Insertion Loss Method

1
Lecture 9

• Insertion Loss Method
• Filter Transformation
• Filter Implementation
• Stepped-Impedance Low-Pass Filter
• Coupled Line Filters
• Filters Using Coupled Resonators

2
Proof of Kurodas Identity

• consider the second identity in which a series
  stub is converted into a shunt stub

3
Proof of Kurodas Identity

• for the short-circuited series stub has an series
  impedance of

4
Proof of Kurodas Identity

• the ABCD matrix of the series stub is
• the ABCD matrix of the transmission line is

5
Proof of Kurodas Identity

• the cascaded ABCD matrix is given by

6
Proof of Kurodas Identity

• the ABCD matrix for the transmission line with a
  characteristic impedance Z1 Zo is given by

7
Proof of Kurodas Identity

• the ABCD matrix of the open-circuit shunt stub is
  given by

8
Proof of Kurodas Identity

• the composite ABCD matrix for the figure on the
  right is given by

9
Proof of Kurodas Identity

• this implies that these two setups are identical
• Design a low-pass third-order maximally flat
  filter using only shunt stubs. The cutoff
  frequency is 8GHz and the impedance is 50 W.
• from table 9.3, g1 0.7654, g2 1.8478,
  g31.8478, g4 0.7654, g5 1

10
Proof of Kurodas Identity

• the lowpass filter prototype
• applying Richards transform

11
Proof of Kurodas Identity

• Add unit elements

12
Proof of Kurodas Identity

• use second Kuroda identity on left,
  (10.765)1.765,
• use first Kuroda identity on right,
  (1/11/1.307)1.765. 1/1.765 0.566,
• 
  ½.3070.433

13
Proof of Kurodas Identity
14
Proof of Kurodas Identity

• use the second Kuroda identity twice,
  (1.8480.567)2.415,
• (0.5670.5672/1.848) 0.741, (10.433)1.433,
  (112/0.433)3.309

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Proof of Kurodas Identity

• scale to 50 W
• all lines are l/8 long at 8 GHz

16
Impedance and Admittance Inverters

• to transform series connected elements to
  shunt-connected elements and vice versa

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Impedance and Admittance Inverters
18
Impedance and Admittance Inverters
19
Impedance and Admittance Inverters
20
Stepped-Impedance Low-Pass Filters

• stepped impedance, or hi-Z, low-Z filters are
  easier to design and take up less space than a
  similar filter using stubs
• limited to application when a sharp cutoff is not
  required

21
Stepped-Impedance Low-Pass Filters

• consider the lowpass filter depicted below
• we will replace the inductance and capacitance
  with a short length of transmission line
• consider the ABCD parameters of a transmission
  line with length

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Stepped-Impedance Low-Pass Filters
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Stepped-Impedance Low-Pass Filters

• AD-BC1, AD,
• for an equivalent T-circuit, its series elements
  are
• 
  )

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Stepped-Impedance Low-Pass Filters

• The shunt element is
• For small ,
• Resembles an inductor when
• Resembles a capicator

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Stepped-Impedance Low-Pass Filters

• therefore a short transmission line with a large
  impedance yields

26
Stepped-Impedance Low-Pass Filters

• a short transmission line with a small impedance
  yields

27
Coupled Line Filters

• a parallel coupled line section is shown below

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Coupled Line Filters

• the even- and odd-mode currents are
• and are the even-mode currents while and
  are the odd-mode currents

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Coupled Line Filters

• we will look at the open-circuit impedance matrix
  which has a bandpass response

30
Coupled Line Filters

• by superposition, we have
• if Ports 1 and 2 are driven by an even-mode
  current when Ports 3 and 4 are open, the
  impedance seen at Ports 1 and 2 is

31
Coupled Line Filters

• the voltage on either conductor due to source
  current i1 can be written as
• , 1 for OC

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Coupled Line Filters

• the voltage at Port 1 or 2 is
• Therefore,
• , similarly, the voltage due to i3 can be written
  as

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Coupled Line Filters

• using the same treatment, for the odd-mode
  excitation, we have

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Coupled Line Filters

• the total voltage is therefore given by the sum
  of all four contributions

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Coupled Line Filters

• from the relation between i and I, we have
• the above equation represent the first row of the
  impedance matrix for the open-circuit from
  symmetry, all other matrix element can be found

36
Coupled Line Filters
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Coupled Line Filters

• a two-port network can be formed from the coupled
  line section by terminating two of the four ports
  in either open or short circuits
• their performances are summarized in Table 9.8
• we will pay more attention to the case that I2
  I4 0

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Coupled Line Filters

• the impedance matrix equation now becomes

39
Coupled Line Filters

• the filter characteristic can be determined from
  the image impedance and the propagation constant
• if the line section is l/4 long

40
Design of Coupled Line Bandpass Filters

• narrowband bandpass filter can be designed by
  cascading open-circuit coupled line sections
• we first show that a single couple line section
  can be approximated by the equivalent circuit

41
Design of Coupled Line Bandpass Filters

• note the the admittance inverter is a
  transmission line of characteristic impedance 1/J
  and electrical length of -90o

42
Design of Coupled Line Bandpass Filters

• we calculate the image impedance and propagation
  constant of the equivalent circuit and show that
  they are approximately equal to those of the
  coupled line section for q p/2, which will
  correspond to the center frequency of the
  bandpass filter

43
Design of Coupled Line Bandpass Filters

• the ABCD matrix of the equivalent circuit is
  given cascading those of three sections of
  transmission line

44
Design of Coupled Line Bandpass Filters

• recall that the image impedance is given by
• 
  as AD

45
Design of Coupled Line Bandpass Filters

• for q p/2
• the propagation constant is

46
Design of Coupled Line Bandpass Filters

• therefore,
  and
• 
  for
• equating these equations yields
• 
  and

47
Design of Coupled Line Bandpass Filters

• we have related the coupled line parameters with
  its equivalent circuit
• Design a four-section coupled line bandpass
  filter with a maximally flat response. The
  passband is 3.00 to 3.50 GHz, and the impedance
  is 50 W. What is the attenuation at 2.9 GHz?

48
Design of Coupled Line Bandpass Filters

• N3,
• transform 2.9 GHz to normalized lowpass filter
  form
• , from Figure 9.26, 10.5 dB

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Design of Coupled Line Bandpass Filters

• the prototype values are given in Table 9.3 and

50
Design of Coupled Line Bandpass Filters

• n gn ZoJn Zoe Zoo
• 1 1.00 0.492 86.7 37.5
• 2 2.00 0.171 60.0 42.9
• 3 1.00 0.171 60.0 42.9
• 4 1.00 0.492 86.7 37.5

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Design of Coupled Line Bandpass Filters

• all lines are l/4 long at 3.25 GHz

52
Filters Using Coupled Resonators

• bandstop and bandpass filter can be designed
  using l/4 shunt transmission line resonators

53
Filters Using Coupled Resonators

• bandstop filter

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Filters Using Coupled Resonators

• Bandpass filter

55
Filters Using Coupled Resonators

• bandpass filter using capacitively-coupled
  resonators

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Filters Using Coupled Resonators

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Filters Using Coupled Resonators