﻿ 基于<i>dq</i>坐标变换的单相逆变器控制技术
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 应用科技  2017, Vol. 44 Issue (3): 54-60  DOI: 10.11991/yykj.201606020 0

### 引用本文

YOU Jiang, LIN Xi, DONG Hao. Research on control strategy of single-phase inverter basedon dq coordinate transform[J]. Applied Science and Technology, 2017, 44(3), 54-60. DOI: 10.11991/yykj.201606020.

### 文章历史

Research on control strategy of single-phase inverter basedon dq coordinate transform
YOU Jiang, LIN Xi, DONG Hao
College of Automation, Harbin Engineering University, Harbin 150001, China
Abstract: Single-phase inverter given signal and feedback signal are sinusoidal, proportional-integral regulator can't be achieved with zero steady-state error and the periodic interference signal can't be achieved with indifferent disturbance in stationary coordinate system, which affects the output steady-state accuracy problems.Single-phase coordinate transformation is adopted to transform AC signals in stationary reference frame to straight flow in a rotating coordinate system.Synchronous reference frame proportional-integral (SRFPI) based outer voltage control and inner inductor current control in stationary reference frame are used in inverter control system.And voltage feedback decoupling is introduced to simplify control system design and improve system robustness.Eventually, Simulink simulation verifies the feasibility of the above theory.
Key words: single-phase inverter    proportional-integral    stationary coordinate system    rotating coordinate system    voltage feedback    decoupling    coordinate transform

1 单相逆变器的参数设计与建模

 $R = \frac{{\bar U_o^2}}{P} = \frac{{{{\left( {311/\sqrt 2 } \right)}^2}}}{{1000}} \approx 50\Omega$ (1)

 $f \ll {f_0} \ll {f_k}$ (2)

 $\left\{ \begin{array}{l} \sqrt {{L_0}{C_0}} = \frac{1}{{2\pi {f_0}}}\\ \sqrt {{L_0}/{C_0}} = \left( {0.5 \sim 0.8} \right)R \end{array} \right.$ (3)

 $\left\{ \begin{array}{l} \Delta {I_{{L_{\max }}}} = \frac{E}{{2L{f_s}}} = 5.25\;{\rm{A}}\\ {f_s} = 1/2\pi \sqrt {{L_0}{C_0}} = 459\;{\rm{Hz}} \end{array} \right.$ (4)

 $G\left( s \right) = \frac{1}{{LC{s^2} + \left( {\frac{L}{R} + rC} \right)s + 1 + \frac{r}{R}}}$ (5)
 图 1 LC滤波器伯德图

1) 电感上的基波电压将使负载上的基波电压产生变化；输出电容的基波电流与负载电流之和将改变逆变器输出电流。

2) 在逆变器带纯电阻负载时，电容中的基波电流将使逆变器桥臂输出电流增大；感性负载时由于电容电流与负载电流方向相反，会使逆变器桥臂输出电流减小[10]

 图 2 单相逆变器拓扑结构图
 $\left\{ \begin{array}{l} L\frac{{{\rm{d}}{i_L}}}{{{\rm{d}}t}} = {K_{{\rm{PWM}}}}{U_{{\rm{DC}}}}-{u_o}-r{i_L}\\ C\frac{{{\rm{d}}{u_o}}}{{{\rm{d}}t}} = {i_C} = {i_L}-{i_o} \end{array} \right.$ (6)

 图 3 单相逆变器系统控制框图

2 电流环控制策略

 图 4 单相逆变器解耦系统控制框图

 图 5 单相逆变器解耦后系统控制框图

 ${G_{zc}} = \frac{{{K_{{\rm{PWM}}}}}}{{Ls + r}}$ (7)

 图 6 电流环控制器伯德图
3 不同坐标系下电压环PI控制策略 3.1 静止坐标系下的电压环控制策略

 ${G_{{\rm{PI}}}} = {K_{\rm{P}}} + \frac{{{K_{\rm{I}}}}}{s}$ (8)

 $\left| {{G_{{\rm{PI}}}}\left( {{\rm{j}}{\omega _f}} \right)} \right| = \sqrt {K_{\rm{P}}^2 + {{\left( {\frac{{{K_{\rm{I}}}}}{{{\omega _f}}}} \right)}^2}}$ (9)

PI环节在s域的表达式可以简化为

 ${G_{{\rm{PI}}}} = {K_{\rm{P}}}\left( {1 + \frac{1}{{{T_{\rm{I}}}s}}} \right)$ (10)

 $\theta = \arctan \left( {\frac{1}{{{T_{\rm{I}}}\omega }}} \right)$ (11)

 图 7 PI控制器伯德图

 图 8 静止坐标系下电压环控制器伯德图

3.2 同步旋转坐标系下的电压环控制策略

 图 9 2种不同移相方式框图

 图 10 虚拟坐标轴传递函数频率特性

 $\frac{{{U_\beta }\left( s \right)}}{{{U_\alpha }\left( s \right)}} = \frac{{\omega _f^2}}{{{s^2} + {\omega _f}s + \omega _f^2}}$ (12)

