topic 4 filters design. basic filters 1 3db (cutoff ) frequency : fc (hz) maximum passband...
TRANSCRIPT
![Page 1: TOPIC 4 Filters Design. Basic Filters 1 3dB (Cutoff ) Frequency : fc (Hz) Maximum Passband Attenuation : 3dB Passband Ripple : Rp (dB) Stopband Frequency](https://reader033.vdocument.in/reader033/viewer/2022061507/5697c0281a28abf838cd7074/html5/thumbnails/1.jpg)
TOPIC 4
Filters Design
![Page 2: TOPIC 4 Filters Design. Basic Filters 1 3dB (Cutoff ) Frequency : fc (Hz) Maximum Passband Attenuation : 3dB Passband Ripple : Rp (dB) Stopband Frequency](https://reader033.vdocument.in/reader033/viewer/2022061507/5697c0281a28abf838cd7074/html5/thumbnails/2.jpg)
Basic Filters1
3dB (Cutoff ) Frequency : fc (Hz)Maximum Passband Attenuation : 3dBPassband Ripple : Rp (dB)Stopband Frequency : fx (Hz)Minimum Stopband Attenuation : Ax
0dB-3dB
-Ax dB
fc fx
Rp dB
0dB
-3dB
-Ax dB
fcfx
Rp dB
Lowpass
Highpass
Passband: 0 — fc (Hz)Stopband: fx—∞ (Hz)
Passband: fc—∞ (Hz)Stopband: 0 — fx (Hz)
![Page 3: TOPIC 4 Filters Design. Basic Filters 1 3dB (Cutoff ) Frequency : fc (Hz) Maximum Passband Attenuation : 3dB Passband Ripple : Rp (dB) Stopband Frequency](https://reader033.vdocument.in/reader033/viewer/2022061507/5697c0281a28abf838cd7074/html5/thumbnails/3.jpg)
Basic of Filters0dB-3dB
-Ax dB
fo fUx
Rp dB
fLx fLp fHp
Lower passband edge = fLp
Upper passband edge = fHp
Lower stopband edge = fLx
Upper stopband edge = fUx
Passband Bandwidth = fHp - fLp
Passband Ripple = Rp dBMaximun Passband Attenuation = 3dBMinimum Stopband Attenuation = AxCenter Frequency = fo = fHp fLp
0dB
-3dB
-Ax dB
fo fUx
Rp dB
fLxfLp fHp
Bandstop
Bandpass
Lower passband edge = fLp
Upper passband edge = fHp
Lower stopband edge = fLx
Upper stopband edge = fUx
Stopband Bandwidth = fUx - fLx
Passband Ripple = Rp dBMaximun Passband Attenuation = 3dBMinimum Stopband Attenuation = AxCenter Frequency = fo = fHp fLp
![Page 4: TOPIC 4 Filters Design. Basic Filters 1 3dB (Cutoff ) Frequency : fc (Hz) Maximum Passband Attenuation : 3dB Passband Ripple : Rp (dB) Stopband Frequency](https://reader033.vdocument.in/reader033/viewer/2022061507/5697c0281a28abf838cd7074/html5/thumbnails/4.jpg)
Technical Parameters of FilterIL: RF insertion loss
0dB
-3dB
-Ax dB
fo fUx
Rp dB
fLx fLp fHp
IL dB
BW
Rejectio
n
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
Rejection: it is parameter according to the specification of a filter
Qulity factor Q: Another parameter describing filter selectivity
![Page 5: TOPIC 4 Filters Design. Basic Filters 1 3dB (Cutoff ) Frequency : fc (Hz) Maximum Passband Attenuation : 3dB Passband Ripple : Rp (dB) Stopband Frequency](https://reader033.vdocument.in/reader033/viewer/2022061507/5697c0281a28abf838cd7074/html5/thumbnails/5.jpg)
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。
![Page 6: TOPIC 4 Filters Design. Basic Filters 1 3dB (Cutoff ) Frequency : fc (Hz) Maximum Passband Attenuation : 3dB Passband Ripple : Rp (dB) Stopband Frequency](https://reader033.vdocument.in/reader033/viewer/2022061507/5697c0281a28abf838cd7074/html5/thumbnails/6.jpg)
