Manipulate

Antenna array design with Chebychev polynomials

March 23, 2015 ece1229 No comments , , , ,

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Prof. Eleftheriades desribed a Chebychev antenna array design method that looks different than the one of the text [1].

Portions of that procedure are like that of the text. For example, if a side lobe level of \( 20 \log_{10} R \) is desired, a scaling factor

\begin{equation}\label{eqn:chebychevSecondMethod:20}
x_0 = \cosh\lr{ \inv{m} \cosh^{-1} R },
\end{equation}

is used. Given \( N \) elements in the array, a Chebychev polynomial of degree \( m = N – 1 \) is used. That is

\begin{equation}\label{eqn:chebychevSecondMethod:40}
T_m(x) = \cos\lr{ m \cos^{-1} x }.
\end{equation}

Observe that the roots \( x_n’ \) of this polynomial lie where

\begin{equation}\label{eqn:chebychevSecondMethod:60}
m \cos^{-1} x_n’ = \frac{\pi}{2} \pm \pi n,
\end{equation}

or

\begin{equation}\label{eqn:chebychevSecondMethod:80}
x_n’ = \cos\lr{ \frac{\pi}{2 m} \lr{ 2 n \pm 1 } },
\end{equation}

The class notes use the negative sign, and number \( n = 1,2, \cdots, m \). It is noted that the roots are symmetric with \( x_1′ = – x_m’ \), which can be seen by direct expansion

\begin{equation}\label{eqn:chebychevSecondMethod:100}
\begin{aligned}
x_{m-r}’
&= \cos\lr{ \frac{\pi}{2 m} \lr{ 2 (m – r) – 1 } } \\
&= \cos\lr{ \pi – \frac{\pi}{2 m} \lr{ 2 r + 1 } } \\
&= -\cos\lr{ \frac{\pi}{2 m} \lr{ 2 r + 1 } } \\
&= -\cos\lr{ \frac{\pi}{2 m} \lr{ 2 ( r + 1 ) – 1 } } \\
&= -x_{r+1}’.
\end{aligned}
\end{equation}

The next step in the procedure is the identification

\begin{equation}\label{eqn:chebychevSecondMethod:120}
\begin{aligned}
u_n’ &= 2 \cos^{-1} \lr{ \frac{x_n’}{x_0} } \\
z_n &= e^{j u_n’}.
\end{aligned}
\end{equation}

This has a factor of two that does not appear in the Balanis design method. It seems plausible that this factor of two was introduced so that the roots of the array factor \( z_n \) are conjugate pairs. Since \( \cos^{-1} (-z) = \pi – \cos^{-1} z \), this choice leads to such conjugate pairs

\begin{equation}\label{eqn:chebychevSecondMethod:140}
\begin{aligned}
\exp\lr{j u_{m-r}’}
&=
\exp\lr{j 2 \cos^{-1} \lr{ \frac{x_{m-r}’}{x_0} } } \\
&=
\exp\lr{j 2 \cos^{-1} \lr{ -\frac{x_{r+1}’}{x_0} } } \\
&=
\exp\lr{j 2 \lr{ \pi – \cos^{-1} \lr{ \frac{x_{r+1}’}{x_0} } } } \\
&=
\exp\lr{-j u_{r+1}}.
\end{aligned}
\end{equation}

Because of this, the array factor can be written

\begin{equation}\label{eqn:chebychevSecondMethod:180}
\begin{aligned}
\textrm{AF}
&= ( z – z_1 )( z – z_2 ) \cdots ( z – z_{m-1} ) ( z – z_m ) \\
&=
( z – z_1 )( z – z_1^\conj )
( z – z_2 )( z – z_2^\conj )
\cdots \\
&=
\lr{ z^2 – z ( z_1 + z_1^\conj ) + 1 }
\lr{ z^2 – z ( z_2 + z_2^\conj ) + 1 }
\cdots \\
&=
\lr{ z^2 – 2 z \cos\lr{ 2 \cos^{-1} \lr{ \frac{x_1′}{x_0} } } + 1 }
\lr{ z^2 – 2 z \cos\lr{ 2 \cos^{-1} \lr{ \frac{x_2′}{x_0} } } + 1 }
\cdots \\
&=
\lr{ z^2 – 2 z \lr{ 2 \lr{ \frac{x_1′}{x_0} }^2 – 1 } + 1 }
\lr{ z^2 – 2 z \lr{ 2 \lr{ \frac{x_2′}{x_0} }^2 – 1 } + 1 }
\cdots
\end{aligned}
\end{equation}

When \( m \) is even, there will only be such conjugate pairs of roots. When \( m \) is odd, the remainding factor will be

\begin{equation}\label{eqn:chebychevSecondMethod:160}
\begin{aligned}
z – e^{2 j \cos^{-1} \lr{ 0/x_0 } }
&=
z – e^{2 j \pi/2} \\
&=
z – e^{j \pi} \\
&=
z + 1.
\end{aligned}
\end{equation}

However, with this factor of two included, the connection between the final array factor polynomial \ref{eqn:chebychevSecondMethod:180}, and the Chebychev polynomial \( T_m \) is not clear to me. How does this scaling impact the roots?

