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青岛市城乡建设局网站,制作手机端网站开发,it培训机构培训排名,如何查网站点击量两种状态平均法在功率变换器建模的应用比较 [!info] Bibliography [1] 高朝晖, 林辉张晓斌 吴小华, “两种状态平均法在功率变换器建模的应用比较,” 计算机仿真, no. 241-244248, 2008. [!note] 状态空间平均法采用直流量近似#xff08;线性系统模型#xff09;…两种状态平均法在功率变换器建模的应用比较 [!info] Bibliography [1] 高朝晖, 林辉张晓斌 吴小华, “两种状态平均法在功率变换器建模的应用比较,” 计算机仿真, no. 241-244248, 2008. [!note] 状态空间平均法采用直流量近似线性系统模型广义状态空间平均采用直流量和基波分量近似。也即状态空间平均法采用0阶傅里叶级数近似广义状态空间平均采用0阶和1阶傅里叶级数近似 应用状态空间平均法分析 Buck变换器 广义状态平均法GSSA 广义状态平均采用傅里叶级数拟合系统状态 x(t)∑n−∞∞⟨x⟩n(t)ejnωtx(t) \sum_{n -\infty}^\infty \langle x \rangle_n(t) e^{j n \omega t} x(t)n−∞∑∞​⟨x⟩n​(t)ejnωt ω2π/T\omega 2\pi / Tω2π/T ⟨x⟩n(t)\langle x \rangle_n(t)⟨x⟩n​(t) 代表傅里叶系数 ⟨x⟩n(t)1T∫t−Ttx(τ)e−jnωτdτ\langle x \ranglen(t) \frac 1T \int{t-T}^t x(\tau) e^{-j n \omega \tau}d\tau ⟨x⟩n​(t)T1​∫t−Tt​x(τ)e−jnωτdτ nnn is AKA index-k average 三角形式傅里叶级数 x(τ)⟨x⟩02∑n1∞(a1cos⁡(nωτ)b1sin⁡(nωτ))x(\tau)\langle x\rangle02 \sum{n1}^{\infty}\left(a_1 \cos (n \omega \tau) b_1 \sin (n \omega \tau)\right) x(τ)⟨x⟩0​2n1∑∞​(a1​cos(nωτ)b1​sin(nωτ)) ⟨x⟩n(t)an−jbn\langle x\rangle_n(t) a_n - j b_n⟨x⟩n​(t)an​−jbn​ an12π∫02πx(ωτ)cos⁡(ωτ)d(ωτ)bn12π∫02πx(ωτ)sin⁡(ωτ)d(ωτ)⟨x⟩01T∫t−Ttx(τ)dτ\begin{aligned} {a}_n \frac{1}{2 \pi} \int_0^{2 \pi} {x}(\omega \tau) \cos (\omega \tau) \mathrm{d}(\omega \tau) \ {b}_n \frac{1}{2 \pi} \int_0^{2 \pi} {x}(\omega \tau) \sin (\omega \tau) \mathrm{d}(\omega \tau) \ \langle{x}\rangle0 \frac{1}{{T}} \int{t-{T}}^t {x}(\tau) \mathrm{d} \tau \end{aligned} an​bn​⟨x⟩0​​2π1​∫02π​x(ωτ)cos(ωτ)d(ωτ)2π1​∫02π​x(ωτ)sin(ωτ)d(ωτ)T1​∫t−Tt​x(τ)dτ​ 性质 d⟨x⟩n(t)dt⟨dxdt⟩n(t)−jnω⟨x⟩n(t)\frac{d \langle x\rangle_n(t)}{d t}\left\langle\frac{d x}{d t}\right\rangle_n(t)-j n \omega\langle x\rangle_n(t)dtd⟨x⟩n​(t)​⟨dtdx​⟩n​(t)−jnω⟨x⟩n​(t) [!note] Proof ⟨x⟩n(t)1T∫0Tx(t−Ts)e−jnω(t−Ts)ds\langle x \rangle_n(t) \frac1T \int_0^T x(t - T s) e^{-j n \omega(t-Ts)}ds⟨x⟩n​(t)T1​∫0T​x(t−Ts)e−jnω(t−Ts)ds ⟨qx⟩n∑i−∞∞⟨q⟩n−i⟨x⟩i\langle qx \ranglen \sum{i -\infty}^\infty \langle q \rangle_{n -i} \langle x \rangle_i⟨qx⟩n​∑i−∞∞​⟨q⟩n−i​⟨x⟩i​