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 + d ) 或 L = 10 log 1 / |S21|2 = 10 log |( a+b+c+d )/2|2
1 01 1 1 0
1/ 10 1 0 1 1
11
11
G
L
G G LL
L
A B R R
RC D j C
R R j C R RR
j CR
• 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。
![Page 7: TOPIC 4 Filters Design. Basic Filters 1 3dB (Cutoff ) Frequency : fc (Hz) Maximum Passband Attenuation : 3dB Passband Ripple : Rp (dB) Stopband Frequency](https://reader033.vdocument.in/reader033/viewer/2022061507/5697c0281a28abf838cd7074/html5/thumbnails/7.jpg)
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
![Page 8: TOPIC 4 Filters Design. Basic Filters 1 3dB (Cutoff ) Frequency : fc (Hz) Maximum Passband Attenuation : 3dB Passband Ripple : Rp (dB) Stopband Frequency](https://reader033.vdocument.in/reader033/viewer/2022061507/5697c0281a28abf838cd7074/html5/thumbnails/8.jpg)
典型滤波器响应最大平坦响应 (butterwoth 响应 )
L = 1 + k2 ( ω /ωc )2N
式中 N 是滤波器阶数, ωc是截止频率,通带为 (0 , ωc ) ,通带边缘损耗为 1 + k2,常选为 -3 dB ,故 k = 1 。 带外衰减随频率增加而单调增加, ω>>ωc 时 , L ≈ ( ω /ωc )2N, 所以衰减以每 10 倍频 20N dB 的速率上升。
![Page 9: TOPIC 4 Filters Design. Basic Filters 1 3dB (Cutoff ) Frequency : fc (Hz) Maximum Passband Attenuation : 3dB Passband Ripple : Rp (dB) Stopband Frequency](https://reader033.vdocument.in/reader033/viewer/2022061507/5697c0281a28abf838cd7074/html5/thumbnails/9.jpg)
典型滤波器响应等波纹响应 (Chebyshev 响应 )
L = 1 + k2 [ TN( ω /ωc ) ]2
因为 x<1 时 , |TN(x)|<1 故通带内波纹为 1 + k2,常选为 -3 dB ,故 k = 1 。 带外衰减随频率增加而单调增加, ω>>ωc 时 , 由 TN(x) 函数性质得到 L ≈ k2/4 ( 2ω /ωc )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) = 2x2 - 1 T3(x) = 4x3 - 3x T4(x) = 8x4 - 8x2 + 1 • • •
![Page 10: TOPIC 4 Filters Design. Basic Filters 1 3dB (Cutoff ) Frequency : fc (Hz) Maximum Passband Attenuation : 3dB Passband Ripple : Rp (dB) Stopband Frequency](https://reader033.vdocument.in/reader033/viewer/2022061507/5697c0281a28abf838cd7074/html5/thumbnails/10.jpg)
Chebyshev Low-Pass Filters Response
![Page 11: TOPIC 4 Filters Design. Basic Filters 1 3dB (Cutoff ) Frequency : fc (Hz) Maximum Passband Attenuation : 3dB Passband Ripple : Rp (dB) Stopband Frequency](https://reader033.vdocument.in/reader033/viewer/2022061507/5697c0281a28abf838cd7074/html5/thumbnails/11.jpg)
Comparison of Frequency response between Butterworht 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 of 7-order chebyshev-type with ripple of 1dB
Comparison between Butterworht and Chebyshev Filters
![Page 12: TOPIC 4 Filters Design. Basic Filters 1 3dB (Cutoff ) Frequency : fc (Hz) Maximum Passband Attenuation : 3dB Passband Ripple : Rp (dB) Stopband Frequency](https://reader033.vdocument.in/reader033/viewer/2022061507/5697c0281a28abf838cd7074/html5/thumbnails/12.jpg)
典型滤波器响应 椭圆滤波器( elliptic filter )是利用椭圆函数( elliptic functio
n )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:
① 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;
② 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; ③ 切比雪夫滤波器的截止特性比巴特沃思滤波器更为陡峭。
