Microwave-Engineering : Impedance Matching

Impedance matching (or tuning) is an important issue for:

• Maximum power is delivered when load is matched to line (assuming the generator is matched)
• Power loss is minimized.
• S/N ratio of receiver components is increased.
• Amplitude and phase errors are reduced.

Whenever Z_L has a nonzero real part, impedance matching is possible with the factors such as:

• Complexity: Simpler one is preferable.
• Wide Bandwidth: Match a load over a band.
• Implementation: Easier one is preferable
• Adjustability: Adjust to match a variable load impedance.

L Networks Matching

The normalized Z_L should be converted to 1+ jx with adding  impedance, then adding –jx the impedance matching will be successful.

Quarter Wave Transformer 

 

It is used for only real load impedance. Complex load impedance can always be transformed to real impedance by an appropriate length of transmission line. But this generally alters the frequency dependency of the equivalent load reducing the bandwidth of the matching. The following relation has to be satisfied as

where

For multiple reflections, totally  is

For each frequency, l has to be equal to  Thus fixed line length is possible only for one frequency.

• Bandwidth performance of QWT for wide band matching: If one set the maximum value of reflection coefficient,  that can be tolerated, then the fractional bandwidth is

This shows that as Z_L becomes closer to Z₀ , the bandwidth increases. This result is valid only for TEM lines. The step changes of reactance effect can be compensated by making a small adjustment in the length of the matching section. The approximate behavior of reflection coefficients is shown at below.

The QWT can be extended as a multisection form for matching of broader bandwidth.

Single Stub Tuning

A single open-circuited (or short-circuited) transmission line is connected either in parallel or series with the feed line at a certain distance from the load. Single Sub Tuning can match any load impedance to the line, but suffer from the disadvantage of requiring a variable length between the load and stub.

Since lumped elements are not required, the single stub is convenient and easy to fabricate in microstrip form. Two adjustable parameters d and susceptance or reactance provided by the stub. Although for microstrip lines open circuit is easy to fabricate since a via-hole is enough, for coax or waveguide short circuit is preferred since open circuit line may be large for radiation. If the impedance has the form of Z₀ + jX at the distance d. Then stub reactance can be chosen as –, resulting for a matching condition. For a given susceptance or reactance, the difference in lengths of open and short-circuited stub is  

Double Stub Tuning

The double stub tuner can not match all load impedances, but load may be the arbitrary distance from the first stub.

The distance between the stubs should be generally chosen as  to reduce the frequency sensitivity.

• Single Section Transformer: The reflection coefficient of single section transformer can be written when discontinuities between the impedances are small as

       

This shows that  is dominated by  and .

• Multisection Transformer: If the applications require more bandwidth, multisection transformers consists of N equal length sections of the lines can be used. The reflection coefficient can be written as

      

The importance of this result is that the desired reflection coefficients response as a function of frequency (kl) can be synthesized by proper choosing of  To obtain passband responses, binominal (maximally flat) and Chebyshev (equal ripple) multisection matching transformers can be used. In the first one: the N – 1 derivatives of  is settled to zero, in the second one:  is equated to Chebyshev polynomial.

 

Tapered Lines

 

The line can be continuously tapered for decreasing the effect of the step changes in characteristic impedance between the discrete sections. The incremental reflection coefficient  

where  from Z(z) can be found. Chancing type of taper (Exponential, Z(z) = e^az, Triangular, Klopfenstein), different band pass characteristic may be applied. Klopfenstein yields the shortest matching section. The Bode-Fano criterion for certain type of canonical load impedances will help us to define a theoretical limit on the minimum reflection with the upper limit of matching performance and provide a benchmark against which a practical design can be compared.

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Team Gradeup.