Time domain stability analysis and improvement method of LLC resonant converter

Abstract

The invention discloses a time domain stability analysis and improvement method for an LLC resonant converter, and the method comprises the steps: solving and generating a Jacobian matrix of the LLC resonant converter according to an extended description function method; solving all characteristic values of the Jacobian matrix; judging the stability of the closed-loop system of the LLC resonant converter according to a Lyapunov indirect method; if the system is stable, outputting a judgment result; if the system is not stable, PI controller parameters are adjusted, or the characteristic value sensitivity of a certain element beta in the matrix is solved, the size of the element beta in the matrix is adjusted based on the characteristic value sensitivity solving result, then the characteristic value is solved again, and the stability is judged. According to the scheme, the system stability influence trend is reflected through time domain analysis, the effects of stability improvement and system instability prevention are achieved, the influence of the load on the stability of the converter is analyzed through the characteristic value sensitivity, the influence trend of the load change on the stability is estimated according to the characteristic value distribution track, the controller acts in advance, and system instability is avoided.

Description

Technical Field

[0001] The present application relates to the technical field of resonant converters, and more particularly to a method for analyzing and improving the time-domain stability of an LLC resonant converter. Background Art

[0002] LLC resonant converter is one of the power topologies that has attracted much attention, especially in servers and electric vehicle power supplies. On the one hand, due to the characteristics of zero voltage switching (ZVS) on the primary side and zero current switching (ZCS) on the secondary side of LLC resonant converter, energy loss is greatly reduced. On the other hand, the high-frequency working characteristics of LLC resonant converter effectively improve the power density.

[0003] In practical applications, if there are defects in the stability of the LLC resonant converter, the output performance of the converter will be lower than the expected target under disturbance conditions, and even cause irreversible damage to the power system. At present, the stability analysis of the LLC resonant converter is still a serious problem. The main reason is that the LLC resonant converter uses pulse frequency modulation instead of traditional pulse width modulation. The characteristic of transferring energy with the fundamental wave and its higher harmonics does not meet the "small ripple" assumption, and the small signal model cannot be established using the state space averaging method, resulting in the failure of the stability analysis relying on the small signal model.

[0004] Although there have been studies on LLC stability, the current research still has the disadvantage of being difficult to apply in practice. The main reasons are: first, LLC converters often assume ideal components and simplified circuit models, but in practice the nonlinear characteristics and parasitic parameters of the components (such as parasitic capacitance, inductance and resistance) may cause deviations from the ideal situation, resulting in instability of the system under certain operating conditions; second, LLC converters usually operate in a higher switching frequency range, and high-frequency operation will introduce some nonlinear and parasitic effects, which may not be fully captured by traditional circuit models, thus affecting stability.

[0005] In summary, how to solve the defects of LLC resonant converter stability analysis and improvement is a problem that needs to be solved urgently. Summary of the invention

[0006] Aiming at the problems existing in the stability analysis and improvement process of LLC resonant converter, this application combines the extended describing function method to establish the time domain model of LLC resonant converter, and determines the stability domain of LLC according to the eigenvalue of the system Jacobian matrix; according to the eigenvalue sensitivity, analyzes the influence trend of controller parameters on stability, and thus obtains a method for improving the stability of LLC converter under variable load conditions. The specific scheme is as follows:

[0007] A time domain stability analysis and improvement method for an LLC resonant converter, comprising:

[0008] S100, establishing a time-domain-based LLC converter model according to an extended describing function method and introducing PI controller parameters into the time-domain model to solve and generate a Jacobian matrix of the LLC resonant converter;

[0009] S200, solving all eigenvalues ​​of the Jacobian matrix and displaying them on a complex plane;

[0010] S300, determine the stability of the LLC resonant converter closed-loop system according to the Lyapunov indirect method:

[0011] S310, if the system is stable, store the time domain model and PI controller parameters and output the determination result;

[0012] S320, if the system is unstable, adjust the PI controller parameters and return to execute steps S200-S300;

[0013] S330, if the system is critically stable, then solve the eigenvalue sensitivity of a certain element β in the matrix, adjust the size of the element β in the matrix based on the eigenvalue sensitivity solution, and return to execute steps S200-S300.

