Filippov stability analysis method for series capacitor buck converter

Abstract

The invention discloses a Filippov stability analysis method for a series capacitor buck converter, and the method comprises the steps: firstly building a differential equation model and a switch switching surface equation of the series capacitor buck converter in different switching states, and calculating the differential equation model into a state transition matrix in a discrete time domain; and calculating a transition matrix corresponding to the switching surface of the switch based on the Filippov thought, and multiplying the state transition matrix and the transition matrix according to the switching sequence to obtain a single-valued matrix corresponding to the system. The stability states of the system under different system parameters and modulation strategies can be obtained by analyzing the characteristic value of the single-valued matrix. According to the method, the problems that a system closed-form solution cannot be obtained through Poincare mapping and the calculation complexity of a multi-mode system is high can be solved, and the system accuracy is improved to 1 / 2 switching frequency; the stability characteristics of the SCBC are accurately analyzed; the method is simple in general calculation and easy to program, and has very high engineering practical value.

Description

Technical Field

[0001] The present invention relates to the technical field of a DC / DC converter connected in series with a capacitor step-down converter, and in particular to a Filippov stability analysis method for a series capacitor converter. Background Art

[0002] Introduced in the early 20th century, the series capacitor buck converter (SCBC) has gained recognition for its high voltage conversion ratio, low switching losses, minimal current ripple, and automatic current balancing mechanism. Multiphase SCBCs further enhance performance by reducing passive component size, handling high currents, and achieving high power density. Consequently, SCBCs are expected to be used in compact DC-DC converters and high-current applications such as CPU voltage regulators and data center power supplies.

[0003] State-space average modeling is the most widely used approach for modeling series-capacitor step-down converters. However, since it only retains the DC component of each state variable within the switching cycle, it can only predict converter dynamics below 1 / 10 to 1 / 5 of the switching frequency when the bandwidth is large relative to the switching frequency. To improve the accuracy of switching-cycle average models for DC-DC systems, continuous-domain modeling methods such as describing function modeling and generalized average modeling, as well as discrete-domain modeling methods using Poincare mapping, have been proposed.

[0004] The core of state-space average modeling is the sliding average operator. State-space average models retain only the DC component of each state variable within a switching cycle. Therefore, they can typically only predict the converter's low-frequency dynamics. Furthermore, switching-cycle average modeling is only applicable to converter systems dominated by DC quantities. It is difficult to directly extract dynamic information from single-phase converters, phase-controlled converters (such as dual-active full-bridge converters and phase-shifted full-bridge converters), and frequency-controlled converters (such as LLC resonant converters).

[0005] The describing function modeling method can be directly applied to nonlinear periodic systems without requiring averaging. The model established using the describing function method is a single-input, single-output linear time-invariant model. However, the describing function modeling process only considers the dominant frequency coupling component. When sideband frequency coupling components are non-negligible (e.g., in asymmetric controllers, unbalanced power grids, or weak power grids), the resulting model deviates significantly from the actual system.

[0006] The generalized average modeling method is based on the principle of harmonic balance. By introducing a generalized average operator, each state variable is transformed into a different frequency space for modeling. The resulting model is a multi-input, multi-output, time-invariant model capable of characterizing the nonlinear behavior of the system and the dynamics of arbitrary-order harmonics. While this method can characterize the dynamics of arbitrary-order harmonics, the model derivation becomes increasingly complex as the number of harmonics considered increases, and the computational complexity makes it unsuitable for practical analytical applications.

[0007] The Poincare mapping method divides space into units representing different dynamic models based on the system's modes and constructs a mapping between each unit to model the state at the beginning and end of a cycle. This method becomes computationally intensive as the number of system modes increases. Furthermore, the Jacobian matrix becomes difficult to solve for stability analysis, and discrete iterative methods cannot obtain a closed-form solution to the system.

[0008] Chinese patent application CN116345913A discloses a stability analysis method for digitally controlled DC / DC converters based on the Filippov method. By establishing a general piecewise smooth dynamic system for the power-stage state variables and a difference equation for the control-stage state variables, this method is suitable for analyzing the low-frequency and high-frequency bifurcation behavior of any digitally controlled DC / DC converter. However, due to its digital control nature, this method is not applicable to analog control systems with fixed time delays. Furthermore, the object under investigation exhibits a zero-order holder effect, meaning that the control signal remains constant within the modulation period.

[0009] At present, the modeling methods for DC / DC converter systems still face problems such as insufficient accuracy, complex derivation and huge computational complexity, lack of versatility, and difficulty in programming implementation, which are not conducive to accurate stability analysis of DC / DC converters.

