_aver , ω s is the switching resonant angular frequency, ω r = 2πf s , i Co is the output capacitor current; since the LLC operates in a steady state, the integral of the output current within half a period is 0. According to the above formula, I rec_averm = πI o / 2 can be calculated. The rectifier current i rec can be regarded as a discontinuous sine half-wave, and it is also necessary to ensure that the average value of the rectifier current within half a switching period is equal to the sum of the current flowing through the output filter capacitor and the load current:

[0044]

[0045] In the above formula, I recm is the peak value of the rectifier current i rec . According to the above formula, I recm = πI o / (2f n ); By联立 the average value of the equivalent rectifier current and the average value of the rectifier current within half a switching period, the following equation can be obtained:

[0046]

[0047] Within half a switching period, to make the integrals of the rectifier current i rec and the equivalent rectifier current i rec_aver equal, only by making ω r and i rec multiplied by f n , the following defines an equivalent resonant angular frequency, denoted as ω re , which is expressed as:

[0048]

[0049] The above equivalent process will not change the impedance Z of the resonant circuit r , combined with the equivalent resonant angular frequency, the equivalent resonant inductance L under under-resonance is obtained re and the equivalent resonant capacitance C re , as shown in the following formula:

[0050] L re = L r / f n ;

[0051] C re = C r / f n ;

[0052] During half a switching period, the average voltage across the magnetizing inductance is V Lm_avg ≈ nV o , and the average magnetizing current within half a period can be calculated based on the expressions of the resonant inductance current in Mode I (t0 - t1) and Mode II (t1 - t2) as follows:

[0053]

[0054] The magnetizing inductance can be modeled as an equivalent magnetizing resistance R Lm , and is expressed as:

[0055]

[0056] The current flowing through the capacitor is equal to i rec - I o , and within the time period t0 - t1, the change in the output voltage of the LLC converter can be calculated by the following formula:

[0057]

[0058] In the above formula, C o represents the value of the output filter capacitor. By combining the above formula and the expression of the equivalent resonant angular frequency, we get:

[0059]

[0060] Through the above formula, the equivalent output filter capacitor C under under-resonance is obtained oe as:

[0061] C oe = C o / f n ;

[0062] According to the above equivalent resonant inductance L re , equivalent resonant capacitance C re , equivalent magnetizing resistance R Lm, the equivalent output filter capacitor C oe and the voltage gain M establish a third-order LLC under-resonant average equivalent circuit.

[0063] Further, in step (2), the third-order LLC under-resonant average equivalent circuit is reduced to obtain a second-order average equivalent circuit.

[0064] For the same step signal applied, the responses of the third-order average model and the second-order average model are the same.

[0065] The step signal v s has an amplitude of V in , which is applied to the third-order average equivalent circuit at the start of a switching period, resulting in the following equation:

[0066]

[0067] In the above equation, i Lre represents the equivalent resonant current, n e represents the transformer turns ratio, n e = n / M. When f n approaches 1, the limit of the above equation is:

[0068]

[0069] Based on C oe >> C re , the above equation can be further expressed as:

[0070]

[0071] Furthermore, the expression for the resonant current i Lre is obtained:

[0072]

[0073] In the above equation, ω re = 1 / sqrt(L re C re ), and the average value of the resonant current i Lre in half a switching period is:

[0074]

[0075] In the above equation, T re = 2πsqrt(L re C re );

[0076] The resonant inductor and the resonant capacitor are equivalent to a resonant element - the resonant inductor L ru, and establish a second-order average value equivalent circuit. Similarly, at the beginning of a switching period, apply a step signal v with an amplitude of V in to the second-order average value equivalent circuit, and obtain the equation: s When f

[0077]

[0078] approaches 1, the limit of the above formula is: n Solve the above formula to obtain the expression of the resonant current i

[0079]

[0080] as: Lru In the above formula,

[0081]

[0082] ω = n ru sqrt(L e C ru )), by obtaining the average value of the resonant current i oe within half a switching period T Lru , we can get: re Let the average value of the resonant current i

[0083]

[0084] be equal to the average value of the resonant current i Lru within half a switching period T Lru . Thus, the resonant inductor L of the second-order equivalent circuit can be obtained as: re Based on C ru >> C oe re , the above formula is expanded using the Taylor series to obtain:

[0087]

[0088] Establish a second-order average value equivalent circuit through the above steps.

