1. Microwave Engineering Chapter 5.7 ~ 5.9
Wonhong Jeong

2. Index Antennas & RF Devices Lab. 5. Impedance matching and tuning
5.7 Chebyshev multi-section matching transformers
5.8 Tapered Lines
5.9 The Bode-Fano Criterion
Antennas & RF Devices Lab.

3. 5.7 Chebyshev multi-section matching transformers
Conclusion
Chebyshev transformer has better performance about passband bandwidth compared with binomial transformer.
However, Chebyshev transformer has rather larger ripple compared with binomial transformer.
Thus, compromising the transformer is required.

4. …… 5.7 Chebyshev multi-section matching transformers
Chebyshev Polynomials
th order Chebyshev polynomial
-1.0
1.0 -2 -4 -6 2 4 6
-0.5
0.5
1.5
-1.5
(5.56) ……
(5.57)
Figure The first four Chebyshev polynomials, Tn (x)

5. 5.7 Chebyshev multi-section matching transformers
Chebyshev Polynomials
-1.0
1.0 -2 -4 -6 2 4 6
-0.5
0.5
1.5
-1.5
For In this range, the Chebyshev polynomials oscillate between This is the equal ripple property, and this region will be mapped to the passband of the matching transformer.
For This region will map to the frequency range outside the passband.
For , the increases faster with x as n increases.
Figure The first four Chebyshev polynomials, Tn (x)

6. 5.7 Chebyshev multi-section matching transformers
Chebyshev Polynomials
Let for , then
Let for , then
(5.58 a)
(5.58 b)
We desire equal ripple for the passband response of the transformer, So it is necessary to map
to and to
Figure Approximate behavior of the reflection coefficient magnitude for a single section quarter-wave transformer operating near its design frequency.

7. 5.7 Chebyshev multi-section matching transformers
Chebyshev Polynomials
This can be accomplished by replacing in (5.58a) with
(5.58 a)
(5.59)
Then for , so over this same range.

8. 5.7 Chebyshev multi-section matching transformers
Design of Chebyshev Transformers
Previous presentation, we already discussed about the reflection coefficient of multisection transformer equation (5.46)
Using this reflection coefficient equation, we can obtain Chebyshev equal ripple passband.
(5.46)
(5.61)
( N : number of sections in the transformer)
the last term
the last term
Find the constant ‘A’ by letting , corresponding to zero frequency.
(5.62)

9. 5.7 Chebyshev multi-section matching transformers
Design of Chebyshev Transformers
If the maximum reflection coefficient in the passband is , the maximum value of in the passband is unity (1). Then we can obtain
(5.62)
Using approximations introduced in Section 5.6, we can determine .
(5.63)
From equation (5.63), can be expressed

10. 5.7 Chebyshev multi-section matching transformers
Design of Chebyshev Transformers
Once is known, the fractional bandwidth can be calculated from equation (5.33)
(5.64)
(5.33)
The characteristic impedances Zn can be found from equation (5.43), although, as in the case of the binomial transformer, accuracy can be improved and self-consistency can be achieved by using the approximation that 1
5/3
1/3
0.1
0.2
0.3
 The above results are approximation basis on small-reflection theory but are general enough to design transformers with an arbitrary ripple level.

11. 5.8 Tapered Line
At the previous study, we discussed about discrete situations of multi section transformer. In this chapter, Continuously tapered line will be introduced.
Consider the continuously tapered line of Figure 5.18 (a) as being made up of a number of incremental sections of length Δz, with an impedance change ΔZ(z) from one section to the next, as shown in Figure 5.18 (b) Then, the incremental reflection coefficient is given by
(Δz = 0)
Figure A tapered transmission line matching section and the model for an incremental length of tapered line. (a) The tapered transmission line matching section (b) Model for and incremental step change in impedance of the tapered line.
(5.66)

12. 5.8 Tapered Line
The total reflection coefficient at can be found by summing all the partial reflections.
(5.67)
If is known, can be found as a function of frequency. Alternatively, if is specified, then in principle can be found.
Figure A tapered transmission line matching section and the model for an incremental length of tapered line. (a) The tapered transmission line matching section (b) Model for and incremental step change in impedance of the tapered line.

13. 5.8 Tapered Line Exponential Taper Reflection coefficient (5.68)
At , , as desired. At ,we wish to have Then we can obtain constant ‘a’.
(5.69)
Length should be greater than λ/2 (βL > π) to minimize the mismatch at low frequencies.
We find by using equation (5.67):
Figure A matching section with an exponential impedance taper. (a) Variation of impedance (b) Resulting reflection coefficient magnitude response.
(5.70)
Observe that this derivation assumes that β, the propagation constant of the tapered line, is not a function of z—an assumption generally valid only for TEM lines.

