Double Stub Matching Using (3λ/8 Spacing)
Double Stub Matching is a practical impedance matching technique used in RF and microwave engineering when the location of matching stubs is fixed. Unlike single-stub matching, two adjustable stubs are employed to achieve a perfect match between the load impedance and the characteristic impedance of the transmission line.
Why double stub impedance matching is used in place of Single Stub Impedance Matching
Double Stub Matching is used when the exact location of a single matching stub cannot be conveniently chosen on a transmission line. In many practical microwave and RF systems, the position of the matching elements is fixed due to mechanical, manufacturing, or space constraints. Under such conditions, a single-stub matching network becomes difficult or impossible to implement because both the stub position and stub length must be adjusted simultaneously.
A double stub matching network overcomes this limitation by using two adjustable stubs separated by a fixed distance. Since only the stub lengths need to be adjusted while the stub locations remain fixed, tuning becomes more practical in real transmission line systems.
The total admittance of a double shunt stub network is:
\[
y_{in}=y_{line}+y_{stub1}+y_{stub2}
\]
or for a double series stub network:
\[
z_{in}=z_{line}+z_{stub1}+z_{stub2}
\]
Advantages of Double Stub Matching
\[
\text{1. Stub locations can remain fixed.}
\]
\[
\text{2. Only stub lengths require adjustment.}
\]
\[
\text{3. Easier implementation in waveguides and coaxial systems.}
\]
\[
\text{4. Suitable when the load position cannot be altered.}
\]
\[
\text{5. Widely used in practical microwave tuning networks.}
\]
Why Double Stub Matching is Inferior to Single Stub Matching
Although double stub matching is more convenient mechanically, it is theoretically less flexible than single stub matching.
In single stub matching, both the stub position and stub length can be selected freely. Therefore, a matching solution always exists for any passive load impedance.
\[
\text{Single Stub: Two adjustable parameters}
\]
\[
d \quad \text{and} \quad l
\]
where
\[
d=\text{distance from load}
\]
\[
l=\text{stub length}
\]
In double stub matching, the spacing between the stubs is fixed.
\[
S=\text{constant}
\]
Only the stub lengths can be varied:
\[
l_1 \quad \text{and} \quad l_2
\]
As a result, certain regions of the Smith Chart become inaccessible and some load impedances cannot be matched for a given stub spacing.
This limitation creates a forbidden region known as the "Double Stub Forbidden Area."
\[
\text{Forbidden Region}
\]
\[
\Longrightarrow
\]
\[
\text{Certain load admittances cannot be transformed}
\]
\[
\text{to the } g=1 \text{ circle}
\]
when the stub spacing is fixed.
Comparison of Single and Double Stub Matching
Single Stub Matching:
\[
\text{Adjustable Position} = \text{Yes}
\]
\[
\text{Adjustable Length} = \text{Yes}
\]
\[
\text{Matching Possible for All Loads} = \text{Yes}
\]
\[
\text{Construction Simplicity} = \text{Moderate}
\]
Double Stub Matching:
\[
\text{Adjustable Position} = \text{No}
\]
\[
\text{Adjustable Length} = \text{Yes}
\]
\[
\text{Matching Possible for All Loads} = \text{No}
\]
\[
\text{Construction Simplicity} = \text{High}
\]
Therefore, single stub matching is theoretically superior because it can match any arbitrary load impedance. Double stub matching is preferred only when practical installation constraints prevent changing the stub location.