Example: Expand \( \textrm{AF} \) for \( N = 4 \).

The roots of \( T_3(x) \) are

\begin{equation}\label{eqn:chebychevSecondMethod:200}
x_n’ \in \setlr{0, \pm \frac{\sqrt{3}}{2} },
\end{equation}

so the array factor is

\begin{equation}\label{eqn:chebychevSecondMethod:220}
\begin{aligned}
\textrm{AF}
&=
\lr{ z^2 + z \lr{ 2 – \frac{3}{x_0^2} } + 1 }\lr{ z + 1 } \\
&=
z^3
+ 3 z^2 \lr{ 1 – \frac{1}{x_0^2} }
+ 3 z \lr{ 1 – \frac{1}{x_0^2} }
+ 1.
\end{aligned}
\end{equation}

With \( 20 \log_{10} R = 30 \textrm{dB} \), \( x_0 = 2.1 \), so this is

\begin{equation}\label{eqn:chebychevSecondMethod:240}
\textrm{AF} = z^3 + 2.33089 z^2 + 2.33089 z + 1.
\end{equation}

With

\begin{equation}\label{eqn:chebychevSecondMethod:260}
z = e^{j (u + u_0) } = e^{j k d \cos\theta + j k u_0 },
\end{equation}

the array factor takes the form

\begin{equation}\label{eqn:chebychevSecondMethod:280}
\textrm{AF}
=
e^{j 3 k d \cos\theta + j 3 k u_0 }
+ 2.33089
e^{j 2 k d \cos\theta + j 2 k u_0 }
+ 2.33089
e^{j k d \cos\theta + j k u_0 }
+ 1.
\end{equation}

This array function is highly phase dependent, plotted for \( u_0 = 0 \) in fig. 1, and fig. 2.

ChebychevSecondMethodPolarFig3pn

fig 1. Plot with u_0 = 0, d = lambda/4

ChebychevSecondMethodSPolarFig4pn

fig 2. Spherical plot with u_0 = 0, d = lambda/4

This can be directed along a single direction (z-axis) with higher phase choices as illustrated in fig. 3, and fig. 4.

 

ChebychevSecondMethodPolarFig1pn

fig 3. Plot with u_0 = 3.5, d = 0.4 lambda

ChebychevSecondMethodSPolarFig2pn

fig 4. Spherical plot with u_0 = 3.5, d = 0.4 lambda

 

These can be explored interactively in this Mathematica Manipulate panel.

References

[1] Constantine A Balanis. Antenna theory: analysis and design. John Wiley \& Sons, 3rd edition, 2005.

Chebychev antenna array design

March 22, 2015 ece1229 No comments , , , ,

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In our text [1] is a design procedure that applies Chebychev polynomials to the selection of current magnitudes for an evenly spaced array of identical antennas placed along the z-axis.

For an even number \( 2 M \) of identical antennas placed at positions \( \Br_m = (d/2) \lr{2 m -1} \Be_3 \), the array factor is

\begin{equation}\label{eqn:chebychevDesign:20}
\textrm{AF}
=
\sum_{m=-N}^N I_m e^{-j k \rcap \cdot \Br_m }.
\end{equation}

Assuming the currents are symmetric \( I_{-m} = I_m \), with \( \rcap = (\sin\theta \cos\phi, \sin\theta \sin\phi, \cos\theta ) \), and \( u = \frac{\pi d}{\lambda} \cos\theta \), this is

\begin{equation}\label{eqn:chebychevDesign:40}
\begin{aligned}
\textrm{AF}
&=
\sum_{m=-N}^N I_m e^{-j k (d/2) ( 2 m -1 )\cos\theta } \\
&=
2 \sum_{m=1}^N I_m \cos\lr{ k (d/2) ( 2 m -1)\cos\theta } \\
&=
2 \sum_{m=1}^N I_m \cos\lr{ (2 m -1) u }.
\end{aligned}
\end{equation}