[0089]

[0090]

[0091] Furthermore, in step (3), the steps to add a small-signal perturbation amount to the second-order average value equivalent circuit and separate the large-signal component and the second-order small-signal perturbation amount to obtain an analytical form of the second-order average value small-signal model are as follows:

[0090] Add a small-signal perturbation to the following variables:

[0091] ​​

[0092]

[0093]

[0094]

[0095] Based on the above second-order average equivalent circuit, the system of differential equations for the LLC resonant converter is as follows:

[0096]

[0097] Applying a small-signal perturbation to the above equation, multiplying both sides of the average value differential equation (a) of the LLC resonant converter by f n By adding a small signal perturbation, we can obtain:

[0098]

[0099]

[0100] In the above formula, F n =F s / f r a = π 2 / 4k, b=1+a, removing the large signal component and the second-order small perturbation component from the above equation, we can obtain:

[0101]

[0102] Substituting the gain expression M into the above equation simplifies the expression, and then dividing both sides of the equation by F. n We can obtain the formula (a) for the average small signal:

[0103]

[0104] In the above formula, N e =n(ba / F) n );

[0105] Multiply both sides of the differential equation system (b) of the LLC resonant converter average value by f. n 2 And by adding a small signal perturbation, we get:

[0106]

[0107]

[0108] Among them, I Lru It can be represented as:

[0109]

[0110] Removing the large signal component and the second-order small perturbation component from the above equation, we get:

[0111]

[0112] M and I Lru Substituting into the above equation further simplifies the process, and dividing both sides of the equation by F. n 2 We obtain the formula (b) for the average small signal:

[0113]

[0114] The average small-signal formulas (a) and (b) represent the average small-signal model of the LLC resonant converter.

[0115] Furthermore, the control-to-output and output impedance transfer functions can be obtained from the above second-order average small-signal model.

[0116] Find the control-to-output transfer function G vf (s):

[0117] The Laplace domain equation is:

[0118]

[0119]

[0120] Combining the above two equations, we obtain the control-to-output transfer function G. vf The analytical expression for (s) is:

[0121]

[0122] Find the output impedance transfer function Z o (s):

[0123] The Laplace domain equation is:

[0124]

[0125]

[0126] Combining the above two equations, we obtain the output impedance transfer function Z. o The analytical expression for (s) is:

[0127]

[0128] Compared with existing technologies, the significant advantages of this solution are:

[0129] The proposed small-signal modeling method for LLC resonant converters combines high accuracy with ease of use, expands the applicable frequency range, and can guide the design and stability analysis of LLC resonant converter controllers. The average-valued small-signal model established through time-domain analysis has high accuracy, and the analytical transfer function is also provided, which is convenient for practical engineering applications. Attached Figure Description

[0130] The accompanying drawings are provided to further illustrate the invention and form part of the specification. They are used in conjunction with embodiments of the invention to explain the invention and do not constitute a limitation thereof. In the drawings:

[0131] Figure 1 This is a waveform diagram illustrating the simplified time-domain analysis working principle in this invention.

[0132] Figure 2 The waveforms of the rectified current and equivalent rectified current in this invention are shown.

[0133] Figure 3 This is the equivalent circuit diagram of the third-order average value of the LLC resonant converter in this invention;

[0134] Figure 4 This is the equivalent circuit diagram of the second-order average value of the LLC resonant converter in this invention;

[0135] Figure 5 This is a schematic diagram of the second-order average small-signal model of the LLC resonant converter in this invention. Detailed Implementation

[0136] The preferred embodiments of the present invention will be described below with reference to the accompanying drawings. It should be understood that the preferred embodiments described herein are for illustration and explanation only and are not intended to limit the present invention.

