一、陷波器在连续域的传递函数
1、最基本的陷波器传函
(1)
其中,wo是所谓“中心频率”,也就是你想要“陷掉”的频率。而 ζ 则是“陷阱”的宽度。
根据公式可以发现,当输入信号频率很小(s=0)或者很大( s=+∞)的时候,上面式子的值是1;当输入信号频率刚好等于 s=jωo的时候,分子是0,所以增益变成0,那这个频率的信号当然就全都被衰减掉了。
由上图可见,ζ越大,则弦波带宽越宽,但弦波频率处的衰减越小。
2、三参数陷波器传函
(2)
其中,ωo是陷波频率(即凹陷的中心频率),ζ1和ζ2是陷波系数
陷波滤波器重点关注的参数一般有三个:
(1)陷波频率(ωo rad/s可转换Hz)
(2)陷波深度(depth为衰减倍数)
例如对于100Hz频率处的衰减深度是100,那经过该滤波器后,幅值衰减100倍。
(3)陷波宽度(△f单位Hz)
即中心频率两侧,幅值衰减-3dB时,对应的两个频率的差值。
ζ1和ζ2与陷波深度depth和陷波宽度△f(Hz)的关系表示如下:
二、陷波器传函在MATLAB中的表达及其离散化
1、以最基本的陷波器传函为例
a、MATLAB中编写如下m文件:
syms w0 s Ts z zeta % 定义符号变量
G1 =(s^2+w0^2)/(s^2+2*w0*zeta*s+w0^2) %传递函数
sys_s2c = 2*(z-1)/Ts/(z+1);
G2 = subs(G1,s,sys_s2c) %离散化 tustin变换
G3 = collect(G2,z) % 将表达式G2中的以z为变量合并相同次幂;
得到连续域和Z域的传递函数如下:
G1 =(s^2 + w0^2)/(s^2 + 2*zeta*s*w0 + w0^2)
G2 =(w0^2 + (2*z - 2)^2/(Ts^2*(z + 1)^2))/(w0^2 + (2*z - 2)^2/(Ts^2*(z + 1)^2) + (2*w0*zeta*(2*z - 2))/(Ts*(z + 1)))
G3 =((Ts^2*w0^2 + 4)*z^2 + (2*Ts^2*w0^2 - 8)*z + Ts^2*w0^2 + 4)/((Ts^2*w0^2 + 4*zeta*Ts*w0 + 4)*z^2 + (2*Ts^2*w0^2 - 8)*z + Ts^2*w0^2 - 4*zeta*Ts*w0 + 4)
根据离散化的方法:
分子分母同时除以A0,得到差分方程系数
差分方程为:
y(k) = b0*x(k) + b1*x(k-1) +b2*x(k-2) - a1*y(k-1) - a2*y(k-2);
则
B0 = Ts^2*w0^2 + 4;
B1 = 2*Ts^2*w0^2 - 8;
B2 = B0; (3)
A0 = Ts^2*w0^2 + 4*zeta*Ts*w0 + 4;
A1 = B1;
A2 = Ts^2*w0^2 - 4*zeta*Ts*w0 + 4;
可得差分方程系数
a0 = 1
b0 = B0/A0
b1 = B1/A0 (4)
b2 = B2/A0
a1 = A1/A0
a2 = A2/A0
b、代入实际参数,求得差分方程系数
f0 = 100; %目标频率
w0 = 2*pi*f0;
Ts = 10e-6; %离散化周期
zeta = 0.5; %带宽
% Control object funciton
gxnum2 = [1 0 w0^2]; % 传函分子
gxden3 = [1 2*w0*zeta w0^2]; % 传函分母
sys2 = tf(gxnum2, gxden3) % 传函
zpk(sys2, 's'); % 将传函用零极点格式表示
dsys2 = c2d(sys2, Ts, 'tustin') % s到z,即,连续到离散域的变换
zpk(dsys2, 'z'); % 将传函用零极点格式表示
[num2, den2] = tfdata(dsys2, 'v'); % 提取传递函数分子、分母的z次幂的系数,并保存到数组
得到传递函数如下:
需要注意的是,在上图Z域传递函数dsys2的各项系数是经四舍五入的,如果直接用这些系数到Simulink或编写程序进行仿真,得到的结果是错误的!!!需要使用更加精确的系数。这些系数可以将式3和式4代入MATLAB中直接求解得到。
b0 = 0.996868276853708 b1 = -1.993697199313698
b2 = 0.996868276853708 a1 = -1.993697199313698
a2 = 0.993736553707416
2、以三参数陷波器为例
三、仿真验证
1、使用经四舍五入的系数仿真
结果错误
2、使用正确的系数
结果正确:
四、C程序验证
1、方式一
直接将差分方程用代码呈现:
double ADValue;
//Notching Filter Coefficient
#define Notching_filter_100Hz_a0 1
#define Notching_filter_100Hz_a1 -1.998704711158930
#define Notching_filter_100Hz_a2 0.998744164397945
#define Notching_filter_100Hz_b0 0.999372082198973
#define Notching_filter_100Hz_b1 -1.998704711158930
#define Notching_filter_100Hz_b2 0.999372082198973
// Control law 2p2z data define
typedef struct IIR_2OR_DATA_TAG{
double coeff_a0;
double coeff_a1;
double coeff_a2;
double coeff_b0;
double coeff_b1;
double coeff_b2;
double filter_out;
double filter_y1;
double filter_y2;
double filter_u1;
double filter_u2;
} IIR_2OR_DATA_DEF;
IIR_2OR_DATA_DEF notching_data = {
