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Chapter 6: Microwave Resonators

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Chapter 6: Microwave Resonators Chapter 6: Microwave Resonators 1) Applications 2) Series and Parallel Resonant Circuits 3) Loaded and Unloaded Q 4) Transmission Line Resonators 5) Rectangular Wave guide Cavities 6) Q of the TE10l Mode 7) Resonator Coupling 8) Gap Coupled Resonator 9) Excitation of Resonators 10) Summary Microwave Resonators Applications Microwave resonators are used in a variety of applications such as: • Filters • Oscillators • Frequency meters • Tuned Amplifier The operation of microwave resonators are very similar to that of the lumped-element resonators of circuit theory, thus we will review the basic of series and parallel RLC resonant circuits first. We will derive some of the basic properties of such circuits. Series Resonant Circuit Near the resonance frequency, a microwave resonator can be modeled as a series or parallel RLC lumped-element equivalent circuit. 1 ZRjLj in C 1 2 1 2 1 Pin I Zin I R j L j 2 2 C 1 Average magnetic energy: W I 2 L m 4 1 1 Average electric energy: W I 2 e 4 2 C Resonance occurs when the average A series RLC resonator and its response. (a) The series RLC circuit. stored magnetic (Wm) and electric (b) The input impedance magnitude energies (We) are equal and Zin is purely versus frequency. real. Series Resonators • The frequency in which 1 o LC is called the resonant frequency. • Another important factor is the Quality Factor Q. Average Energy Stored Q Energy Loss / Second L 1 BW 1/ Q Q o R o RC Near the resonance o 1 22 ZRjL(1 ) RjL (o ) in 22LC A series RLC resonator and its 2RQ response. (a) The series RLC circuit. Rj (b) The input impedance magnitude o 2 2 versus frequency. o o o 2 ZRjLin 2 Series Resonators The input impedance magnitude versus frequency. L 1 Q o R o RC BW Fractional Bandwidth is defined as: BWQ 1/ o 2 and happens when the average (real) power delivered to the circuit is one-half that delivered at the resonance. • Bandwidth increases as R increases. • Narrower bandwidth can be achieved at higher quality factor (Smaller R). Parallel Resonators 1 11 ZjCin RjL Resonance occurs when the average stored magnetic and electric energies are equal and Zin is purely real. The input impedance at resonance is equal to R. Average Energy Stored Q Energy Loss / Second 1 R o QRC o A parallel RLC resonator and its LC o L response. (a) The parallel RLC circuit. (b) The input impedance magnitude Note: Resonance frequency is equal to the versus frequency. series resonator case. Q is inversed. Parallel Resonators The input impedance magnitude versus frequency. R QRCo BW 1/ Q o L • Bandwidth reduces as R increases. • Narrower bandwidth can be achieved at higher quality factor (Larger R). 1 1 R Close to the resonance frequency: Zin 2 j C R 1 2 j Q / o o 1 R Zin jC2( o ) Loaded and Unloaded Q Factor The Q factors that we have calculated were based on the characteristic of the resonant circuit itself, in the absence of any loading effect (Unloaded Q). A resonant circuit connected to an external load, RL. In practice a resonance circuit is always connected to another circuitry, which will always have the effect of lowering the overall Q (Loaded Q). o L for series connection RL Qe R L for parallel connection o L Loaded Q Factor If the resonator is a series RLC and coupled to an external load resistor RL, the effective resistance is: R e R R L If the resonator is a parallel RLC and coupled to an external load resistor RL, the effective resistance is: R e R R L / R R L Then the loaded Q can be written as: 111 QQQL e Note: Loaded Q factor is always smaller than Unloaded Q. Transmission Line Resonators (Short Circuited) • Ideal lumped element (R, L and C) are usually impossible to find at microwave frequencies. • We can design resonators with transmission line sections with different lengths and terminations (Open or Short). • Since we are interested in the Q of these resonators we will consider the Lossy Transmission Line. For the special case of : / 2 Z in Z o tanh j tanh j tan Z Z A short-circuited length of lossy in o 1 j tanh tanh transmission line. L Q o R 2 Transmission Line Resonators (Short Circuited) n • The resonance occurs for n 1,2,3, 2 ZZin o[/] j o L Q o R 2 A short-circuited length of lossy transmission line and the voltage distributions for n=1, 2 Transmission Line resonators Zo Z in j /2 o Z R o 1 C L 2 4 ooZ o C Q 2 An open-circuited length of lossy transmission line, and the voltage distributions for n = 1 resonators. Rectangular waveguide cavity resonator We can look at them as short circuit section of transmission line 22 2 mn mn ab Boundry conditions enforced EExy0 for z 0, d l mn dl mn l 1,2,3... d 2 222 lmn 2 dab 2 2 2 2f l m n c d a b then the resonance frequency A rectangular resonant cavity, and the electric field 222 distributions for the TE and TE resonant cl m n 101 102 f modes. 2 dab if b a d the lowest and dominant resonant TE (resp TM) mode will be TE101 (resp. TM110) Rectangular waveguide cavity resonator Q factor for TE10l 1 11 Q QQcd 1 Q d tan 3 2ad 1 Q c 2 23 3 23 3 2 Rs 22lab bd lad ad Rso 2 Resonators - coupling R Critical coupling occurs when 2Q Q Q L e u Z Zo i n Q If we define coupling coefficient g u then we have Qe series RLC parallel RLC • undercoupled resonator if g<1 Z0 R • critically coupled resonator if g=1 (resonator matched to the feed line) g g R Z • overcoupled resonator if g>1 0 g 1 g 1 g 1 Series RLC circuit Gap coupled microstrip resonator resonators 2 Serial RLC coupling capacitor acts as impedance inverter Z tan l b c z j where bc ZoC Zo bc tan l Resonance occurs when tanlb c 0 The coupling of the feed line to the resonator lowers its resonant frequency Gap coupled microstrip resonator resonators () o ZZ() o 22 j 2Qbcoc b RZ o 2 bc 2Qbc 2Q Zo the coupling coeficient g R 2 2Qbuc b for critical coupling Zo R c 2Q for undercoupled resonator bc 2Q for overcoupled resonator bc 2Q .
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