Transmission Line Matching Quarter Wave Transformers. Resistive Pads.
Transmission lines are commonly used as components in communication systems. In this capacity, the TL functions as a conduit for an electrical signal as it propagates from one subsystem to another. In such an application, we want to transfer as much of the signal’s electrical energy to its destination as possible. However, reflections from the load (i.e., the stination) may prevent this from happening. Consider the sinusoidal steady state excitation of a TL:
As we have already seen in previous lectures, there will be no reflection from the load if Z_{L} = Z_{0} . This is called a matched TL. A matched TL has a VSWR = 1. In general, though, an equivalent circuit at the input to the above TL is
According to the maximum power transfer theorem and a fixed Z_{s} , maximum power is delivered to the load on a lossless TL when Z_{in} = Z_{s} ^{*} . Furthermore, for almost all high frequency equipment Z _{s} = Z_{0} . Now, if Z_{ L } = Z_{0} it is straightforward to show that Z_{ in } = Z_{L}. Hence, Z_{ in } = Z_{s}^{*} [because Z_{s}^{*}=(z_{0})^{*}= Z =0 and Z_{0}= Z_{L} ]. In summary, on a lossless and matched TL we have shown that the maximum power transfer condition is met.
Matching Networks
In the more common situation when Z_{L} ¹ Z _{0}, matching a load to a TL is commonly accomplished using additional circuitry attached to the TL. This additional circuitry is called a matching network.
We will discuss three types of matching networks in this course:
1. Quarter-wave transformer
2. Resistive pads
3. Single-stub tuner.
Quarter-Wave Transformer
We’ve seen in a couple of lectures now that for a TL in the sinusoidal steady state with an arbitrary load
the total impedance at position d from the load is
With the generalized reflection coefficient given as
Now suppose that the length L of the TL at some frequency is exactly l / 4. Then,
There fore
By definition, we know that
Substituting this result in (2) gives
Or
This result in (3) is an interesting characteristic of a TL that is exactly l /4 long at a given frequency. We can harness this characteristic to design a matching network using a l /4-long section of TL. Consider the following structure composed of an impedance load interconnected to a TL (we’ll call this TL #1) through a l /4-section of a second TL:
Imagine we wish to match an arbitrary load L Z to the TL #1. We can use (3) to design this l /4 section of TL by adjusting its characteristic impedance, Z_{0} . In particular, to match Z_{L} to TL #1 we require Z_{in,1} = Z_{0,1} . For this example, the applicable quantities for (3) are
Using these values in (3) gives
In other words, if the l /4 section of TL has the special characteristic impedance given in (4) – a geometric mean of Z _{0,1}and Z_{L} then TL #1 will be matched to the load. This type of matching network is called a quarter-wave transformer (QWT). It transforms the impedance from one value to another from the output of the l /4 section to its input, according to (3).
QWT Discussion
Ideally, a matching network should not consume (much) power. In (4) we can deduce that Z_{0} will generally be a complex quantity indicating that for a perfect match, the QWT will need to include a lossy TL. However, if the load is purely resistive we can connect the QWT directly to the load and use a lossless TL. Otherwise if Z_{ L } is not resistive, we can insert a third TL (TL #2 in the figure below of length d_{q} ) such that Z_{in,2} is real. [Think of the Smith chart (!) and the values of Z_{in,2} at |V_{ d}|_{max} or |V_{ d}|_{min}]
Two disadvantages of this QWT are that:
1. We would need to cut the TL to insert it, and
2. It works perfectly only for one load at one frequency.
Actually, it produces some bandwidth of “acceptable” VSWR on the TL, as do all real-life matching networks.
Resistive Pads
A resistive pad is essentially an attenuator that provides a good deal of matching capabilities. Unfortunately, this matching capability comes at a price: reduced signal level. Moreover, these pads are sometimes relatively expensive. A pi-structure resistive pad has the following topology:
The insertion loss (IL) of a resistive pad is defined as
It can be shown that the design equations for the pi-structure resistive pad are
A sample of a resistive pad design and its performance is shown in the VisualEM “Section C.4.4 and Problem C.4.13” worksheet