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Lecture 9. Insertion Loss Method Filter Transformation Filter Implementation Stepped-Impedance Low-Pass Filter Coupled Line Filters Filters Using Coupled Resonators. Proof of Kuroda’s Identity. consider the second identity in which a series stub is converted into a shunt stub.

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lecture 9
Lecture 9
  • Insertion Loss Method
  • Filter Transformation
  • Filter Implementation
  • Stepped-Impedance Low-Pass Filter
  • Coupled Line Filters
  • Filters Using Coupled Resonators

Microwave Technique

proof of kuroda s identity
Proof of Kuroda’s Identity
  • consider the second identity in which a series stub is converted into a shunt stub

Microwave Technique

proof of kuroda s identity3
Proof of Kuroda’s Identity
  • for the short-circuited series stub has an series impedance of

Microwave Technique

proof of kuroda s identity4
Proof of Kuroda’s Identity
  • the ABCD matrix of the series stub is
  • the ABCD matrix of the transmission line is

Microwave Technique

proof of kuroda s identity5
Proof of Kuroda’s Identity
  • the cascaded ABCD matrix is given by

Microwave Technique

proof of kuroda s identity6
Proof of Kuroda’s Identity
  • the ABCD matrix for the transmission line with a characteristic impedance Z1 + Zo is given by

Microwave Technique

proof of kuroda s identity7
Proof of Kuroda’s Identity
  • the ABCD matrix of the open-circuit shunt stub is given by

Microwave Technique

proof of kuroda s identity8
Proof of Kuroda’s Identity
  • the composite ABCD matrix for the figure on the right is given by

Microwave Technique

proof of kuroda s identity9
Proof of Kuroda’s 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, g3=1.8478, g4 = 0.7654, g5 = 1

Microwave Technique

proof of kuroda s identity10
Proof of Kuroda’s Identity
  • the lowpass filter prototype
  • applying Richard’s transform

Microwave Technique

proof of kuroda s identity11
Proof of Kuroda’s Identity
  • Add unit elements

Microwave Technique

proof of kuroda s identity12
Proof of Kuroda’s Identity
  • use second Kuroda identity on left, (1+0.765)=1.765,
  • use first Kuroda identity on right, (1/1+1/1.307)=1.765. 1/1.765 = 0.566,

½.307=0.433

Microwave Technique

proof of kuroda s identity13
Proof of Kuroda’s Identity

Microwave Technique

proof of kuroda s identity14
Proof of Kuroda’s Identity
  • use the second Kuroda identity twice, (1.848+0.567)=2.415,
  • (0.567+0.5672/1.848)= 0.741, (1+0.433)=1.433, (1+12/0.433)=3.309

Microwave Technique

proof of kuroda s identity15
Proof of Kuroda’s Identity
  • scale to 50 W
  • all lines are l/8 long at 8 GHz

Microwave Technique

impedance and admittance inverters
Impedance and Admittance Inverters
  • to transform series connected elements to shunt-connected elements and vice versa

Microwave Technique

stepped impedance low pass filters
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

Microwave Technique

stepped impedance low pass filters21
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

Microwave Technique

stepped impedance low pass filters23
Stepped-Impedance Low-Pass Filters
  • AD-BC=1, A=D,
  • for an equivalent T-circuit, its series elements are

)

Microwave Technique

stepped impedance low pass filters24
Stepped-Impedance Low-Pass Filters
  • The shunt element is
  • For small ,
  • Resembles an inductor when
  • Resembles a capicator

Microwave Technique

stepped impedance low pass filters25
Stepped-Impedance Low-Pass Filters
  • therefore a short transmission line with a large impedance yields

Microwave Technique

stepped impedance low pass filters26
Stepped-Impedance Low-Pass Filters
  • a short transmission line with a small impedance yields

Microwave Technique

coupled line filters
Coupled Line Filters
  • a parallel coupled line section is shown below

Microwave Technique

coupled line filters28
Coupled Line Filters
  • the even- and odd-mode currents are
  • and are the even-mode currents while and are the odd-mode currents

Microwave Technique

coupled line filters29
Coupled Line Filters
  • we will look at the open-circuit impedance matrix which has a bandpass response

Microwave Technique

coupled line filters30
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

Microwave Technique

coupled line filters31
Coupled Line Filters
  • the voltage on either conductor due to source current i1 can be written as
  • , = 1 for OC

Microwave Technique

coupled line filters32
Coupled Line Filters
  • the voltage at Port 1 or 2 is
  • Therefore,
  • , similarly, the voltage due to i3 can be written as

Microwave Technique

coupled line filters33
Coupled Line Filters
  • using the same treatment, for the odd-mode excitation, we have

Microwave Technique

coupled line filters34
Coupled Line Filters
  • the total voltage is therefore given by the sum of all four contributions

Microwave Technique

coupled line filters35
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

Microwave Technique

coupled line filters36
Coupled Line Filters

Microwave Technique

coupled line filters37
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

Microwave Technique

coupled line filters38
Coupled Line Filters
  • the impedance matrix equation now becomes

Microwave Technique

coupled line filters39
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

Microwave Technique

design of coupled line bandpass filters
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

Microwave Technique

design of coupled line bandpass filters41
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

Microwave Technique

design of coupled line bandpass filters42
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

Microwave Technique

design of coupled line bandpass filters43
Design of Coupled Line Bandpass Filters
  • the ABCD matrix of the equivalent circuit is given cascading those of three sections of transmission line

Microwave Technique

design of coupled line bandpass filters44
Design of Coupled Line Bandpass Filters
  • recall that the image impedance is given by
  • as A=D

Microwave Technique

design of coupled line bandpass filters45
Design of Coupled Line Bandpass Filters
  • for q = p/2
  • the propagation constant is

Microwave Technique

design of coupled line bandpass filters46
Design of Coupled Line Bandpass Filters
  • therefore, and

for

  • equating these equations yields
  • and

Microwave Technique

design of coupled line bandpass filters47
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?

Microwave Technique

design of coupled line bandpass filters48
Design of Coupled Line Bandpass Filters
  • N=3,
  • transform 2.9 GHz to normalized lowpass filter form:
  • , from Figure 9.26, = 10.5 dB

Microwave Technique

design of coupled line bandpass filters49
Design of Coupled Line Bandpass Filters
  • the prototype values are given in Table 9.3 and

Microwave Technique

design of coupled line bandpass filters50
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

Microwave Technique

design of coupled line bandpass filters51
Design of Coupled Line Bandpass Filters
  • all lines are l/4 long at 3.25 GHz

Microwave Technique

filters using coupled resonators
Filters Using Coupled Resonators
  • bandstop and bandpass filter can be designed using l/4 shunt transmission line resonators

Microwave Technique

filters using coupled resonators53
Filters Using Coupled Resonators
  • bandstop filter

Microwave Technique

filters using coupled resonators54
Filters Using Coupled Resonators
  • Bandpass filter

Microwave Technique

filters using coupled resonators55
Filters Using Coupled Resonators
  • bandpass filter using capacitively-coupled resonators

Microwave Technique