# 2.6: $$T$$ or Chain Scattering Parameters of Cascaded Two-Port Networks

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The $$T$$ parameters, also known as chain scattering parameters, are a cascadable form of scattering parameters. They are similar to regular $$S$$ parameters and can be expressed in terms of the $$a$$ and $$b$$ root power waves or traveling voltage waves. Two two-port networks, A and B, in cascade are shown in Figure $$\PageIndex{1}$$. Here $$(A)$$ and $$(B)$$ are used as superscripts to distinguish the parameters of each two-port network, but the subscripts $$A$$ and $$B$$ are used for matrix quantities. Since

$\label{eq:1}a_{2}^{(A)}=b_{1}^{(B)}\quad\text{and}\quad b_{2}^{(A)}=a_{1}^{(B)}$

Figure $$\PageIndex{1}$$: Two cascaded two-ports.

it is convenient to put the $$a$$ and $$b$$ parameters in cascadable form, leading to the following two-port chain matrix or $$T$$ matrix representation (with respect to Figure 2.4.3):

$\label{eq:2} \left[\begin{array}{c}{a_{1}}\\{b_{1}}\end{array}\right] =\left[\begin{array}{cc}{T_{11}}&{T_{12}}\\{T_{21}}&{T_{22}}\end{array}\right]\left[\begin{array}{c}{b_{2}}\\{a_{2}}\end{array}\right]$

where the $$\text{T}$$ matrix or chain scattering matrix is

$\label{eq:3}\mathbf{T}=\left[\begin{array}{cc}{T_{11}}&{T_{12}}\\{T_{21}}&{T_{22}}\end{array}\right]$

$$\mathbf{T}$$ is very similar to the scattering transfer matrix $$(\:^{S}\mathbf{T})$$ of Section 2.3.7. The only difference is the ordering of the $$a$$ and $$b$$ components. You will come across both forms, so be careful that you understand which is being used. Both forms are used for the same function—cascading two-port networks. The relationships between $$\mathbf{T}$$ and $$\mathbf{S}$$ are given by

$\label{eq:4}\mathbf{T}=\left[\begin{array}{cc}{S_{21}^{-1}}&{-S_{21}^{-1}S_{22}}\\{S_{21}^{-1}S_{11}}&{S_{12}-S_{11}S_{21}^{-1}S_{22}}\end{array}\right]$

$\label{eq:5}\mathbf{S}=\left[\begin{array}{cc}{T_{21}T_{11}^{-1}}&{T_{22}-T_{21}T_{11}^{-1}T_{12}}\\{T_{11}^{-1}}&{-T_{11}^{-1}T_{12}}\end{array}\right]$

For a two-port network, using Equations $$\eqref{eq:1}$$ and $$\eqref{eq:2}$$,

$\label{eq:6}\left[\begin{array}{c}{a_{1}^{(A)}}\\{b_{1}^{(A)}}\end{array}\right]=\mathbf{T}_{A}\left[\begin{array}{c}{b_{2}^{(A)}}\\{a_{2}^{(A)}}\end{array}\right]\quad\text{and}\quad\left[\begin{array}{c}{a_{1}^{(B)}}\\{b_{1}^{(B)}}\end{array}\right]=\mathbf{T}_{B}\left[\begin{array}{c}{b_{2}^{(B)}}\\{a_{2}^{(B)}}\end{array}\right]$

thus

$\label{eq:7}\left[\begin{array}{c}{a_{1}^{(A)}}\\{b_{1}^{(A)}}\end{array}\right]=\mathbf{T}_{A}\mathbf{T}_{B}\left[\begin{array}{c}{b_{2}^{(B)}}\\{a_{2}^{(B)}}\end{array}\right]$

For $$n$$ cascaded two-port networks, Equation $$\eqref{eq:7}$$ generalizes to

$\label{eq:8}\left[\begin{array}{c}{a_{1}^{(1)}}\\{b_{1}^{(1)}}\end{array}\right]=\mathbf{T}_{1}\mathbf{T}_{2}\ldots\mathbf{T}_{n}\left[\begin{array}{c}{b_{2}^{(n)}}\\{a_{2}^{(n)}}\end{array}\right]$

and so the $$\mathbf{T}$$ matrix of the cascaded network is the matrix product of the $$\mathbf{T}$$ matrices of the individual two-ports.

Previously, in Section 2.3.7, the scattering transfer parameters were introduced. Both the chain scattering parameters and scattering transfer parameters are referred to as $$T$$ parameters. Be careful to denote which form is being used.