This is a sum of only odd cosines, and can be expanded as a sum that includes all the odd powers of \( \cos u \). Suppose for example that this is a four element array with \( N = 2 \). In this case the array factor has the form

\begin{equation}\label{eqn:chebychevDesign:60}
\begin{aligned}
\textrm{AF}
&=
2 \lr{ I_1 \cos u + I_2 \lr{ 4 \cos^3 u – 3 \cos u } } \\
&=
2 \lr{ \lr{ I_1 – 3 I_2 } \cos u + 4 I_2 \cos^3 u }.
\end{aligned}
\end{equation}

The design procedure in the text sets \( \cos u = z/z_0 \), and then equates this to \( T_3(z) = 4 z^3 – 3 z \) to determine the current amplitudes \( I_m \). That is

\begin{equation}\label{eqn:chebychevDesign:80}
\frac{ 2 I_1 – 6 I_2 }{z_0} z + \frac{8 I_2}{z_0^3} z^3 = -3 z + 4 z^3,
\end{equation}

or

\begin{equation}\label{eqn:chebychevDesign:100}
\begin{aligned}
\begin{bmatrix}
I_1 \\
I_2
\end{bmatrix}
&=
{\begin{bmatrix}
2/z_0 & -6/z_0 \\
0 & 8/z_0^3
\end{bmatrix}}^{-1}
\begin{bmatrix}
-3 \\
4
\end{bmatrix} \\
&=
\frac{z_0}{2}
\begin{bmatrix}
3 (z_0^2 -1) \\
z_0^2
\end{bmatrix}.
\end{aligned}
\end{equation}

The currents in the array factor are fully determined up to a scale factor, reducing the array factor to

\begin{equation}\label{eqn:chebychevDesign:140}
\textrm{AF} = 4 z_0^3 \cos^3 u – 3 z_0 \cos u.
\end{equation}

The zeros of this array factor are located at the zeros of

\begin{equation}\label{eqn:chebychevDesign:120}
T_3( z_0 \cos u ) = \cos( 3 \cos^{-1} \lr{ z_0 \cos u } ),
\end{equation}

which are at \( 3 \cos^{-1} \lr{ z_0 \cos u } = \pi/2 + m \pi = \pi \lr{ m + \inv{2} } \)

\begin{equation}\label{eqn:chebychevDesign:160}
\cos u = \inv{z_0} \cos\lr{ \frac{\pi}{3} \lr{ m + \inv{2} } } = \setlr{ 0, \pm \frac{\sqrt{3}}{2 z_0} }.
\end{equation}

showing that the scaling factor \( z_0 \) effects the locations of the zeros. It also allows the values at the extremes \( \cos u = \pm 1 \), to increase past the \( \pm 1 \) non-scaled limit values. These effects can be explored in this Mathematica notebook, but can also be seen in fig. 1.

ChebyChevThreeScaledFig2pn

fig 1. T_3( z_0 x) for a few different scale factors z_0.

 

The scale factor can be fixed for a desired maximum power gain. For \( R
\textrm{dB} \), that will be when

\begin{equation}\label{eqn:chebychevDesign:180}
20 \log_{10} \cosh( 3 \cosh^{-1} z_0 ) = R \textrm{dB},
\end{equation}

or

\begin{equation}\label{eqn:chebychevDesign:200}
z_0 = \cosh \lr{ \inv{3} \cosh^{-1} \lr{ 10^{\frac{R}{20}} } }.
\end{equation}

For \( R = 30 \) dB (say), we have \( z_0 = 2.1 \), and

\begin{equation}\label{eqn:chebychevDesign:220}
\textrm{AF}
= 40 \cos^3 \lr{ \frac{\pi d}{\lambda} \cos\theta } – 6.4 \cos \lr{ \frac{\pi d}{\lambda} \cos\theta }.
\end{equation}

These are plotted in fig. 2 (linear scale), and fig. 3 (dB scale) for a couple values of \( d/\lambda \).

ChebychevT3FittingFig3pn

fig 2. T_3 fitting of 4 element array (linear scale).

ChebychevT3FittingDbFig4pn

fig 3. T_3 fitting of 4 element array (dB scale).

To explore the \( d/\lambda \) dependence try this Mathematica notebook.

References

[1] Constantine A Balanis. Antenna theory: analysis and design. John
Wiley & Sons, 3rd edition, 2005.

Mathematica CDF notebooks for ece1229 (antenna theory)

February 8, 2015 ece1229 No comments , ,

TaiAndPereiraSampleFieldThetaCapComponentFig1pn

infinitesimalDipoleErRealFig1pn

I put together a couple of cool Manipulate notebooks for some radiation plots.

I am not able to share these directly as blog posts since the CDF plugin that I am using in my wordpress instance appears to have a new plugin or version incompatibility, and is no longer working. The link above has some plain html javascript wrappers for these notebooks, and works at least with chrome and firefox on windows 7.