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。
![Page 13: TOPIC 4 Filters Design. Basic Filters 1 3dB (Cutoff ) Frequency : fc (Hz) Maximum Passband Attenuation : 3dB Passband Ripple : Rp (dB) Stopband Frequency](https://reader033.vdocument.in/reader033/viewer/2022061507/5697c0281a28abf838cd7074/html5/thumbnails/13.jpg)
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,极点全部落在阻带 >1 内,具有如下形式
)(Cn
))((
))(()(
24
222
2
23
221
2
BCn
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。
1 3 2 4
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。
210lg 1 ( )LA Cn
椭圆函数滤波器的衰减特性为:
![Page 14: TOPIC 4 Filters Design. Basic Filters 1 3dB (Cutoff ) Frequency : fc (Hz) Maximum Passband Attenuation : 3dB Passband Ripple : Rp (dB) Stopband Frequency](https://reader033.vdocument.in/reader033/viewer/2022061507/5697c0281a28abf838cd7074/html5/thumbnails/14.jpg)
椭圆函数滤波器响应
![Page 15: TOPIC 4 Filters Design. Basic Filters 1 3dB (Cutoff ) Frequency : fc (Hz) Maximum Passband Attenuation : 3dB Passband Ripple : Rp (dB) Stopband Frequency](https://reader033.vdocument.in/reader033/viewer/2022061507/5697c0281a28abf838cd7074/html5/thumbnails/15.jpg)
典型滤波器响应线性相位响应 Φ(ω) = A ω[ 1 + p (ω /ωc )2N]
式中 Φ(ω) 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。
![Page 16: TOPIC 4 Filters Design. Basic Filters 1 3dB (Cutoff ) Frequency : fc (Hz) Maximum Passband Attenuation : 3dB Passband Ripple : Rp (dB) Stopband Frequency](https://reader033.vdocument.in/reader033/viewer/2022061507/5697c0281a28abf838cd7074/html5/thumbnails/16.jpg)
低通原型滤波器器件参数的确定
L
C~
1
R
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 g0 = 1 , ωc = 1 。
由微波网络级联可得此电路的响应为 L=1+[(1-R)2+(C2R2+ L2- 2LCR2)ω2 +L2C2R2ω4]/4R
最平坦响应为 L=1+ k2ω4 k=1 ω=1 时衰减 3dB
得到 R=1, L = C = 21/2
等波纹响应为 L=1+ k2(2ω 2 - 1)2 k=1 波纹 3dB 得到 R=5.81, L=3.1 C = 0.53
对于 N = 2 的低通原型,
其结构图如右图所示:
![Page 17: TOPIC 4 Filters Design. Basic Filters 1 3dB (Cutoff ) Frequency : fc (Hz) Maximum Passband Attenuation : 3dB Passband Ripple : Rp (dB) Stopband Frequency](https://reader033.vdocument.in/reader033/viewer/2022061507/5697c0281a28abf838cd7074/html5/thumbnails/17.jpg)
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。
rL=gN + 1 = 1
rG=g0 =1
~
L2=g2 Ln=gn
C3=g3C1=g1
shunt capacitance series inductance
L1=g1 L3=g3
Cn=gnC2=g2~
rG=g0 =1
rL=gN + 1 = 1
series inductance shunt capacitance
图中器件的编号从信号源端的 g0 一直到负载端的 g
N+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。
![Page 18: TOPIC 4 Filters Design. Basic Filters 1 3dB (Cutoff ) Frequency : fc (Hz) Maximum Passband Attenuation : 3dB Passband Ripple : Rp (dB) Stopband Frequency](https://reader033.vdocument.in/reader033/viewer/2022061507/5697c0281a28abf838cd7074/html5/thumbnails/18.jpg)
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:
![Page 19: TOPIC 4 Filters Design. Basic Filters 1 3dB (Cutoff ) Frequency : fc (Hz) Maximum Passband Attenuation : 3dB Passband Ripple : Rp (dB) Stopband Frequency](https://reader033.vdocument.in/reader033/viewer/2022061507/5697c0281a28abf838cd7074/html5/thumbnails/19.jpg)
