Nicola De Zanche^{1}

This lecture covers the basic concepts of RF power transfer over transmission lines. Tools such as scattering parameters and the Smith chart are also discussed.

Connections that exceed or approach a substantial fraction of a wavelength are made using uniform transmission lines, such as coaxial cables, whose cross-sectional geometry does not change along the length. This allows the length of the transmission line to be divided into infinitesimal sections $$$dz$$$ (Figure 2a), from which the telegrapher equations are derived using Kirchhoff’s laws [1]:

$$\frac{{dV\left(z\right)}}{{dz}}=-i\omega LI\left(z\right)$$

$$\frac{{dI\left(z\right)}}{{dz}}=-i\omega CV\left(z\right).$$

Here $$$L$$$ and $$$C$$$ represent, respectively, the inductance and capacitance per unit length of the transmission line. Differentiating these equations and substituting for the first derivatives yields

$$\frac{{{d^2}V\left(z\right)}}{{d{z^2}}}+{\omega ^2}LCV\left(z\right)=0$$

$$\frac{{{d^2}I\left(z\right)}}{{d{z^2}}}+{\omega^2}LCI\left(z\right)=0,$$

which are one-dimensional forms of the Helmholtz equation (phasor wave equation) and thus describe waves that propagate with phase velocity $$${v_p}=1/\sqrt{LC}$$$ in either direction:

$$V\left(z\right)=V_0^+{e^{-i\omega z/{v_p}}}+V_0^-{e^{i\omega z/{v_p}}}$$

$$I\left(z\right)=I_0^+{e^{-i\omega z/{v_p}}}+I_0^-{e^{i\omega z/{v_p}}}.$$

For each direction the ratio of voltage and current is a constant known as the characteristic impedance

$${Z_0}=\frac{{V_0^+}}{{I_0^+}}=\frac{{V_0^-}}{{I_0^-}}=\sqrt{\frac{L}{C}},$$

and allows the general solution to be written as a function of only voltage or current waves. Terminating the transmission line with a load impedance $$$Z_L$$$ (Figure 2b) imposes a boundary condition at $$$z=0$$$ and thus a relationship between incident and reflected waves

$$V_0^-=\frac{{{Z_L}-{Z_0}}}{{{Z_L}+{Z_0}}}V_0^+={\rm{\Gamma }}V_0^+,$$

where $$$\Gamma$$$ is known as the voltage reflection coefficient and $$$\left|{\rm{\Gamma }}\right|\le1$$$ (for a passive $$$Z_L$$$). The condition for maximum power transfer to the load is $$$\Gamma=0$$$, i.e., $$$Z_L=Z_0$$$. If the reflection coefficient is known the impedance is calculated as

$$\frac{{{Z_L}}}{{{Z_0}}}=\frac{{1+{\rm{\Gamma}}}}{{1-{\rm{\Gamma}}}},$$

where we have explicitly normalized it relative to $$$Z_0$$$.
It may be shown that at other points ($$$z$$$) along the transmission line the *phase *of the reflection coefficient
changes in proportion to the length

$${\rm{\Gamma }}\left(z\right)={\rm{\Gamma}}\left(0\right){e^{-2i\omega z/{v_p}}},$$

making a full rotation in a distance of one half wavelength. Consequently, the impedance of the load is transformed by the line according to

$$\frac{{{Z_{in}}}}{{{Z_0}}} = \frac{{1 + {\rm{\Gamma }}\left( 0 \right){e^{ - 2i\omega z/{v_p}}}}}{{1 - {\rm{\Gamma }}\left( 0 \right){e^{ - 2i\omega z/{v_p}}}}}.$$

To obtain admittances the inverse of this equation is used.

Scattering Parameters

Scattering (or S) parameters [1] are an extension of the definition of reflection coefficient to devices that have more than one connection (port), e.g., transmission lines, amplifiers, attenuators, power splitters, circulators, T/R switches, etc. The reflected waves ($$$V_i^ - $$$) are obtained from the incident waves ($$$V_i^ +$$$) by multiplication with the S matrix

$$\left[ {\begin{array}{*{20}{c}}{V_1^ - }\\{V_2^ - }\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{{S_{11}}}&{{S_{12}}}\\{{S_{21}}}&{{S_{22}}}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{V_1^ + }\\{V_2^ + }\end{array}} \right].$$

Each element of the S matrix represents the reflected wave when only one port is driven and all ports are matched. If this condition is satisfied then the diagonal elements of S are the actual reflection coefficients seen at the respective ports. The off-diagonal elements are coupling (transmission) coefficients between ports. Scattering parameters are provided for most RF devices or measured using a vector network analyzer (VNA).

The Smith chart (Figure 3) is a convenient graphical tool that represents the coordinate transformation between $$$\Gamma$$$ inside the unit circle and the normalized impedance ($$$Z/Z_0$$$) or admittance ($$$YZ_0$$$). The transformed impedance is obtained graphically as follows [2]:

- find the normalized load impedance in curvilinear coordinates and identify the corresponding magnitude of $$$\Gamma$$$ (e.g., set the radius of a compass)
- starting from that point, use the compass to draw an arc of a circle (centered at the origin) for a total rotation equal to half the number of wavelengths ($$$\omega z/{v_p}$$$)
- read the normalized load impedance in curvilinear coordinates at the end of the arc.

With this procedure it is readily observed that a half-wavelength line (or any multiple thereof) transforms any impedance into itself, which is useful when matching to $$$Z_0$$$ is not desired or practical. A quarter-wave line (plus any number of half wavelengths) inverts the normalized impedance

$$\frac{{{Z_{in}}}}{{{Z_0}}}=\frac{{{Z_0}}}{{{Z_L}}},$$

and is used to perform impedance matching as well as to transform an infinite impedance to a short and vice versa.

The Smith chart also allows seeing the effect (on the reflection coefficient) of adding impedances to the load. Matching (achieving $$$\Gamma=0$$$) typically consists of adding reactances in series (parallel) to follow the constant-resistance (-conductance) circles on the Smith chart. Figure 10.2 of Ref. 2 illustrates the range of use of the 8 possible L-type matching networks.

- David M. Pozar, Microwave Engineering, Fourth Edition. 2012 Wiley.
- Phillip H. Smith, Electronic Applications of the Smith Chart, Second Edition. 2000 SciTech Publishing.
- Reykowski A, Wright SM, Porter JR. Design of matching networks for low noise preamplifiers. Magn Reson Med 1995;33:848–852.
- Frankel S. Reactance networks for coupling between unbalanced and balanced circuits. Proc IRE 1941;29:486–493.

Figure 1: lumped element representation of a generator connected to a load a) in the case of negligible connection length, and b) including parasitic inductance and capacitance due to longer cables.

Figure 2: a generator connected to a load through a transmission line. a) represents an infinitesimally short section of the line and b) is the full schematic.

Figure 3: the Smith chart (https://commons.wikimedia.org/wiki/File:Smith_chart_gen.svg).