1. Chapter 8.1 Chapter 8.2 PERIODIC STRUCTURES
FILTER DESIGN BY THE IMAGE PARAMETER METHOD
Hogil Lee

2. 8.1 PERIODIC STRUCTURES
An infinite transmission line or waveguide periodically loaded with reactive elements.
Periodic structures support slow-wave propagation (slower than the phase velocity of the unloaded line), and have passband and stopband characteristics similar to those of filters;
they find application in traveling-wave tubes, masers*, phase shifters, and antennas.
*Maser : microwave amplification by stimulated emission of radiation

3. Analysis of Infinite Periodic Structures
𝑗𝑏=𝑗𝐵/ 𝑌 0
Where 𝜃=𝑘𝑑.
𝐴𝐷−𝐵𝐶=1. ∴ Reciprocal network.

4. Analysis of Infinite Periodic Structures
Wave propagating in the +𝑧 direction
When 𝑉 𝑛+1 and 𝐼 𝑛+1 have a solution that is not 0, the determinant is 0.

5. Analysis of Infinite Periodic Structures
𝑐𝑜𝑠ℎ𝛾𝑑 is real,
∴𝛼=0 𝑜𝑟 𝛽=0
Case 1:𝛼=0, 𝛽≠0. This case corresponds to a nonattenuated propagating wave on the periodic structure, and defines the passband of the structure.
𝑐𝑜𝑠𝜃− 𝑏 2 𝑠𝑖𝑛𝜃 ≤1
Case 2:𝛼≠0, 𝛽=0,𝜋. In this case the wave does not propagate, but is attenuated along the line; this defines the stopband of the structure.
𝛼>0, positively traveling waves; 𝛼<0, negatively traveling waves.
If 𝑐𝑜𝑠𝜃− 𝑏 2 𝑠𝑖𝑛𝜃≤−1, 𝛽=𝜋, d=𝜆/2.

6. Analysis of Infinite Periodic Structures
Depending on the frequency and normalized susceptance values, the periodically loaded line will exhibit either passbands or stopbands, and so can be considered as a type of filter.
These waves are similar to the elastic waves (Bloch waves) that propagate through periodic crystal lattices.
Bloch wave (Bloch state)
If the potential is a periodic function,
𝑈 𝑟 =𝑈 𝑟+𝑇
The wave function of the Schrodinger equation can be written as
𝜓 𝑘 𝑟 = 𝑢 𝑘 𝑟 𝑒 𝑖𝑘∙𝑟
<

7. Analysis of Infinite Periodic Structures
We can define a characteristic impedance (Bloch impedance) at the unit cell terminals as
𝐴=𝐷 for symmetric unit cells.
The ± solutions correspond to the characteristic impedance for positively and negatively traveling waves, respectively.

8. Analysis of Infinite Periodic Structures
Case 1:𝛼=0, 𝛽≠0. passband.
𝑐𝑜𝑠ℎ𝛾𝑑=𝐴≤1, 𝑍 𝐵 will be real.
Case 2:𝛼≠0, 𝛽=0. stopband.
𝑐𝑜𝑠ℎ𝛾𝑑=𝐴≥1, 𝑍 𝐵 will be imaginary.
This situation is similar to that for the wave impedance of a waveguide, which is real for propagating modes and imaginary for cutoff, or evanescent, modes.

9. Terminated Periodic Structures
A truncated periodic structure terminated in a load impedance 𝑍 𝐿 .
Replaced γ z in (8.3) with jβnd since we are interested only in terminal quantities.

10. Terminated Periodic Structures
A truncated periodic structure terminated in a load impedance 𝑍 𝐿 .
The reflection coefficient at the load
If the unit cell network is symmetric (A = D), 𝑍 𝐵 + = 𝑍 𝐵 −
In order to avoid reflections on the terminated periodic structure we must have 𝑍 𝐿 = 𝑍 𝐵 .

11. 𝒌-𝜷 Diagrams and Wave Velocities
When studying the passband and stopband characteristics of a periodic structure, it is useful to plot the propagation constant, β, versus the propagation constant of the unloaded line, k (or ω). Such a graph is called a k-β diagram, or Brillouin diagram.
The k-β diagram can be plotted from (8.9a),
𝑘<𝑘 𝑐 , non real solution for 𝛽, non propagating
𝑘>𝑘 𝑐 , real solution for 𝛽, propagating
𝑘 approaches 𝛽 for large values of 𝛽