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:
![Page 20: TOPIC 4 Filters Design. Basic Filters 1 3dB (Cutoff ) Frequency : fc (Hz) Maximum Passband Attenuation : 3dB Passband Ripple : Rp (dB) Stopband Frequency](https://reader033.vdocument.in/reader033/viewer/2022061507/5697c0281a28abf838cd7074/html5/thumbnails/20.jpg)
低通原型滤波器器件参数的确定 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) :
LAr = 0.01dB
n g1 g2 g3 g4 g5 g6 g7 g8 g9 g10 g11
1 0.0960 1.0000
2 0.4488 0.4077 1.1007
3 0.6291 0.9702 0.6291 1.0000
4 0.7128 1.2003 1.3212 0.6476 1.1007
5 0.7563 1.3049 1.5773 1.3049 0.7563 1.0000
6 0.7813 1.3600 1.6896 1.5350 1.4970 0.7098 1.1007
7 0.7969 1.3924 1.7481 1.6331 1.7481 1.3924 0.7969 1.0000
8 0.8072 1.4130 1.7824 1.6833 1.8529 1.6193 1.5554 0.7333 1.1007
9 0.8144 1.4270 1.8043 1.7125 1.9057 1.7125 1.8043 1.4270 0.8144 1.0000
10 0.8196 1.4369 1.8192 1.7311 1.9362 1.7590 1.9055 1.6527 1.5817 0.7446 1.1007
![Page 21: TOPIC 4 Filters Design. Basic Filters 1 3dB (Cutoff ) Frequency : fc (Hz) Maximum Passband Attenuation : 3dB Passband Ripple : Rp (dB) Stopband Frequency](https://reader033.vdocument.in/reader033/viewer/2022061507/5697c0281a28abf838cd7074/html5/thumbnails/21.jpg)
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:
![Page 22: TOPIC 4 Filters Design. Basic Filters 1 3dB (Cutoff ) Frequency : fc (Hz) Maximum Passband Attenuation : 3dB Passband Ripple : Rp (dB) Stopband Frequency](https://reader033.vdocument.in/reader033/viewer/2022061507/5697c0281a28abf838cd7074/html5/thumbnails/22.jpg)
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:
n g1 g2 g3 g4 g5 g6 g7 g8 g9 g10 g11
1 2.000 1.0000
2 1.5774 0.4226 1.0000
3 1.255 0.5528 0.1922 1.0000
4 1.0598 0.5116 0.3181 0.1104 1.0000
5 0.9303 0.4577 0.3312 .2090 0.0718 1.0000
6 0.8377 0.4116 0.31586 .2364 .1480 0.0505 1.00
7 0.7677 0.3744 0.2944 .2378 .1778 .1104 0.0375 1.0000
8 0.7125 .3446 0.2735 .2297 .1867 .1387 .0855 0.0289 1.000
9 0.6678 0.3203 0.2547 .2184 .1859 .1506 .1111 0.0682 0.0230 1.0000
10 0.6305 0.3002 0.23842 .2066 .1808 .15390 .1240 0.0911 0.0557 0.0187 1.0000
![Page 23: TOPIC 4 Filters Design. Basic Filters 1 3dB (Cutoff ) Frequency : fc (Hz) Maximum Passband Attenuation : 3dB Passband Ripple : Rp (dB) Stopband Frequency](https://reader033.vdocument.in/reader033/viewer/2022061507/5697c0281a28abf838cd7074/html5/thumbnails/23.jpg)
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
![Page 24: TOPIC 4 Filters Design. Basic Filters 1 3dB (Cutoff ) Frequency : fc (Hz) Maximum Passband Attenuation : 3dB Passband Ripple : Rp (dB) Stopband Frequency](https://reader033.vdocument.in/reader033/viewer/2022061507/5697c0281a28abf838cd7074/html5/thumbnails/24.jpg)
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)
Butterworth LowPass Filters1
/10
/10
10 10.5 log log
10 1
Axx
Apc
fN
f
NKN
KgK ,....,2,1,
2
)12(sin2
Step 2: Determine the Number of elements , N is a integer
Step 3: Calculate Prototype Element Values , gK 。
Step1 : Specification
