Numerical 10 Shunt Stub with S Matrix

A single-stub tuner is to match a lossless line to a load of an antenna. Design the stub with any assumed placement and length and Metion all the design steps with proper reasoning amd Provide/derive S matrix for both Before and after the design.

A single-stub tuner is to match a lossless line to a load of an antenna. Design the stub with any assumed placement and length and Metion all the design steps with proper reasoning amd Provide/derive S matrix for both Before and after the design.

Smith Chart Shunt Stub Matching Numerical

Given

\[
Z_L = 60 + j80 \ \Omega
\]

\[
Z_0 = 100 \ \Omega
\]

Step1: Normalize the load by dividing it by the characteristic’s impedance by the line

                \[
Z_{in}=\frac{Z_L}{Z_0}=\frac{60+j80}{100}=0.6+j0.8
\]

Step 2: Plot Zn in the smith Chart

Step3: Draw the SWR Circle using prime center as the pivot point and Zn as a point of circumference of circle

                SWR = 4:1

Step3: Draw a line from the normalized load impedance through the prime center of the chart out to the wavelength scale towards generator scale

WTGb = 0.436 λ

Furthermore, Note where this line crosses the swr circle opposite from the normalized load. This is normalized admittance Yn labelled as Point B on the Chart

                Yn = 0.3 – j0.4

Step 5: From the normalized admittance point B. move clockwise around the SWR circle until it crosses the R=1 circle for the first time. The is point C on the Chart and is denoted as normalized Susceptance (Bn)

Yc = 1 + j 1.5

Step 6: Draw a line from the perimeter center of chart through the susceptance point C to the wave length scale. Note the reading on wavelength towards generator scale.

WTGc = 0.176 λ

Step 7: The difference in wavelength between the reading of step 4 WTGb and the recorded in Step 6 WTGC moving clockwise distance from the load terminals to point where the matching sub will be connected.

\[
D_s=(0.176-0.436)\lambda+0.5\lambda
\]

\[
D_s=-0.26\lambda+0.5\lambda
\]

\[
D_s=0.240\lambda
\]

Alternatively,

\[
D_s=(0.5-0.436)\lambda+0.176\lambda
\]

\[
D_s=0.064\lambda+0.176\lambda
\]

\[
D_s=0.240\lambda
\]

Step 8: The reactive portion recorded in step 5 normailized susceptance of (j1.5) must be cancelled. Find the opoosite value (-j1.5) on the R- Circle on the chart noted as point D. Record the wavelength towards the generator reading at point D

                WTGd = 0.344 λ

Step 9: For shorted stub (admittance case), the length of the shorted stub is measured from the point of infinity () to point D

\[
L_s=(0.344-0.25)\lambda
\]

\[
L_s=0.094\lambda
\]

 

For open shunt stub the length of stub is measured from the point of 0 wavelength to point D in clockwise directions

\[
L_o=0.344\lambda
\]

 

Before Single-Stub Matching: Unmatched Load

Before the single-stub tuner is connected, the antenna load is generally not matched to the characteristic impedance of the transmission line. Therefore, the load impedance is different from the characteristic impedance, such that \(Z_L\neq Z_0\). When the antenna is considered as a one-port network, its input reflection coefficient is equal to the load reflection coefficient.

  \[   S_{11}=\Gamma_L=\frac{Z_L-Z_0}{Z_L+Z_0}\neq0   \]  

The corresponding one-port scattering matrix is simply

  \[   [S_{\text{before}}]=[\Gamma_L]   \]  

A nonzero value of \(S_{11}\) indicates that part of the incident power is reflected from the unmatched antenna load.

Unmatched Load as a Two-Port Transmission-Line Section

If the transmission line and the unmatched load section are modeled as a two-port network, the scattering matrix can be represented in the general form

  \[   [S_{\text{before}}]=   \begin{bmatrix}   \Gamma_L &   \sqrt{1-\lvert\Gamma_L\rvert^2}e^{-j\phi}\\   \sqrt{1-\lvert\Gamma_L\rvert^2}e^{-j\phi} &   \Gamma_L'   \end{bmatrix}   \]  

Here, \(\Gamma_L\) represents the reflection seen at the input reference plane, while \(\Gamma_L'\) represents the reflection behavior associated with the other port under the chosen two-port model. The phase term \(e^{-j\phi}\) represents the phase accumulated during propagation through the transmission-line section.

