6.2: Matching Networks

Matching networks are constructed using lossless elements such as lumped capacitors, lumped inductors and transmission lines and so have, ideally, no loss and introduce no additional noise. This section discusses matching objectives and the types of matching networks.

Figure \(\PageIndex{1}\): A source with Thevenin equivalent impedance \(Z_{S}\) and load with impedance \(Z_{L}\) interfaced by a matching network presenting an impedance \(Z_{\text{in}}\) to the source.

Reflection-less match	Maximum power transfer
\(Z_{\text{in}}=Z_{S}\)	\(\begin{aligned}Z_{\text{in}}&=Z_{S}^{\ast}\\ \Gamma_{\text{in}}&=\Gamma_{S}^{\ast}\end{aligned}\nonumber\)

Table \(\PageIndex{1}\): Matching conditions with reference to Figure \(\PageIndex{1}\). \(\Gamma_{\text{in}},\: \Gamma_{S}\), must be with respect to a real \(Z_{\text{REF}}\).

6.2.1 Matching for Zero Reflection or for Maximum Power Transfer

With RF circuits the aim of matching is to achieve maximum power transfer. With reference to Figure \(\PageIndex{1}\) the condition for maximum power transfer is \(Z_{\text{in}} = Z_{S}^{\ast}\) (see Section 2.6.2 of [1]). An alternative matching objective, used most commonly with digital circuits, is a reflection-less match. The reflection-less match and the maximum power transfer match are only equivalent if \(Z_{S}\) and \(Z_{L}\) are real. Nearly always in RF design the matching objective is maximum power transfer, and this is assumed unless the reflection-less match is specifically indicated.

The maximum power transfer matching condition can also be specified in terms of reflection coefficients with respect to an real reference impedance \(Z_{\text{REF}}\). (In the definition of reflection coefficient \(Z_{\text{REF}}\) can be complex, but many of the manipulations in this book only apply when \(Z_{\text{REF}}\) is real and this should be assumed unless it is specifically stated that it can be complex.) The condition for maximum power transfer is \(Z_{\text{in}} = Z_{S}^{\ast}\) which is equivalent to \(\Gamma_{\text{in}} = \Gamma_{S}^{\ast}\). The proof is as follows:

\[\label{eq:1}\Gamma_{\text{in}}=\left(\frac{Z_{\text{in}}-Z_{\text{REF}}}{Z_{\text{in}}+Z_{\text{REF}}}\right) \]

and for maximum power transfer \(Z_{\text{in}} = Z_{S}^{\ast}\), so

\[\begin{align} \Gamma_{\text{in}}^{\ast}=\frac{Z_{\text{in}}-Z_{\text{REF}}}{Z_{\text{in}}+Z_{\text{REF}}}&=\left(\frac{Z_{S}^{\ast}-Z_{\text{REF}}}{Z_{S}^{\ast}+Z_{\text{REF}}}\right)^{\ast}=\frac{(Z_{S}^{\ast}-Z_{0})}{(Z_{S}^{\ast}+Z_{0})^{\ast}} \nonumber \\ \label{eq:2}&=\frac{(Z_{S}^{\ast})^{\ast}-Z_{\text{REF}}^{\ast}}{(Z_{S}^{\ast})^{\ast}+Z_{\text{REF}}^{\ast}}=\frac{Z_{S}-Z_{\text{REF}}^{\ast}}{Z_{S}+Z_{\text{REF}}^{\ast}}=\Gamma_{S} \end{align} \]

If \(Z_{\text{REF}}\) is real, \(Z_{\text{REF}}^{\ast} = Z_{\text{REF}}\) and then the condition for maximum power transfer is

\[\label{eq:3}\Gamma_{\text{in}}^{\ast}=\frac{Z_{S}-Z_{\text{REF}}}{Z_{S}+Z_{\text{REF}}}=\Gamma_{S} \]

Thus, provided that \(Z_{\text{REF}}\) is real, the condition for maximum power transfer in terms of reflection coefficients is \(\Gamma_{\text{in}}^{\ast} =\Gamma_{S}\) or \(\Gamma_{\text{in}} =\Gamma_{S}^{\ast}\). The matching conditions are summarized in Table \(\PageIndex{1}\).

