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Lecture 8

Lecture 8. Periodic Structures Image Parameter Method Insertion Loss Method Filter Transformation. Periodic Structures. periodic structures have passband and stopband characteristics and can be employed as filters. Periodic Structures.

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Lecture 8

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  1. Lecture 8 • Periodic Structures • Image Parameter Method • Insertion Loss Method • Filter Transformation Microwave Technique

  2. Periodic Structures • periodic structures have passband and stopband characteristics and can be employed as filters Microwave Technique

  3. Periodic Structures • consider a microstrip transmission line periodically loaded with a shunt susceptance b normalized to the characteristic impedance Zo: Microwave Technique

  4. Periodic Structures • the ABCD matrix is composed by cascading three matrices, two for the transmission lines of length d/2 each and one for the shunt susceptance, Microwave Technique

  5. Periodic Structures • i.e. Microwave Technique

  6. Periodic Structures • q = kd, and k is the propagation constant of the unloaded line • AD-BC = 1 for reciprocal networks • assuming the the propagation constant of the loaded line is denoted by g, then Microwave Technique

  7. Periodic Structures • therefore, • or Microwave Technique

  8. Periodic Structures • for a nontrivial solution, the determinant of the matrix must vanish leading to • recall that AD-CB = 0 for a reciprocal network, then • Or Microwave Technique

  9. Periodic Structures • Knowing that, the above equation can be written as • since the right-hand side is always real, therefore, either a or b is zero, but not both Microwave Technique

  10. Periodic Structures • if a=0, we have a passband, b can be obtained from the solution to • if the the magnitude of the rhs is less than 1 Microwave Technique

  11. Periodic Structures • if b=0, we have a stopband, a can be obtained from the solution to • as cosh function is always larger than 1, a is positive for forward going wave and is negative for the backward going wave Microwave Technique

  12. Periodic Structures • therefore, depending on the frequency, the periodic structure will exhibit either a passband or a stopband Microwave Technique

  13. Periodic Structures • the characteristic impedance of the load line is given by • , + for forward wave and - for backward wavehere the unit cell is symmetric so that A = D • ZB is real for the passband and imaginary for the stopband Microwave Technique

  14. Periodic Structures • when the periodic structure is terminated with a load ZL , the reflection coefficient at the load can be determined easily Microwave Technique

  15. Periodic Structures • Which is the usual result Microwave Technique

  16. Periodic Structures • it is useful to look at the k-b diagram (Brillouin) of the periodic structure Microwave Technique

  17. Periodic Structures • in the region where b < k, it is a slow wave structure, the phase velocity is slow down in certain device so that microwave signal can interacts with electron beam more efficiently • when b = k, we have a TEM line Microwave Technique

  18. Filter Design by the Image Parameter Method • let us first define image impedance by considering the following two-port network Microwave Technique

  19. Filter Design by the Image Parameter Method • if Port 2 is terminated with Zi2, the input impedance at Port 1 is Zi1 • if Port 1 is terminated with Zi1, the input impedance at Port 2 is Zi2 • both ports are terminated with matched loads Microwave Technique

  20. Filter Design by the Image Parameter Method • at Port 1, the port voltage and current are related as • the input impedance at Port 1, with Port 2 terminated in , is Microwave Technique

  21. Filter Design by the Image Parameter Method • similarly, at Port 2, we have • these are obtained by taking the inverse of the ABCD matrix knowing that AB-CD=1 • the input impedance at Port 2, with Port 1 terminated in , is Microwave Technique

  22. Filter Design by the Image Parameter Method • Given and , we have • , , • if the network is symmetric, i.e., A = D, then Microwave Technique

  23. Filter Design by the Image Parameter Method • if the two-port network is driven by a voltage source Microwave Technique

  24. Filter Design by the Image Parameter Method • Similarly we have, , A = D for symmetric network • Define , Microwave Technique

  25. Filter Design by the Image Parameter Method • consider the low-pass filter Microwave Technique

  26. Filter Design by the Image Parameter Method • the series inductors and shunt capacitor will block high-frequency signals • a high-pass filter can be obtained by replacing L/2 by 2C and C by L in T-network Microwave Technique

