Lecture 5

1. Lecture 5 • Microwave Resonators • Series and Parallel Resonant Circuits • Transmission Line Resonators • A Gap-Coupled Microstrip Resonator EE 41139

2. Microwave Resonators • microwave resonators are used in many applications • filters, oscillators, frequency meters, tuned amplifiers • its operations are very similar to the series and parallel RLC resonant circuits EE 41139

3. Microwave Resonators • we will review the series and parallel RLC ciruits and discuss the implementation of the microwave resonators using distributive elements such as the microstrip line, rectangular and circular cavities, etc. EE 41139

4. Series and Parallel RLC Circuits • consider the series RLC resonator shown below: EE 41139

5. Series and Parallel RLC Circuits • the input impedance Zin is given by • -------(1) • the average complex power delivered to the resonator is EE 41139

6. Series and Parallel RLC Circuits • the average power dissipated by the resistor is • recall that the energy stored in the inductor is • the time-averaged energy stored in the inductor is EE 41139

7. Series and Parallel RLC Circuits • similarly, the time-averaged energy stored in the capacitor is • the input impedance can then be expressed as follows: • -----(2) EE 41139

8. Series and Parallel RLC Circuits • at resonance, the average stored magnetic and electric energies are equal, therefore, we have • and the resonance frequency is defined as EE 41139

9. Series and Parallel RLC Circuits • the quality factor is defined as the product of the angular frequency and the ratio of the average energy stored to energy loss per second • Q is a measure of loss of a resonant circuit, lower loss implies higher Q and high Q implies narrower bandwidth EE 41139

10. Series and Parallel RLC Circuits • at resonance We = Wm and we have •       ----(3) • when R decreases Q increases as R dictates the power loss EE 41139

11. Series and Parallel RLC Circuits • the input impedance can be rewritten in the following form: EE 41139

12. Series and Parallel RLC Circuits • near by the resonance, i.e., , we can define and • ----(4) EE 41139

13. Series and Parallel RLC Circuits • the above form is useful for finding equivalent circuit near the resonance, for example, we can find out the resistance at resonance and so as L • for practical resonators, the loss is small; therefore, we can start with the lossless case and include the effect of the loss afterward EE 41139

14. Series and Parallel RLC Circuits • Consider the equation • As • ------(5) EE 41139

15. Series and Parallel RLC Circuits • From the EQ.4, when R = 0 for the lossless case, therefore, we can define a complex effective frequency • ----(6) so that, • --- (7) to incorporate the loss EE 41139

16. Series and Parallel RLC Circuits • From EQ.4 we have • the half-power fractional bandwidth (Zin=R/ ) is • And therefore Q = 1/BW ---(8) EE 41139

17. Series and Parallel RLC Circuits • now let us turn our attention to the parallel RLC resonator: EE 41139

18. Series and Parallel RLC Circuits • The input impedance is equal to • -----(9) • At resonance, and • , same results as in series RLC EE 41139

19. Series and Parallel RLC Circuits • the quality factor, however, is different EE 41139

20. Series and Parallel RLC Circuits • contrary to series RLC, the Q of the parallel RLC increases as R increases • similar to series RLC, we can derive an approximate expression for parallel RLC near resonance EE 41139

21. Series and Parallel RLC Circuits • Given EE 41139

22. Series and Parallel RLC Circuits • ----(11) EE 41139

23. Series and Parallel RLC Circuits • similar to the series RLC case, the effect of the loss can be incorporated into the lossless result by defining a complex frequency equal to • -----(12) • as in the series case, the half-power bandwidth is given by BW=1/Q --- (13) EE 41139

24. Loaded and Unloaded Q • Q defined above is a characteristic of the resonant circuit, this will change when the circuit is connected to a load EE 41139

25. Loaded and Unloaded Q • if the load is connected with the series RLC, the resistance in the series RLC is given by R’=R+RL, the corresponding quality factor QL becomes EE 41139

