6.4: Symmetric Coupled Transmission Lines

In this section, even and odd modes are considered as defining independent transmission lines. The development is restricted to a symmetrical pair of coupled lines. Thus the strips have the same self-inductance, \(L_{s} = L_{11} = L_{22}\), and self-capacitance, \(C_{s} = C_{11} = C_{22}\), where the subscript \(s\) stands for “self.” \(L_{m} = L_{12} = L_{21}\) and \(C_{m} = C_{12} = C_{21}\) are the mutual inductance and capacitance of the lines, and the subscript \(m\) stands for “mutual.” The coupled transmission line equations are

\[\label{eq:1}\frac{dV_{1}(x)}{dx}=-\jmath\omega L_{s}I_{1}(x)-\jmath\omega L_{m}I_{2}(x) \]

\[\label{eq:2}\frac{dV_{2}(x)}{dx}=-\jmath\omega L_{m}I_{1}(x)-\jmath\omega L_{s}I_{2}(x) \]

\[\label{eq:3}\frac{dI_{1}(x)}{dx}=-\jmath\omega C_{s}V_{1}(x)-\jmath\omega C_{m}V_{2}(x) \]

\[\label{eq:4}\frac{dI_{2}(x)}{dx}=-\jmath\omega C_{m}V_{1}(x)-\jmath\omega C_{s}V_{2}(x) \]

The even mode is defined as the mode corresponding to both conductors being at the same potential and carrying the same currents:\(^{1}\)

\[\label{eq:5}V_{1}=V_{2}=V_{2}\quad\text{and}\quad I_{1}=I_{2}=I_{2} \]

The odd mode is defined as the mode corresponding to the conductors being at opposite potentials relative to the reference conductor and carrying currents of equal amplitude but of opposite sign:\(^{2}\)

\[\label{eq:6}V_{1}=-V_{2}=V_{o}\quad\text{and}\quad I_{1}=-I_{2}=I_{o} \]

The characteristics of the two possible modes of the coupled transmission lines are now described. For the even mode, from Equations \(\eqref{eq:1}\) and \(\eqref{eq:2}\),

\[\label{eq:7}\frac{d}{dx}\left[V_{1}(x)+V_{2}(x)\right] =-\jmath\omega\left[ L_{m}+L_{s}\right]\left[ I_{1}(x)+I_{2}(x)\right] \]

which becomes

\[\label{eq:8}\frac{dV_{e}(x)}{dx}=-\jmath\omega (L_{s}+L_{m})I_{e}(x) \]

Similarly, using Equations \(\eqref{eq:3}\) and \(\eqref{eq:4}\),

\[\label{eq:9}\frac{d}{dx}\left[I_{1}(x)+I_{2}(x)\right]=-\jmath\omega (C_{s}+C_{m})\left[V_{1}(x)+V_{2}(x)\right] \]

which in turn becomes

\[\label{eq:10}\frac{dI_{e}(x)}{dx}=-\jmath\omega (C_{s}+C_{m})V_{e}(x) \]

Defining the even-mode inductance and capacitance, \(L_{e}\) and \(C_{e}\), respectively, as

\[\label{eq:11}L_{e}=L_{s}+L_{m}=L_{11}+L_{12}\quad\text{and}\quad C_{e}=C_{s}+C_{m}=C_{11}+C_{12} \]

leads to the even-mode telegrapher’s equations:

\[\label{eq:12}\frac{dV_{e}(x)}{dx}=-\jmath\omega L_{e}I_{e}(x) \]

and

\[\label{eq:13}\frac{dI_{e}(x)}{dx}=-\jmath\omega C_{e}V_{e}(x) \]

From these, the even-mode characteristic impedance can be found,

\[\label{eq:14}Z_{0e}=\sqrt{\frac{L_{e}}{C_{e}}}=\sqrt{\frac{L_{s}+L_{m}}{C_{s}+C_{m}}} \]

and also the even-mode phase velocity,

\[\label{eq:15}v_{pe}=\frac{1}{\sqrt{L_{e}C_{e}}} \]

The characteristics of the odd-mode operation of the coupled transmission line can be determined in a similar procedure to that used for the even mode. Using Equations \(\eqref{eq:1}\)–\(\eqref{eq:4}\), the odd-mode telegrapher’s equations become

\[\label{eq:16}\frac{dV_{o}(x)}{dx}=-\jmath\omega (L_{s}-L_{m})I_{o}(x) \]

and

\[\label{eq:17}\frac{dI_{o}(x)}{dx}=-\jmath\omega (C_{s}-C_{m})V_{o}(x) \]

Defining \(L_{o}\) and \(C_{o}\) for the odd mode such that

\[\label{eq:18}L_{o}=L_{s}-L_{m}=L_{11}-L_{12}\quad\text{and}\quad C_{o}=C_{s}-C_{m}=C_{11}-C_{12} \]

then the odd-mode characteristic impedance is

\[\label{eq:19}Z_{0o}=\sqrt{\frac{L_{o}}{C_{o}}}=\sqrt{\frac{L_{s}-L_{m}}{C_{s}-C_{m}}} \]

and the odd-mode phase velocity is

\[\label{eq:20}v_{po}=\frac{1}{\sqrt{L_{o}C_{o}}} \]

Now for a sanity check. If the individual strips are widely separated, \(L_{m}\) and \(C_{m}\) will become very small and \(Z_{0e}\) and \(Z_{0o}\) will be almost equal. As the strips become closer, \(L_{m}\) and \(C_{m}\) will become larger and \(Z_{0e}\) and \(Z_{0o}\) will diverge. This is as expected.

Footnotes

[1] Here \(I_{e}=(I_{1}+I_{2})/2\) and \(V_{e}=(V_{1}+V_{2})/2\).

[2] Here \(I_{o}=(I_{1}-I_{2})/2\) and \(V_{o}=(V_{1}-V_{2})/2\).