  27. Filter Design by the Image Parameter Method • the ABCD matrix is given by • Image impedance Microwave Technique

  28. Filter Design by the Image Parameter Method • Propagation constant • For the above T-network, Microwave Technique

  29. Filter Design by the Image Parameter Method • Define a cutoff frequency as, • a nominal characteristic impedance Ro • , k is a constant Microwave Technique

  30. Filter Design by the Image Parameter Method • the image impedance is then written as • the propagation factor is given as Microwave Technique

  31. Filter Design by the Image Parameter Method • For , is real and which imply a passband • For , is imaginary and we have a stopband Microwave Technique

  32. Filter Design by the Image Parameter Method • this is a constant-k low pass filter, there are two parameters to choose (L and C) which are determined by wc and Ro • when , the attenuation is slow, furthermore, the image impedance is not a constant when frequency changes Microwave Technique

  33. Filter Design by the Image Parameter Method • the m-derived filter section is designed to alleviate these difficulties • let us replace the impedances Z1 with Microwave Technique

  34. Filter Design by the Image Parameter Method • we choose Z2 so that ZiT remains the same • therefore, Z2 is given by Microwave Technique

  35. Filter Design by the Image Parameter Method • recall that Z1 = jwL and Z2 = 1/jwC, the m-derived components are Microwave Technique

  36. Filter Design by the Image Parameter Method • the propagation factor for the m-derived section is Microwave Technique

  37. Filter Design by the Image Parameter Method • if we restrict 0 < m < 1, is real and >1 , for w > the stopband begins at w =as for the constant-k section • When w = , where • e becomes infinity and the filter has an infinite attenuation Microwave Technique

  38. Filter Design by the Image Parameter Method • when w > , the attenuation will be reduced; in order to have an infinite attenuation when , we can cascade a the m-derived section with a constant-k section to give the following response Microwave Technique

  39. Filter Design by the Image Parameter Method • the image impedance method cannot incorporate arbitrary frequency response; filter design by the insertion loss method allows a high degree of control over the passband and stopband amplitude and phase characteristics Microwave Technique

  40. Filter Design by the Insertion Loss Method • if a minimum insertion loss is most important, a binomial response can be used • if a sharp cutoff is needed, a Chebyshev response is better • in the insertion loss method a filter response is defined by its insertion loss or power loss ratio Microwave Technique

  41. Filter Design by the Insertion Loss Method • , IL = 10 log • , , M and N are real polynomials Microwave Technique

  42. Filter Design by the Insertion Loss Method • for a filter to be physically realizable, its power loss ratio must be of the form shown above • maximally flat (binomial or Butterworth response) provides the flattest possible passband response for a given filter order N Microwave Technique

  43. Filter Design by the Insertion Loss Method • The passband goes from to , beyond , the attenuation increases with frequency • the first (2N-1) derivatives are zero for and for , the insertion loss increases at a rate of 20N dB/decade Microwave Technique

  44. Filter Design by the Insertion Loss Method • equal ripple can be achieved by using a Chebyshev polynomial to specify the insertion loss of an N-order low-pass filter as Microwave Technique

  45. Filter Design by the Insertion Loss Method • a sharper cutoff will result; (x) oscillates between -1 and 1 for |x| < 1, the passband response will have a ripple of 1+ in the amplitude • For large x, and therefore for Microwave Technique

  46. Filter Design by the Insertion Loss Method • therefore, the insertion loss of the Chebyshev case is times of the binomial response for • linear phase response is sometime necessary to avoid signal distortion, there is usually a tradeoff between the sharp-cutoff response and linear phase response Microwave Technique

  47. Filter Design by the Insertion Loss Method • a linear phase characteristic can be achieved with the phase response Microwave Technique

  48. Filter Design by the Insertion Loss Method • a group delay is given by • this is also a maximally flat function, therefore, signal distortion is reduced in the passband Microwave Technique

  49. Filter Design by the Insertion Loss Method • it is convenient to design the filter prototypes which are normalized in terms of impedance and frequency • the designed prototypes will be scaled in frequency and impedance • lumped-elements will be replaced by distributive elements for microwave frequency operations Microwave Technique

  50. Filter Design by the Insertion Loss Method • consider the low-pass filter prototype, N=2 Microwave Technique

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