Synopsis

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.

Introduction

Lumped Circuits and Matching Conditions

Uniform Transmission Lines

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).

Smith Chart

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.

Matching Circuits and Baluns

Acknowledgements

References

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