# 5.3: Coupled Transmission Line Theory

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In this section coupled transmission line theory is developed in terms of the quantities shown in Figure 5.2.4. The voltages and currents shown here are phasors that vary along the line and are functions of \(x\).

The quasi-TEM mode of propagation is also assumed, and the transmission line system is completely lossless with perfect conductors and insulators. In phasor form, the generalized telegrapher’s equations for a pair of coupled lines are\(^{1}\)

\[\begin{align}\label{eq:1}\frac{dV_{1}(x)}{dx}&=-\jmath\omega L_{11}I_{1}(x)-\jmath\omega L_{12}I_{2}(x) \\ \label{eq:2}\frac{dV_{2}(x)}{dx}&=-\jmath\omega L_{21}I_{1}(x)-\jmath\omega L_{22}I_{2}(x) \\ \label{eq:3}\frac{dI_{1}(x)}{dx}&=-\jmath\omega C_{11}V_{1}(x)-\jmath\omega C_{12}V_{2}(x) \\ \label{eq:4}\frac{dI_{2}(x)}{dx}&=-\jmath\omega C_{21}V_{1}(x)-\jmath\omega C_{22}V_{2}(x)\end{align} \]

These are generalizations of the telegrapher’s equations of a single transmission line, and for reciprocity, \(C_{12} = C_{21}\) and \(L_{12} = L_{21}\). Compaction of the equations is obtained by introducing the per unit length inductance matrix, \(\mathbf{L}\), and the per unit length capacitance matrix, \(\mathbf{C}\), defined as

\[\label{eq:5}\mathbf{L}=\left[\begin{array}{cc}{L_{11}}&{L_{12}}\\{L_{12}}&{L_{22}}\end{array}\right] \]

and

\[\label{eq:6}\mathbf{C}=\left[\begin{array}{cc}{C_{11}}&{C_{12}}\\{C_{12}}&{C_{22}}\end{array}\right] \]

The next step is to express the voltage on the pair of coupled transmission lines in vector form as

\[\label{eq:7}\mathbf{V}(x)=\left[\begin{array}{c}{V_{1}(x)}\\{V_{2}(x)}\end{array}\right]=[V_{1}(x)V_{2}(x)]^{\text{T}} \]

where \(\text{T}\) indicates transpose and converts the row vector into a column vector. Similarly the vector of currents on the coupled transmission lines is

\[\label{eq:8}\mathbf{I}(x)=[I_{1}(x)I_{2}(x)]^{\text{T}} \]

Using the above relations, the telegrapher’s equation, from Equations \(\eqref{eq:1}\)–\(\eqref{eq:4}\), is represented in matrix form as

\[\label{eq:9}\frac{d}{dx}\mathbf{V}(x)=-\jmath\omega\mathbf{LI}(x) \]

and

\[\label{eq:10}\frac{d}{dx}\mathbf{I}(x)=-\jmath\omega\mathbf{CV}(x) \]

Rearranging Equations \(\eqref{eq:9}\) and \(\eqref{eq:10}\), taking derivatives, and after substitution, the final wave-equation form is obtained:

\[\label{eq:11}\frac{d^{2}}{dx}^{2}\mathbf{V}(x)+\omega^{2}\mathbf{LCV}(x)=0 \]

and

\[\label{eq:12}\frac{d^{2}}{dx}^{2}\mathbf{I}(x)+\omega^{2}\mathbf{LCI}(x)=0 \]

Solving these second-order differential equations yields descriptions of the propagation characteristics. With confidence, a solution in the form of propagating waves can be assumed:

\[\label{eq:13}\mathbf{V}(x)=\mathbf{V}_{0}\mathbf{β}=\mathbf{V}_{0}\text{diag}(e^{-\jmath\beta_{1}x},e^{-\jmath\beta_{2}x})=\mathbf{V}_{0}\left[\begin{array}{cc}{e^{-\jmath\beta_{1}x}}&{0}\\{0}&{e^{-\jmath\beta_{2}x}}\end{array}\right] \]

Substituting this into Equation \(\eqref{eq:11}\) yields

\[\label{eq:14}-\mathbf{β}^{2}\mathbf{V}_{0}+\omega^{2}\mathbf{LCV}_{0}=0 \]

For a nontrivial solution of Equation \(\eqref{eq:14}\), the determinant of the matrix equation should be zero:

\[\label{eq:15}\det\left(\mathbf{LC}-\frac{\mathbf{β}^{2}}{\omega^{2}}\mathbf{U}\right)=0,\quad\text{where the unit matrix}\quad\mathbf{U}=\left[\begin{array}{cc}{1}&{0}\\{0}&{1}\end{array}\right] \]

Equation \(\eqref{eq:15}\) is the characteristic equation that can be solved to determine the phase constant, \(\beta\). There are many possible solutions, and weighted linear combinations of the solutions are also solutions.