Suppose that x(t)x(t)x(t) and q(t)q(t)q(t) can be approximated by 0- and 1-order Fourier series (即直流和基波量) q(t)≈⟨q⟩0⟨q⟩−1e−jωt⟨q⟩1ejωtx(t)≈⟨x⟩0⟨x⟩−1e−jωt⟨x⟩1ejωt\begin{aligned} q(t) \approx \langle q \rangle0 \langle q \rangle{-1} e^{- j \omega t} \langle q \rangle_1 e^{j \omega t} \ x(t) \approx \langle x \rangle0 \langle x \rangle{-1} e^{- j \omega t} \langle x \rangle_1 e^{j \omega t} \end{aligned} q(t)x(t)​≈⟨q⟩0​⟨q⟩−1​e−jωt⟨q⟩1​ejωt≈⟨x⟩0​⟨x⟩−1​e−jωt⟨x⟩1​ejωt​ [!note] 这里注意三角形式傅里叶级数中的直流和基波项(a0/2a1cos⁡(ωt)b1sin⁡(ωt)a_0/2 a_1 \cos(\omega t) b_1 \sin(\omega t)a0​/2a1​cos(ωt)b1​sin(ωt))对应于复数形式的傅立叶级数中 000、±1±1±1次项(c0c−1e−iωtc1eiωtc0 c{-1}e^{-i \omega t} c_{1} e^{i\omega t}c0​c−1​e−iωtc1​eiωt) xxx 和 qqq 乘积可表示为 ⟨qx⟩0⟨q⟩0⟨x⟩0⟨q⟩−1⟨x⟩1⟨q⟩1⟨x⟩−1⟨qx⟩1⟨q⟩0⟨x⟩1⟨q⟩1⟨x⟩0⟨qx⟩−1⟨q⟩0⟨x⟩−1⟨q⟩−1⟨x⟩0\begin{aligned} \langle qx \rangle_0 \langle q \rangle_0 \langle x \rangle0 \langle q \rangle{-1} \langle x \rangle_1 \langle q \rangle1 \langle x \rangle{-1} \ \langle qx \rangle_1 \langle q \rangle_0 \langle x \rangle1 \langle q \rangle{1} \langle x \rangle0\ \langle qx \rangle{-1} \langle q \rangle0 \langle x \rangle{-1} \langle q \rangle_{-1} \langle x \rangle_0 \end{aligned} ⟨qx⟩0​⟨qx⟩1​⟨qx⟩−1​​⟨q⟩0​⟨x⟩0​⟨q⟩−1​⟨x⟩1​⟨q⟩1​⟨x⟩−1​⟨q⟩0​⟨x⟩1​⟨q⟩1​⟨x⟩0​⟨q⟩0​⟨x⟩−1​⟨q⟩−1​⟨x⟩0​​ 正负平均指数互为共轭⟨x⟩1⟨x⟩−1∗\langle x \rangle1 \langle x \rangle{-1}^⟨x⟩1​⟨x⟩−1∗​ ⟨q⟩1⟨q⟩1Rj⟨q⟩1I⟨q⟩−1∗(⟨q⟩−1R⟨q⟩−1I)∗\langle q \rangle_1 \langle q \rangle_1^R j \langle q \rangle1^I \langle q \rangle{-1}^ \left(\langle q \rangle{-1}^R \langle q \rangle{-1}^I\right)^⟨q⟩1​⟨q⟩1R​j⟨q⟩1I​⟨q⟩−1∗​(⟨q⟩−1R​⟨q⟩−1I​)∗⟨x⟩1⟨x⟩1Rj⟨x⟩1I⟨x⟩−1∗(⟨x⟩−1R⟨x⟩−1I)∗\langle x \rangle_1 \langle x \rangle_1^R j \langle x \rangle1^I \langle x \rangle{-1}^ \left(\langle x \rangle{-1}^R \langle x \rangle{-1}^I\right)^*⟨x⟩1​⟨x⟩1R​j⟨x⟩1I​⟨x⟩−1∗​(⟨x⟩−1R​⟨x⟩−1I​)∗ 于是 ⟨qx⟩0⟨q⟩0⟨x⟩02(⟨q⟩1R⟨x⟩1R⟨q⟩1I⟨x⟩1I)⟨qx⟩1R⟨q⟩0⟨x⟩1R⟨q⟩1R⟨x⟩0⟨qx⟩1I⟨q⟩0⟨x⟩1I⟨q⟩1I⟨x⟩0\begin{aligned} \langle qx \rangle_0 \langle q \rangle_0 \langle x \rangle0 2\left(\langle q \rangle{1}^R \langle x \rangle_1^R \langle q \rangle1^I \langle x \rangle{1}^I\right) \ \langle qx \rangle_1^R \langle q \rangle_0 \langle x \rangle1^R \langle q \rangle{1}^R \langle x \rangle0\ \langle qx \rangle{1}^I \langle q \rangle0 \langle x \rangle{1}^I \langle q \rangle_{1}^I \langle x \rangle_0 \end{aligned} ⟨qx⟩0​⟨qx⟩1R​⟨qx⟩1I​​⟨q⟩0​⟨x⟩0​2(⟨q⟩1R​⟨x⟩1R​⟨q⟩1I​⟨x⟩1I​)⟨q⟩0​⟨x⟩1R​⟨q⟩1R​⟨x⟩0​⟨q⟩0​⟨x⟩1I​⟨q⟩1I​⟨x⟩0​​ GSSA 建模 BUCK 定义开关函数 q(t){0关1开q(t) \left{ \begin{matrix} 0 关 \ 1 开 \end{matrix} \right. q(t){01​关开​ BUCK system model: LdiLdtvinq(t)−voCdvodtiL−voR\begin{aligned} L \frac{d {iL}}{d t}v{i n} q(t)-v_o \ C \frac{d v_o}{d t}i_L-\frac{v_o}{R} \end{aligned} ​LdtdiL​​vin​q(t)−vo​Cdtdvo​​iL​−Rvo​​​ 0 平均指数模型 Ld⟨iL⟩0dtVin⟨q⟩0−⟨vo⟩0Cd⟨vo⟩0dt⟨iL⟩0−⟨vo⟩0R\begin{aligned} L\frac{d\langle i_L\rangle0}{dt}V{in} \langle q \rangle_0 - \langle v_o \rangle_0 \ C\frac{d\langle v_o\rangle_0}{dt}\langle{i_L}\rangle_0-\frac{\left\langle\mathrm{v}_o\right\rangle_0}{{R}} \end{aligned} ​Ldtd⟨iL​⟩0​​Vin​⟨q⟩0​−⟨vo​⟩0​Cdtd⟨vo​⟩0​​⟨iL​⟩0​−R⟨vo​⟩0​​​ 1平均指数模型 d⟨iL⟩1dt−jω⟨iL⟩11L(Vin⟨q⟩1−⟨vo⟩1)d⟨vo⟩1dt−jω⟨vo⟩11C(⟨iL⟩0−⟨vo⟩0R)\begin{aligned} \frac{d\langle i_L\rangle_1}{dt} -j\omega \langle i_L\rangle1 \frac1L \left( V{in} \langle q \rangle_1 - \langle v_o \rangle_1 \right)\ \frac{d\langle v_o\rangle_1}{dt} -j\omega \langle v_o \rangle_1 \frac 1C \left( \langle{i_L}\rangle_0-\frac{\left\langle\mathrm{v}_o\right\rangle_0}{{R}} \right) \end{aligned} dtd⟨iL​⟩1​​dtd⟨vo​⟩1​​​−jω⟨iL​⟩1​L1​(Vin​⟨q⟩1​−⟨vo​⟩1​)−jω⟨vo​⟩1​C1​(⟨iL​⟩0​−R⟨vo​⟩0​​)​ 考虑共轭关系1平均指数模型的实部虚部分别可以写作 