![Page 25: TOPIC 4 Filters Design. Basic Filters 1 3dB (Cutoff ) Frequency : fc (Hz) Maximum Passband Attenuation : 3dB Passband Ripple : Rp (dB) Stopband Frequency](https://reader033.vdocument.in/reader033/viewer/2022061507/5697c0281a28abf838cd7074/html5/thumbnails/25.jpg)
)arccos(
1arccos 22
2
c
x
f
f
MagMag
N
110
1010/2
10/2
rp
AxMag
)(sin22
NK
BK
,...,2,1,2
)12(sin NK
NK
AK
2sinh
N
37.17cothln
rp
1cosh
1cosh 1
N
Chebyshev LowPass 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 output impedance
Step1 : Specification
Step 3: Calculate Prototype Element Values , gK 。
Bg
AAg
KK
KKK
4
11
21
Ag
2 11
gN+1=1 N 奇数gN+1=coth2(β/4) N 偶数
![Page 26: TOPIC 4 Filters Design. Basic Filters 1 3dB (Cutoff ) Frequency : fc (Hz) Maximum Passband Attenuation : 3dB Passband Ripple : Rp (dB) Stopband Frequency](https://reader033.vdocument.in/reader033/viewer/2022061507/5697c0281a28abf838cd7074/html5/thumbnails/26.jpg)
椭圆函数滤波器低通原型
s 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。
C1
C2C3 C5 C1 C2
C4
L2 L4 L1 L3
L2 L4
L5
a )电容输入 b )电感输入
两种椭圆函数低通滤波器原型电路
s
L5C4L4L3C2L2L1
0.8851.1000.2831.7261.1800.11781.044501.6900.8360.9640.4011.6571.2280.14001.032451.5400.7660.8750.5301.5861.1930.17701.010401.4140.7010.7040.7421.4881.1390.23000.977351.309
C5L4C4C3L2C2C1 LAs
(dB)
s
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
![Page 27: TOPIC 4 Filters Design. Basic Filters 1 3dB (Cutoff ) Frequency : fc (Hz) Maximum Passband Attenuation : 3dB Passband Ripple : Rp (dB) Stopband Frequency](https://reader033.vdocument.in/reader033/viewer/2022061507/5697c0281a28abf838cd7074/html5/thumbnails/27.jpg)
椭圆函数滤波器技术参数
0
p s13 24
AL
AsL
ArL
LAs :阻带抑制LAr :通带波纹 :通带截止频率p
:阻带抑制频率s
![Page 28: TOPIC 4 Filters Design. Basic Filters 1 3dB (Cutoff ) Frequency : fc (Hz) Maximum Passband Attenuation : 3dB Passband Ripple : Rp (dB) Stopband Frequency](https://reader033.vdocument.in/reader033/viewer/2022061507/5697c0281a28abf838cd7074/html5/thumbnails/28.jpg)
Frequency transformations from normalized LPF to others
C=gK
L=gk
Lowpass lowpass highpass bandpass bandstop Prototype pratical pratical pratical praticalValue value value value value
U Lo
2 U LBW -
L
c
C
c
L1c
C1c
o C 2
LBW o L 2
BW
CBW
BW
LBW
L
o2
BW
1
CBW
1
C
o2
BW
![Page 29: TOPIC 4 Filters Design. Basic Filters 1 3dB (Cutoff ) Frequency : fc (Hz) Maximum Passband Attenuation : 3dB Passband Ripple : Rp (dB) Stopband Frequency](https://reader033.vdocument.in/reader033/viewer/2022061507/5697c0281a28abf838cd7074/html5/thumbnails/29.jpg)
g1 g2 g3 g4 g52.2072 1.1279 3.1025 1.1279 2.2072
C1 L2 C3 L4 C5
Cal. value 93.658pF 119.67nH 131.65pF 119.67nH 93.658pF
Practical 94pF 120nH 132pF 120nH 94pF
Examples of LPF design
Impedance: Zo (ohm)=50 Cutoff Frequency: fc (MHz)=75 Stopband Frequency: fx (MHz)=100 Maximum 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 20dB attenuation at 100MHz
Solution:
N=5
Step 4 : Select shunt capacitance series inductance