[0137] The small-signal modeling method for LLC resonant converters based on time-domain correction includes the following steps:

[0138] (1) By simplifying the time domain analysis, the equivalent models of each component of the LLC resonant converter and the third-order average equivalent circuit are established;

[0139] (2) Reduce the third-order average equivalent circuit obtained in step (1) to a second-order equivalent circuit;

[0140] (3) Add a disturbance to the second-order equivalent circuit obtained in step (2) and separate the large signal component and the second-order small signal disturbance to obtain the analytical form of the second-order average small signal model.

[0141] In step (1), the equivalent models of each component of the LLC resonant converter and the third-order average equivalent circuit are established through simplified time-domain analysis. Figure 1To simplify the waveform diagram of the working principle of the time-domain analysis, the following assumptions are made regarding the simplified time-domain analysis:

[0142] The duration of the LC resonance phase is half the resonance period, 0.5T. r T r The resonant period is represented by the LLC converter, which includes four operating modes within one period; this is because the LLC converter operates in one switching period T. s The internal working waveform is symmetrical, so only the first two modes need to be analyzed.

[0143] Analyze the first two modes:

[0144] In mode I (t0-t1), both the resonant inductor and the resonant capacitor resonate. The time-domain equation for this mode is expressed as:

[0145]

[0146]

[0147]

[0148] In the above formula, i Lr (t), v Cr (t) and i Lm (t) represent the resonant current, resonant capacitor voltage, and magnetizing inductor current, respectively. p The peak value of the resonant current; Lr and Cr represent the values ​​of the resonant inductance and resonant capacitance, respectively; ω r It is the resonant angular frequency, and Z is the initial phase angle of the resonant current; r It is the resonant impedance, and Based on the time-domain equations of mode I (t0-t1), the values ​​of the resonant current, resonant capacitor voltage, and magnetizing inductor current at times t0 and t1 can be obtained as follows:

[0149]

[0150]

[0151]

[0152] In the above formula, n represents the transformer change; V i V is the input voltage. o For output voltage; I L0 and I L1 The resonant currents at times t0 and t1 are respectively; V C0 and V C1 t0 and t1 are the resonant capacitor voltages, respectively; Lm represents the excitation capacitor value;

[0153] In Mode II (t1-t2), the resonant inductor, resonant capacitor, and magnetizing inductor resonate together. The resonant inductor current and the magnetizing inductor current are equal and change approximately linearly. The resonant capacitor voltage in this mode is expressed as:

[0154]

[0155] In the above formula, I L1 Let be the resonant current at time t2. Based on the resonant capacitor voltage of mode II (t1-t2), the value of the resonant capacitor voltage at time t2 can be obtained as follows:

[0156]

[0157] In the above formula, V C2 Let be the resonant capacitor voltage at time t2. Based on the half-cycle symmetry of the waveform, the values ​​of the resonant current, resonant capacitor voltage, and magnetizing inductor current at time t2 are as follows:

[0158] i Lr (t2)=I L2 =-i Lr (t0)=-I L0 ;

[0159] v Cr (t2)=V C2 =-v Cr (t0)=-V C0 ;

[0160] i Lm (t2)=I L2 =-i Lm (t0) = -I0;

[0161] By simultaneously solving the equations, we can obtain the values ​​of the resonant current, resonant capacitor voltage, and magnetizing inductor current at times t0, t1, and t2. The solution is as follows:

[0162]

[0163]

[0164] In the above formula, k is the ratio of the magnetizing inductance to the resonant inductance, k = L m / L r ;

[0165] Divide both sides of the above equation by nV o The simplified form of the gain expression can be obtained as follows:

[0166]

[0167] In the above formula, M represents the voltage gain, f n To standardize the switching frequency, f n =f s / f r ;f s f is the switching frequency. s =1 / T s f r The resonant frequency;

[0168] To facilitate the explanation of the equivalent modeling of components in an LLC converter, Figure 2 The diagram shows the waveforms of the rectified current and the equivalent rectified current. Based on the simplified time-domain analysis above, for the resonant inductor L... r and resonant capacitor C r The process of performing average equivalent modeling is as follows:

[0169] The rectified current i rec The equivalent rectified current i is in the form of a continuous sinusoidal half-wave. rec_aver To ensure the correctness of the equivalence, the average value of the equivalent rectified current within half a switching cycle must be equal to the sum of the current flowing through the output filter capacitor and the load current, i.e., satisfying:

