Filters. A RF/microwave filter is (typically) a passive, reciprocal, 2- port linear device. Filter

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1 3/1/25 Filters.do 1/8 Filters A RF/mirowave filter is (typially) a passive, reiproal, 2- port linear devie. Pin Filter Pout If port 2 of this devie is terminated in a mathed load, then we an relate the inident and output power as: out 2 21 P = S P in We define this power transmission through a filter in terms of the power transmission oeffiient T: T P P out in = S 21 2 Sine mirowave filters are typially passive, we find that: 1 in other words, P out P in.

2 3/1/25 Filters.do 2/8 Q: What happens to the missing power P in P out? A: Two possibilities: the power is either absorbed (P abs ) by the filter (onverted to heat), or is refleted (P r ) at the input port. I.E.: P abs Pin Filter Pout P r Thus, by onservation of energy: P = P + P + P in r abs out Now ideally, a mirowave filter is lossless, therefore P abs = and: Pin = Pr + Pout whih alternatively an be written as: Pin Pr + Pout = Pin Pin Pr P 1 = + P P in out in

3 3/1/25 Filters.do 3/8 Reall that P out P in =, and we an likewise define P r P in as the power refletion oeffiient: Γ Pr P in = S 11 2 We again emphasize that the filter output port is terminated in a mathed load. Thus, we an onlude that for a lossless filter: 1 = Γ + Whih is simply another way of saying for a lossless devie that = S + S. Now, here s the important part! For a mirowave filter, the oeffiients Γ and are funtions of frequeny! I.E.,: Γ ( ) and ( ) The behavior of a mirowave filter is desribed by these funtions!

4 3/1/25 Filters.do 4/8 We find that for most signal frequenies s, these funtions will have a value equal to one of two different approximate values. Either: Γ ( = ) and ( = ) 1 s s or Γ ( = ) 1 and ( = ) s s In the first ase, the signal frequeny s is said to lie in the pass-band of the filter. Almost all of the inident signal power will pass through the filter. In the seond ase, the signal frequeny s is said to lie in the stop-band of the filter. Almost all of the inident signal power will be refleted at the input almost no power will appear at the filter output.

5 3/1/25 Filters.do 5/8 Consider then these four types of funtions of Γ ( ) and ( ): 1. Low-Pass Filter ( ) 1 1 Γ ( ) Note for this filter: 1 < < ( ) = ( ) = > 1 > Γ This filter is a low-pass type, as it passes signals with frequenies less than, while rejeting signals at frequenies greater than. Q: This frequeny seems to be very important! What is it?

6 3/1/25 Filters.do 6/8 A: Frequeny is a filter parameter known as the utoff frequeny; a value that approximately defines the frequeny region where the filter pass-band transitions into the filter stop band. Aording, this frequeny is defined as the frequeny where the power transmission oeffiient is equal to ½: ( = ) =.5 Note for a lossless filter, the utoff frequeny is likewise the value where the power refletion oeffiient is ½: 2. High-Pass Filter Γ ( = ) =.5 ( ) 1 1 Γ ( )

7 3/1/25 Filters.do 7/8 Note for this filter: < 1 < ( ) = ( ) = 1 > > Γ This filter is a high-pass type, as it passes signals with frequenies greater than, while rejeting signals at frequenies less than. 3. Band-Pass Filter ( ) 1 1 Γ ( ) Note for this filter: < < ( ) = Γ( ) = > 2 1 < 2

8 3/1/25 Filters.do 8/8 This filter is a band-pass type, as it passes signals within a frequeny bandwidth, while rejeting signals at all frequenies outside this bandwidth. In addition to filter bandwidth, a fundamental parameter of bandpass filters is, whih defines the enter frequeny of the filter bandwidth. 3. Band-Stop Filter ( ) Γ ( ) 1 1 Note for this filter: < < ( ) = Γ( ) = > 2 < 2 This filter is a band-stop type, as it rejets signals within a frequeny bandwidth, while passing signals at all frequenies outside this bandwidth.