# Coupled Transmission Line Analysis

In this section, we focus on the simple theoretical analysis of a CTL loaded with inductive or capacitive elements and derive the associated dispersion relation. To formulate the problem, two identical transmission lines with similar constitutive parameters (L,C) are considered, as illustrated in Fig. 3.4. For the moment, only inductive coupling is considered. The specific CTL is illustrated in Fig. 3.4.

**Figure 3.4 **Coupled transmission lines with its unit cell to the left described using identical lumped circuit elements (*L,C*). A common ground is used forming a typical three-phase power transmission system.

## Derivation

Let us assume that the two identical transmission lines (*i.e *straight wires) have the same per unit inductance of L, and per unit capacitance of C. Thus, individually, each transmission line supports a wave that has the same phase velocity, *1/yfhC* , ideally the speed of light for free-space propagation. When the pair of transmission lines is close to each other, they exhibit coupling. This coupling can be described by an inductor placed between the two transmission lines when forming their equivalent circuit. Using this unit cell model, the telegrapherâ€™s equation takes this form:

In the above, the voltages *V _{12}* and currents

*I*refer to the excitation values as measured between each transmission line and the ground lines. We also note the presence of additional terms due to

_{12}*L*These terms do not exist in the telegrapherâ€™s equations and will be responsible for the more complex dispersion curve.

_{M}.To solve Eq. (3.6), we shall assume a time dependence of e^{j(wt-bz)}, where *w* is angular frequency and *b* is wavenumber along the direction (z-axis) of propagation. Applying the time dependence on a pair of equations (3.6a and 3.6b), the resulting dispersion relation takes the form

The solution to this fourth-order equation leads to four possible constants:

The corresponding phase velocities are:

Clearly, the presence of L_{M} leads to slow wave formation as depicted in Fig. 3.5. But more importantly, this *p _{2}* implies a much slower wave and, therefore, more control to phase velocity and more control in designing smaller antennas, couplers, RF signal dividers, smaller TWTs, and even low- and high-power BWOs. Clearly, the added mutual inductance has led to the realization of slow waves using circuit parameters rather than actual dielectrics. Therefore, this analysis demonstrates that the dielectric behavior can be emulated by CTLs.

**Figure 3.5 **Coupled transmission lines can support slow waves. The plotted dispersion lines (red and blue lines) provide an approach for artificial dielectrics. The CTL parameters are *(L,C)* = (1.61722 pH, 6.8798 pF) and L_{M} = 0.37308 pH.