d⟨iL⟩1Rdtω⟨iL⟩1I1L(Vin⟨q⟩1R−⟨vo⟩1R)d⟨iL⟩1Idt−ω⟨iL⟩1R1L(Vin⟨q⟩1I−⟨vo⟩1I)d⟨vo⟩1Rdtω⟨vo⟩1I1C(⟨iL⟩1R−⟨vo⟩1RR)d⟨vo⟩1Idt−ω⟨vo⟩1R1C(⟨iL⟩1I−⟨vo⟩1IR)\begin{aligned} \frac{d\langle i_L\rangle_1^R}{dt} \omega \langle i_L\rangle1^I \frac1L \left( V{in} \langle q \rangle_1^R - \langle v_o \rangle_1^R \right)\ \frac{d\langle i_L\rangle_1^I}{dt} -\omega \langle i_L\rangle1^R \frac1L \left( V{in} \langle q \rangle_1^I - \langle v_o \rangle_1^I \right)\ \frac{d\langle v_o\rangle_1^R}{dt} \omega \langle v_o \rangle_1^I \frac 1C \left( \langle{i_L}\rangle_1^R-\frac{\left\langle\mathrm{v}_o\right\rangle_1^R}{{R}} \right)\ \frac{d\langle v_o\rangle_1^I}{dt} -\omega \langle v_o \rangle_1^R \frac 1C \left( \langle{i_L}\rangle_1^I -\frac{\left\langle\mathrm{v}_o\right\rangle_1^I}{{R}} \right) \end{aligned} dtd⟨iL​⟩1R​​dtd⟨iL​⟩1I​​dtd⟨vo​⟩1R​​dtd⟨vo​⟩1I​​​ω⟨iL​⟩1I​L1​(Vin​⟨q⟩1R​−⟨vo​⟩1R​)−ω⟨iL​⟩1R​L1​(Vin​⟨q⟩1I​−⟨vo​⟩1I​)ω⟨vo​⟩1I​C1​(⟨iL​⟩1R​−R⟨vo​⟩1R​​)−ω⟨vo​⟩1R​C1​(⟨iL​⟩1I​−R⟨vo​⟩1I​​)​ 选取状态变量x[⟨iL⟩1R,⟨iL⟩1I,⟨vo⟩1R,⟨vo⟩1I,⟨iL⟩0R,⟨vo⟩0R]Tx [\langle i_L\rangle_1^R, \langle i_L\rangle_1^I, \langle v_o \rangle_1^R, \langle v_o \rangle_1^I,\langle i_L \rangle_0^R, \langle v_o \rangle_0^R]^Tx[⟨iL​⟩1R​,⟨iL​⟩1I​,⟨vo​⟩1R​,⟨vo​⟩1I​,⟨iL​⟩0R​,⟨vo​⟩0R​]T, 可以得到状态空间方程 x˙AxBu\begin{aligned} \dot x Ax Bu\ \end{aligned} x˙​AxBu​ A[0ω−1/L000−ω00−1/L001/C0−1/(RC)ω0001/C−ω−1/(RC)0000000−1/L00001/C−1/(RC)]A \left[\begin{matrix} 0 \omega -1/L 0 00 \ -\omega 0 0 -1/L 0 0\ 1/C 0 -1/(RC) \omega 0 0\ 0 1/C -\omega -1/(RC) 0 0\ 0 0 0 0 0 -1/L \ 0 0 0 0 1/C -1 / (RC) \end{matrix}\right]A​0−ω1/C000​ω001/C00​−1/L0−1/(RC)−ω00​0−1/Lω−1/(RC)00​000001/C​0000−1/L−1/(RC)​​B[⟨q⟩1RL,⟨q⟩1IL,0,0,⟨q⟩0L,0]TB \left[\frac{\langle q \rangle_1^R}L, \frac{\langle q \rangle_1^I}L, 0, 0, \frac{\langle q \rangle0}L,0 \right]^TB[L⟨q⟩1R​​,L⟨q⟩1I​​,0,0,L⟨q⟩0​​,0]TuVinu V{in}uVin​ 开关函数 q(t)q(t)q(t) 的 0 阶和 1 阶傅里叶系数 ⟨q⟩1R12π∫02πDcos⁡(ωτ)d(ωτ)12πsin⁡(2πD)⟨q⟩1I−12π∫02πDsin⁡(ωτ)d(ωτ)12π[cos⁡(2πD)−1]⟨q⟩0D\begin{aligned} \langle q \rangle_1^R \frac 1{2\pi} \int_0^{2\pi D} \cos (\omega \tau ) d(\omega \tau) \frac1{2\pi} \sin (2\pi D)\\langle q \rangle_1^I -\frac 1{2\pi} \int_0^{2\pi D} \sin (\omega \tau ) d(\omega \tau) \frac1{2\pi} [\cos (2\pi D) - 1]\ \langle q \rangle_0 D\ \end{aligned} ⟨q⟩1R​⟨q⟩1I​⟨q⟩0​​2π1​∫02πD​cos(ωτ)d(ωτ)2π1​sin(2πD)−2π1​∫02πD​sin(ωτ)d(ωτ)2π1​[cos(2πD)−1]D​ 根据广义状态模型的解 xxx可以得到系统输出为 iL2x1cos⁡(ωt)−2x2sin⁡(ωt)x5vo2x3cos⁡(ωt)−2x4sin⁡(ωt)x6\begin{aligned} i_L 2x_1 \cos (\omega t) - 2x_2 \sin (\omega t) x_5\ v_o 2x_3 \cos (\omega t) - 2x_4 \sin (\omega t) x_6\ \end{aligned} iL​vo​​2x1​cos(ωt)−2x2​sin(ωt)x5​2x3​cos(ωt)−2x4​sin(ωt)x6​​ namely: iL2⟨iL⟩1Rcos⁡(ωt)−2⟨iL⟩1Isin⁡(ωt)⟨iL⟩0vo2⟨vo⟩1Rcos⁡(ωt)−2⟨vo⟩1Isin⁡(ωt)⟨vo⟩0\begin{aligned} i_L 2\langle i_L \rangle_1^R \cos (\omega t) - 2\langle i _L\rangle_1^I \sin (\omega t) \langle i_L\rangle_0\ v_o 2\langle v_o \rangle_1^R \cos (\omega t) - 2\langle v_o \rangle_1^I \sin (\omega t) \langle v_o \rangle0\ \end{aligned} iL​vo​​2⟨iL​⟩1R​cos(ωt)−2⟨iL​⟩1I​sin(ωt)⟨iL​⟩0​2⟨vo​⟩1R​cos(ωt)−2⟨vo​⟩1I​sin(ωt)⟨vo​⟩0​​ 另外普通状态平均的解其实是 iLSSAx5,voSSAx6i{L_{SSA}} x5, v{o_{SSA}} x_6 iLSSA​​x5​,voSSA​​x6​ 仿真研究 MATLAB Code: %————————————— % This is the simulation from
% 两种状态平均法在功率变换器建模的应用比较 % % hu 2023-03-03 %—————————————clc,clear,close allVin 20; R 10; L 1e-3; C 1e-6; f 40e3; w 2 * pi * f; T 1 / f; sim my_power_BuckConverter D .5; q1R 1 / 2 / pi * sin(2 * pi * D); q1I 1 / 2 / pi * (cos(2 * pi * D) - 1); q0 D;A [0 w -1/L 0 0 0-w 0 0 -1/L 0 01/C 0 -1/(R*C) w 0 00 1/C -w -1/(R*C) 0 00 0 0 0 0 -1/L0 0 0 0 1/C -1/(R*C)]; B [q1R / L, q1I / L, 0, 0, q0 / L, 0]; x zeros(6,1); u Vin;ts 0; h T/500; tf 1e-3;iLGSSAout []; voGSSAout []; iLSSAout []; voSSAout [];for tts:h:tfxdot A * x B * u;x x xdot * h;iLGSSA 2 * x(1) * cos(w * t) - 2 * x(2) * sin(w * t) x(5);voGSSA 2 * x(3) * cos(w * t) - 2 * x(4) * sin(w * t) x(6);iLSSA x(5);voSSA x(6);iLGSSAout [iLGSSAout;iLGSSA];voGSSAout [voGSSAout;voGSSA];iLSSAout [iLSSAout;iLSSA];voSSAout [voSSAout;voSSA]; endt ts:h:tf; pos mypos(8); i 1; linewidth 1.5; fontsize 12; tSIM iLSIMout.Time; iLGSSAout_interp interp1(t,iLGSSAout,tSIM); iLSSAout_interp interp1(t,iLSSAout,tSIM); voGSSAout_interp interp1(t,voGSSAout,tSIM); voSSAout_interp interp1(t,voSSAout,tSIM);figure plot(tSIM,iLGSSAout_interp,tSIM,iLSSAoutinterp,tSIM,iLSIMout.Data,linewidth,linewidth); h legend($i{L{GSSA}}\(,\)i{L{SSA}}\(,\)i{L_{SIM}}\(); h.Interpreter latex; h.FontSize fontsize; h.Location southeast; h.Orientation horizon; set(gcf,position,pos{i}); i i 1; grid onfigure plot(tSIM,voGSSAout_interp,tSIM,voSSAout_interp,tSIM,voSIMout.Data,linewidth,linewidth); h legend(\)v{o{GSSA}}\(, \)v{o{SSA}}\(, \)v{o{SIM}}$); h.Interpreter latex; h.FontSize fontsize; h.Location southeast; h.Orientation horizon; set(gcf,position,pos{i}); i i 1; grid onmypos function: function pos mypos(i,figs1,figs2) % mypos.m 给定 figure 对象个数求解合适的摆放位置向量以防止图片堆叠 % i figure 个数 % figs1,figs2 figure 对象的长和高 % pos mypos(i,figs1,figs2) 求出 i 个 figure 对象的合理摆放位置且大小设置为[figs1,figs2] % 输出 pos 是元胞数组使用规范已生成figure对象后set(gcf,position,pos{i}) % Remark 更方便的绘图程序见 myplot.m% hu 2018-6-11 % hu 2018-8-8 Modified Remark is added % hu 2018-11-3 Modified Description is updatedif nargin ~ 3figs [400,300]; %default size is 560*420, 500*280 is suitable for paper shows elsefigs [figs1,figs2]; end if i 8disp(too many figures! The maximum number is 8) end scr get(0,screensize); for k 1:iif k 4pos{k} [scr(1) (k - 1) * figs(1),scr(2) scr(4) / 2,figs];endif k 4pos{k} [scr(1) (k - 5) * scr(3) / 4,scr(2) 30,figs];end end end真实数据用 SIMULINK 2018a 得到PWM 频率 f40kHzf 40kHzf40kHz可以看到 GSSA 基本和 SIMULINK 数据重合SSA 仅代表了其直流分量 仿真文件: 链接: https://pan.baidu.com/s/1ftQQ68H0nHVZ3LQkPNEObw?pwdmgf4 提取码: mgf4