[0010] Therefore, developing a new stability analysis method is a difficult problem that needs to be solved urgently by those skilled in the art. Summary of the Invention

[0011] In view of the above-mentioned problems existing in the prior art, the purpose of the present invention is to provide a Filippov stability analysis method for series capacitor step-down converters to solve the problems of current modeling methods for DC / DC converter systems, such as insufficient accuracy, complex derivation and huge computational complexity, lack of versatility, and difficulty in programming implementation.

[0012] To achieve the above objectives, the present invention provides a Filippov stability analysis method for a series capacitor step-down converter, the method comprising the following steps:

[0013] Step 1: Establish the state space equation of the series capacitor buck converter. Determine the converter's operating mode under different switching states based on different switching modes. Establish a differential equation system based on the corresponding operating mode to describe the SCBC power stage state variable dynamics model.

[0014] (1) Select the two-phase inductor current i L1 ,i L2 , series capacitor voltage v cb , output capacitor voltage v c is the circuit state variable xc =[i L1 i L2 v cb v c ] T Build a state-space model:

[0015]

[0016] The superscript 'T' represents the transpose of the vector, i∈{1,2,3,4}, t i Indicates the switching moment, V in is the input DC voltage, the state matrix With the input matrix Can write

[0017]

[0018] Where α=1+R c / R o , S1, S2, S3 and S4 are complementary switches, C b is a capacitor, L1 and L2 are inductors, C o is the output capacitor, R o are load resistors; R1, R2, and R c Represent the inductors L1, L2 and output capacitor C o parasitic resistance;

[0019] (2) Select the integral of the error ξ as the control state variable, and write the differential equation as

[0020]

[0021] where u mod V is the modulation signal used to generate the duty cycle compared with the carrier signal. ref is the output voltage reference, ξ is the integral of the error, K p is the proportional gain, K i is the integral gain, v out is the output voltage and it can be described as

[0022]

[0023] (3) Combine the circuit equation and the control equation and select the state variable as x=[i L1 i L2 v cb v c ξ] T , the input variable is u=[V in V ref ] T , then the state space equation is established as

[0024]

[0025] Among them A i and B i for

[0026]

[0027] Step 2: Solve the steady-state trajectory of the system based on the state space equation established by Newton-Raphson;

[0028] x k+1 =x k =Φ(d)x k +Ψ(d)

[0029]

[0030] The Newton-Raphson method can be used to numerically solve the above equations to obtain the system at each switching time t i Steady-state solution of the system state X i , T s represents the switching cycle;

[0031] Step 3: Calculate the state transition matrix (STM) of the system

[0032] φ i (t i ,t i-1 )=e AiTi

[0033] Where T i =t i -t i-1 is the time when the system is in the i-th mode;

[0034] Step 4: Calculate the system's transition matrix at four switching moments. The calculation formula is as follows:

[0035] First switch

[0036]

[0037] Where I is the 5×5 identity matrix, X i is SCBC at t = t i The steady-state solution of the switching manifold is

[0038] (n1) T =【K p c v -K i ]T s

[0039] Second switch

[0040] S2=I

[0041] The third switch

[0042]

[0043] Where T is a local single-value matrix, representing the mapping from the first to the third switching;

[0044] T=φ3(t3,t2)·S2·φ2(t2,t1)·S1

[0045] The fourth switch

[0046] S4=I

[0047] Step 5: Calculate the system's unique matrix

[0048]

[0049] Represents the mapping of state disturbance before and after a switching cycle;

[0050] Step 6: Calculate the eigenvalue of the system. The characteristic equation of the system is det(zI-M r )=0, and finally the stability of the system can be evaluated by the eigenvalue |z i |, i = {1, 2, 3, 4, 5} modulus to evaluate;

[0051] Step 7: Set appropriate parameters, select different working conditions, traverse all parameters and calculate the eigenvalues ​​to obtain the stable region of the system for different parameters.

[0052] As an implementation manner of the present invention, in step 1, the series capacitor buck converter operates in a continuous conduction mode.

[0053] As an embodiment of the present invention, the series capacitor step-down converter in step 1 is controlled by a proportional-integral controller to adjust the output voltage v out .

[0054] As an implementation manner of the present invention, the modulus of the eigenvalue in step six is ​​less than 1, and the system is stable.

[0055] As an implementation manner of the present invention, if the maximum value of the eigenvalue norm in step six is ​​greater than 1, the system is unstable.

