Microwave Circuits 1

Microwave Filters (8) Periodic Structures

where the ABCD matrix represents the cascade of a section of length , a shunt susceptance , and another transmission line section of length .

where . Microwave Circuits 2

Assume the voltages and currents satisfy wave equation

Then,

or

For a nontrivial solution to exist, the following must be satisfied

That is . The of this wave is defined as

Also

Then Microwave Circuits 3

For symmetrical unit cells, , therefore

The solutions correspond to the positively and negatively traveling waves, respectively.

Due to reciprocity, , the above equation becomes . Let , then

1. . a. Nonattenuating, propagating wave. b. Passband.

c.

Has multiple solutions only when d. is pure real. 2. . a. Attenuating, nonpropagating wave. b. Stopband. c. Microwave Circuits 4

Has two solutions only when d. is pure imaginary.

Terminated Periodic Structures

if .

Example 8.1

, slow wave. Microwave Circuits 5

Filter Design by The Image Parameter Method(8.2)

= input impedance at port 1 when port is terminated with . = input impedance at port 1 when port is terminated with . From ABCD matrix, we have

Solving and gives

and . If symmetric, and . If port 2 is terminated in , Voltage ratio equals Microwave Circuits 6

Current ratio

Define propagation factor as or

Constant-k Filter Sections

Low pass filters

From Table 8.1, the image impedance equals

where and Microwave Circuits 7

1. : Passband. is real. is imaginary. . 2. : Stopband. is imaginary. is real. . Attenuation rate for is 40 dB/decade.

Disadvantages: 1. Slow cutoff. 2. Image impedance is a function of frequency.

High Pass Filters Microwave Circuits 8 Microwave Circuits 9 m-Derived Filter Sections

Let

In order to keep the image impedance the same

For a low pass filter, and . Then

where Microwave Circuits 10

If , is real, and for . Stopband. When , , implying infinite attenuation.

Disadvantages: 1. Slow attenuation when . 2. Image impedance is a function of frequency. Microwave Circuits 11

Erecta: 1. p. 434, in Table 8.1, the propagation

constant should be .

2. p. 434, Eq. 8.36, add absolute value to the term .

3. p. 437, Eq. 8.43, add absolute value to the term . Microwave Circuits 12 m-derived - section

Goals: 1. Provide the same form of the propagation factor of an m-derived T-section. 2. Provide a less frequency dependent image impedance.

Procedures: 1. Use Table 8.1, to find out the equivalent - section that give the same the propagation constants as the T-section. 2. Compute the image impedance.

Choose m=0.6 to minimize the variation. Microwave Circuits 13

3. Now . Split the - section to two parts to match . It can be proved that the image impedances equals to and respectively. 4. Let the of the original - section be and the split - section . It is obviously .

Composite Filters 1. From the given impedance and the cut-off frequency , determine the values of the and the capacitance by .

Then, the constant-k T section can be implemented. 2. Determine the m value from the infinite attenuation pole by Microwave Circuits 14

Then m-derived T section can be implemented. 3. Choose m=0.6 as the m value of the bisected - section.

Example 8.2: LOW-PASS COMPOSITE FILTER DESIGN Microwave Circuits 15

From Table 8.2 1. Constant-k T section

2. m-derived T section

3. m=0.6 matching section Microwave Circuits 16

Filter Design by the Insertion Loss Method (8.3)

Benefits: can synthesize a desired response systematically. Disadvantage: trade-off between the ideal and real responses.

Define: insertion loss or power loss ratio, by

Since is an even function,

Maximally flat:

1. Order: N 2. Low pass, binomial or Butterworth response. 3. Flattest passband. 4. : 5. Passband: . 6. Power loss ratio at : . For , 3-dB loss.

7. When , .

8. Attenuation rate: 9. The first (2N-1) derivatives are zero at

Equal ripple:

1. Order: N Microwave Circuits 17

2. Low pass. 3. is the Chebyshev polynomial of order N. 4. Equal ripple of amplitude .

5. When , .

6. Sharper cutoff than maximally flat types by a factor .

7. Attenuation rate: .

Linear Phase:

Then the group delay

is maximally flat. Microwave Circuits 18

Maximally Flat Low-Pass Filter Prototype

Consider the above circuit,

Since

We have

In order to fit the above equation to a maximally flat low-pass filter with , , and , that is, , It is required that , , and .

Solving for , and , we have Microwave Circuits 19

.

Procedures: 1. Referring to Fig. 8.26, determine the order from the required attenuation characteristics. 2. By looking up Table 8.3, determine the normalized component values by the following rules: a. is the generator resistance if the ladder circuit starts with a shunt capacitance, or generator conductance if starts with a series . b. is the inductance for series inductors, or capacitance for shunt . c. the load resistance if is a shunt , or load conductance if is a series . 3. Impedance and frequency scaling. Let and be the real generator resistance and cutoff frequency respectively, then Microwave Circuits 20

Example 8.3 Design a maximal flat low-pass filter with a cutoff frequency of 2 GHz, impedance of , and at least 15 dB insertion loss at 3 GHz Microwave Circuits 21 Microwave Circuits 22

Equal Ripple Low-Pass Filter Prototype Procedures: 1. Referring to Fig. 8.27, determine the order from the required attenuation characteristics and choose the desired ripple level. 2. Decide the component values by Table 8.4 and follow the same procedure in the maximal flat prototype. Note when N is even, the generator resistance and the load resistance are not equal. 3. Impedance and frequency scaling. Use the same procedure in the maximal flat prototype. Microwave Circuits 23 Microwave Circuits 24

Linear Phase Low-Pass Filter Prototypes Procedure: Same as previous prototypes except using Table 8.5.