[0014] The above technical scheme is based on the Lyapunov stability analysis method. First, the time domain model of the LLC resonant converter is established in combination with the extended describing function method. The influence of load and PI controller parameters on stability is analyzed by combining the eigenvalue sensitivity and Lyapunov theory. By adjusting the PI control parameters, the system can obtain better stability performance when the load changes.

[0015] Furthermore, the stability criterion of the Lyapunov indirect method includes:

[0016] When at least one eigenvalue of the Jacobian matrix has a positive real part, the system is unstable;

[0017] When the real part of at least one eigenvalue of the Jacobian matrix is ​​0, the system is critically stable;

[0018] When the eigenvalues ​​of the Jacobian matrix all have negative real parts, the system is stable;

[0019] The method of determining the stability of the LLC resonant converter closed-loop system according to the Lyapunov indirect method includes:

[0020] A300, determine whether the eigenvalue has a positive real part:

[0021] A310, if the eigenvalue contains a positive real part, the system is considered unstable;

[0022] A320, if the eigenvalues ​​do not have positive real parts, determine whether all eigenvalues ​​have negative real parts:

[0023] A321, if all eigenvalues ​​have negative real parts, the system is considered stable;

[0024] A322, if not all eigenvalues ​​have negative real parts, the system is judged to be critically stable.

[0025] Furthermore, if the system is unstable, adjust the PI controller parameters, including:

[0026] Let k p =k p +k, observe and determine the changing trend of eigenvalues ​​on the complex plane:

[0027] If the eigenvalue moves to the left, the increase of k is conducive to the stability of the system, and the new PI controller parameters are brought in to re-solve the eigenvalue;

[0028] If the above eigenvalue moves to the right, let k = -p × k, p>1, and then solve the new k p value; p is not clearly defined and only represents the proportionality coefficient;

[0029] Among them, k p is the proportional coefficient of the PI controller, and k represents the direction of increase or decrease of the PI controller parameters.

[0030] Furthermore, solving the eigenvalue sensitivity of a certain element β in the matrix includes:

[0031] The sensitivity of a certain eigenvalue of the Jacobian matrix corresponding to the LLC converter closed-loop system to a certain element β in the matrix can be expressed by the partial derivative of the eigenvalue with respect to the element β, as shown in formula (1):

[0032]

[0033] Among them, J F is the Jacobian matrix of LLC resonant converter, λ i is the eigenvalue of the LLC Jacobian matrix, m i is the left eigenvector of the Jacobian matrix, n i is the right eigenvector of the matrix, that is:

[0034]

[0035] is the sensitivity of a certain eigenvalue of the Jacobian matrix to a certain element β in the matrix, let p and q are the real and imaginary parts of the eigenvalue sensitivity;

[0036] The calculation result of formula (1) represents the movement of the eigenvalue on the real axis and the imaginary axis when the element β in the matrix changes.

[0037] Furthermore, according to the eigenvalue sensitivity of a certain parameter β calculated by formula (1), combined with the stability criterion of the Lyapunov indirect method, if p<0, it means that the element β makes the eigenvalue move to the left half plane of the complex plane, which is beneficial to improving the stability of the system; if p>0, it means that the element β makes the eigenvalue move to the right half plane of the complex plane, which is not conducive to improving the stability of the system;

[0038] The adjusting the size of the element β in the matrix based on the eigenvalue sensitivity solution result includes:

[0039] Determine whether the eigenvalue sensitivity of element β has a positive real part:

[0040] If the real part of the eigenvalue sensitivity of the element β is greater than 0, then the parameter corresponding to the element β is determined to be beneficial to the stability of the system, and β = β + I, I>0; otherwise, β = β - I, I>0, and the new element β is brought into the Jacobian matrix to re-determine its eigenvalue;

[0041] Here, I indicates the direction in which β increases or decreases.

[0042] Furthermore, the extended description function method includes: converting the nonlinear switching link into a linear link through Fourier series, and then using the two-angle sum and difference formula in the trigonometric identity to approximate the variable frequency signal into a variable amplitude signal with a fixed frequency and a variable amplitude.