[0056] As an implementation manner of the present invention, in step six, the eigenvalue leaves the unit circle in the negative real part, and the system undergoes period-doubling bifurcation.

[0057] As an embodiment of the present invention, in step six, when the eigenvalues ​​leave the unit circle in the form of complex number pairs, a Hopf bifurcation occurs.

[0058] Compared with the prior art, the present invention has the following beneficial effects:

[0059] 1. The object of this invention is analog control system, not digital control;

[0060] 2. The research object of the present invention has a zero-order holder effect, that is, the control signal still changes within the modulation period.

[0061] 3. The Filippov method can solve the problem that the Poincare mapping cannot obtain a closed-form solution to the system and the high computational complexity of multimodal systems.

[0062] 4. Solve the problem that the state space model is only effective at low frequencies and improve the system accuracy to 1 / 2 the switching frequency.

[0063] 5. The problem of the difficulty in accurately modeling a time-delay switching system using an SCBC with fixed phase shift modulation is solved. This method can be used to accurately analyze the stability characteristics of the SCBC. It is suitable for analog control systems with fixed delays, where conventional methods are not applicable.

[0064] 6. It solves the problem of complex calculation of generalized average modeling. This method is simple in general calculation and easy to program and implement, and has high engineering practical value. BRIEF DESCRIPTION OF THE DRAWINGS

[0065] Figure 1 This is the main circuit diagram of SCBC.

[0066] Figure 2 This is a diagram showing the period-doubling bifurcation of the system when the eigenvalue leaves the unit circle in the negative real part.

[0067] Figure 3 is a graph showing the Hopf bifurcation that occurs when the eigenvalues ​​leave the unit circle in the form of complex pairs.

[0068] Figure 4 This is a flow chart of stability analysis based on the Filippov method.

[0069] Figure 5 This is a diagram of the application scenario of SCBC with small volume and high power density. DETAILED DESCRIPTION

[0070] The present invention is described in detail below with reference to the accompanying drawings and specific embodiments. This embodiment is implemented based on the technical solution of the present invention, and provides a detailed implementation method and specific operation process, but the protection scope of the present invention is not limited to the following embodiments.

[0071] The main circuit diagram of SCBC is as follows Figure 1 shown.

[0072] A Filippov stability analysis method for a series capacitor step-down converter, the method steps are as follows:

[0073] Step 1: Establish the state-space equations for the series capacitor buck converter. Determine the converter's operating mode under different switching conditions based on the different switching modes. Establish a set of differential equations based on the corresponding operating modes to describe the SCBC power stage state variable dynamics model.

[0074] Select the two-phase inductor current i L1 ,i L2 , series capacitor voltage v cb , output capacitor voltage v c is the circuit state variable x c =[i L1 i L2 v cb v c ] T Build a state-space model.

[0075]

[0076] The superscript 'T' represents the transpose of the vector, i∈{1,2,3,4}, t i represents the switching moment, the state matrix With the input matrix Can write

[0077]

[0078] The output voltage can be written as

[0079]

[0080] Without loss of generality, the SCBC operates in continuous conduction mode (CCM). In the SCBC, α = 1 + R c / R o , bridge arm A consists of two complementary switches S1 and S2, capacitor C b and inductor L1. Bridge arm B consists of two complementary switches S3 and S4 and inductor L2. These two phases work together to provide a o and the load resistor R o In addition, R1, R2 and R c They represent the parasitic resistance of inductors L1, L2 and output capacitor, respectively, and the input DC voltage V in is constant.

[0081] SCBC is controlled by a proportional-integral (PI) controller to adjust the output voltage v outIn steady state, the switch of phase B operates with the same duty cycle as the switch of phase A, but the phase shift is 180°. For the PI controller, the integral of the error ξ is selected as the control state variable, and the differential equation is written as

[0082]

[0083] where u mod V is the modulation signal used to generate the duty cycle compared with the carrier signal. ref is the output voltage reference, is the integral of the error, K p is the proportional gain, K i Is the integral gain. Combine the circuit equation with the control equation and select the state variable as x=[i L1 i L2 v cb v c ξ] T , the input variable is u=[V in V ref ] T , then the state space equation is established as

[0084]

[0085] Among them A i and B i for

[0086]

[0087] Step 2: Solve the system steady-state trajectory based on the established state-space equation.

[0088] x k+1 =x k =Φ(d)x k +Ψ(d)

[0089]

[0090] The Newton-Raphson method can be used to numerically solve the above equations to obtain the system at each switching time t i The steady-state solution X i , T s Indicates the switching cycle.

