Microwave Resonators in EMLAB
Explore the world of microwave resonators, their design, performance, and applications with detailed insights into series and parallel resonant circuits, bandwidth considerations, energy resonance, unloaded Q-factor, and loaded Qs of series resonators. Discover the nuances of microwave resonator technology through comprehensive analysis and visual aids provided in the EMLAB resources.
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Presentation Transcript
1 Chapter 6. Microwave resonators EMLAB
Resonators 2 ?/2 ?01 ?01 ?02 1 2 2 1 =?02 ?01 ?02+ ?01 2?? = ?? (? = 1,2,3, ) 2 1 ?21? ??? 12? ????12 (1 + )? ???( )? ???(1 ) EMLAB
Resonators-, 3 ? = (1 + )? ???(1 ) +(1 + )? ???( )( )? 2???(1 ) + =(1 + )? ???(1 ) 1 2? 2??? =(1 2)? ??? 1 2? 2??? ? = + (1 + )? ???( )? ???(1 ) +(1 + )? ???( )? ???( )? ???( )? ???(1 ) + = +(1 2)? 2???( ) 1 2? 2??? = 1 (1 2)? 2??? 1 2? 2??? 1 ? 0.8 0.6 0.4 ? 0.2 0 0 2 4 6 8 10 12 14 16 18 EMLAB
6.1 Series & parallel resonant circuits 4 S-parameter of series resonator ? 1 ?0 ?0 ? = ? ?? ?? Characteristic impedance ?/?0 2 + ?/?0 2 2 + ?/?0 2 2 + ?/?0 ?/?0 2 + ?/?0 1 ?11= ? = ??? + ??? ?12= ? ? ? ? 1 0 1 ? 1 = ?21= ? = ??? + ?22= ??? EMLAB
Bandwidth of series resonator 5 2 1 ?? 1 ?21= = ?21 ? 2 + ?/?0 1 + 2?0 ?? 1 2=1 2= ?21 2 1 2?? 1 1 + ?? 4?0 1 ?? ?2?? 2??0? 1 = 0 ? ?0 ??= 2?0 ?2+?0 2= 0 ?? + ?0 ? ?0 Standard form of 2ndorder eq. 2 ? ?0 2 +1 ? 2 ?0?0? 1 = 0 + 1 = 0 ? ?0 ? ?0 ?0?0?2+ 1 = ?0?0? 2?0?0?=?0? 1 ?3?? ?0 = 2?0?0? =1 ???????= 2?0 ? EMLAB
Energy resonance 6 ?? ??= ?? Resonance : ? 1 4? ?2=1 1 1 ?2??2 ? ? = ?2? 4 ?? Unloaded Q-factor : resonator ? ??+ ?? ????? Energy stored Average Power loss= ?0 ? = ?0 ??=1 4? ?2 Average magnetic energy stored Average electric energy stored ??=1 4? ?2 1 4? ?2+1 1 1 2? +1 1 ?2??2 =?0? 2 ?2? 4 2 ? ??=1 ? =1 1 ? = ?0 = ?0 ?2??2 4? 1 2?2? ? ?? 4 EMLAB
7 = ?2?0 ?0 ?:1 1:? ?0 Z ?0 Source load transformer Z0 . EMLAB
Series resonator Qloaded 8 1 Z0= 100 ???????= 2?0?0? 0 -10 dB(S(2,1)) dB(S(1,1)) Q -20 -30 -40 1E7 1E8 1E9 1E10 5E10 freq, Hz Z0= 10 Q 0 -10 dB(S(2,1)) dB(S(1,1)) -20 -30 -40 1E7 1E8 1E9 1E10 5E10 freq, Hz EMLAB
Qloaded,Qunloaded of a series resonator 9 Unloaded Q resonator Q, loaded Q source Q. ?? ?? ?? +1 ? ??? ??? ????? ??= ?? = = ??? + ?2?? + 1= ? ? + ?? +1 ?2+? ?? ?? ?? ?2+?0 ?0 ?=? ? =?0? ? 2= 0, ?? + ?0 ? ? ? ???=?0? Q 2?0 ?????????=?0? Resonator Q ? 1 =2?0+ ? ?0? 1 1 = + External Q Q ??????? ??? ?? EMLAB
S-parameter of parallel resonators 10 1 ?0 ?0= ?0 ? 1 ? = ? ?? ?? ?/?0 2 + ?/?0 2 2 + ?/?0 2 2 + ?/?0 ?/?0 2 + ?/?0 1 ?11= ? = ??? + ??? ?12= ? ? ? ? 1 ? 1 ??? 0 1 ?21= = ? = ??? + ?22= EMLAB
Bandwidth of parallel resonator 11 2 1 ?? 1 ?21= = ? 2 + ?/?0 1 + 2?0 ?? 1 2=1 2= ?21 2 1 2?? 1 1 + ?? 4?0 1 ?? ?2?? 2??0? 1 = 0 ? ?0 ??= 2?0 ?21 2 ? ?0 2?0?0? 1 = 0 ? ?0 ?3?? ?0 ?0?0?2+ 1 = ?0?0? = 2?0?0? =1 ? 2?0?0?=?0? 1 ???????= 2?0 EMLAB
