Double stub impedance matching

Double stub impedance matching

Impedance matching can be achieved by inserting two stubs at specified locations along transmission line as shown below

There are two design parameters for double stub matching:
The length of the first stub line Lstub1
The length of the second stub line Lstub2
In the double stub configuration, the stubs are inserted at predetermined locations. In this way, if the load impedance is changed, one simply has to replace the stubs with another set of different length.

The drawback of double stub tuning is that a certain range of load admittances cannot be matched once the stub locations are fixed.
Three stubs are necessary to guarantee that match is always possible. The length of the first stub is selected so that the admittance at the location of the second stub (before the second stub is inserted) has real part equal to the characteristic admittance of the line

If one moves from the location of the second stub back to the load, the circle of the allowed normalized admittances is mapped into another circle, obtained by pivoting the original circle about the center of the chart.

At the location of the first stub, the allowed normalized admittances are found on an auxiliary circle which is obtained by rotating the unitary conductance circle counterclockwise, by an angle

Given the load impedance, we need to follow these steps to complete the double stub design:

(a) Find the normalized load impedance and determine the corresponding location on the chart.
(b) Draw the circle of constant magnitude of the reflection coefficient |Γ| for the given load.
(c) Determine the normalized load admittance on the chart.
This is obtained
by rotating -180° on the constant |Γ| circle, from the load impedance point. From now on, all values read on the chart are normalized admittances. (d) Find the normalized admittance at location dstub1 by moving clockwise on the constant |Γ| circle.

(e) Draw the auxiliary circle
(f) Add the first stub admittance so that the normalized admittance point on the Smith chart reaches the auxiliary circle (two possible solutions). The admittance point will move on the corresponding conductance circle, since the stub does not alter the real part of the admittance

(g) Map the normalized admittance obtained on the auxiliary circle to the location of the second stub dstub2. The point must be on the unitary conductance circle
(h) Add the second stub admittance so that the total parallel admittance equals the characteristic admittance of the line to achieve exact matching condition

As mentioned earlier, a double stub configuration with fixed stub location may not be able to match a certain range of load impedances.
This is easily seen on the Smith chart. If the normalized admittance of the line, at the first stub location, falls inside a certain forbidden conductance circle tangent to the auxiliary circle (and always contained inside the unitary conductance circle), it is not possible to find a value for the first stub that can bring the normalized admittance to the auxiliary circle.

Therefore, it is

impossible to position the normalized admittance of the second stub location on the unitary conductance circle.
When this condition occurs, the location of one of the stubs must be changed appropriately. Alternatively, a third stub could be added.

Examples of forbidden regions follow.