[0170]

[0171] In the above formula, I rec_averm The equivalent rectified current i rec_aver The peak value, ω s ω is the resonant angular frequency of the switch. r =2πf s i Co The output capacitor current is given. Since the LLC operates in steady state, the integral of the output current within half a cycle is 0. Therefore, I can be calculated using the above formula. rec_averm =πI o / 2, rectified current i rec This can be viewed as a discontinuous half-sine wave, and it is also necessary to ensure that the average value of the rectified current within half a switching cycle is equal to the sum of the load current flowing through the output filter capacitor:

[0172]

[0173] In the above formula, I recm The rectified current i rec The peak value of I can be calculated according to the above formula. recm =πI o / (2f n By combining the equivalent average rectified current and the average rectified current over half a switching cycle, we can obtain the following equation:

[0174]

[0175] Within half a switching cycle, we want to make the rectified current i rec and equivalent rectified current i rec_aver The integrals are equal only if ω is set to ω. r and i rec Multiply by f n Here, we define an equivalent resonant angular frequency, denoted as ω. re , is represented as:

[0176]

[0177] The equivalent process described above will not change the impedance Z of the resonant circuit. r By combining the equivalent resonant angular frequency, the equivalent resonant inductance L under underresonance is obtained. re and equivalent resonant capacitance C re As shown in the formula below:

[0178] L re =L r / f n ;

[0179] C re =C r / f n ;

[0180] During half a switching cycle, the average voltage across the magnetizing inductor is V. Lm_avg ≈nV o Based on the expressions for the resonant inductor current in mode I (t0-t1) and mode II (t1-t2), the average excitation current over half a cycle can be calculated as follows:

[0181]

[0182] The magnetizing inductor can be modeled as an equivalent magnetizing resistance R. Lm And expressed as:

[0183]

[0184] The current flowing through the capacitor is equal to i rec -I o The change in output voltage of the LLC converter during the time period t0-t1 can be calculated using the following formula:

[0185]

[0186] In the above formula, C o This represents the output filter capacitor value. Combining the above equation with the expression for the equivalent resonant angular frequency, we obtain:

[0187]

[0188] The equivalent output filter capacitor C under underresonance is obtained from the above equation. oe for:

[0189] C oe =C o / f n ;

[0190] Based on the above equivalent resonant inductance L re Equivalent resonant capacitance C re Equivalent excitation resistance R Lm Equivalent output filter capacitor C oe And establish a voltage gain M as Figure 3 The equivalent circuit of the third-order LLC underresonant average value is shown.

[0191] Figure 4 This is represented by the second-order average value equivalent circuit diagram of an LLC resonant converter, as shown below. Figure 4 The derivation of order reduction for the third-order average equivalent circuit is presented. The theoretical basis for this order reduction is that the dynamic response of a linear circuit can be obtained by analyzing the response when a step signal is applied and the circuit has no initial state. For the same step signal, the response of the third-order average model is the same as that of the second-order average model.

[0192] Step signal v s The amplitude is V in When applied to the third-order average equivalent circuit at the beginning of a switching cycle, the following equation is obtained:

[0193]

[0194] In the above formula, i Lre Represents the equivalent resonant current, n e n represents the transformer turns ratio. e =n / M, when f n When the expression approaches 1, the limit of the above expression is:

[0195]

[0196] Because in actual circuits, the output filter capacitor value is much larger than the resonant capacitor value. Based on C oe >>C re The above formula can be further expressed as:

[0197]

[0198] This leads to the resonant current i. Lre The expression:

[0199]

[0200] In the above formula, ω re =1 / sqrt(L re C re ), thus obtaining the resonant current i Lre The average value over half a switching cycle is:

[0201]

[0202] In the above formula, T re =2πsqrt(L re C re );

[0203] The resonant inductor and resonant capacitor are equivalent to a single resonant element—the resonant inductor L. ru A second-order average equivalent circuit is established, and the third-order average equivalent circuit is identical to the second-order average equivalent circuit except for the resonant element. Similarly, at the beginning of a switching cycle, an amplitude of V is applied to the second-order average equivalent circuit. in The step signal v s The resulting equation is:

[0204]

[0205] When f n When the expression approaches 1, the limit of the above expression is:

[0206]

[0207] Solving the above equation yields the resonant current i. Lru The expression is:

[0208]

[0209] In the above formula, ω ru =n e sqrt(L ru C oe By determining the resonant current i Lru In half a switching cycle T re The average value within the range can be obtained as follows:

[0210]

[0211] To make the third-order model and the second-order reduced-order model completely equivalent, let the resonant current i Lru and resonant current i Lru In half a switching cycle T re Since their average values ​​are equal, the resonant inductance L of the second-order equivalent circuit can be obtained. ru for:

[0212]

[0213] Based on C oe >>C re Applying Taylor expansion to the above equation yields:

[0214]

[0215] The second-order average value equivalent circuit is established through the above steps.

[0216] Further, in step (3), the steps of adding a small-signal perturbation to the second-order average equivalent circuit and separating the large-signal component and the second-order small-signal perturbation to obtain the analytical form of the second-order average small-signal model are as follows:

[0217] Add a small signal to the following variables:

[0218]

[0219]

[0220]

[0221]

[0222] Based on the above second-order average equivalent circuit, the system of differential equations for the LLC resonant converter is as follows:

[0223]

[0224] Applying a small-signal perturbation to the above equation, multiplying both sides of the average value differential equation (a) of the LLC resonant converter by f n By adding a small signal perturbation, we can obtain:

[0225]

[0226]

[0227] In the above formula, F n =F s / f r a = π 2 / 4k, b=1+a, removing the large signal component and the second-order small perturbation component from the above equation, we can obtain:

[0228]

[0229] Substituting the gain expression M into the above equation simplifies the expression, and then dividing both sides of the equation by F. n We can obtain the formula (a) for the average small signal:

[0230]

[0231] In the above formula, N e =n(ba / F) n );

[0232] Multiply both sides of the differential equation system (b) of the LLC resonant converter average value by f. n 2 And by adding a small signal perturbation, we get:

[0233]

[0234]

[0235] Among them, I Lru It can be represented as:

[0236]

[0237] Removing the large signal component and the second-order small perturbation component from the above equation, we get:

[0238]

[0239] M and I Lru Substituting into the above equation further simplifies the process, and dividing both sides of the equation by F. n 2 We obtain the formula (b) for the average small signal:

[0240]

[0241] The average small-signal formulas (a) and (b) represent the average small-signal model of the LLC resonant converter and can be plotted as follows: Figure 5 The LLC resonant converter shown is a second-order average small-signal model.

[0242] Furthermore, the control-to-output and output impedance transfer functions can be obtained from the above second-order average small-signal model.

[0243] Find the control-to-output transfer function G vf (s):

[0244] The Laplace domain equation is:

[0245]

[0246]

[0247] Combining the above two equations, we obtain the control-to-output transfer function G. vfThe analytical expression for (s) is:

[0248]

[0249] Find the output impedance transfer function Z o (s):

[0250] The Laplace domain equation is:

[0251]

[0252]

[0253] Combining the above two equations, we obtain the output impedance transfer function Z. o The analytical expression for (s) is:

[0254]

[0255] Compared with existing technologies, the significant advantages of this solution are:

[0256] The proposed small-signal modeling method for LLC resonant converters combines high accuracy with ease of use, expands the applicable frequency range, and can guide the design and stability analysis of LLC resonant converter controllers. The average-valued small-signal model established through time-domain analysis has high accuracy, and the analytical transfer function is also provided, which is convenient for practical engineering applications.