14. 5.8 Tapered Line Triangular Taper Reflection coefficient λ/4 (5.71)
So that the derivative is triangular in form:
Length should be greater than λ/2 (βL >2 π) to minimize the mismatch at low frequencies.
(5.72)
We find by using equation (5.67):
The peaks of the triangular taper are lower than the corresponding peaks of the exponential case.
(5.73)
Figure A matching section with an triangular taper for d(ln Z/Z0)/dz. (a) Variation of impedance (b) Resulting reflection coefficient magnitude response.

15. 5.8 Tapered Line Klopfenstein Taper
A number of methods for choosing an impedance matching taper are exist. In various matching taper techniques, Klopfenstein taper is logical method to optimize the impedance matching.
Reflection coefficient
(5.74) [1], [4]
Where the function is defined as
(5.75)
( : modified Bessel function)
The resulting reflection coefficient is given by
: reflection coefficient at zero frequency
(5.77)
(5.76)
References: [1] R.E. Collin, Foundations for Microwave Engineering, 2nd edition, McGraw-Hill, New York, 1992
[4] R.E. Collin, “The Optimum Tapered Transmission Line Matching Section,” Proceedings of the IRE, Vol. 44, pp , Apr. 1956

16. 5.8 Tapered Line Klopfenstein Taper
The maximum ripple in the passband is
(5.78)
* What is merits of using the Klopfenstein Taper?
1. It has minimum reflection coefficient over the passband.
2. It yields the shortest matching section.

17. 5.8 Tapered Line Example 5.8 Design of tapered matching sections
Exponential taper, Triangular taper, Klopfenstein taper to match a 50 Ω to 100 Ω line.
0.4
100
Triangular taper
Klopfenstein taper 90
0.3
Exponential taper 80
0.2 70
Passband of Klopfenstein taper
0.1 60 50
0.2
0.4
0.6
0.8
1.0
(a)
(b)
Figure Solution to example 5.8. (a) impedance variations for the triangular, exponential, and Klopfenstein tapers. (b) Resulting reflection coefficient magnitude vs frequency for the tapers of (a).

18. 5.9 The Bode-Fano Criterion
Figure 5.1 A lossless network matching an arbitrary load impedance to a transmission line.
We limit our discussion to the circuit of Figure 5.1, where a lossless network is used to match an arbitrary complex load, generally over a nonzero bandwidth.
We might raise the following questions in regard to this problems.
- Can we achieve a perfect match (zero reflection) over a specified bandwidth?
- If not, how well can we do? What is the trade-off between , the maximum
allowable reflection in the passband, and the bandwidth?
- How complex must the matching network be for a given specification?
These questions can be answered by the Bode-Fano criterion. The Bode–Fano criterion represents an optimum result that can be ideally achieved, even though such a result may only be approximated in practice. [7], [8]
References: [7] H. W. Bode, Network Analysis and Feedback Amplifier Design, Ban Nostrand, N.Y., 1945
[8] R. M. Fano, “Theoretical Limitations on the Broad Band Matching of Arbitrary Impedances,” Journal of the Franklin Institute, vol. 249,
pp , Jan. 1950, and pp , Feb. 1950

19. 5.9 The Bode-Fano Criterion
Lossless
Matching
network
(a)
Lossless
Matching
network
(b)
Lossless
Matching
network
(c)
Lossless
Matching
network
(d)
( : center frequency of the matching bandwidth)
Figure The Bode-Fano limits for RC and RL loads matched with passive and lossless networks (ω0 is the center frequency of the matching bandwidth). (a) Parallel RC. (b) Series RC. (c) Parallel RL. (d) Series RL.

20. 5.9 The Bode-Fano Criterion
Assume that we desire to synthesize a matching network with a reflection coefficient response like that shown in Figure 5.23 (a).
Lossless
Matching
network
(5.80)
Which leads to the following conclusions
For a given load (a fixed RC product), a broader
bandwidth (Δω) can be achieved only at the expense
of a higher reflection coefficient in the passband (Γm).
The passband reflection coefficient, Γm, cannot be
zero unless Δ ω = 0. Thus a perfect match can be
achieved only at a finite number of discrete frequencies,
as illustrated in Figure 5.23 (b).
As R and/or C increases, the quality of the match
(Δω and/or 1/Γm) must decrease.
Realizable
Not realizable
Figure Illustrating the Bode-Fano criterion. (a) A possible reflection coefficient response (b) Nonrealizable and realizable reflection coefficient responses.

21. Thank you for your attention