Example $$\PageIndex{1}$$: Development of Chain Scattering Parameters

Derive the $$T$$ parameters of the two-port network to the right using a reference impedance $$Z_{0}$$.

Figure $$\PageIndex{2}$$

Solution

$$\underline{\text{Derivation of }T_{11}}$$

Terminate Port $$\mathsf{2}$$ in a matched load so that $$a_{2} = 0$$. Then Equation $$\eqref{eq:2}$$ becomes $$a_{1} = T_{11}b_{2}$$ and $$b_{1} = T_{21}b_{2}$$ so

Figure $$\PageIndex{3}$$

$\label{eq:9}\Gamma_{\text{in}}=b_{1}/a_{1}=T_{21}/T_{11}$

Now $$Z_{\text{in}}=2r+Z_{0}$$, therefore

\begin{aligned}\Gamma_{\text{in}}&=\frac{Z_{\text{in}}-Z_{0}}{Z_{\text{in}}+Z_{0}} \\ &=\frac{2r+Z_{0}-Z_{0}}{2r+Z_{0}+Z_{0}}=\frac{r}{r+Z_{0}}\end{aligned}\nonumber

Thus

$\label{eq:10}\Gamma_{\text{in}}=\frac{T_{21}}{T_{11}}=\frac{r}{r+Z_{0}}$

The next stage is relating $$b_{2}$$ to $$a_{1}$$ and since both ports have the same reference impedance $$a$$ and $$b$$ can be replaced by traveling voltages

\begin{aligned}V_{1}=(V_{1}^{+}+V_{1}^{-})&=V_{1}^{+}(1+\Gamma_{\text{in}}) \\ &=V_{1}^{+}\frac{2r+Z_{0}}{r+Z_{0}}\end{aligned}\nonumber

Using voltage division (and since $$V_{2}^{+} = a_{2} = 0$$)

\begin{align}V_{2}=V_{2}^{-}&=\frac{Z_{0}}{2r+Z_{0}}V_{1}\nonumber \\ &=V_{1}^{+}\frac{Z_{0}}{2r+Z_{0}}\frac{2r+Z_{0}}{r+Z_{0}}\nonumber \\ V_{2}^{-}&=V_{1}^{+}\frac{Z_{0}}{r+Z_{0}}\nonumber \\ \label{eq:11}T_{11}&=\frac{V_{1}^{+}}{V_{2}^{-}}=\frac{a_{1}}{b_{2}}=\frac{r+Z_{0}}{Z_{0}}\end{align}

$$\underline{\text{Derivation of }T_{21}}$$

Combining Equations $$\eqref{eq:10}$$ and $$\eqref{eq:11}$$

$\label{eq:12}T_{21}=\Gamma_{\text{in}}T_{11}=\frac{r}{r+Z_{0}}\frac{r+Z_{0}}{Z_{0}}=\frac{r}{Z_{0}}$

$$\underline{\text{Derivation of }T_{22}}$$

Terminate Port $$\mathsf{2}$$ in a matched load so that $$a_{1} = 0$$. Then Equation $$\eqref{eq:2}$$ becomes

Figure $$\PageIndex{4}$$

\begin{align}0&=T_{11}b_{2}+T_{12}a_{2}\nonumber \\ \label{eq:13}\Gamma_{\text{in}}&=\frac{b_{2}}{a_{2}}=\frac{-T_{12}}{T_{11}}\to T_{12}=-T_{11}\Gamma_{\text{in}} \\ b_{1}&=T_{21}b_{2}+T_{22}a_{2} \\ \label{eq:14}\frac{b_{1}}{a_{2}}&=T_{21}\frac{b_{2}}{a_{2}}+T_{22}\end{align}

Now $$Z_{\text{in}} = 2r + Z_{0}$$ and so

$\label{eq:15}\Gamma_{\text{in}}=\frac{2r+Z_{0}-Z_{0}}{2r+Z_{0}+Z_{0}}=\frac{r}{r+Z_{0}}$

Using voltage division

\begin{align}V_{1}&=V_{1}^{-}=\frac{Z_{0}}{2r+Z_{0}}V_{2}=\frac{Z_{0}}{2r+Z_{0}}V_{2}^{+}(1+\Gamma_{\text{in}})\nonumber \\ &=V_{2}^{+}\frac{Z_{0}}{2r+Z_{0}}\frac{2r+Z_{0}}{r+Z_{0}}=V_{2}^{+}\frac{Z_{0}}{r+Z_{0}}\nonumber \\ \label{eq:16}\frac{V_{1}^{-}}{V_{2}^{+}}&=\frac{b_{1}}{a_{2}}=\frac{Z_{0}}{r+Z_{0}}\end{align}