[0257] Finally, it should be noted that the above descriptions are merely preferred embodiments of the present invention and are not intended to limit the present invention. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art can still modify the technical solutions described in the foregoing embodiments or make equivalent substitutions for some of the technical features. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A small-signal modeling method for LLC resonant converters based on time-domain correction, characterized in that, Includes the following steps: (1) Establish the equivalent models of each component of the LLC resonant converter and the third-order average equivalent circuit; (2) Reduce the third-order average equivalent circuit obtained in step (1) to a second-order equivalent circuit; (3) Add a disturbance to the second-order equivalent circuit obtained in step (2), and separate the large-signal component and the second-order small-signal disturbance to obtain the analytical form of the second-order average small-signal model; Based on simplified time-domain analysis, for resonant inductors and resonant capacitor The process of performing average equivalent modeling is as follows: Rectified current Equivalent to an equivalent rectified current in the form of a continuous sinusoidal half-wave. To ensure the correctness of the equivalence, the average value of the equivalent rectified current within half a switching cycle must be equal to the sum of the current flowing through the output filter capacitor and the load current, i.e., satisfying: ; In the above formula, Equivalent rectified current peak value The resonant angular frequency of the switch. This is the output capacitor current; since the LLC operates in steady state, the output current integral over half a cycle is 0. Therefore, it can be calculated using the above formula. rectified current This can be viewed as a discontinuous half-sine wave, and it is also necessary to ensure that the average value of the rectified current within half a switching cycle is equal to the sum of the load current flowing through the output filter capacitor: ; In the above formula, Rectified current The peak value can be calculated according to the above formula. By combining the equivalent average rectified current and the average rectified current over half a switching cycle, we can obtain the following equation: ; Within half a switching cycle, the rectified current is to be... and equivalent rectified current The integrals are equal, only if we let and Multiply The following defines an equivalent resonant angular frequency, denoted as . , represented as: ; The equivalent process described above will not change the impedance of the resonant circuit. By combining the equivalent resonant angular frequency, the equivalent resonant inductance under underresonance is obtained. and equivalent resonant capacitance As shown in the formula below: ; ; in: This marks the start of half a switching cycle; This refers to the switching time of the resonant cavity's operating modes; This is the end time of half a switching cycle; This is the resonant inductance value; This is the value of the resonant capacitor; For switching cycles; The resonant period; The switching frequency; The resonant frequency; To standardize the switching frequency, .

2. The small-signal modeling method for LLC resonant converters based on time-domain correction according to claim 1, characterized in that, In step (1), the equivalent models of each component of the LLC resonant converter and the third-order average equivalent circuit are established through simplified time-domain analysis. The following assumptions are made for the simplified time-domain analysis: The duration of the LC resonance phase is half of the resonance period. ,in The resonant period is represented by the number of resonant modes. Within one period, the LLC includes four operating modes. Analyze the first two modes: Modal When both the resonant inductor and resonant capacitor resonate, the time-domain equation for this mode is expressed as: ; ; ; In the above formula, and These represent the resonant current, the resonant capacitor voltage, and the magnetizing inductor current, respectively. This is the peak value of the resonant current; and These represent the values ​​of the resonant inductance and resonant capacitance, respectively. It is the resonant angular frequency, and The initial phase angle of the resonant current; It is the resonant impedance, and According to the mode The time-domain equations can be used to obtain the resonant current, resonant capacitor voltage, and magnetizing inductor current. and The values ​​at time points are as follows: ; ; ; In the above formula, For transformer turns ratio; Input voltage; This refers to the output voltage. and They are respectively and The resonant current at a given moment; and They are respectively and The resonant capacitor voltage at time t; Indicates the value of the excitation capacitor; Modal The resonant inductor, resonant capacitor, and magnetizing inductor resonate together. The resonant inductor current and the magnetizing inductor current are equal and change approximately linearly. The resonant capacitor voltage in this mode is expressed as: ; In the above formula, for The resonant current at a given time, according to the mode The resonant capacitor voltage can be obtained from the resonant capacitor voltage at... The value at time: ; In the above formula, for Based on the half-cycle symmetry of the waveform, the resonant current, resonant capacitor voltage, and magnetizing inductor current at time t can be obtained. The values ​​at time points are as follows: ; ; ; The resonant current, resonant capacitor voltage, and magnetizing inductor current can be obtained by simultaneously solving the equations. and The value at time t is obtained by solving for: ; ; In the above formula, This is the ratio of the magnetizing inductance to the resonant inductance. ; Divide both sides of the above equation by The simplified form of the gain expression can be obtained as follows: ; In the above formula, Indicates voltage gain. To standardize the switching frequency, For switching frequency, The resonant frequency; During half a switching cycle, the average voltage across the magnetizing inductor is According to the resonant inductor current in mode and modality The expression can be used to calculate the average excitation current over half a cycle as follows: ; The magnetizing inductor can be modeled as an equivalent magnetizing resistance. And expressed as: ; The current flowing through the capacitor is equal to ,exist The change in output voltage of the LLC converter over a given time period can be calculated using the following formula: ; In the above formula, This represents the output filter capacitor value. Combining the above equation with the expression for the equivalent resonant angular frequency, we obtain: ; The equivalent output filter capacitor under underresonance conditions is obtained from the above equation. for: ; Based on the above equivalent resonant inductance Equivalent resonant capacitance Equivalent excitation resistance Equivalent output filter capacitor and voltage gain Construct a third-order LLC underresonant average value equivalent circuit.

