# 2.5: Scattering Parameter Matrices of Common Two-Ports

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RF and microwave circuits can generally be represented as interconnected two-ports, as most RF and microwave circuit designs involve cascaded functional blocks such as amplifiers, matching networks, filters, etc.$$^{1}$$ (see Figure 2.4.4). Thus there is great interest in various manipulations that can be performed on two-ports as well as the network parameters of common two-port circuit topologies. As an example, consider the matching networks in Figure 2.4.4. These are used to achieve maximum power transfer in an amplifier by acting as impedance transformers. Matching networks assume a variety of forms, as shown in Figure 2.4.5, and all can be viewed as two-port networks and a combination of simpler components. In this section, strategies are presented for developing the $$S$$ parameters of two-ports.

### Transmission Line

The traveling waves on a transmission line (Figure $$\PageIndex{1}$$(a)) have a phase that depends on the electrical length, $$\theta$$, of the line. The transmission line has a characteristic impedance, $$Z_{0}$$, and length, $$\ell$$, which in general is different from

Figure $$\PageIndex{1}$$: Two-ports: (a) section of transmission line; and (b) series element in the form of a two-port.

the system reference impedance, here $$Z_{01}$$. Thus

$\label{eq:1}S_{11}=S_{22}=\frac{\Gamma(1-\text{e}^{-2\jmath\theta})}{1-\Gamma^{2}\text{e}^{-2\jmath\theta}}\quad\text{and}\quad S_{21}=S_{12}=\frac{(1-\Gamma^{2})\text{e}^{-\jmath\theta}}{1-\Gamma^{2}\text{e}^{-2\jmath\theta}}$

where $$\theta =\beta\ell$$ and

$\label{eq:2}\Gamma=\frac{Z_{0}-Z_{01}}{Z_{0}+Z_{01}}$

If the reference impedance is the same as the characteristic impedance of the line, i.e. $$Z_{01} = Z_{0}$$ and $$\Gamma =0$$, the scattering parameters of the line are

$\label{eq:3}\mathbf{S}=\left[\begin{array}{cc}{0}&{\text{e}^{-\jmath\theta}}\\{\text{e}^{-\jmath\theta}}&{0}\end{array}\right]$

### Shunt Element

The $$S$$ parameters of the shunt element (Figure 2.3.5) were developed in Section 2.3.5. In a slightly different form these are

$\label{eq:4}S_{11}=S_{22}=-\frac{\overline{y}}{(\overline{y}+2)}\quad\text{and}\quad S_{12}=S_{21}=\frac{2}{(\overline{y}+2)}$

where $$\overline{y} = Y/Y_{0}$$ is the admittance normalized to the system reference admittance $$(Y_{0} = 1/Z_{0})$$.

### Series Element

The $$S$$ parameters of the series element (Figure $$\PageIndex{1}$$(b)) are

$\label{eq:5}S_{11}=S_{22}=\frac{\overline{z}}{(\overline{z}+2)}\quad\text{and}\quad S_{12}=S_{21}=\frac{2}{(\overline{z}+2)}$

where $$\overline{z} = Z/Z_{0}$$ is the normalized impedance.

## Footnotes

[1] This arrangement tends to maximize bandwidth, minimize losses, and maximize efficiency. Lower-frequency analog design utilizes more complex arrangements; for example, feedback high in the circuit hierarchy improves the reliability and robustness of design but comes at the cost of reduced bandwidth and lower power efficiency.

2.5: Scattering Parameter Matrices of Common Two-Ports is shared under a CC BY-NC license and was authored, remixed, and/or curated by LibreTexts.