 图 11 SRFPI控制结构框图

 $\left[\begin{array}{l} {U_d}\\ {U_q} \end{array} \right] = \left[\begin{array}{l} \cos \;\theta \;\;\;\;\;\sin \;\theta \\ -\sin \;\theta \;\;\;\cos \;\theta \end{array} \right]\left[\begin{array}{l} {U_\alpha }\\ {U_\beta } \end{array} \right]$ (13)
 $\begin{array}{l} \frac{{\rm{d}}}{{{{\rm{d}}_t}}}\left[\begin{array}{l} {U_d}\\ {U_q} \end{array} \right] = \frac{1}{C}\left[\begin{array}{l} {I_d}\\ {I_q} \end{array} \right] - \frac{1}{{RC}}\left[\begin{array}{l} {U_d}\\ {U_q} \end{array} \right] + \left[\begin{array}{l} 0\;\;\;\;\;\omega \\ -\omega \;\;0 \end{array} \right]\left[\begin{array}{l} {U_d}\\ {U_q} \end{array} \right]\\ \frac{{\rm{d}}}{{{{\rm{d}}_t}}}\left[\begin{array}{l} {I_d}\\ {I_q} \end{array} \right] = \frac{{{U_{{C_f}}}}}{L}\left[\begin{array}{l} {D_d}\\ {D_q} \end{array} \right] - \frac{1}{L}\left[\begin{array}{l} {U_d}\\ {U_q} \end{array} \right] + \left[\begin{array}{l} 0\;\;\;\;\;\omega \\ -\omega \;\;0 \end{array} \right]\left[\begin{array}{l} {U_d}\\ {U_q} \end{array} \right] \end{array}$ (14)

 图 12 单相逆变器dq解耦控制框图

 图 13 单相逆变器双闭环dq电压反馈解耦控制框图

 $\begin{array}{l} \left[\begin{array}{l} i_\alpha ^*\left( t \right)\\ i_\beta ^*\left( t \right) \end{array} \right] = \left[\begin{array}{l} \cos \left( {\omega t} \right)\;\;\;\;-\sin \left( {\omega t} \right)\\ \sin \left( {\omega t} \right)\;\;\;\;\;\;\cos \left( {\omega t} \right) \end{array} \right]\\ \left\{ {\left[\begin{array}{l} {G_{{\rm{PI}}}}\left( t \right)\;\;\;\;0\\ \;\;\;0\;\;\;\;\;\;{G_{{\rm{PI}}}}\left( t \right) \end{array} \right]\left\{ {\left[\begin{array}{l} \cos \left( {\omega t} \right)\;\;\;\sin \left( {\omega t} \right)\\ -\sin \left( {\omega t} \right)\;\;\;\cos \left( {\omega t} \right) \end{array} \right]\left[\begin{array}{l} {U_\alpha }\left( t \right)\\ {U_\beta }\left( t \right) \end{array} \right]} \right\}} \right\} \end{array}$ (15)

 $\left[\begin{array}{l} {i_\alpha }\left( s \right)\\ {i_\beta }\left( s \right) \end{array} \right] = \frac{1}{2}\left[\begin{array}{l} {G_{{\rm{PI}}}}\left( {s + {\rm{j}}\omega } \right)-{\rm{j}}{G_{{\rm{PI}}}}\left( {s + {\rm{j}}\omega } \right)\\ + {G_{{\rm{PI}}}}\left( {s-{\rm{j}}\omega } \right) + {\rm{j}}{G_{{\rm{PI}}}}\left( {s-{\rm{j}}\omega } \right)\\ + {\rm{j}}{G_{{\rm{PI}}}}\left( {s + {\rm{j}}\omega } \right){G_{{\rm{PI}}}}\left( {s + {\rm{j}}\omega } \right)\\ - {\rm{j}}{G_{{\rm{PI}}}}\left( {s - {\rm{j}}\omega } \right) + {G_{{\rm{PI}}}}\left( {s - {\rm{j}}\omega } \right) \end{array} \right]\left[\begin{array}{l} {U_\alpha }\left( s \right)\\ {U_\beta }\left( s \right) \end{array} \right]$ (16)

GPI(s)=KP+KI/s代入式(16) 化简得到

 $\left[\begin{array}{l} i_\alpha ^*\left( s \right)\\ i_\beta ^*\left( s \right) \end{array} \right] = \left[\begin{array}{l} {K_{\rm{P}}} + \frac{{{K_{\rm{I}}}s}}{{{s^2} + \omega _0^2}}-\frac{{{K_{\rm{I}}}{\omega _0}}}{{{s^2} + \omega _0^2}}\\ \;\;\;\;\frac{{{K_{\rm{I}}}{\omega _0}}}{{{s^2} + \omega _0^2}}\;\;\;{K_{\rm{P}}} + \frac{{{K_{\rm{I}}}s}}{{{s^2} + \omega _0^2}} \end{array} \right]\left[\begin{array}{l} {u_\alpha }\left( s \right)\\ {u_\beta }\left( s \right) \end{array} \right]$ (17)

 ${G_v} = \frac{{{a_4}{s^4} + {a_3}{s^3} + {a_2}{s^2} + {a_1}s + {a_0}}}{{{s^4} + {\omega _f}{s^3} + 2\omega _f^2{s^2} + \omega _f^3s + \omega _f^4}}$ (18)

 图 14 转换函数Gv伯德图

 图 15 电压环被控对象
 ${G_{zv}} = \frac{{{K_{{\rm{PWM}}}}{G_c}{G_1}{G_2}}}{{1 + {K_{{\rm{PWM}}}}{G_c}{G_1} + {G_2}/R}}$ (19)

 图 16 电压环控制器伯德图
4 仿真及实验结果分析

 图 17 SRFPI控制双环仿真
 图 18 2种情况的THD分析

 图 19 突加负载两种情况的输出电压波形
5 结论

1) 本文利用电压反馈解耦简化了控制系统的设计，提高了系统的鲁棒性。

2) 提出电压环在dq坐标系下的PI控制电流环在静止坐标系下的比例控制的双闭环控制策略。通过计算将旋转坐标系下的PI控制可以转换为静止坐标系下的比例谐振控制，在基波频率处获得较大的幅值增益，对正弦波进行更好的跟踪的能力。