03-01-2013, 03:07 PM
Impedance Matching
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Conjugate and Reflectionless Matching
The Th´evenin equivalent circuits depicted in Figs. 10.11.1 and 10.11.3 also allow us to
answer the question of maximum power transfer. Given a generator and a length-d
transmission line, maximum transfer of power from the generator to the load takes
place when the load is conjugate matched to the generator
Thus, the conjugate match condition can be phrased in terms of the input quantities
and the equivalent circuit of Fig. 10.9.1. More generally, there is a conjugate match at
every point along the line.
Indeed, the line can be cut at any distance l from the load and its entire left segment
including the generator can be replaced by a Th´evenin-equivalent circuit. The conjugate
matching condition is obtained by propagating Eq. (12.1.3) to the left by a distance l, or
equivalently
Quarter-Wavelength Chebyshev Transformers
Quarter-wavelength Chebyshev impedance transformers allow the matching of realvalued
load impedances ZL to real-valued line impedances Z0 and can be designed to
achieve desired attenuation and bandwidth specifications.
The design method has already been discussed in Sec. 6.8. The results of that section
translate verbatim to the present case by replacing refractive indices ni by line
admittances Yi = 1/Zi. Typical design specifications are shown in Fig. 6.8.1.
In an M-section transformer, all segments have equal electrical lengths, Li = li/λi =
nili/λ0 = 1/4 at some operating wavelength λ0.
Two-Section Dual-Band Chebyshev Transformers
Recently, a two-section sixth-wavelength transformer has been designed [976,977] that
achieves matching at a frequency f1 and its first harmonic 2f1. Each section has length
λ/6 at the design frequency f1. Such dual-band operation is desirable in certain applications,
such as GSM and PCS systems. The transformer is depicted in Fig. 12.4.1.
Here, we point out that this design is actually equivalent to a two-section quarterwavelength
Chebyshev transformer whose parameters have been adjusted to achieve
reflectionless notches at both frequencies f1 and 2f1.
Quarter-Wavelength Transformer With Series Section
One limitation of the Chebyshev quarter-wavelength transformer is that it requires the
load to be real-valued. The method can be modified to handle complex loads, but generally
the wide bandwidth property is lost. The modification is to insert the quarterwavelength
transformer not at the load, but at a distance from the load corresponding
to a voltage minimum or maximum.
Quarter-Wavelength Transformer With Shunt Stub
Two other possible methods of matching a complex load are to use a shorted or opened
stub connected in parallel with the load and adjusting its length or its line impedance
so that its susceptance cancels the load susceptance, resulting in a real load that can
then be matched by the quarter-wave section.
In the first method, the stub length is chosen to be either λ/8 or 3λ/8 and its
impedance is determined in order to provide the required cancellation of susceptance.
In the second method, the stub’s characteristic impedance is chosen to have a convenient
value and its length is determined in order to provide the susceptance cancellation.
These methods are shown in Fig. 12.6.1. In practice, they are mostly used with
microstrip lines that have easily adjustable impedances. The methods are similar to the
stub matching methods discussed in Sec. 12.8 in which the stub is not connected at the
load but rather after the series segment.