6.2.2 Types of Matching Networks

Up to a few gigahertz, lumped inductors and capacitors can be used in matching networks. Above a few gigahertz, distributed parasitics (losses and additional capacitive or inductive effects) can render lumped-element networks impractical. Inductors in particular have high loss and high parasitic capacitances at microwave frequencies. Lumped capacitors are useful circuit elements at much higher frequencies than are lumped inductors. Segments of transmission lines are used in matching networks and are used instead of lumped elements when loss must be kept to a minimum, power levels are high, or the parasitics of lumped elements render them unusable. This is because the loss of an appropriate transmission line component is always much less than the loss of a lumped inductor.

Many factors affect the selection of the components of a matching network. The most important of these are size, complexity, bandwidth, and adjustability. The impact on a circuit of a distributed element is directly related to its length compared to one-quarter of a wavelength \((\lambda /4)\). At \(1\text{ GHz}\) a circuit board, for example, with a relative permittivity of \(4\) has \(\lambda /4=3.75\text{ cm}\). This is far too large to fit in consumer wireless products operating at \(1\text{ GHz}\).

Performance of lumped-element capacitors and inductors refers to both the self-resonant frequency of an element and its loss. Inductors are a particular case in point. An actual inductor must be modeled using capacitive elements, capturing inter-winding capacitance as well as the primary inductive element. At some frequency the inductive and capacitive elements will resonate and this is called the self-resonant frequency of the element. The self-resonant frequency is the maximum operating frequency of the element. With regards to complexity, the simplest matching network design is most often preferred. A simpler matching network is usually more reliable, less lossy, and cheaper than a complex design. The pursuit of small size and/or higher bandwidth, however, may necessitate that the simplest circuit not be selected. Any type of matching network can ideally give, ignoring resistive loses, a perfect match at a single frequency. Away from this center frequency the match will not be ideal. From experience, achieving a reasonable match over a \(5\%\) fractional bandwidth based on a single-frequency match can usually be achieved with simple matching networks.

An impedance matching network may consist of

1. Lumped elements only. These are the smallest networks, but have the most stringent limit on the maximum frequency of operation. The relatively high resistive loss of an inductor is the main limiting factor limiting performance. The self resonant frequency of an inductor limits operation to low microwave frequencies.
2. Distributed elements (microstrip or other transmission line circuits) only. These have excellent performance, but their size restricts their use in systems to above a few gigahertz.
3. A hybrid design combining lumped and distributed elements, primarily small sections of lines with capacitors. These lines are shorter than in a design with distributed elements only, but the hybrid design has higher performance than a lumped-element-only design.
4. Adhoc solutions (suggested by input impedance behavior and features of various components).

This chapter concentrates on matching network design. One emphasis here is on the development of design equations and synthesis of desired results. The second emphasis is on graphical techniques for matching network design based on using A Smith chart. This is particularly power as it enables

Figure \(\PageIndex{2}\): A transformer as a matching network. Port \(\mathsf{1}\) is on the left or primary side and Port \(\mathsf{2}\) is on the right or secondary side.

typologies to traded-off. An alternative design approach used by some designers is to choose a circuit topology and then use a circuit optimizer to arrive at circuit values that yield the desired characteristics. This is sometimes a satisfactory design technique, but is not a good solution for new designs as it does not provide insight and does not help in choosing new topologies.

6.2.3 Summary

There are many design choices in the type of matching network to be developed but the common guidelines are to minimize losses and to keep the matching network compact. These objectives are not always compatible. Matching network design in this chapter is based on perfect matching at one frequency and design decisions are made to maximize, or otherwise manipulate, the bandwidth of the match. True broadband matching network design is more like filter design.