After Single-Stub Matching: Perfectly Matched System

After the single-stub tuner is properly positioned and its length is correctly selected, the stub cancels the reactive component of the transformed load. The input admittance of the system becomes equal to the characteristic admittance of the transmission line:

  \[   Y_{\text{in}}=Y_0   \]  

Consequently, the input reflection coefficient becomes zero:

  \[   S_{11}=\Gamma_{\text{in}}=0   \]  

When the matched antenna system is considered as a one-port network, its scattering matrix is therefore

  \[   [S_{\text{after}}]=[0]   \]  

This represents a perfectly matched one-port network in which no incident power is reflected at the input.

Matched System as a Two-Port Network

When the matched transmission-line system is modeled as a two-port network, both ports are matched under the ideal condition, so the reflection coefficients at the two ports are zero:

  \[   S_{11}=S_{22}=0   \]  

For a lossless matched transmission line, the magnitude of the forward and reverse transmission coefficients is unity:

  \[   \lvert S_{21}\rvert=\lvert S_{12}\rvert=1   \]  

The transmission through the line only introduces a phase shift. Therefore, the scattering matrix of the ideal matched line can be written as

  \[   [S_{\text{after}}]=   \begin{bmatrix}   0 & e^{-j\beta l}\\   e^{-j\beta l} & 0   \end{bmatrix}   \]  

where \(\beta\) is the phase constant of the transmission line and \(l\) is the length of the matched line section. Thus, after single-stub matching, the system has zero reflection while the incident signal is transmitted through the lossless line with only a phase shift.

For Simplicity

Ideal Two-Port Representation After Matching

The mathematical condition gives

\[ \lvert S_{12}\rvert=\lvert S_{21}\rvert=1 \]

For a perfectly matched two-port network, there is no reflection at either port. Therefore,

\[ S_{11}=S_{22}=0 \]

Hence, the scattering matrix can be written in the form

\[ [S_{\text{after}}]= \begin{bmatrix} 0&S_{12}\\ S_{21}&0 \end{bmatrix} \]

Since the network is reciprocal, the transmission coefficients in both directions are equal. Therefore,

\[ S_{12}=S_{21} \]

Substituting this reciprocity condition into the scattering matrix gives

\[ [S_{\text{after}}]= \begin{bmatrix} 0&S_{12}\\ S_{12}&0 \end{bmatrix} \]

For a lossless network, the scattering matrix satisfies the condition

\[ [S][S]^T=[I] \]

Therefore,

\[ \begin{bmatrix} 0&S_{12}\\ S_{12}&0 \end{bmatrix} \begin{bmatrix} 0&S_{12}\\ S_{12}&0 \end{bmatrix} = \begin{bmatrix} 1&0\\ 0&1 \end{bmatrix} \]

Multiplying the matrices gives

\[ \begin{bmatrix} S_{12}^{2}&0\\ 0&S_{12}^{2} \end{bmatrix} = \begin{bmatrix} 1&0\\ 0&1 \end{bmatrix} \]

Thus, the transmission coefficient has unit magnitude, and for the ideal case its magnitude is

\[ \lvert S_{12}\rvert=\lvert S_{21}\rvert=1 \]

Therefore, the ideal matched two-port scattering matrix can be represented as

\[ [S_{\text{after}}]= \begin{bmatrix} 0&1\\ 1&0 \end{bmatrix} \]

More generally, a transmission coefficient with unit magnitude can be represented in phase form as

\[ S_{12}=e^{j\theta} \]

and similarly for \(S_{21}\). Thus, for an ideal matched and lossless transmission line, the transmission coefficient has unit magnitude, while the transmission through the line introduces only a phase shift. Therefore, the scattering matrix of the ideal matched line can be written as

\[ [S_{\text{after}}]= \begin{bmatrix} 0&e^{-j\beta l}\\ e^{-j\beta l}&0 \end{bmatrix} \]

where \(\beta\) is the phase constant of the transmission line and \(l\) is the length of the line section. The factor \(e^{-j\beta l}\) represents the phase shift introduced by propagation through the line, while its magnitude remains unity:

\[ \left\lvert e^{-j\beta l}\right\rvert=1 \]

Although this result is mathematically valid for an ideal lossless 2-port network, it does not provide an appropriate model for a standard single antenna. The reason is that free space is not a conventional guided network port in the same sense as the input port of a transmission line, waveguide, or other microwave network. The radiation from the antenna propagates into the surrounding electromagnetic space and is not normally represented as a single second guided port in the antenna's basic S-parameter model.