3. The small-signal modeling method for LLC resonant converters based on time-domain correction according to claim 2, characterized in that, In step (2), the third-order LLC underresonant average equivalent circuit is reduced in order to obtain the second-order average equivalent circuit; When the same step signal is applied, the response of the third-order average model is the same as that of the second-order average model. Step signal The amplitude is When applied to the third-order average equivalent circuit at the beginning of a switching cycle, the following equation is obtained: ; In the above formula, Represents the equivalent resonant current. Indicates the transformer turns ratio. ,when When the expression approaches 1, the limit of the above expression is: ; based on The above formula can be further expressed as: ; This leads to the resonant current. The expression: ; In the above formula, The resonant current is obtained. The average value over half a switching cycle is: ; In the above formula, ; The resonant inductor and resonant capacitor can be considered as a single resonant element—the resonant inductor. A second-order average equivalent circuit is established, and similarly, an amplitude of [value missing] is applied to the second-order average equivalent circuit at the beginning of a switching cycle. step signal The resulting equation is: ; when When the expression approaches 1, the limit of the above expression is: ; Solving the above equation yields the resonant current. The expression is: ; In the above formula, By determining the resonant current In half a switching cycle The average value within the range can be obtained as follows: ; Let the resonant current and resonant current In half a switching cycle Since their average values ​​are equal, the resonant inductance of the second-order equivalent circuit can be obtained. for: ; based on Applying Taylor expansion to the above equation yields: ; The second-order average value equivalent circuit is established through the above steps.

4. The small-signal modeling method for LLC resonant converters based on time-domain correction according to claim 3, characterized in that, In step (3), the steps of adding a small-signal perturbation to the second-order average equivalent circuit and separating the large-signal component and the second-order small-signal perturbation to obtain the analytical form of the second-order average small-signal model are as follows: Add a small signal to the following variables: ; ; ; ; Based on the above second-order average equivalent circuit, the system of differential equations for the LLC resonant converter is as follows: ; Applying a small-signal perturbation to the above equations will change the average value differential equations of the LLC resonant converter. Multiply both sides of the equation by By adding a small signal perturbation, we can obtain: ; ; In the above formula, Removing the large signal component and the second-order small perturbation component from the above equation, we get: ; Gain expression M Substitute into the above equation to simplify, and divide both sides of the equation by... The formula for the average small signal (a) can be obtained as follows: ; In the above formula, ; The average value differential equations of the LLC resonant converter Multiply both sides of the equation by And by adding a small signal perturbation, we get: ; ; in, It can be represented as: ; Removing the large signal component and the second-order small perturbation component from the above equation, we get: ; Will M and Substituting into the above equation further simplifies the process, and dividing both sides of the equation by... The formula for the average small signal (b) is obtained as follows: ; The average small-signal formulas (a) and (b) represent the average small-signal model of the LLC resonant converter.

5. The small-signal modeling method for LLC resonant converters based on time-domain correction according to claim 4, characterized in that, The second-order average small-signal model described above can be used to obtain the control-to-output and output impedance transfer functions; Find the control-to-output transfer function : The Laplace domain equation is: ; ; Combining the above two equations, we obtain the control-to-output transfer function. The parsing expression is: ; Find the output impedance transfer function : The Laplace domain equation is: ; ; Combining the above two equations, we obtain the output impedance transfer function. The parsing expression is: 。

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