![Page 30: TOPIC 4 Filters Design. Basic Filters 1 3dB (Cutoff ) Frequency : fc (Hz) Maximum Passband Attenuation : 3dB Passband Ripple : Rp (dB) Stopband Frequency](https://reader033.vdocument.in/reader033/viewer/2022061507/5697c0281a28abf838cd7074/html5/thumbnails/30.jpg)
Result
![Page 31: TOPIC 4 Filters Design. Basic Filters 1 3dB (Cutoff ) Frequency : fc (Hz) Maximum Passband Attenuation : 3dB Passband Ripple : Rp (dB) Stopband Frequency](https://reader033.vdocument.in/reader033/viewer/2022061507/5697c0281a28abf838cd7074/html5/thumbnails/31.jpg)
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 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,
1
21
2
2
2
1
XXX
PassXU
oXUX
PassXL
XL
oX
MIN
BWf
ff
BWf
f
f
PLPUpassPUPLo ffBWfff ,
X
Ap
Ax
Nlog
110110
log5.0 10/
10/
( 1 ) For Butterworth Type
( 2 ) For Chebyshev Type
110
1010/2
10/2
rp
AxMag
)arccos(
1arccos 22
2
X
Mag
Mag
N
![Page 32: TOPIC 4 Filters Design. Basic Filters 1 3dB (Cutoff ) Frequency : fc (Hz) Maximum Passband Attenuation : 3dB Passband Ripple : Rp (dB) Stopband Frequency](https://reader033.vdocument.in/reader033/viewer/2022061507/5697c0281a28abf838cd7074/html5/thumbnails/32.jpg)
Design of BandPass Filters2
ZoBW
gCp
BW
ZogLs
pass
eveneven
pass
oddodd
2,
2
pass
eveneven
pass
oddodd BW
ZogLs
ZoBW
gCp
2
,2
Cp
Lp
LsCs
Cp
Ls
oo
So
S
Po
P
f
LC
CL
2
1
1
2
2
prototype bandpassTransforma-tion fomu
la
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 values of bpf 。 Transformate the lowpass prototype element values to the bandpass ones according the right transformation table
![Page 33: TOPIC 4 Filters Design. Basic Filters 1 3dB (Cutoff ) Frequency : fc (Hz) Maximum Passband Attenuation : 3dB Passband Ripple : Rp (dB) Stopband Frequency](https://reader033.vdocument.in/reader033/viewer/2022061507/5697c0281a28abf838cd7074/html5/thumbnails/33.jpg)
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 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 Attenuation at passband: rp = 0.1 dB Minimum Attenuation at stopband : Ax = 30dB
)arccos(
1arccos 22
2
X
MagMag
N
Step 2 : determine the order of elements , N=3
778.2),( 21 XXX MIN
778.21
,333.31
2
2
2
1
PassXU
oXUX
PassXL
XL
oX BWf
ff
BWf
ff
MHzfff PUPLo ,83.74 MHzffBW PLPUpass 10
![Page 34: TOPIC 4 Filters Design. Basic Filters 1 3dB (Cutoff ) Frequency : fc (Hz) Maximum Passband Attenuation : 3dB Passband Ripple : Rp (dB) Stopband Frequency](https://reader033.vdocument.in/reader033/viewer/2022061507/5697c0281a28abf838cd7074/html5/thumbnails/34.jpg)
Result
C1 456pF L2 1268nH C3 456pFTransformated values of BPF
L1 10nH C2 3.6pF L3 10nH
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 according the transformation table 。
rL=gN + 1 = 1
rG=g0 =1
~
L2=1.5937
C1=1.4329 C3=1.4329
![Page 35: TOPIC 4 Filters Design. Basic Filters 1 3dB (Cutoff ) Frequency : fc (Hz) Maximum Passband Attenuation : 3dB Passband Ripple : Rp (dB) Stopband Frequency](https://reader033.vdocument.in/reader033/viewer/2022061507/5697c0281a28abf838cd7074/html5/thumbnails/35.jpg)
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