1. Chebyshev Multi-section Matching
The Chebyshev transformer is optimum bandwidth to allow ripple within the passband response, and is known as equally ripple.
Larger bandwidth than that of binomial matching.
The Chebyshev characteristics

3. Chebyshev Transformer Design

4. Example5.10: Design a three-section Chebyshev transformer to match a 100 load to a 50 line, with m=0.05?
Solution 

6. Using table design for N=3 and ZL/Z0=2 can find coefficient as 1
Using table design for N=3 and ZL/Z0=2 can find coefficient as , , and So Z1=57.37, Z2=70.71, and Z3=87.15.

7. Tapered Lines Matching
The line can be continuously tapered instead of discrete multiple sections to achieve broadband matching.
Changing the type of line can obtain different passband characteristics.
Relation between characteristic impedance
and reflection coefficient
Three type of tapered line
will be considered here
1) Exponential
2)Triangular
3) Klopfenstein

8. Exponential Taper
The length (L)of line should be greater than /2(l>) to minimize the mismatch at low frequency.

9. Triangular Taper
The peaks of the triangular taper are lower than the corresponding peaks of the exponential case.
First zero occurs at l=2

10. Klopfenstein Taper
For a maximum reflection coefficient specification in the passband, the Klopfenstein taper yields the shortest matching section (optimum sense).
The impedance taper has steps at z=0 and L, and so does not smoothly join the source and load impedances.

11. Example5.11: Design a triangular, exponential, and Klopfenstein tapers to match a 50 load to a 100 line?
Solution
Triangular taper 
Exponential taper

12. Klopfenstein taper

13. Bode-Fano Criterion
The criterion gives a theoretical limit on the minimum reflection magnitude (or optimum result) for an arbitrary matching network
The criterion provide the upper limit of performance to tradeoff among reflection coefficient, bandwidth, and network complexity.
For example, if the response ( as the left hand side of next page) is needed to be synthesized, its function is given by applied the criterion of parallel RC
For a given load, broader bandwidth , higher m.
m  0 unless =o. Thus a perfect match can be achieved only at a finite number of frequencies.
As R and/or C increases, the quality of the match ( and/or m) must decrease. Thus higher-Q circuits are intrinsically harder to match than are lower-Q circuits.

15. Chapter 6
Microwave Resonators

16. RLC Series Resonant Circuit
Microwave resonators are used in a variety of applications, including filters, oscillators, frequency meters, and tuned amplifiers.
The operation of microwave resonators is very similar to that of the lumped-element resonators (such as parallel and series RLC resonant circuits) of circuit theory.
RLC Series Resonant Circuit
Vout
Vin