- Telegraph Equations

A coaxial cable, which is composed of an inner conductor as a signal line and an outer conductor as a ground shield, forms capacitors between two conductors and inductors with a magnetic field around the inner conductor as shown in Figure 20a. With inductance *L* ¯ and capacitance *C* ¯ per unit length, the system is discretized as the LC ladder composed of *Li*+ 12 = *L* ¯ Δ*z* and *Ci* = *C* ¯ Δ*z* in a unit length Δ*z* as shown in Figure 20b. In the *i*-th segment, Kirchhoff's voltage and current laws are, respectively, expressed as

$$L\_{i-\frac{1}{2}} \frac{\mathbf{d}I\_{i-\frac{1}{2}}}{\mathbf{d}t} = V\_{i-1} - V\_{i\prime} \tag{14}$$

$$\mathbf{C}\_{i}\frac{\mathbf{d}V\_{i}}{\mathbf{d}t} = I\_{i-\frac{1}{2}} - I\_{i+\frac{1}{2}}.\tag{15}$$

In the continuum limit of Δ*z* → 0, the above equations become

$$L\frac{\partial I}{\partial t} = -\frac{\partial V}{\partial z},\tag{16}$$

$$
\mathcal{L}\frac{\partial V}{\partial t} = -\frac{\partial I}{\partial z'}\tag{17}
$$

which are well known as the telegraph equations. Generally, *L* ¯(*z*) and *<sup>C</sup>*¯(*z*) can depend on *z*.

**Figure 20.** (**a**) coaxial cable and (**b**) its circuit model.


Here, we see that the self-dual transmission line does not cause backscattering. If a transmission line has a constant impedance *Z* = *<sup>L</sup>*¯(*z*)/*C*¯(*z*) independent of *z*, then the transmission line becomes self-dual: the telegraph equations with a uniform *Z* are invariant under duality transformation (*<sup>V</sup>*, *I*) ↔ (*<sup>R</sup>*ref*I*, *<sup>V</sup>*/*R*ref) with *R*ref = *Z*. In terms of the new variables *η*± = *V* ± *Z I*, the self-dual telegraph equations are written as

$$\frac{\partial \eta\_{\pm}}{\partial z} \pm \frac{1}{v\_0} \frac{\partial \eta\_{\pm}}{\partial t} = 0,\tag{18}$$

where *<sup>v</sup>*0(*z*) = 1/√*L*¯*C*¯. With arbitrary functions *f*<sup>±</sup>, the general expression of the solution can be provided by *η*±(*<sup>t</sup>*, *z*) = *f*±(*<sup>t</sup>* ∓ *z* <sup>d</sup>*z*/*<sup>v</sup>*0(*z*)), which corresponds to a propagating wave with velocity <sup>±</sup>*v*0(*z*) without backscatterng. If *<sup>Z</sup>*(*z*) depends on *z*, self-duality is broken and backscattering may be observed.


Next, we give a circuit-theoretical derivation of zero backscattering for a self-dual system. A transmission line composed of LC ladder circuits is shown in Figure 21a, where inductances

*Li*+ 12 and capacitances *Ci* may depend on the position *i*. The dual circuit with respect to a global resistance *R*ref is illustrated as shown in Figure 21b. The transformed circuit elements are given as

$$L\_i^\star = \mathbb{C}\_i (R\_{\text{ref}})^2, \quad \mathbb{C}\_{i+\frac{1}{2}}^\star = L\_{i+\frac{1}{2}} / (R\_{\text{ref}})^2. \tag{19}$$

In the limit of small *Li* and *<sup>C</sup>i*+ 12 , each capacitance can be shifted to next position because the potential difference across each inductance is negligibly small. By repeating the shift of the capacitances *Ci* → *Ci*+<sup>1</sup> in the original circuit as shown in Figure 21c, we obtain the same circuit as Figure 21b (*Li* = *Li*+ 12and *<sup>C</sup>i*+ 12= *Ci*) under the following condition for any *i*:

$$R\_{\rm ref} = \sqrt{\frac{L\_{i + \frac{1}{2}}}{C\_i}}.\tag{20}$$

In other words, the LC ladder network with constant *Li*+ 12 /*Ci* is self-dual for impedance inversion with respect to *R*ref defined by Equation (20). In this way, the *global* parameter *R*ref is linked with the *local* impedance *Zi* = *Li*+ 12/*Ci*, which is defined by *Li*+ 12and *Ci* in each site.

Next, we consider excitation of the self-dual LC ladder by a voltage source *V* connected at the left-hand side as shown in Figure 21d. The dual circuit is illustrated in Figure 21e, where the current source is given by *I*s = *V*s/*R*ref. By repeating the shift of the capacitances *Ci* → *Ci*+<sup>1</sup> as depicted in Figure 21f, Figure 21e can be identified with Figure 21f. As discussed in Section 2.6, the self-dual circuit is characterized by the effective resistance *R*ref as shown in Figure 21g. Imagine that we connect a uniform transmission line (T1) with a characteristic impedance of *R*ref to the half-infinite circuit (T2). When the signal propagates from T1 to T2, any backscattering does not appear due to the impedance matching.

**Figure 21.** (**a**) LC ladder circuit and (**b**) its dual circuit; (**c**) capacitors in the circuit (**a**) are shifted; (**d**) LC ladder circuit excited by a voltage source and (**e**) its dual circuit; (**f**) capacitors in the circuit (**d**) are shifted; (**g**) input impedance of the LC ladder circuit.