12. 𝒌-𝜷 Diagrams and Wave Velocities
Phase velocity
Group velocity

13. EXAMPLE 8.1 ANALYSIS OF A PERIODIC STRUCTURE
𝒁 𝟎 =𝟓𝟎 𝛀
𝒅=𝟏.𝟎 𝒄𝒎
𝑪 𝟎 =𝟐.𝟔𝟔𝟔 𝒑𝑭
𝒇=𝟑.𝟎 𝑮𝑯𝒛
𝒌= 𝒌 𝟎
Passband
(0 ≤𝑘 0 𝑑≤15)
𝑐𝑜𝑠𝜃− 𝑏 2 𝑠𝑖𝑛𝜃 ≤1
Stopband

14. EXAMPLE 8.1 ANALYSIS OF A PERIODIC STRUCTURE

15. EXAMPLE 8.1 ANALYSIS OF A PERIODIC STRUCTURE
At 3 GHz,
𝛽𝑑=1.5 and 𝛽=150 rad/m
(Slow-wave)
Bloch impedance

16. 8.2 FILTER DESIGN BY THE IMAGE PARAMETER METHOD
The image parameter method of filter design involves the specification of passband and stopband characteristics for a cascade of simple two-port networks, and so is related in concept to the periodic structures of Section 8.1.
Matched
Matched

17. Image impedances and Transfer Functions for Two-Port Networks.

18. Image impedances and Transfer Functions for Two-Port Networks
𝑍 𝑖2 = 𝐷 𝑍 𝑖1 𝐴 . If the network is symmetric, then 𝐴=𝐷 and 𝑍 𝑖1 = 𝑍 𝑖2 as expected.

19. Image impedances and Transfer Functions for Two-Port Networks
𝑉 2 =𝐷 𝑉 1 − 𝐵𝐼 1 = 𝐷− 𝐵 𝑍 𝑖1 𝑉 1 (8.28a) since 𝑉 1 = 𝐼 1 𝑍 𝑖1
𝐼 2 =−𝐶 𝑉 1 + 𝐴𝐼 1 = −𝐶 𝑍 𝑖1 +𝐴 𝐼 1 (8.28b)
Propagation factor

20. Constant-k Filter Sections
Low-pass T-network

21. Constant-k Filter Sections
Since ,
Passband or stopband is determined according to the sign

22. Constant-k Filter Sections
Case 1:ω< 𝜔 𝑐 , passband.
𝑍 𝑖𝑇 is real, 𝛾 is imaginary, 𝑒 𝛾 =1.
Case 2: ω> 𝜔 𝑐 , stopband.
𝑍 𝑖𝑇 is imaginary, 𝛾 is real, 1< 𝑒 𝛾 <0.

23. Constant-k Filter Sections
Low-pass 𝝅-network
High-pass network
𝑍 𝑖𝜋 = 𝐿 𝐶 1− 𝜔 2 𝐿𝐶 4 = 𝑅 − 𝜔 2 𝜔 𝑐 2

24. m-Derived Filter Sections
Constant-k filter section suffers from the disadvantages of a relatively slow attenuation rate past cutoff, and a nonconstant image impedance. The m-derived filter section is a modification of the constant-k section designed to overcome these problems.
Value to obtain the same value of 𝑍 𝑖𝑇

25. m-Derived Filter Sections
Low-pass Filter
0<𝑚<1,
𝜔> 𝜔 𝑐 , stopband.
𝛾 is real, 𝑒 𝛾 >1.
When 𝜔= 𝜔 ∞ , where
1+ 𝑍 1 ′ 4 𝑍 2 ′ =∞, 𝑒 𝛾 =∞.

26. m-Derived Filter Sections
𝑍 2 ′ =0 (Resonance)
Sharp cutoff response
Attenuation decreases
The m-derived section can be cascaded with a constant-k section to give the composite attenuation response

27. m-Derived Filter Sections
Independent of m
Dependent of m
Chosen value to obtain the same value of 𝑍 𝑖𝑇

28. m-Derived Filter Sections
A value of m = 0.6 generally minimize the variation of 𝑍 𝑖𝜋

29. m-Derived Filter Sections

30. Composite Filters
By combining in cascade the constant-k, m-derived sharp cutoff and the m-derived matching sections we can realize a filter with the desired attenuation and matching properties.
≈ 𝑅 0

31. Composite Filters

32. EXAMPLE 8.2 LOW-PASS COMPOSITE FILTER DESIGN
𝑓 𝑐 =2 𝑀𝐻𝑧, 𝑅 0 =50 𝛺, 𝑓 ∞ =2.05 𝑀𝐻𝑧 (for the attenuation pole)
Constant-k section
m-derived sharp-cutoff section
m=0.6 matching sections

33. EXAMPLE 8.2 LOW-PASS COMPOSITE FILTER DESIGN
Sharp dip at f=2.05 MHz