Microwave Filter Design
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1 3/1/2005 Microwave Filter Design.doc 1/7 Microwave Filter Design Recall that a lossless filter can be described in terms of or its power either its power transmission coefficient ( ) reflection coefficient Γ ( ), as the two values are completely dependent: ( ) = 1 ( ) Γ Ideally, these functions would be quite simple: 1. ( ) = 1 and ( ) = 0 Γ for all frequencies within the passband. 2. ( ) = 0 and ( ) = 1 Γ for all frequencies within the stopband. For example, the ideal low-pass filter would be: 1 1 Γ c c
2 3/1/2005 Microwave Filter Design.doc 2/7 Add to this a linear phase response, and you have the perfect microwave filter! There s just one small problem with this perfect filter It s impossible to build! Now, if we consider only possible (i.e., realizable) filters, we must limit ourselves to filter functions that can be expressed as finite polynomials of the form: = b + b + b + + b ( ) 2 a0 + a1+ a The order N of the (denominator) polynomial is likewise the order of the filter. N N There are many different types of polynomials that result in good filter responses, and each type has its own set of characteristics. The type of polynomial likewise describes the type of microwave filter. Let s consider three of the most popular types: 1. Elliptical Elliptical filters have three primary characteristics: a) They exhibit very steep roll-off, meaning that the transition from pass-band to stop-band is very rapid.
3 3/1/2005 Microwave Filter Design.doc 3/7 b) They exhibit ripple in the pass-band, meaning that the value of will vary slightly within the pass-band. 1 c) They exhibit ripple in the stop-band, meaning that the value of will vary slightly within the stop-band. We find that we can make the roll-off steeper by accepting more ripple. 2. Chebychev Chebychev filters are also known as equal-ripple filters, and have two primary characteristics a) Steep roll-off (but not as steep as Elliptical). b) Pass-band ripple (but not stop-band ripple).
4 3/1/2005 Microwave Filter Design.doc 4/7 1 We likewise find that the roll-off can be made steeper by accepting more ripple. 3. Butterworth Also known as maximally flat filters, they have two primary characteristics a) Gradual roll-off. 1 b) No ripple not anywhere.
5 3/1/2005 Microwave Filter Design.doc 5/7 Q: So we always chose elliptical filters; since they have the steepest roll-off, they are closest to ideal right? A: Ooops! I forgot to talk about the phase response S ( ) of these filters. Let s examine S ( ) for each filter type before we pass judgment. Butterworth S ( ) Close to linear phase. Chebychev S ( ) Not very linear. Elliptical S ( ) A big non-linear mess! Thus, it is apparent that as a filter roll-off improves, the phase response gets worse (watch out for dispersion!). There is no such thing as the best filter type! Q: So, a filter with perfectly linear phase is impossible to construct? A: No, it is possible to construct a filter with near perfect linear phase but it will exhibit a horribly poor roll-off!
6 3/1/2005 Microwave Filter Design.doc 6/7 Now, for any type of filter, we can improve roll-off (i.e., increase stop-band attenuation) by increasing the filter order N. However, be aware that increasing the filter order likewise has these deleterious effects: 1. It makes phase response S ( ) worse (i.e., more nonlinear). 2. It increases filter cost, weight, and size. 3. It increases filter insertion loss (this is bad). 4. It makes filter performance more sensitive to temperature, aging, etc. From a practical viewpoint, the order of a filter should typically be kept to 10 N <. Q: So exactly what are these filter polynomials ( )? How do we determine them? A: Fortunately, radio engineers do not need to determine specific filter polynomials in order to specify (to filter manufacturers) what they want built. Instead, radio engineers simply can specify the type and order of a filter, saying things like:
7 3/1/2005 Microwave Filter Design.doc 7/7 or or I need a 3 rd -order Chebychev filter! Get me a 5 th -order Butterworth filter! I wish I d paid more attention in EECS 622! Thus, the most important filter specifications are: 1. Filter bandwidth and center frequency 2. Filter type and order. However, there are many more important filter specifications!
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