In single-stub matching, the antenna is instead treated through its input impedance \(Z_L\) or, equivalently, its input admittance \(Y_L\). The objective is to transform this input impedance through the transmission line and add a suitable shunt stub so that the input impedance seen by the source becomes equal to the characteristic impedance \(Z_0\). Therefore, the matching analysis concerns the antenna's 1-port input behavior, rather than treating free-space radiation as a second guided network port.

\[ Z_{\text{in}}=Z_0 \]

Under the ideal matched condition, the input reflection coefficient becomes

\[ \Gamma_{\text{in}} = \frac{Z_{\text{in}}-Z_0} {Z_{\text{in}}+Z_0} =0 \]

and consequently

\[ S_{11}=0 \]

Thus, for single-stub matching of a conventional antenna, the important S-parameter is the input reflection coefficient \(S_{11}\). The antenna should not be represented as a conventional lossless \(2\times2\) guided network merely by assigning free space as Port 2.

Before Single-Stub Matching: Unmatched Load

Before the single-stub tuner is connected, the antenna load is generally not matched to the characteristic impedance of the transmission line. Therefore, the load impedance is different from the characteristic impedance, such that \(Z_L\neq Z_0\). When the antenna is considered as a one-port network, its input reflection coefficient is equal to the load reflection coefficient.

  \[   S_{11}=\Gamma_L=\frac{Z_L-Z_0}{Z_L+Z_0}\neq0   \]  

The corresponding one-port scattering matrix is simply

  \[   [S_{\text{before}}]=[\Gamma_L]   \]  

A nonzero value of \(S_{11}\) indicates that part of the incident power is reflected from the unmatched antenna load.

Unmatched Load as a Two-Port Transmission-Line Section

If the transmission line and the unmatched load section are modeled as a two-port network, the scattering matrix can be represented in the general form

  \[   [S_{\text{before}}]=   \begin{bmatrix}   \Gamma_L &   \sqrt{1-\lvert\Gamma_L\rvert^2}e^{-j\phi}\\   \sqrt{1-\lvert\Gamma_L\rvert^2}e^{-j\phi} &   \Gamma_L'   \end{bmatrix}   \]  

Here, \(\Gamma_L\) represents the reflection seen at the input reference plane, while \(\Gamma_L'\) represents the reflection behavior associated with the other port under the chosen two-port model. The phase term \(e^{-j\phi}\) represents the phase accumulated during propagation through the transmission-line section.

After Single-Stub Matching: Perfectly Matched System

After the single-stub tuner is properly positioned and its length is correctly selected, the stub cancels the reactive component of the transformed load. The input admittance of the system becomes equal to the characteristic admittance of the transmission line:

  \[   Y_{\text{in}}=Y_0   \]  

Consequently, the input reflection coefficient becomes zero:

  \[   S_{11}=\Gamma_{\text{in}}=0   \]  

When the matched antenna system is considered as a one-port network, its scattering matrix is therefore

  \[   [S_{\text{after}}]=[0]   \]  

This represents a perfectly matched one-port network in which no incident power is reflected at the input.

Matched System as a Two-Port Network

When the matched transmission-line system is modeled as a two-port network, both ports are matched under the ideal condition, so the reflection coefficients at the two ports are zero:

  \[   S_{11}=S_{22}=0   \]  

For a lossless matched transmission line, the magnitude of the forward and reverse transmission coefficients is unity:

  \[   \lvert S_{21}\rvert=\lvert S_{12}\rvert=1   \]  

The transmission through the line only introduces a phase shift. Therefore, the scattering matrix of the ideal matched line can be written as

  \[   [S_{\text{after}}]=   \begin{bmatrix}   0 & e^{-j\beta l}\\   e^{-j\beta l} & 0   \end{bmatrix}   \]  

where \(\beta\) is the phase constant of the transmission line and \(l\) is the length of the matched line section. Thus, after single-stub matching, the system has zero reflection while the incident signal is transmitted through the lossless line with only a phase shift.

