Periodic Structures

1
Lecture 8

• Periodic Structures
• Image Parameter Method
• Insertion Loss Method
• Filter Transformation

2
Periodic Structures

• periodic structures have passband and stopband
  characteristics and can be employed as filters

3
Periodic Structures

• consider a microstrip transmission line
  periodically loaded with a shunt susceptance b
  normalized to the characteristic impedance Zo

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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Periodic Structures

• i.e.

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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

7
Periodic Structures

• therefore,
• or

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

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

10
Periodic Structures

• if a0, we have a passband, b can be obtained
  from the solution to
• if the the magnitude of the rhs is less than 1

11
Periodic Structures

• if b0, 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

12
Periodic Structures

• therefore, depending on the frequency, the
  periodic structure will exhibit either a passband
  or a stopband

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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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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Periodic Structures

• Which is the usual result

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Periodic Structures

• it is useful to look at the k-b diagram
  (Brillouin) of the periodic structure

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Periodic Structures

• in the region where b lt 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

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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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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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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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-CD1
• the input impedance at Port 2, with Port 1
  terminated in , is

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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Filter Design by the Image Parameter Method

• if the two-port network is driven by a voltage
  source

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Filter Design by the Image Parameter Method

• Similarly we have,
  , A D for symmetric network
• Define ,

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Filter Design by the Image Parameter Method

• consider the low-pass filter

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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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Filter Design by the Image Parameter Method

• the ABCD matrix is given by
• Image impedance

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Filter Design by the Image Parameter Method

• Propagation constant
• For the above T-network,

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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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Filter Design by the Image Parameter Method

• the image impedance is then written as
• the propagation factor is given as

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Filter Design by the Image Parameter Method

• For , is real and
  which imply a passband
• For , is imaginary and
  we have a stopband

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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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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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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Filter Design by the Image Parameter Method

• recall that Z1 jwL and Z2 1/jwC, the
  m-derived components are

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Filter Design by the Image Parameter Method

• the propagation factor for the m-derived section
  is

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Filter Design by the Image Parameter Method

• if we restrict 0 lt m lt 1, is real and
  
• gt1 , for w gt 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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Filter Design by the Image Parameter Method

• when w gt , 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

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

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

41
Filter Design by the Insertion Loss Method

• , IL 10 log
• , , M and N are
  real polynomials

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

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 and for
  , the insertion loss increases at a
  rate of 20N dB/decade

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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Filter Design by the Insertion Loss Method

• a sharper cutoff will result (x)
  oscillates between -1 and 1 for x lt 1, the
  passband response will have a ripple of 1
  in the amplitude
• For large x,
  and therefore
  for

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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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Filter Design by the Insertion Loss Method

• a linear phase characteristic can be achieved
  with the phase response

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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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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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Filter Design by the Insertion Loss Method

• consider the low-pass filter prototype, N2

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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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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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Filter Design by the Insertion Loss Method

• compare this equation with the maximally flat
  equation, we have
• R1,
  which implies C L as R 1
• which implies C L

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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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Filter Design by the Insertion Loss Method

• we have or
• note that R is not unity, a mismatch will result
  if the load is R1 a quarter-wave transformer
  can be used to match the load

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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

57
Filter Transformations

• impedance and frequency scaling
• the source impedance is , the impedance scaled
  quantities are

58
Filter Transformations

• both impedance and frequency scaling
• low-pass to high-pass transformation
• ,

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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

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

61
Filter Implementation

• we need to replace lumped-elements by
  distributive elements

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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

63
Filter Implementation

• let us concentrate on the first two
• a shunt capacitor can be converted to a series
  inductor