Substituting Equations $$\eqref{eq:15}$$ and $$\eqref{eq:16}$$ in Equation $$\eqref{eq:14}$$ and rearranging

$T_{22}=\frac{Z_{0}}{r+Z_{0}}-T_{21}\frac{r}{r+Z_{0}}\nonumber$

Combining this with Equation $$\eqref{eq:12}$$

$\label{eq:17}T_{22}=\frac{Z_{0}}{r+Z_{0}}-\frac{r}{Z_{0}}\frac{r}{r+Z_{0}}=\frac{Z_{0}^{2}-r^{2}}{Z_{0}(r+Z_{0})}$

$$\underline{\text{Derivation of }T_{12}}$$

Combining Equations $$\eqref{eq:11}$$, $$\eqref{eq:13}$$ and $$\eqref{eq:15}$$

$\label{eq:18}T_{12}=-\frac{r+Z_{0}}{Z_{0}}\frac{r}{r+Z_{0}}=-\frac{r}{Z_{0}}$

$$\underline{\text{Summary}}$$:

The chain scattering matrix $$(T)$$ parameters are given in Equations $$\eqref{eq:11}$$, $$\eqref{eq:12}$$, $$\eqref{eq:17}$$, and $$\eqref{eq:18}$$.

Example $$\PageIndex{2}$$: Cascading Chain Scattering Parameters

Develop the chain scattering parameters of the network to the right. Use a reference impedance $$Z_{\text{REF}} = Z_{0}$$, the characteristic impedance of the lines.

Figure $$\PageIndex{5}$$

Solution

The network comprises a transmission line of electrical length $$\theta$$ in cascade with a resistive network and another line of the same electrical length. From Equation (2.5.3) the scattering parameters of the transmission line are

$\label{eq:19}\mathbf{S}_{L}=\left[\begin{array}{cc}{0}&{\text{e}^{-\jmath\theta}}\\{\text{e}^{-\jmath\theta}}&{0}\end{array}\right]$

From Equation $$\eqref{eq:4}$$ the chain scattering matrix of the line is

$\label{eq:20}\mathbf{T}_{L}=\left[\begin{array}{cc}{\text{e}^{\jmath\theta}}&{0}\\{0}&{\text{e}^{-\jmath\theta}}\end{array}\right]$

From Example $$\PageIndex{1}$$ the chain scattering matrix of the resistive network is

$\label{eq:21}\mathbf{T}_{r}=\frac{1}{Z_{0}}\left[\begin{array}{cc}{gr+Z_{0}}&{-r}\\{r}&{(Z_{0}^{2}-r^{2})/(r+Z_{0})}\end{array}\right]$

Yielding the chain scattering matrix of the cascade;

$\label{eq:22}\begin{array}{l}{T_{LrL}=T_{L}T_{r}T_{L}} \\ {\frac{1}{Z_{0}}\left[\begin{array}{cc}{(r+Z_{0})\text{e}^{2\jmath\theta}}&{-r}\\{r}&{(Z_{0}^{2}-r^{2})/(r+Z_{0})\text{e}^{-2\jmath\theta}}\end{array}\right]}\end{array}$

Example $$\PageIndex{3}$$: Input Reflection Coefficient of a Terminated Two-Port

What is the input reflection coefficient in terms of chain scattering parameters of the terminated two-port to the right where the two-port is described by its chain scattering parameters.

Figure $$\PageIndex{6}$$

Solution

The chain scattering parameter relations are

$\left[\begin{array}{c}{a_{1}}\\{b_{1}}\end{array}\right]=\left[\begin{array}{cc}{T_{11}}&{T_{12}}\\{T_{21}}&{T_{22}}\end{array}\right]\left[\begin{array}{c}{b_{2}}\\{a_{22}}\end{array}\right]\nonumber$

Expanding this matrix equation and using the substitution $$a_{2} = \Gamma_{L}b_{2}$$

\begin{align}\label{eq:23}a_{1}&=T_{11}b_{2}+T_{12}\Gamma_{L}b_{2} \\ \label{eq:24}b_{1}&+T_{21}b_{2}+T_{22}\Gamma_{L}b_{2}\end{align}

