Even Order Stop Bands and Pass Bands
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Even Order Stop Bands and Pass Bands |
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Description of the Problem The even order filters of Chebyshev I, Chebyshev II, Hourglass, and Elliptic create a special problem in that their standard forms are not realizable with passive RCL circuits when their source and load resistances are comparable. Filter Solutions solves this problem by modifying these standard filters to a form that is realizable with such circuits. In general, for an equally terminated filter to be realizable, a low pass filter frequency response must have a reflection zero at 0Hz and a transmission zero at infinity. For high pass filters the reflection zero must be at infinity, and the transmission zero must be at 0Hz. For band pass filters, the reflection zero must be at the center frequency, and the transmission zero at 0Hz and infinity. For band stop filters the reflection zero must be at 0Hz and infinity, and the transmission zero must be at the center frequency. Filter Solutions displays the Even Order Mod option in the lower right part of the Control Panel which, when checked, modifies the filter frequency response to conform to both these requirements. This even order modification is very important when designing diplexers because it minimizes the number of inductors needed to create the diplexer. Below is a graphical description of the problem for low pass filters and the frequency modifying solution. |
| Reflection Zero not at Zero Hz
Chebyshev I, Occasionally Hourglass, and Elliptic filters are affected. |
| This 6th order Chebyshev I filter illustrates the problem. The lowest frequency reflection zero for this 6th order filter is between two and three Hz, and therefore is not realizable with an RCL circuit with comparable source and load resistances.
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| This 5th order Chebyshev I filter is realizable. The lowest frequency reflection zero is at 0Hz, so this filter is readily realizable with RCL circuits with any source resistance.
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| To realize the 6th order Chebyshev I circuit, Filter Solution modifies the frequency response such that the lowest frequency reflection zero is mapped to 0Hz. The result is no longer a true Chebyshev I filter, but is rather a modified Chebyshev I filter that is realizable with RCL circuits of comparable source and load resistances.
As you can see, the lowest frequency reflection zero is now at 0Hz. It is still a 6th order filter with greater attenuation that the 5th order filter.
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| Transmission Zero not at Infinity
The stop band filters of Chebyshev II, Hourglass, and Elliptic filters are affected. |
| This 6th order Chebyshev II filter illustrates the problem. The highest frequency transmission zero for this 6th order filter is slightly less than five Hz, and therefor is not realizable with an RCL circuit with comparablesource and load resistances.
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| This 5th order Chebyshev II filter is realizable. The highest frequency transmission zero is at infinity, so this filter is readily realizable with RCL circuits with any source resistance.
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| To realize the 6th order Chebyshev II circuit, Filter Solution modifies the frequency response such that the highest frequency transmission zero is mapped to infinity. The result is no longer a true Chebyshev II filter, but is rather a modified Chebyshev II filter that is realizable with RCL circuits of comparable source and load resistances.
As you can see, the highest frequency transmission zero is now at infinity. It is still a 6th order filter with greater attenuation that the 5th order filter.
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