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1、TOPIC 4,Filters Design,Basic Filters1,3dB (Cutoff ) Frequency : fc (Hz) Maximum Passband Attenuation : 3dB Passband Ripple : Rp (dB) Stopband Frequency : fx (Hz) Minimum Stopband Attenuation : Ax,Lowpass,Highpass,Passband: 0 fc (Hz) Stopband: fx (Hz),Passband: fc (Hz) Stopband: 0 fx (Hz),Basic of Fi
2、lters,Bandstop,Bandpass,Technical Parameters of Filter,IL: RF insertion loss,Q = f0 / BW,Rp: Ripple in the passband,BW: Difference between upper and lower freqencies at which the attenuation is 3 dB,SF: Describing the sharpness of the response with the ratio between the Ax dB and the 3 dB bandwiths,
3、Rejection: it is parameter according to the specification of a filter,Qulity factor Q: Another parameter describing filter selectivity,微波网络综合法设计滤波器,微波网络综合法设计滤波器时，将整个滤波器看成是多级二端口网络的级联，实际中这些二端口网络是串连电感并联电容。,一般先设计低通原型滤波器，实际的低通高通带通带阻滤波器可由低通原型变换得到。,微波网络综合法设计滤波器,由转移参量可以得到整个滤波器的频率响应特性。 S21= 2 / ( a + b + c +
4、 d ) 或 L 10 log 1 / |S21|2 10 log |( a+b+c+d )/2|2,使频率响应满足指定的响应特性得到串连电感并联电容的大小。,典型滤波器响应,实际的滤波器响应有以下几种：,最大平坦响应(Butterwoth响应) 等波纹响应(Chebyshev响应) 椭圆函数响应 线性相位响应,典型滤波器响应,最大平坦响应(butterwoth响应) L 1 k2 ( /c )2N,式中N是滤波器阶数， c是截止频率，通带为(0， c )，通带边缘损耗为 1 k2，常选为-3 dB，故 k1。 带外衰减随频率增加而单调增加， c 时, L ( /c )2N, 所以衰减以每10
5、倍频 20N dB的速率上升。,典型滤波器响应,等波纹响应(Chebyshev响应) L 1 k2 TN( /c ) 2,因为xc 时, 由TN(x)函数性质得到 L k2/4 ( 2 /c )2N, 所以衰减也以每10倍频 20N dB的速率上升。但其衰减比最平坦响应大 22N/4,式中TN(x)是Chebyshev函数，其多项式表示为 T1(x) x T2(x) 2x21 T3(x) 4x33x T4(x) 8x4 8x2 1 ,Chebyshev Low-Pass Filters Response,Comparison of Frequency response between Butt
6、erworht and Chebyshev Filters,where B(3, ): attennuation response of 3-order butterworth-type T( 0.25, 3, ) ): attennuation response of 3-order chebyshev-type with ripple of 0.25dB T( 0.5, 5, ) ): attennuation response of 5-order chebyshev-type with ripple of 0.5dB T(1, 7, ) ): attennuation response
7、 of 7-order chebyshev-type with ripple of 1dB,Comparison between Butterworht and Chebyshev Filters,典型滤波器响应,椭圆滤波器（elliptic filter）是利用椭圆函数（elliptic function）的双周期函数性质设计的。,就低通滤波器而言，如将巴特沃思滤波器与切比雪夫滤波器的幅频特性加以比较，它们具有以下特点： 在巴特沃思滤波器中，无论是通带还是阻带均表现为单调衰减，并且不产生波纹； 在切比雪夫滤波器中，通带内产生波纹，但阻带则为单调衰减； 切比雪夫滤波器的截止特性比巴特沃思滤波器
8、更为陡峭。,因而可以这样设想，如果在通带和阻带两方面都允许波纹存在，就能得到截止特性比切比雪夫滤波器更为陡峭的滤波器。基于这种思路的滤波器，就是由W.Cauer提出的椭圆滤波器。,典型滤波器响应,其中， 为 的分式有理多项式，其零点全部在通带 1内，具有如下形式,其中 为零衰减频率， 为无穷衰减频率，零衰减频率的个数与 无穷衰减频率的个数相等。,这种衰减特性与契比雪夫滤波器衰减特性相比，有如下特点： （1）通带内仍有契比雪夫滤波器响应的等波纹特性； （2）阻带内增加了有限频率上的极点，也呈现等波纹特性；（3）过渡段区域的斜率更为陡峭。,椭圆函数滤波器的衰减特性为：,椭圆函数滤波器响应,典型滤波
9、器响应,线性相位响应 () A 1 p ( /c )2N,式中() 滤波器电压转移函数的相位，p为常数。,通常良好的截止响应特性与良好的相位响应是一对矛盾。,还可以有其他的响应，上述4种是最常用的。,低通原型滤波器器件参数的确定,低通原型滤波器器件参数的确定是一个道理简单计算复杂的过程。在低通原型滤波器中，一般取g01， c1。,由微波网络级联可得此电路的响应为 L=1+(1-R)2+(C2R2+ L2- 2LCR2)2 +L2C2R24/4R,最平坦响应为 L=1+ k24 k=1 =1时衰减3dB 得到 R=1, L = C = 21/2,等波纹响应为 L=1+ k2(2 21)2 k=1
10、 波纹3dB 得到 R=5.81, L=3.1 C = 0.53,对于N2的低通原型，其结构图如右图所示：,低通原型滤波器器件参数的确定,一般低通原型滤波器的两种结构如下图所示。,图中器件的编号从信号源端的g0一直到负载端的gN+1. 两个电路同一编号的器件取值相同，给出同样的频响。因此它们互为对偶电路。,低通原型滤波器器件参数的确定,原则上，可求任意N阶低通原型滤波器的器件参数值。但工程应用时，N过大不实际。对于最平坦响应的低通原型滤波器。前人将至10阶滤波器的参数值列表如下：,低通原型滤波器器件参数的确定,最平坦响应的低通原型滤波器至15阶时的衰减曲线如下：,低通原型滤波器器件参数的确定,
11、对于等波纹响应的低通原型滤波器，至10阶的滤波器参数值列表如下(带内波纹0.01dB)：,低通原型滤波器器件参数的确定,等波纹响应的低通原型滤波器至15阶时的衰减曲线如下：,低通原型滤波器器件参数的确定,对于线性相位响应低通原型滤波器，因为转移参量的相位不像幅度那样有较简单的表达式，器件参数求解更复杂。至10阶的滤波器参数值列表如下：,低通原型滤波器器件参数的确定,最大平坦响应和等波纹响应低通原型滤波器经常用到。有时通过查衰减曲线及查表得不到相应的阶数及器件参数值，这时可依据滤波器相关指标，由公式计算得到N及gn,Impedance: Zo (ohm) Cutoff Frequency: fc
12、 (Hz) Stopband Frequency: fx (Hz) Maximum Attenuation at cutoff frequency: Ap (dB) Minimum Attenuation at stopband frequency：Ax(dB),Butterworth LowPass Filters1,Step 2: Determine the Number of elements，N is a integer,Step 3: Calculate Prototype Element Values，gK。,Step1： Specification,Chebyshev LowPa