[0043] The beneficial effects of this application are:

[0044] This application proposes a stability analysis method for LLC resonant converter based on Lyapunov stability theory, and combines eigenvalue sensitivity to provide a stability improvement method under variable load conditions. This method determines the stability range of controller parameters based on the eigenvalues ​​of the Jacobian matrix, and intuitively reflects the stability change trend of the LLC resonant converter when the controller parameters change. Combined with eigenvalue sensitivity analysis, a method for improving system stability under variable load conditions is provided. According to the eigenvalue sensitivity, the influence trend of load size on stability can be known, which is conducive to implementation in actual engineering. BRIEF DESCRIPTION OF THE DRAWINGS

[0045] Figure 1 A schematic diagram of the steps of the method of this application;

[0046] Figure 2 A schematic diagram of the process of this application method;

[0047] Figure 3 The distribution of the 8 eigenvalues ​​of the Jacobian matrix is ​​shown, that is, the distribution of the eigenvalues ​​of the Jacobian matrix when kp changes;

[0048] Figure 4The main eigenvalues ​​(λ6, λ7, λ8) of the Jacobian matrix are shown, that is, the distribution of λ6, λ7, and λ8 as kp changes under different loads. DETAILED DESCRIPTION

[0049] The embodiments of the present application are described in detail below, and examples of the embodiments are shown in the attached Figure 1-4 Shown in.

[0050] In the description of this specification, the description with reference to the terms "certain embodiments", "one embodiment", "some embodiments", "illustrative embodiments", "examples", "specific examples", or "some examples" means that the specific features, structures, materials, or characteristics described in conjunction with the embodiments or examples are included in at least one embodiment or example of the present application. In this specification, the schematic representations of the above terms do not necessarily refer to the same embodiment or example. Moreover, the specific features, structures, materials, or characteristics described may be combined in any one or more embodiments or examples in a suitable manner.

[0051] like Figure 1-4 As shown, a time domain stability analysis and improvement method of an LLC resonant converter described in an embodiment of the present application mainly includes the following steps:

[0052] S100, solving the Jacobian matrix of the LLC resonant converter according to the extended describing function method;

[0053] S200, solving all eigenvalues ​​of the Jacobian matrix and displaying them on a complex plane;

[0054] S300, determine the stability of the LLC resonant converter closed-loop system according to the Lyapunov indirect method:

[0055] S310, if the system is stable, store the time domain model and PI controller parameters and output the determination result;

[0056] S320, if the system is unstable, adjust the PI controller parameters and return to execute steps S200-S300;

[0057] S330, if the system is critically stable, then solve the eigenvalue sensitivity of a certain element β in the matrix, adjust the size of the element β in the matrix based on the eigenvalue sensitivity solution, and return to execute steps S200-S300.

[0058] In the above step S100, the extended description function method is the main method for LLC small signal modeling. This method first converts the nonlinear switching link into a linear link through the Fourier series, and then uses the two-angle sum and difference formula in the trigonometric identity to approximate the variable frequency signal as a variable amplitude signal with a fixed frequency and a variable amplitude, thereby eliminating the nonlinear calculation problem caused by the frequency change. In step S100, firstly, the time domain model of the LLC resonant converter is established in combination with the extended description function method, and the range of PI controller parameters under stable conditions is given, and simulation verification is performed to establish an LLC converter model based on the time domain and introduce the PI controller parameters into the above time domain model.

[0059] Combined with the above Jacobian matrix, the stability criterion of the Lyapunov indirect method can be expressed as:

[0060] a, when at least one eigenvalue of the Jacobian matrix has a positive real part, the system is unstable;

[0061] b, when the real part of at least one eigenvalue of the Jacobian matrix is ​​0, the system is critically stable;

[0062] c. When the eigenvalues ​​of the Jacobian matrix all have negative real parts, the system is stable.

[0063] Therefore, in step S300, the stability of the LLC resonant converter closed-loop system is determined according to the Lyapunov indirect method, combined with Figure 2 As shown, it is preferred to include the following steps and perform them in the following order:

[0064] A300, determining whether the eigenvalue of the matrix contains a positive real part:

[0065] A310, if the eigenvalue contains a positive real part, the system is considered unstable;

[0066] A320, if the eigenvalues ​​do not have positive real parts, determine whether all eigenvalues ​​have negative real parts:

[0067] A321, if all eigenvalues ​​have negative real parts, the system is considered stable;

[0068] A322, if not all eigenvalues ​​have negative real parts, the system is judged to be critically stable.