Multiplying Equation $$\eqref{eq:23}$$ by $$(T_{21} + T_{22}\Gamma_{L})$$ and Equation $$\eqref{eq:24}$$ by $$(T_{11} +T_{12}\Gamma_{L})$$ and then subtracting

\begin{aligned}(T_{21} + T_{22}\Gamma_{L})a_{1} &= (T_{21} + T_{22}\Gamma_{L})(T_{11}b_{2} + T_{12}\Gamma_{L})b_{2}\nonumber \\ (T_{11} + T_{12}\Gamma_{L})b_{1} &= (T_{11} + T_{12}\Gamma_{L})(T_{21}b_{2} + T_{22}\Gamma_{L})b_{2}\nonumber \\ (T_{21} + T_{22}\Gamma_{L})a_{1} &= (T_{11} + T_{12}\Gamma_{L})b_{1}\nonumber\end{aligned} \nonumber

Thus the input reflection coefficient is

$\label{eq:25}\Gamma_{\text{in}}=\frac{b_{1}}{a_{1}}=\frac{T_{21}+T_{22}\gamma_{L}}{T_{11}+T_{12}\gamma_{L}}$

Now consider a transmission line of electrical length $$\theta$$ and characteristic impedance $$Z_{0}$$ (see the figure b) in cascade with the two-port.

Figure $$\PageIndex{7}$$

The input reflection coefficient now is found by first determining the total chain scattering matrix of the cascade:

\begin{aligned}^{\mathbf{T}}\mathbf{T}&=\mathbf{T}_{\mathbf{L}}\mathbf{T}=\left[\begin{array}{cc}{\text{e}^{\jmath\theta}}&{0}\\{0}&{\text{e}^{-\jmath\theta}}\end{array}\right]\left[\begin{array}{cc}{T_{11}}&{T_{12}}\\{T_{21}}&{T_{22}}\end{array}\right]\nonumber \\ &=\left[\begin{array}{cc}{T_{11}\text{e}^{\jmath\theta}}&{T_{12}\text{e}^{\jmath\theta}}\\{T_{21}\text{e}^{-\jmath\theta}}&{T_{22}\text{e}^{-\jmath\theta}}\end{array}\right]\left[\begin{array}{cc}{^{T}T_{11}}&{^{T}T_{12}}\\{^{T}T_{21}}&{^{T}T_{22}}\end{array}\right]\nonumber\end{aligned} \nonumber

Thus the input reflection coefficient is now

\begin{align}\Gamma_{\text{in}}&=\frac{b_{1}}{a_{1}}=\frac{^{T}T_{21}+\:^{T}T_{22}\Gamma_{L}}{^{T}T_{11}+\:^{T}T_{12}\Gamma_{L}}\nonumber \\ &=\frac{T_{21}\text{e}^{-\jmath\theta}+T_{22}\text{e}^{-\jmath\theta}\Gamma_{L}}{T_{11}\text{e}^{\jmath\theta}+T_{12}\text{e}^{\jmath\theta}\Gamma_{L}}\nonumber \\ \label{eq:26}&=\text{e}^{-2\jmath\theta}\left(\frac{T_{21}+T_{22}\Gamma_{L}}{T_{11}+T_{12}\Gamma_{L}}\right)\end{align}

If the load is now a short circuit $$\Gamma_{L} = −1$$ and the input reflection coefficient of the line and two-port cascade is

$\label{eq:27}\Gamma_{\text{in}}=\text{e}^{-2\jmath\theta}\left(\frac{T_{21}-T_{22}}{T_{11}-T_{12}}\right)$

## 2.6.1 Terminated Two-Port Network

A two-port with Port 2 terminated in a load with reflection coefficient $$\Gamma_{L}$$ is shown in Figure 2.7.1 and $$a_{2} =\Gamma_{L}b_{2}$$. Substituting this in Equation $$\eqref{eq:2}$$ leads to

$a_{1} = (T_{11} + T_{12}\Gamma_{L})b_{2}\quad\text{and}\quad b_{1} = (T_{21} + T_{22}\Gamma_{L})b_{2}\nonumber$

eliminating $$b_{2}$$ results in

$\label{eq:28}\Gamma_{\text{in}}=\frac{b_{1}}{a_{1}}=\frac{T_{21}+T_{22}\Gamma_{L}}{T_{11}+T_{12}\Gamma_{L}}$

2.6: $$T$$ or Chain Scattering Parameters of Cascaded Two-Port Networks is shared under a CC BY-NC license and was authored, remixed, and/or curated by LibreTexts.