26. Loaded and Unloaded Q --- (13)  • on the other hand, if the load is connected with the parallel RLC, we have 1/R’=1/R+1/RL EE 41139

27. Loaded and Unloaded Q -----------(14) EE 41139

28. Transmission Line Resonators • we discuss the use of transmission lines to realize the RLC resonator • for a resonator, we are interested in Q and therefore, we need to consider lossy transmission lines EE 41139

29. Short-Circuited l/2 Line • Consider the transmission line equation • for the transmission line shown below: • for a short-circuited line EE 41139

30. Short-Circuited l/2 Line • Given , • , • And tanh(jx)=jtan(x) • -------(15) EE 41139

31. Short-Circuited l/2 Line • note that • tanh(A+B)=(tanh A + tanh B)/(1+ tanh A tanh B), EQ. (15) becomes • our goal here is to compare the above equation with either EQ. (4) or EQ. (11) so that we can find out the corresponding R, L and C EE 41139

32. Short-Circuited l/2 Line • Assuming a TEM line so that • , for • We have • knowing that tan d = d when d is small EE 41139

33. Short-Circuited l/2 Line • note that the loss is usually very small and therefore, the input impedance can be rewritten as: • --------(16) EE 41139

34. Short-Circuited l/2 Line • comparing EQ. (16) and EQ. (4) where • ,we have • ----------(17) • at resonance, Zin = R and this will occur for l = nl/2---(18) EE 41139

35. Short-Circuited l/2 Line • The quality factor is given by • ----(19) • Q increases as the attenuation decreases EE 41139

36. Short-Circuited l/4 Line • recall that the input impedance of a short-circuited line is given by EE 41139

37. Short-Circuited l/4 Line • For , we have EE 41139

38. Short-Circuited l/4 Line • The input impedance can be written as, • --------(20) EE 41139

39. Short-Circuited l/4 Line • comparing EQ. (16) and EQ. (11) where • ,we have • ---(21) EE 41139

40. Short-Circuited l/4 Line • The quality is given by, • ---(22) • same as short-circuited l/2 line, Q increases as the attenuation decreases EE 41139

41. Open-Circuited l/2 Line • Consider the transmission line equation, • for the transmission line shown below: EE 41139

42. Open-Circuited l/2 Line • for an open-circuited line EE 41139

43. Open-Circuited l/2 Line • the input impedance for the open-circuited l/2 line can be rewritten as: • ------(23) EE 41139

44. Open-Circuited l/2 Line • comparing EQ. (23) and EQ. (11) where • , we have • -----(24) EE 41139

45. Open-Circuited l/2 Line • at resonance, Zin = R and this will occur for l = nl/2---(25) • the quality factor is given by • ----(26) • Q increases as the attenuation decreases, same as the short-circuited l/2 line EE 41139

46. Coupling to Resonators • to obtain maximum power transfer between a resonator and a feed line, the resonator must be matched to the feed at the resonant frequency • let us consider a series resonant circuit with the input impedance given by • EQ. (4), EE 41139

47. Coupling to Resonators • The unload Q is given by   and for a matched load, R = Zo • the external Qe is given by where RL = Zo • for the resonator is matched to the feed at the resonant frequency, the external Qe is equal to Q, this is what we refer as critical coupling EE 41139

48. Coupling to Resonators • a coupling coefficient g is defined as • ----(27) • g < 1, undercoupled • g = 1, critically coupled • g > 1, overcoupled EE 41139

49. Design of a Gap-Coupled Microstrip Resonator • Objectives: • to design and fabricate a l/2 open-circuited microstrip resonator with a gap coupled microstrip feed line, the resonance frequency should be close to 5 GHz. EE 41139

50. Design of a Gap-Coupled Microstrip Resonator • to investigate overcoupled, critically coupled and undercoupled resonators • to use closed-form formulas for circuit design and simulations EE 41139