13、ss Filters2,Impedance: Zo (ohm) Cutoff Frequency: fc (Hz) Stopband Frequency: fx (Hz) Maximum Attenuation at cutoff frequency: Ap (dB) Minimum Attenuation at stopband frequency：Ax(dB),Step 2: Determine the Number of elements，N is an odd integer that is to avoid differrence between the input and outp
14、ut impedance,Step1： Specification,Step 3: Calculate Prototype Element Values，gK。,gN+1=1 N奇数,gN+1=coth2(/4) N偶数,椭圆函数滤波器低通原型,由滤波器的设计指标LAs(dB), 和LAr(dB),得到上述原型电路的系数，需要用雅可比椭圆函数的保角变换技术，其数学推导和计算都比较繁琐。现已有图标曲线，可供设计此类滤波器时查用。,两种椭圆函数低通滤波器原型电路,下表给出了N=5 带内波纹衰减Lar=0.1的椭圆函数低通滤波器的系数,椭圆函数滤波器技术参数,LAs：阻带抑制,LAr：通带波纹,：通
15、带截止频率,：阻带抑制频率,Frequency transformations from normalized LPF to others,Lowpass lowpass highpass bandpass bandstop Prototype pratical pratical pratical pratical Value value value value value,Examples of LPF design,Impedance: Zo (ohm)=50 Cutoff Frequency: fc (MHz)=75 Stopband Frequency: fx (MHz)=100 Ma
16、ximum Attenuation at cutoff frequency: 3 (dB) Minimum Attenuation at stopband frequency：20(dB),Step 2: Determine the Number of elements,Step1： Specification,Step 3: Calculate Prototype Element Values，gK。,Design a LC 1 dB ripple Chebyshev-type LPF(Zo=50 ohm) with 75MHz cutoff frequency and at least 2
17、0dB attenuation at 100MHz,Solution:,N=5,Step 4：Select shunt capacitance series inductance,Result,Impedance: Zo (ohm) upper passband edge frequency: fPU (Hz) lower passband edge frequency: fPL (Hz) upper stopband edge frequency: fXU (Hz) lower stopband edge frequency: fXL (Hz) Maximum Attenuation at
18、passband: Ap (dB) Minimum Attenuation at stopband：Ax(dB),Step 2: Determine the Number of elements，N is an odd integer that is to avoid differrence between the input and output impedance,Step1： Specification,Design of BandPass Filters,（1）For Butterworth Type,（2）For Chebyshev Type,Design of BandPass F
19、ilters2,Step 3: Calculate Prototype Element Values，gK, as before. Select series induct-ance shunt capacitance or shunt capacitance series inductance, then calculate the values of C and L 。,a) series inductance shunt capacitance,b) shunt capacitance series inductance,Step 4: Calculate the component v
20、alues of bpf。Transformate the lowpass prototype element values to the bandpass ones according the right transformation table,Example of BPF design,Design a 0.1 dB ripple Chebyshev-type BPF(Zo=50 ohm) with bandpass of 10MHz and central frequency at 75MHz, the Minimum Attenuation at stopband has to be
21、 30dB with 30MHz stopband,Step1： Specification Impedance）: Zo = 50 ohm upper passband edge frequency: fPU = 75 + 5 = 80 MHz lower passband edge frequency: fPL = 75 5 = 70 MHz upper stopband edge frequency: fXU = 75 + 15 = 90 MHz lower stopband edge frequency : fXL = 75 15 = 60 MHz Maximum Attenuatio
22、n at passband: rp = 0.1 dB Minimum Attenuation at stopband：Ax = 30dB,Step 2： determine the order of elements，N=3,Result,Step 3: Calculate Prototype Element Values，gK. Select shunt capacitance series inductance type. Calculate the values of L and C,Step 4: Calculate the component values of bpf accord
23、ing the transformation table。,Home work,1) Design a 0.5 dB ripple Chebyshev-type LPF(Zo=50 ohm) with bandpass of 10MHz and central frequency at 75MHz, the Minimum Attenuation at stopband has to be 20dB with 30MHz stopband, design a Butterworth-type LPF with the same specification and do comparison between them,2) Design a LC 0.1 dB ripple elliptic function LPF(Zo=50 ohm) with 75MHz cutoff frequency and at least 35dB attenuation at 98MHz. and calculate its frequency responding curve by using ABCD matrix,