# Narrowband Matching

A transmission line of characteristic impedance is matched to a load when : not incident waves upon the load are reflected back at the source. A matching network is used to achieve these conditions, placed between the load and the line. Examples of matching networks include

• The transformer
• A transmission line in series of length
• A capacitor/inductor in shunt
• A short circuit stub in parallel

Note that for lines of length , since , the input impedance of the line is equal to the load impedance and the line does not modify the impedance of the load to which it is connected.

## Transformer

The input impedance of a line of length is This eliminates reflections at to make .

\Gamma = \frac{Z_{in} - Z_0}{Z_{in_} + Z_0}

At the frequency for which the transformer is a perfect , there is a perfect match, and . However, as we deviate from the match frequency, the performance degrades: We use as an acceptable maximum reflection coefficent, for which the bandwith is defined:

Solving for from the above equations gives:

Assuming TEM lines, where is the designed frequency, we can then link with , the max/min frequency at which our match has an acceptable performance:

The fractional bandwith of a matching section (where is derived above):

The smaller the load mismatch, the larger the bandwidth.

## Lumped Element Matching Networks

An L-secion uses two reactive elements to match a load impedance to a transmission line. If falls within the circle on the Smith chart, then the left configuration is used, else the right configuration is used. Let where (inside circle). For an impedance match:

Solving for and :

Two solutions are possible, and both are physically realisable with capacitors/inductors.

Conside the alternative where (outside circle):

## Shunt Lumped Element Matching

We use a lumped element in parallel with the load to achieve matching as shown in the figure. As the element is in shunt, we work in the admittance domain. Assuming , the aim is at terminal to transform to , and to . Assuming and , the aim is to choose a length and value of to match of the feedline to , given by the sum of and .