For Simplicity

Ideal Two-Port Representation After Matching

The mathematical condition gives

\[ \lvert S_{12}\rvert=\lvert S_{21}\rvert=1 \]

For a perfectly matched two-port network, there is no reflection at either port. Therefore,

\[ S_{11}=S_{22}=0 \]

Hence, the scattering matrix can be written in the form

\[ [S_{\text{after}}]= \begin{bmatrix} 0&S_{12}\\ S_{21}&0 \end{bmatrix} \]

Since the network is reciprocal, the transmission coefficients in both directions are equal. Therefore,

\[ S_{12}=S_{21} \]

Substituting this reciprocity condition into the scattering matrix gives

\[ [S_{\text{after}}]= \begin{bmatrix} 0&S_{12}\\ S_{12}&0 \end{bmatrix} \]

For a lossless network, the scattering matrix satisfies the condition

\[ [S][S]^T=[I] \]

Therefore,

\[ \begin{bmatrix} 0&S_{12}\\ S_{12}&0 \end{bmatrix} \begin{bmatrix} 0&S_{12}\\ S_{12}&0 \end{bmatrix} = \begin{bmatrix} 1&0\\ 0&1 \end{bmatrix} \]

Multiplying the matrices gives

\[ \begin{bmatrix} S_{12}^{2}&0\\ 0&S_{12}^{2} \end{bmatrix} = \begin{bmatrix} 1&0\\ 0&1 \end{bmatrix} \]

Thus, the transmission coefficient has unit magnitude, and for the ideal case its magnitude is

\[ \lvert S_{12}\rvert=\lvert S_{21}\rvert=1 \]

Therefore, the ideal matched two-port scattering matrix can be represented as

\[ [S_{\text{after}}]= \begin{bmatrix} 0&1\\ 1&0 \end{bmatrix} \]

More generally, a transmission coefficient with unit magnitude can be represented in phase form as

\[ S_{12}=e^{j\theta} \]

and similarly for \(S_{21}\). Thus, for an ideal matched and lossless transmission line, the transmission coefficient has unit magnitude, while the transmission through the line introduces only a phase shift. Therefore, the scattering matrix of the ideal matched line can be written as

\[ [S_{\text{after}}]= \begin{bmatrix} 0&e^{-j\beta l}\\ e^{-j\beta l}&0 \end{bmatrix} \]

where \(\beta\) is the phase constant of the transmission line and \(l\) is the length of the line section. The factor \(e^{-j\beta l}\) represents the phase shift introduced by propagation through the line, while its magnitude remains unity:

\[ \left\lvert e^{-j\beta l}\right\rvert=1 \]

Although this result is mathematically valid for an ideal lossless 2-port network, it does not provide an appropriate model for a standard single antenna. The reason is that free space is not a conventional guided network port in the same sense as the input port of a transmission line, waveguide, or other microwave network. The radiation from the antenna propagates into the surrounding electromagnetic space and is not normally represented as a single second guided port in the antenna's basic S-parameter model.

In single-stub matching, the antenna is instead treated through its input impedance \(Z_L\) or, equivalently, its input admittance \(Y_L\). The objective is to transform this input impedance through the transmission line and add a suitable shunt stub so that the input impedance seen by the source becomes equal to the characteristic impedance \(Z_0\). Therefore, the matching analysis concerns the antenna's 1-port input behavior, rather than treating free-space radiation as a second guided network port.

\[ Z_{\text{in}}=Z_0 \]

Under the ideal matched condition, the input reflection coefficient becomes

\[ \Gamma_{\text{in}} = \frac{Z_{\text{in}}-Z_0} {Z_{\text{in}}+Z_0} =0 \]

and consequently

\[ S_{11}=0 \]

Thus, for single-stub matching of a conventional antenna, the important S-parameter is the input reflection coefficient \(S_{11}\). The antenna should not be represented as a conventional lossless \(2\times2\) guided network merely by assigning free space as Port 2.