topic 4 filters design. basic filters 1 3db (cutoff ) frequency : fc (hz) maximum passband...
TRANSCRIPT
TOPIC 4
Filters Design
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)
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
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
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 + 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。
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 ( 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。
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 • • •
Chebyshev Low-Pass Filters Response
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
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。
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:
椭圆函数滤波器响应
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。
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:
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。
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 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
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:
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
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 (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
)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
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
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
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
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
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 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
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
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
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
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