[0091] Step 3: Calculate the state transition matrix (STM) of the system

[0092]

[0093] Where T i =t i -t i-1 is the time when the system is in the i-th mode.

[0094] Step 4: Calculate the system's transition matrix at four switching moments. The calculation formula is as follows:

[0095] First switch

[0096]

[0097] Where I is the 5×5 identity matrix, X i is SCBC at t = t i The steady-state solution of the switching manifold is

[0098] (n1) T =[K p c v -K i ]T s

[0099] Second switch

[0100] S2=I

[0101] The third switch

[0102]

[0103] Where T is a local single-value matrix that describes the mapping from the first to the third switching. This recursive method can handle the time lag problem caused by the inherent delay in the phase-shift modulation process.

[0104] T=φ3(t3,t2)·S2·φ2(t2,t1)·S1

[0105] The fourth switch

[0106] S4=I

[0107] The jump matrix incorporates the controller's time disturbance into the calculation by constructing the state disturbance before and after the switch, so that the closed-form solution of the system can be obtained.

[0108] Step 5: Calculate the system's unique matrix

[0109]

[0110] Describes the mapping of state disturbances before and after a switching cycle

[0111] Step 6: Calculate the eigenvalue of the system. The characteristic equation of the system is det(zI-M r )=0. Finally, the stability of the system can be evaluated by evaluating the eigenvalue |z i|, i = {1, 2, 3, 4, 5}. If the modulus of all eigenvalues ​​is less than 1, the system is stable; if the maximum eigenvalue norm is greater than 1, the system is unstable. Furthermore, the type of bifurcation behavior can be identified by the point at which the eigenvalue crosses the unit circle as it deviates from it. When an SCBC undergoes an unstable bifurcation, the frequency dominated by the eigenvalue that leaves the unit circle is the oscillation frequency.

[0112] Based on the above discussion:

[0113] 1) The modulus of the eigenvalue determines the stability of the system;

[0114] 2) The type of bifurcation behavior can be identified by the position where the eigenvalue crosses the unit circle. Specifically, when the eigenvalue leaves the unit circle in the negative real part, the system undergoes a period-doubling bifurcation, such as Figure 2 As shown; when the eigenvalues ​​leave the unit circle in the form of complex pairs, Hopf bifurcation occurs, as shown in Figure 3 shown.

[0115] 3) The eigenvalues ​​outside the unit circle can be used to predict the oscillation frequency of SCBC.

[0116] Step 7: Set appropriate parameters, select different working conditions, traverse all parameters and calculate the eigenvalues ​​to obtain the stable region of the system for different parameters.

[0117] The flow chart of SCBC stability analysis based on the above analysis is as follows Figure 4 shown.

[0118] The series capacitor buck converter (SCBC) combines a two-phase interleaved buck converter with a switched capacitor front end, allowing high-frequency operation in the MHz range and achieving better system dynamics while reducing component stress. It has a high step-down conversion ratio and, through a multi-phase interleaved design, can significantly reduce the size of passive components and achieve higher power density. At the same time, it has an automatic current balancing mechanism to ensure load balance between phases, avoid local overheating and performance degradation, and thus improve the overall performance and reliability of the system. It is very suitable for applications with strict requirements on size, weight, efficiency and fast response.

[0119] like Figure 5As shown, in the field of mobile electronic devices, the high power density and miniaturization of SCBC are very suitable for mobile phones, tablets and other portable electronic products that have strict requirements on size and weight. High-frequency operation allows for faster charging and higher efficiency, which helps to extend battery life. In the field of electric vehicles, in electric vehicles (EVs), hybrid electric vehicles (HEVs) and electric bicycles, SCBC can be used to optimize the power management system and improve charging speed and efficiency. Reducing the size and weight of passive components is crucial to increasing the vehicle's range. In the field of data centers and servers, data centers have high requirements for energy efficiency and thermal management. SCBC can reduce heat dissipation requirements and improve energy conversion efficiency, thereby reducing operating costs. Rapid response capabilities help to cope with sudden changes in server load and maintain system stability.

[0120] The SCBC is a switching nonlinear system. Switching events induce rich and complex nonlinear dynamic behaviors, including various bifurcations and chaotic motions. Furthermore, its topology constitutes a multi-resonant system, including inline resonance and output resonance. Furthermore, the fixed-delay phase-shift modulation strategy employed during its modulation leads to stability issues. This technology is primarily used to analyze the stability of the SCBC and determine the system's dynamic indicators, thereby guiding the design of SCBC control and modulation strategies.