If only the quasi-TEM modes are considered, then there are two possible sets of solutions for the phase constant, with one set of solutions being

\[\label{eq:16}\beta_{1}=\pm\omega S_{1} \]

and the other

\[\label{eq:17}\beta_{2}=\pm\omega S_{2} \]

The different signs here are physically interpreted as referring to the forward- and backward-traveling waves. Thus the coupled pair of conductors supports two unique families of modes (each family comprising forward- and backward-traveling waves) with each family relating to a particular field configuration on the coupled line system.\(^{2}\) That is, \(S_{1}\) and \(S_{2}\) are each single numbers and, just considering the forward-traveling waves, \(\omega S_{1}\) is the propagation constant of one mode and \(\omega S_{2}\) is the propagation constant of the second mode.

The circuit-level model of a coupled line pair is developed by considering the calculation of the \(\mathbf{L}\) and \(\mathbf{C}\) matrices. The elements of the capacitance matrix are obtained in two simulations in which the line charge is calculated. A variety of commercially available software packages exist for the extraction of the per unit length \(\mathbf{L}\) and \(\mathbf{C}\) matrices. The matrix \(\mathbf{C}\) is calculated, in most packages, from the solutions of a two-dimensional electrostatic problem. The steps involve solving for the charges on the lines with voltages set on the conductors. With total voltages \(V_{1}\) and \(V_{2}\) on lines \(\mathsf{1}\) and \(\mathsf{2}\), the charges on lines \(Q_{1}\) and \(Q_{2}\) are

\[\label{eq:18}Q_{1}=C_{11}V_{1}+C_{12}V_{2} \]

and

\[\label{eq:19}Q_{2}=C_{12}V_{1}+C_{22}V_{2} \]

**Simulation 1**: With \(V_{1} = 1\) and \(V_{2} = 0\), the charges are calculated with the result that

\[\label{eq:20}C_{11}=Q_{1}>0 \]

and

\[\label{eq:21}C_{12}=Q_{2}<0 \]

(Note that \(C_{ij}\), \(i\neq j\), is negative.)

**Simulation 2**: With \(V_{1} = 0\) and \(V_{2} = 1\), the charges are recalculated, and now

\[\label{eq:22}C_{22}=Q_{2}\quad\text{and}\quad C_{12}=Q_{1} \]

The characterization of the lines is completed by determining the elements of the inductance matrix. This is done by calculating the capacitances with and without the dielectric. The principle effect of the dielectric is to alter the configuration and magnitude of the electric field. The dielectric has little effect on the magnitude and orientation of the magnetic field. With the same current on the coupled lines, the same magnetic energy is stored, and the inductances of the coupled line are unchanged by the dielectric. The other assumption is that with a TEM mode on the lines, and in the absence of a dielectric, the velocity of propagation is \(c = 1/\sqrt{LC}\). Specifically, the assumption is that for a TEM mode and without a dielectric, the phase velocity is just \(c\). This is a very good approximation and is exact if the conductors have infinite conductivity. (If the conductors have finite conductivity there would be field inside the conductors, and the wave slows down slightly.) Determining the capacitance matrix without the dielectric enables the inductance matrix to be calculated:

\[\label{eq:23}\mathbf{L}=\mathbf{L}_{0}=\frac{1}{c^{2}}\mathbf{C}_{0}^{-1} \]

where the subscript \(0\) indicates free space (but a subscript \(0\) on \(Z\), e.g., \(Z_{0}\), indicates characteristic impedance).

## 5.3.1 Summary

This section introduced the telegrapher’s equations for a pair of coupled lines in a form that is an extension of the telegrapher’s equations of a single line but with the \(L\) and \(C\) of a single line replaced by \(2\times 2\text{ L}\) and \(\text{C}\) matrices. It is no longer necessary to deal with fields and a circuit model can be used.

## Footnotes

[1] In Figure 5.2.4(a) \(V_{1}\) and \(V_{2}\) are the voltages at terminals 1 and 2 of lines 1 and 2. They are also port voltages referenced to the ground immediately below the appropriate terminal. \(V_{1}(x)\) and \(V_{2}(x)\) are the total voltage anywhere on lines 1 and 2, respectively, as shown in Figure 5.2.4(b). Similar notation will be used for voltages and traveling-wave components.

[2] Also it is up to the user to decide the form of the families to use. The most convenient family in microwave analysis is to use even and odd modes. In general, a system with \(N\) active conductors (and one reference conductor) will support \(2\times N\) (quasi-) TEM modes.