3-7-2Chebyshev Arrays 《天线与电波传播》课件
相关主题
- 1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
- 2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
- 3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。
For a symmetrical broadside array with 2N1elements
a1x1coskd b 1x1
1coskd
1coskd
来自百度文库
If d 2 these constants become
ax11 bx11
2
2
If we specify the ratio of the main-beam maximum to
Chebyshev Arrays
One design criterion that is often chosen is that which will produce the narrowest possible beam width for a given side-lobe level or conversely that will produce the smallest side-lobe level for a given beam width. If we had a polynomial with these properties it would be easy to determine the required current distribution in the elements of the array. Fortunately there exists a series of polynomials known as the Chebyshev polynomials that can be adapted to the design of optimum arrays according to the criterion given above. The method was first introduced by Dolph, so this type of array is also called a Dolph-chebyshev array. The theory of the hebyshev array is readily developed by using the Fourier series representation of the array factor. Before presenting this theory we will summarize the basic properties of the Chebyshev polynomials.
acosuz x1coskd b 1xz
cosuz coskd
cosuz coskd
(3.81)
TN(x1)TN(ab)RcoshNcosh1(ab) (3.82)
T2(abcosu)(2a2b21)4abcosub2cos2u
which is also a finite Fourier series. In general, Tn(abcous) is a finite
Fourier series, with terms up to cosNu and may therefore be identified as an array factor for an array with 2N1 elements.
The Chebyshev polynomials are defined by the following relations:
T1(x ) x T2(x) 2 x2 1 T3(x) 4 x3 3x T4(x) 8x4 8x2 1
Tn ( x ) 2 xTn1( x ) Tn2 ( x )
If the beam null is placed at z then the corresponding value of u
Is uzkdcozsand x z is given by
x z a b cu z o a b s ck o d cs z o )(s
Now x z , the zero closest to one, is also given by Eq. (3.72) as
these polynomials also satisfy the relationship
Tn(co )scons
and when is complex
Tn(co)shconsh
e jx e jx cos x
2 cosh x chx ex ex
2
Similarly, upon using 4c3 o u s3 co u c so 3 uswe find that T 3 ( a b cu o ) 4 ( s a b cu o ) 3 3 s ( a b cu o ) s
the side-lobe level to be R
x1cos1 hcoN s1c ho (1sRh )
In some designs the beam width is given. The main beam extends
from the last zero of TN ( x) before xreaches the value of 1 up to x 1
xz co2sNabcouzs
This equation, along with the requirement that a bco kd s ) ( 1
can be solved for aand b to give
acouszx1coksd couszcoksd
b 1xz cousz coksd
a1x1coskd b 1x1
1coskd
1coskd
来自百度文库
If d 2 these constants become
ax11 bx11
2
2
If we specify the ratio of the main-beam maximum to
Chebyshev Arrays
One design criterion that is often chosen is that which will produce the narrowest possible beam width for a given side-lobe level or conversely that will produce the smallest side-lobe level for a given beam width. If we had a polynomial with these properties it would be easy to determine the required current distribution in the elements of the array. Fortunately there exists a series of polynomials known as the Chebyshev polynomials that can be adapted to the design of optimum arrays according to the criterion given above. The method was first introduced by Dolph, so this type of array is also called a Dolph-chebyshev array. The theory of the hebyshev array is readily developed by using the Fourier series representation of the array factor. Before presenting this theory we will summarize the basic properties of the Chebyshev polynomials.
acosuz x1coskd b 1xz
cosuz coskd
cosuz coskd
(3.81)
TN(x1)TN(ab)RcoshNcosh1(ab) (3.82)
T2(abcosu)(2a2b21)4abcosub2cos2u
which is also a finite Fourier series. In general, Tn(abcous) is a finite
Fourier series, with terms up to cosNu and may therefore be identified as an array factor for an array with 2N1 elements.
The Chebyshev polynomials are defined by the following relations:
T1(x ) x T2(x) 2 x2 1 T3(x) 4 x3 3x T4(x) 8x4 8x2 1
Tn ( x ) 2 xTn1( x ) Tn2 ( x )
If the beam null is placed at z then the corresponding value of u
Is uzkdcozsand x z is given by
x z a b cu z o a b s ck o d cs z o )(s
Now x z , the zero closest to one, is also given by Eq. (3.72) as
these polynomials also satisfy the relationship
Tn(co )scons
and when is complex
Tn(co)shconsh
e jx e jx cos x
2 cosh x chx ex ex
2
Similarly, upon using 4c3 o u s3 co u c so 3 uswe find that T 3 ( a b cu o ) 4 ( s a b cu o ) 3 3 s ( a b cu o ) s
the side-lobe level to be R
x1cos1 hcoN s1c ho (1sRh )
In some designs the beam width is given. The main beam extends
from the last zero of TN ( x) before xreaches the value of 1 up to x 1
xz co2sNabcouzs
This equation, along with the requirement that a bco kd s ) ( 1
can be solved for aand b to give
acouszx1coksd couszcoksd
b 1xz cousz coksd