[0069] According to the above Lyapunov indirect method, combined with Figure 2 As shown, in step S320, if the system is unstable, adjusting the PI controller parameters includes:

[0070] Let k p =k p +k, observe and determine the changing trend of eigenvalues ​​on the complex plane:

[0071] If the eigenvalue moves to the left, the increase of k is conducive to the stability of the system, and the new PI controller parameters are brought in to re-solve the eigenvalue; if the above eigenvalue moves to the right, let k = -p×k, p>1, and then solve the new k p Value. Among them, k p is the proportional coefficient of the PI controller. In the modeling process of the extended describing function method, the influence of the PI controller is considered. Therefore, adjusting the PI controller parameters will have an important impact on the stability of the system. k represents the direction of increase or decrease of the PI controller parameters.

[0072] In step S330, when the system is critically stable, the eigenvalue sensitivity of a certain element β in the matrix is ​​solved, which specifically includes:

[0073] The sensitivity of a certain eigenvalue of the Jacobian matrix corresponding to the LLC converter closed-loop system to a certain element β in the matrix can be expressed by the partial derivative of the eigenvalue with respect to the element β, as shown in formula (1):

[0074]

[0075] Among them, J F is the Jacobian matrix of the LLC resonant converter. This matrix is ​​an 8th-order matrix, which can describe the linearized dynamic behavior of the system and is an important tool for analyzing the stability of the system. The superscript T represents the matrix transpose.

[0076] λ i is the eigenvalue of the LLC Jacobian matrix. The matrix has 8 eigenvalues, among which the eigenvalue closest to the real axis is λ 6 , 7 It is the main eigenvalue and has the greatest impact on the stability of the system;

[0077] m i is the left eigenvector of the Jacobian matrix, n i is the right eigenvector of the matrix. Based on the above calculation formula, we have:

[0078]

[0079] in, is the sensitivity of a certain eigenvalue of the Jacobian matrix to a certain element β in the matrix, let p and q are the real and imaginary parts of the eigenvalue sensitivity. The superscript T indicates the matrix transpose.

[0080] The calculation result of formula (1) represents the movement of the eigenvalue on the real axis and the imaginary axis when the element β in the matrix changes. According to the eigenvalue sensitivity of a certain parameter β calculated by formula (1), combined with the stability criterion of the Lyapunov indirect method, if p<0, it means that the element β makes the eigenvalue move to the left half plane of the complex plane, which is beneficial to improve the stability of the system; if p>0, it means that the element β makes the eigenvalue move to the right half plane of the complex plane, which is not conducive to improving the stability of the system.

[0081] In step S330, Figure 2 As shown, the size of the element β in the matrix is ​​adjusted based on the eigenvalue sensitivity solution result, including:

[0082] Determine whether the eigenvalue sensitivity of element β has a positive real part:

[0083] If the real part of the eigenvalue sensitivity of the element β is greater than 0, then the parameter corresponding to the element β is determined to be beneficial to the stability of the system, and β=β+I, I>0; otherwise, β=β-I, I>0, and the new element β is brought into the Jacobian matrix to re-determine its eigenvalue, where I represents the direction in which β increases or decreases.

[0084] The solution proposed in the implementation mode of the present application is based on the Lyapunov stability analysis method. First, the time domain model of the LLC resonant converter is established in combination with the extended describing function method. The influence of the load and PI controller parameters on the stability is analyzed by combining the eigenvalue sensitivity and Lyapunov theory. By adjusting the PI control parameters, the system can obtain better stability performance when the load changes.

[0085] Although the embodiments of the present application have been shown and described above, it can be understood that the above embodiments are exemplary and cannot be understood as limitations on the present application. Ordinary technicians in this field can change, modify, replace and modify the above embodiments within the scope of the present application.