Why \(3\lambda/8\) Corresponds to a \(270^\circ\) Rotation For a stub spacing of
\[ l = \frac{3\lambda}{8} \] the phase change becomes \[ \Delta\phi = 2\left(\frac{2\pi}{\lambda}\right) \left(\frac{3\lambda}{8}\right) \] \[ = \frac{12\pi}{8} \] \[ = \frac{3\pi}{2} \] Converting to degrees: \[ \frac{3\pi}{2} = 270^\circ \] Therefore, \[ \boxed{\frac{3\lambda}{8} \rightarrow 270^\circ} \] on the Smith Chart. Relationship Between the Two A full revolution on the Smith Chart corresponds to \[ 360^\circ \] and the chart repeats every \[ \frac{\lambda}{2} \] Since \[ 270^\circ = 360^\circ - 90^\circ \] a rotation of \(270^\circ\) clockwise reaches the same relative position as a rotation of \(90^\circ\) counterclockwise. This is why \[ \frac{\lambda}{8} \] and \[ \frac{3\lambda}{8} \] produce very similar double-stub matching behavior and have the same forbidden-region character
The Smith Chart provides a graphical method for determining the required stub lengths and matching locations.
Steps for Double Stub Impedance Matching
Step 1: Normalize the Load Impedance
Normalize the load impedance by dividing it by the characteristic impedance of the transmission line.
\[
z_N=\frac{Z_L}{Z_0}
\]
Plot the normalized impedance on the Smith Chart and construct the SWR circle.
Record:
\[
z_N=?
\]
\[
\text{SWR}=?
\]
Convert the normalized impedance into normalized admittance.
\[
y_N=\frac{1}{z_N}
\]
Record:
\[
y_N=?
\]
\[
\text{WTG}_B=?
\]
Step 2: Construct the 3/8 Spacing Circle
Construct the spacing circle corresponding to a stub spacing of:
\[
\frac{3\lambda}{8}
\]
The spacing circle intersects the stub circle at Points C and D.
Join Points C and D through the chart center.
Starting from the wavelength reading associated with:
\[
y_N
\]
move clockwise around the wavelength scale until the load line falls between the radial boundaries defined by Points C and D.
The arc between Points C and D determines the region where the SWR circle must reside on the spacing circle.
Step 3: Locate the First Stub Position
Assume the distance from the load to the first matching stub is:
\[
D_1=0.4\lambda
\]
Move clockwise around the SWR circle by:
\[
0.4\lambda
\]
until reaching Point F.
Record:
\[
y_F=?
\]
\[
D_1=0.4\lambda
\]
Step 4: Move Along the Reactance Circle
Follow the reactance circle passing through Point F toward smaller reactance values.
Move left until reaching Point G on the edge of the spacing circle.
Record:
\[
y_G=?
\]
Step 5: Determine the Required Stub Susceptance
Calculate the susceptance difference between Points F and G.
Record:
\[
\text{Distance}=?
\]
To cancel the negative susceptance component, locate the equal positive susceptance value.
\[
+j0.94
\]
Mark this location as Point H.
Record:
\[
\text{WTG}_H=?
\]
Step 6: Calculate the First Stub Length
The stub length is determined using the same procedure as single-stub matching.
Short-Circuited Stub
Measure clockwise from the infinite-admittance point.
\[
L_s=?
\]
Open-Circuited Stub
Measure clockwise from the zero-admittance point.
\[
L_o=?
\]
Step 7: Construct the Second SWR Circle
Using Point G as a point on the circumference and the chart center as the pivot point, construct a second SWR circle.
Record:
\[
\text{SWR}=?
\]
Step 8: Determine the Second Stub Length
Starting from Point G, move clockwise along the second SWR circle until reaching the:
\[
R=1
\]
circle located inside the spacing circle.
Mark this location as Point I.
Record:
\[
y_I=?
\]
\[
\text{WTG}_I=?
\]
Determine the susceptance required to cancel the remaining reactive component and locate the corresponding point on the susceptance circle.
Record:
\[
\text{WTG}_S=?
\]
Short-Circuited Stub
\[
L_s=?
\]
Open-Circuited Stub
\[
L_o=?
\]
Double Stub Matching employs two adjustable stubs separated by a fixed spacing to achieve impedance matching when single-stub matching is impractical. The Smith Chart simplifies the process by allowing graphical determination of admittance transformations, spacing-circle constraints, SWR circles, and the required stub lengths for both short-circuited and open-circuited stubs.