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Chebyshev polynominals in low side-lobe antenna array

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chunweimin

Junior Member level 1
Chebyshev Array

Hi,

I am working on linear array analysis and try to use the Chebyshev polynomials to achieve a low side-lobe pattern of the array. I have calculated the excitation coefficients of each element. When it comes to implementation using unequal power splitters, the equations require the "power ratio" in order to calculate the corresponding impedances of the lines. Do I need to square the excitation coefficients before I substitute them into the impedance equations or I apply them directly? Thanks very much for your help.

Chunwei

Chebyshev Array

Hi Chunwei,

If you calculated the coefficients as amplitudes, you have to square them in order to have the power ratios needed for synthesis of splitters.
Check your array synthesis method in order to be sure that the coefficients are in magnitude (and eventually phase as well) and not in power.
Usually the synthesis gives results in amplitude or amplitude-phase, and the scaling is such that the sum of the squared amplitudes is unity.
Regards

Z

chunweimin

Points: 2
Re: Chebyshev Array

Hello Zorro,

Thanks very much for your reply, it is very helpful. Please refer to the design procedure on the notes attached, which I followed.
I am actually designing a 1x8 Chebyshev linear array with side-lobe level (SLL) of 40dB. The SLL was converted to magnitude by SLL="20"log(R), then substitute R into the Chebyshev polynomial. So the obtained coefficients are in voltage sense rather than power right?

If so, then the following are the calculated amplitude excitation coefficients defined in the E-field of each element:

6.3, 17.9, 32.6, 42.9, 42.9, 32.6, 17.9, 6.3

So I square them, then normalize by the peak value 42.9^2
This then yields

0.02, 0.18, 0.58, 1, 1, 0.58, 0.18, 0.02 (Please shout if I was wrong)

Would you kindly suggest any clever way to unequally split the power like the above? (2% power seems to be quite challenging to split using passive splitters) I believe a passive reactive power splitter will do the job, but we won't have isolation between the outputs. Maybe using Wilkinson power splitters? or branch line couplers? or better use tunable amplifiers? Thanks very much.

Best regards,

Chunwei

Re: Chebyshev Array

Hi Chunwei

I checked the result with the amplitudes you give:
6.3, 17.9, 32.6, 42.9, 42.9, 32.6, 17.9, 6.3
and the result is (within +/- 1 dB error for having the amplitudes expressed with only 2 decimal digits) of a 40 dB side lobe level.
The values
0.02, 0.18, 0.58, 1, 1, 0.58, 0.18, 0.02
express relative power levels.

A structure for manage the power splitting in this case is (only left half shown):

\/././
.\/./
..\/
...\
.....\/

Use of unequal Wilkinson splitters is good. Nevertheless, the 9 dB ratio at the extremes is challenging. There is a variation of Wilkinson splitter suitable for high ratios, that uses the split-and-recombination concept. I think you can find it in the literature.

Regards

Z

chunweimin

Points: 2
Re: Chebyshev Array

Hi Zorro,

Thanks very much for your confirmations. This really gives me confidence to carry on.
Would you possibly have some more information about the split-and-recombination concept for WPS, perhaps some papers in hand for me to read if that would not bother you much?
Thanks very much.

Best regards,

Chunwei

Chebyshev Array

chunweimin said:
Would you possibly have some more information about the split-and-recombination concept for WPS, perhaps some papers in hand for me to read if that would not bother you much?
The idea is to split a small part of the power using two cascaded stages, and then recombine the resting two branches.
I think I have somewhere a paper. I will check.
Regards

Z

Added after 1 hours 37 minutes:

Hi Chunwei,

Note that the structure described there is a compacted form of three associated Wilkinson power splitters, two used as dividers and one as a combiner.
Patent # 5563558 (Reentrant power coupler) can be useful too.
Regards

Z

Points: 2