Qloaded,Qunloaded of a parallel resonator 12 ?? ??? ??? +1 ?? ???? ????= = = ?2?? +?? ?0 ?= 1 ?+ ?? +1 ?2+1 ?+ 1 ?? ?? ? ?2+?0 1 2= 0, ?? + ?0 ? = ?0?? ?? ?0 2 ???= ?0? ?????????= ?0?? 1 ?0+1 ???????= ?0? 1 ?0+1 ? 1 ?0+1 ?0+1 ?0? 1 ?? 1 1 ? = = + External Q Q ??? ?? EMLAB
Parallel resonator Qloaded 13 ?0 2 ???????= ?0? Z0= 10 0 Q -10 dB(S(2,1)) dB(S(1,1)) -20 -30 -40 1E7 1E8 1E9 1E10 5E10 freq, Hz Z0= 100 Q 0 -10 dB(S(2,1)) dB(S(1,1)) -20 -30 -40 -50 1E7 1E8 1E9 1E10 5E10 freq, Hz EMLAB
Series resonator (Qunloaded) 14 Losses in resonator Insertion loss . R = 30 [Ohm] Series resonator S21 . (loss ) R = 1 [Ohm] 0 0 -10 -10 dB(S(2,1)) dB(S(1,1)) dB(S(2,1)) dB(S(1,1)) -20 -20 -30 -30 -40 -40 1E7 1E8 1E9 1E10 5E10 1E7 1E8 1E9 1E10 5E10 freq, Hz freq, Hz EMLAB
Parallel resonator 15 Insertion loss . R = 100 [Ohm] Parallel resonator S21 . (loss ) R = 1k [Ohm] 0 0 -10 -10 dB(S(2,1)) dB(S(1,1)) dB(S(2,1)) dB(S(1,1)) -20 -20 -30 -30 -40 -40 1E7 1E8 1E9 1E10 5E10 1E7 1E8 1E9 1E10 5E10 freq, Hz freq, Hz EMLAB
Transmission line resonator : /2 + short 16 ??+ ??0tan?? ?0+ ???tan?? ???= ??0tan?? ???= ?0 ? = ?/2 ?? =2?? ?0 2=2?? ? = ?? = ? + ? ? ? ? 2?0 ?0 ?0 ???= ??0tan?? = ??0tan ? + ? ? = ??0tan ? ? ??0? ? tan? ? ( ? << 1) ?0 ?0 ?0 ? ?0+ 1 ? ?0 1 ???= ? + ?(? ?? + 1)(? ?? 1) ? ?0 ?0 1 ???= ? + ??? + ? + ? ?? ?0? ? + 2? = ? + 2?? ? ?0 = ? + 2?? ? 2? EMLAB
Transmission line resonator : /2 + short 17 tanh?? = ?tan? tan?? = ?tanh? : for lossy transmission line ???= ??0tan? ? ?? = ?0tanh(? + ??)? ? = ? + ?? short ? = ?/2 tanh?? + ?tan?? 1 + tanh?? ?tan?? Transform to series resonant circuit ???= ?0tanh(? + ??)? = ?0 1 ???= ? + ??? + ??? ? + 2?? ? Near resonance ? = ?0?? ? =?0? ?? =2?? ?0 2= ?? = ? + ? ? 2?0 1 ?0 ? ?0 tan?? = tan? ? ?0 ? = ? ? 2? tanh?? ??, ?0 ?0 ?0? 2 ?0??= ??? ?0 ?? + ? ? ? =?0? ? ? 2? = 2??= ?0 ? EMLAB
18 Figure 6.4 (p. 273) A short-circuited length of lossy transmission line, and the voltage distributions for n = 1 resonators. ( = ?/2) and n = 2( = ?) EMLAB
Transmission line resonator : /4 + short 19 short ? = ?/4 tanh?? + ?tan?? 1 + tanh?? ?tan?? ???= ?0tanh(? + ??)? = ?0 Transform to parallel resonant circuit 1 1 Near resonance, ???= = 1 ?+ ??2?? 1 1 1 ?+ ?2? ? ?0 1 ?+ ??? + 1 ?2 ?0 1 ?? =2?? ?0 4=? ? ?0 =? 2+? ? ?0 ??? ?? ? 2 2 = 2 1 ?+ ?? ? 2+? ? ?0 = cot? ? ?0 ?0 tan?? = tan 2 2 ? ?0 ? ?0 tanh?? ?cot? 1 + ?tanh?? tan? ? =?0 ? 1 2? 2 2 ? ?0 ???= ?0 = ?0 ??, ? = 4?0?0, ? = ? ?0 1 tanh?? ?cot? tanh?? + ?tan? ?0 2 2 ? ? 2? ?0 ? = ?0?? = 4??= ???= ?? +?? ? ?0 2 EMLAB
Open-circuited /2 line 20 ?0 ???= ?? + ? ? ? ?0 ? ? 2? ? = ?0?? = 2??= Figure 6.5 (p. 276) An open-circuited length of lossy transmission line, and the voltage distributions for n = 1 resonators. ( = ?/2) and n = 2( = ?) EMLAB
Coupling of resonators 21 EMLAB
Resonators- 22 0 -10 -20 dB(S(1,1)) dB(S(2,1)) -30 -40 -50 -60 0 50 100 150 200 250 300 freq, MHz 100 50 phase(S(1,1)) phase(S(2,1)) 0 -50 -100 Parallel resonant circuit -150 -200 0 50 100 150 200 250 300 freq, MHz EMLAB