[0121] The basic principles, main features, and advantages of the present invention are shown and described above. Those skilled in the art should understand that the present invention is not limited to the above embodiments. The above embodiments and descriptions are merely illustrative of the principles of the present invention. Various changes and modifications may be made to the present invention without departing from the spirit and scope of the present invention. Such changes and modifications are intended to fall within the scope of the present invention. The scope of protection claimed in the present invention is defined by the appended claims and their equivalents.

Claims

1. A Filippov stability analysis method for series capacitor step-down converters, characterized in that: The method steps are as follows: Step 1: Establish the state space equation of the series capacitor buck converter. Determine the converter's operating mode under different switching states based on different switching modes. Establish a differential equation system based on the corresponding operating mode to describe the SCBC power stage state variable dynamics model. (1) Select the two-phase inductor current i L1 ,i L2 , series capacitor voltage v cb , output capacitor voltage v c is the circuit state variable x c =[i L1 i L2 v cb v c ] T Build a state-space model: The superscript 'T' represents the transpose of the vector, i∈{1,2,3,4}, t i Indicates the switching moment, V in is the input DC voltage, the state matrix With the input matrix can write; Where α=1+R c / R o , S1, S2, S3 and S4 are complementary switches, C b is a capacitor, L1 and L2 are inductors, C o is the output capacitor, R o are load resistors; R1, R2, and R c Represent the inductors L1, L2 and output capacitor C o parasitic resistance; (2) Select the integral of the error ξ as the control state variable and write the differential equation; where u mod V is the modulation signal used to generate the duty cycle compared with the carrier signal. ref is the output voltage reference, ξ is the integral of the error, K p is the proportional gain, K i is the integral gain, v out is the output voltage and it can be described as; (3) Combine the circuit equation and the control equation and select the state variable as x=[i L1 i L2 v cb v c ξ] T , the input variable is u=[V in V ref ] T , then the state space equation is established as; Among them A i and B i for; Step 2: Solve the steady-state trajectory of the system based on the state space equation established by Newton-Raphson; The Newton-Raphson method can be used to numerically solve the above equations to obtain the system at each switching time t i Steady-state solution of the system state X i , T s represents the switching cycle; Step 3: Calculate the state transition matrix of the system; Where T i =t i -t i-1 is the time when the system is in the i-th mode; Step 4: Calculate the system's transition matrix at four switching moments. The calculation formula is as follows: First switch; Where I is the 5×5 identity matrix, X i is SCBC at t = t i The steady-state solution of the switching manifold is (n1) T =[K p c v -K i ]T s ; Secondary switching; S2=I; The third switch Where T is a local single-value matrix, representing the mapping from the first to the third switching; T=φ3(t3,t2)·S2·φ2(t2,t1)·S1; The fourth switch S4=I; Step 5: Calculate the system's single-value matrix Monodromy Represents the mapping of state disturbance before and after a switching cycle; Step 6: Calculate the eigenvalue of the system. The characteristic equation of the system is det(zI-M r )=0, and finally the stability of the system can be evaluated by the eigenvalue |z i |, i = {1, 2, 3, 4, 5} modulus to evaluate; Step 7: Set appropriate parameters, select different working conditions, traverse all parameters and calculate the eigenvalues ​​to obtain the stable region of the system for different parameters.

2. The Filippov stability analysis method for series capacitor step-down converter according to claim 1, characterized in that: In step 1, the series capacitor step-down converter operates in a continuous conduction mode.

3. The Filippov stability analysis method for series capacitor step-down converter according to claim 1, characterized in that: The series capacitor step-down converter in step 1 is controlled by a proportional-integral controller to adjust the output voltage v out .

4. The Filippov stability analysis method for series capacitor step-down converter according to claim 1, characterized in that: If the modulus of the eigenvalue in step six is ​​less than 1, the system is stable.

5. The Filippov stability analysis method for series capacitor step-down converter according to claim 1, characterized in that: If the maximum value of the eigenvalue norm in step six is ​​greater than 1, the system is unstable.

6. The Filippov stability analysis method for series capacitor step-down converter according to claim 1, characterized in that: In step six, the eigenvalue leaves the unit circle in the negative real part, and the system undergoes period-doubling bifurcation.

7. The Filippov stability analysis method for series capacitor step-down converter according to claim 1, characterized in that: In step 6, when the eigenvalues ​​leave the unit circle in the form of complex number pairs, a Hopf bifurcation occurs.

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