Claims

1. A time domain stability analysis and improvement method for LLC resonant converter, characterized in that: include: S100, establishing a time-domain-based LLC converter model according to an extended describing function method and introducing PI controller parameters into the time-domain model to solve and generate a Jacobian matrix of the LLC resonant converter; S200, solving all eigenvalues ​​of the Jacobian matrix and displaying them on a complex plane; S300, determine the stability of the LLC resonant converter closed-loop system according to the Lyapunov indirect method: S310, if the system is stable, store the time domain model and PI controller parameters and output the determination result; S320, if the system is unstable, adjust the PI controller parameters and return to execute steps S200-S300; S330, if the system is critically stable, then solve the eigenvalue sensitivity of a certain element β in the matrix, adjust the size of the element β in the matrix based on the eigenvalue sensitivity solution, and return to execute steps S200-S300.

2. The method according to claim 1, characterized in that The stability criteria of the Lyapunov indirect method include: When at least one eigenvalue of the Jacobian matrix has a positive real part, the system is unstable; When the real part of at least one eigenvalue of the Jacobian matrix is ​​0, the system is critically stable; When the eigenvalues ​​of the Jacobian matrix all have negative real parts, the system is stable; The method of determining the stability of the LLC resonant converter closed-loop system according to the Lyapunov indirect method includes: A300, determine whether the eigenvalue has a positive real part: A310, if the eigenvalue contains a positive real part, the system is considered unstable; A320, if the eigenvalues ​​do not have positive real parts, determine whether all eigenvalues ​​have negative real parts: A321, if all eigenvalues ​​have negative real parts, the system is considered stable; A322, if not all eigenvalues ​​have negative real parts, the system is judged to be critically stable.

3. The method according to claim 1, characterized in that If the system is unstable, adjust the PI controller parameters, including: Let k p =k p +k, observe and determine the changing trend of eigenvalues ​​on the complex plane: If the eigenvalue moves to the left, the increase of k is conducive to the stability of the system, and the new PI controller parameters are brought in to re-solve the eigenvalue; If the above eigenvalue moves to the right, let k = -p × k, p>1, and then solve the new k p value; Among them, k p is the proportional coefficient of the PI controller, and k represents the direction of increase or decrease of the PI controller parameters.

4. The method according to claim 1, characterized in that: Solve the eigenvalue sensitivity of an element β in the matrix, including: The sensitivity of a certain eigenvalue of the Jacobian matrix corresponding to the LLC converter closed-loop system to a certain element β in the matrix can be expressed by the partial derivative of the eigenvalue with respect to the element β, as shown in formula (1): Among them, J F is the Jacobian matrix of LLC resonant converter, λ i is the eigenvalue of the LLC Jacobian matrix, m i is the left eigenvector of the Jacobian matrix, n i is the right eigenvector of the matrix, that is: is the sensitivity of a certain eigenvalue of the Jacobian matrix to a certain element β in the matrix, let p and q are the real and imaginary parts of the eigenvalue sensitivity; The calculation result of formula (1) represents the movement of the eigenvalue on the real axis and the imaginary axis when the element β in the matrix changes.

5. The method according to claim 4, characterized in that According to the eigenvalue sensitivity of a parameter β calculated by formula (1), combined with the stability criterion of the Lyapunov indirect method, if p<0, it means that the element β makes the eigenvalue move to the left half plane of the complex plane, which is beneficial to improving the stability of the system; if p>0, it means that the element β makes the eigenvalue move to the right half plane of the complex plane, which is not conducive to improving the stability of the system; The adjusting the size of the element β in the matrix based on the eigenvalue sensitivity solution result includes: Determine whether the eigenvalue sensitivity of element β has a positive real part: If the real part of the eigenvalue sensitivity of the element β is greater than 0, then the parameter corresponding to the element β is determined to be beneficial to the stability of the system, and β = β + I, I>0; otherwise, β = β - I, I>0, and the new element β is brought into the Jacobian matrix to re-determine its eigenvalue; Here, I indicates the direction in which β increases or decreases.

6. The method according to claim 1, characterized in that The extended description function method includes: converting the nonlinear switch link into a linear link through Fourier series, and then using the two-angle sum and difference formula in the trigonometric identity to approximate the variable frequency signal into a variable amplitude signal with fixed frequency and variable amplitude.

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