Filter Transformation (8.4)

Goals: 1. Low pass  high pass. 2. Low pass  band pass. 3. Low pass  band stop.

Low Pass  High Pass Let , then the low pass response becomes a high pass Microwave Circuits 25 response. Procedure: 1. Convert the in the low pass prototypes to capacitances as follows

2. Convert the capacitances in the low pass prototypes to inductances as follows

3. Include the effect of impedance scaling

Low Pass  Band Pass

Let , where and . Then,

which is a band pass response with cutoff at and .

An inductance in the low pass prototype will converts to a series inductance and a series capacitance as follow Microwave Circuits 26

An capacitance in the low pass prototype will converts to a parallel inductance and a parallel capacitance as follow

Notice that both pairs of inductances and capacitance have the same resonant frequency .

Low Pass  Band Stop

Similarly, Let .

Convert an inductance in the low pass prototype to a parallel inductance and a parallel capacitance as follow

Convert an capacitance in the low pass prototype to a series inductance and a series capacitance as follow Microwave Circuits 27

To sum up Microwave Circuits 28

Example 8.4: BANDPASS FILTER DESIGN N=3, 0.5 dB equal ripple, , , . Microwave Circuits 29

Filter Implementation (8.5)

Richard’s Transformation

Choose at such that and . A zero occur at . Kuroda’s identities • Physically separate transmission line stubs. • Transform series stubs into shunt stubs, or vice versa. • Change impractical characteristic impedance into more realizable ones. Microwave Circuits 30

Consider Table 8.7(a), Microwave Circuits 31 where .

Since the left and the right ABCD matrix must be the same, we have

Example 8.5 LOW-PASS FILTER DESIGN USING STUBS , 3 dB equal ripple. Microwave Circuits 32

Impedance and Inverters Microwave Circuits 33

Stepped-Impedance Low-Pass Filters (8.6)

Convert the ABCD matrix of a short transmission line to Z- parameters, we have

Using T equivalent circuit, we have

If is large

. If is small

To sum up, Microwave Circuits 34

Inductor:

Capacitor:

Example 8.7 STEPPED IMPEDANCE FILTER DESIGN , 20 dB attenuation at 4 GHz. Maximal flat. . Substrate: Microwave Circuits 35

Filters Using Coupled Resonators (8.8)

Bandstop and Bandpass Filters Using Quarter-wave Resonators

For a open-circuit transmission line, where for . Let , where . Assume ideal transmission line, then . Now

For a series LC circuit near resonance

Thus Microwave Circuits 36

For the circuit Microwave Circuits 37

Which should equal to the bandstop prototype

To make the two values the same, we need

Solving for the above equations, we have

Then,

where .

To sum up, bandstrop filter: open stub with , Microwave Circuits 38

bandpass filter: short stub with

Example 8.8 BANDSTOP FILTER DESIGN , N=3, 0.5 dB equal ripple. n

1 1.5963 265.9 2 1.0967 387.0 3 1.5963 265.9 Microwave Circuits 39

Coupled Line Filters

Even-Odd Mode Analysis Denote the left side as port 1 and the right side port 2. Let be the even-odd mode voltages and currents. Goal: find out the impedance matrices 1. Even mode: Microwave Circuits 40

2. Odd mode:

To sum up,

Since

We have Microwave Circuits 41

Let port 2 and 4 open, we have

Computing the image impedance

If ,  which is real and positive since

. Microwave Circuits 42

. Cutoff frequency

Propagation constant

Passband is real for

The image impedance of the above is

At , . The propagation constant is

Comparing to the original equations, we have

with . Solve for the even and odd mode impedance to give Microwave Circuits 43 Microwave Circuits 44

The shunt impedance

For a parallel LC circuit near resonance

Matching to the bandpass prototype, we have

In general: Microwave Circuits 45 Microwave Circuits 46

Example 8.7 COUPLED LINE BANDPASS FILTER DESIGN , N=3, 0.5 dB equal ripple. n

1 1.5963 0.3137 70.61 39.24 2 1.0967 0.1187 56.64 44.77 3 1.5963 0.1187 56.64 44.77 4 1.0000 0.3137 70.61 39.24 Microwave Circuits 47

Bandpass Filters Using Capacitive Coupled Resonators Microwave Circuits 48

Example 8.10 CAPACITIVELY COUPLED SERIES RESONATOR BANDPASS FILTER DESIGN , N=3, 0.5 dB equal ripple. n

1 1.5963 0.3137 0.554 155.8 2 1.0967 0.1187 0.192 166.5 3 1.5963 0.1187 0.192 155.8 4 1.0000 0.3137 0.554 Microwave Circuits 49

Bandpass Filters Using Capacitively Coupled Shunt Resonators Microwave Circuits 50

Similar to 8.8 bandstop filter,

From Fig. 8.38 Microwave Circuits 51

Example 8.10 CAPACITIVELY COUPLED SHUNT RESONATOR BANDPASS FILTER DESIGN N=3, 0.5 dB equal-ripple, , , . n

1 1.5963 0.2218 0.2896 -0.3652 -0.04565 73.6 2 1.0967 0.0594 0.0756 -0.1512 -0.0189 83.2 3 1.5963 0.0594 0.0756 -0.3652 -0.04565 73.6 4 1.0000 0.2218 0.2896