1. Lecture 8 Periodic Structures Image Parameter Method
Insertion Loss Method
Filter Transformation
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2. Periodic Structures
periodic structures have passband and stopband characteristics and can be employed as filters
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3. Periodic Structures
consider a microstrip transmission line periodically loaded with a shunt susceptance b normalized to the characteristic impedance Zo:
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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,
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5. Periodic Structures
i.e.
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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
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7. Periodic Structures
therefore, or
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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
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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
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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
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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
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12. Periodic Structures
therefore, depending on the frequency, the periodic structure will exhibit either a passband or a stopband
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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
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14. Periodic Structures
when the periodic structure is terminated with a load ZL , the reflection coefficient at the load can be determined easily
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15. Periodic Structures Which is the usual result EE 41139
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16. Periodic Structures
it is useful to look at the k-b diagram (Brillouin) of the periodic structure
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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
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18. Filter Design by the Image Parameter Method
let us first define image impedance by considering the following two-port network
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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
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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
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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
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22. Filter Design by the Image Parameter Method
Given and , we have
, ,
if the network is symmetric, i.e., A = D, then
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23. Filter Design by the Image Parameter Method
if the two-port network is driven by a voltage source
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24. Filter Design by the Image Parameter Method
Similarly we have, , A = D for symmetric network
Define ,
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25. Filter Design by the Image Parameter Method
consider the low-pass filter
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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
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27. Filter Design by the Image Parameter Method
the ABCD matrix is given by
Image impedance
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28. Filter Design by the Image Parameter Method
Propagation constant
For the above T-network,
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29. Filter Design by the Image Parameter Method
Define a cutoff frequency as,
a nominal characteristic impedance Ro
, k is a constant
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30. Filter Design by the Image Parameter Method
the image impedance is then written as
the propagation factor is given as
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31. Filter Design by the Image Parameter Method
For , is real and which imply a passband
For , is imaginary and we have a stopband
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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
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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
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34. Filter Design by the Image Parameter Method
we choose Z2 so that ZiT remains the same
therefore, Z2 is given by
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35. Filter Design by the Image Parameter Method
recall that Z1 = jwL and Z2 = 1/jwC, the m-derived components are
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36. Filter Design by the Image Parameter Method
the propagation factor for the m-derived section is
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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
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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
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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
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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
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41. Filter Design by the Insertion Loss Method
, IL = 10 log
, , M and N are real polynomials
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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
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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
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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
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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 in the amplitude
For large x, and therefore for
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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
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47. Filter Design by the Insertion Loss Method
a linear phase characteristic can be achieved with the phase response
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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
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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
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50. Filter Design by the Insertion Loss Method
consider the low-pass filter prototype, N=2
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51. Filter Design by the Insertion Loss Method
assume a source impedance of 1 W and a cutoff frequency
the input impedance is given by
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52. Filter Design by the Insertion Loss Method
the reflection coefficient at the source impedance is given by
the power loss ratio is given by
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53. Filter Design by the Insertion Loss Method
compare this equation with the maximally flat equation, we have
R=1, which implies C = L as R = 1
which implies C = L =
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54. Filter Design by the Insertion Loss Method
for equal-ripple prototype, we have the power loss ratio
Since
Compare this with
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55. Filter Design by the Insertion Loss Method
we have or
note that R is not unity, a mismatch will result if the load is R=1; a quarter-wave transformer can be used to match the load
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56. Filter Design by the Insertion Loss Method
if N is odd, R = 1 as there is a unity power loss ratio at w = 0 of N being odd
Table 9.4 can be used for equal-ripple low-pass filter prototypes
Table 9.5 can be used for maximally flat time delay low-pass filter prototypes
after the filter prototypes have been designed, we need to perform impedance and frequency scaling
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57. Filter Transformations
impedance and frequency scaling
the source impedance is , the impedance scaled quantities are:
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58. Filter Transformations
both impedance and frequency scaling
low-pass to high-pass transformation ,
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59. Filter Transformations
Bandpass transmission
As a series indicator , is transformed into a series LC with element values
A shunt capacitor, , is transformed into a shunt LC with element values
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60. Filter Transformations
bandstop transformation
A series indicator, , is transformed into a parallel LC with element values
A shunt capacitor, , is transformed into a series LC with element values
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61. Filter Implementation
we need to replace lumped-elements by distributive elements:
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62. Filter Implementation
there are four Kuroda identities to perform any of the following operations:
physically separate transmission line stubs
transform series stubs into shunt stubs, or vice versa
change impractical characteristic impedances into more realizable ones
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63. Filter Implementation
let us concentrate on the first two
a shunt capacitor can be converted to a series inductor
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