Notching_filter_100Hz_a0,
Notching_filter_100Hz_a1,
Notching_filter_100Hz_a2,
Notching_filter_100Hz_b0,
Notching_filter_100Hz_b1,
Notching_filter_100Hz_b2,
0, 0, 0, 0 ,0
};
double iir_2or_func(IIR_2OR_DATA_DEF *filter_date, double target)
{
//
//y(out) = b0*x(k) + b1*x(k-1) +b2*x(k-2) - a1*y(k-1) - a2*y(k-2);
//
filter_date->filter_out = (filter_date->coeff_b0 * (target)) \
+ (filter_date->coeff_b1 * filter_date->filter_u1) \
+ (filter_date->coeff_b2 * filter_date->filter_u2) \
- (filter_date->coeff_a1 * filter_date->filter_y1) \
- (filter_date->coeff_a2 * filter_date->filter_y2);
//
// Update last data
//
filter_date->filter_y2 = filter_date->filter_y1;
filter_date->filter_y1 = filter_date->filter_out;
filter_date->filter_u2 = filter_date->filter_u1;
filter_date->filter_u1 = target;
//
// Return Value
//
return(filter_date->filter_out);
}
得到结果正确:
2、方式二
参考文献3中的方法,增加一个中间变量w,来实现。
double ADValue;
//Notching Filter Coefficient
#define Notching_filter_100Hz_GAIN 1
#define Notching_filter_100Hz_A1 -1.998704711158930
#define Notching_filter_100Hz_A2 0.998744164397945
#define Notching_filter_100Hz_B0 0.999372082198973
#define Notching_filter_100Hz_B1 -1.998704711158930
#define Notching_filter_100Hz_B2 0.999372082198973
// Control law 2p2z data define
typedef struct IIR_2OR_DATA_TAG{
double coeff_GAIN;
double coeff_B0;
double coeff_B1;
double coeff_B2;
double coeff_A1;
double coeff_A2;
double filter_out;
double filter_W0;
double filter_W1;
double filter_W2;
} IIR_2OR_DATA_DEF;
IIR_2OR_DATA_DEF notching_data = {
Notching_filter_100Hz_GAIN,
Notching_filter_100Hz_B0,
Notching_filter_100Hz_B1,
Notching_filter_100Hz_B2,
Notching_filter_100Hz_A1,
Notching_filter_100Hz_A2,
0, 0, 0, 0
};
double iir_2or_func(IIR_2OR_DATA_DEF *filter_date, double target)
{
//
// w0 = x(0) - A1 * W1 - A2 * W2
//
filter_date->filter_W0 = (target) - filter_date->coeff_A1 * filter_date->filter_W1 \
- filter_date->coeff_A2 * filter_date->filter_W2;
//
// Y(0) = Gain * (B0 * W0 + B1 * W1 + B2 * W2)
//
filter_date->filter_out = filter_date->coeff_GAIN * ( filter_date->coeff_B0 * filter_date->filter_W0 \
+ filter_date->coeff_B1 * filter_date->filter_W1 \
+ filter_date->coeff_B2 * filter_date->filter_W2 );
//
// Update last data
//
filter_date->filter_W2 = filter_date->filter_W1;
filter_date->filter_W1 = filter_date->filter_W0;
//
// Return Value
//
return(filter_date->filter_out);
}
仿真结果:
与方法一完全一样。
参考文献:
1、Simulink 窄带陷波滤波器(Notch filter)仿真到代码生成
2、温故知新(五)——三参数陷波滤波器离散化推导及MATLAB实现
3、陷波器及其算法(基于C语言)文章来源:https://www.toymoban.com/news/detail-648234.html
PLECS仿真:notch_filter.plecs文章来源地址https://www.toymoban.com/news/detail-648234.html
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