# 2.3 Transmission-Line Equations

A transmission line usually connects a source on one end to a load on the other. Before considering the complete circuit, however, we will develop general equations that describe the voltage across and current carried by the transmission line as a function of time t and spatial position z. Using the lumped-element model of Fig. 2-6(c), we begin by considering a differential length Δ z as shown in Fig. 2-8. The quantities υ(z,t) and i(z,t) denote the instantaneous voltage and current at the left end of the differential section (node N), and similarly υ(z+Δ z,t) and i(z+Δ z,t) denote the same quantities at node (N+1), located at the right end of the section. Application of Kirchhoff’s voltage law accounts for the voltage drop across the series resistance R′Δ z and inductance L′Δ z:

υ(z,t)−R′Δ z i(z,t)−L′Δ z∂i(z,t)∂t−υ(z+Δ z,t)=0.(2.12)

Upon dividing all terms by Δ z and rearranging them, we obtain

−υ(z+Δ z, t)−υ(z,t)Δ z=R′i(z,t)+L′∂i(z,t)∂t.(2.13)

In the limit as Δ z→0, Eq. (2.13) becomes a differential equation:

# Figure 2-8 Equivalent circuit of a two-conductor transmission line of differential length Δ z.

Figure 2 8 Full Alternative Text

Similarly, Kirchhoff’s current law accounts for current drawn from the upper line at node (N+1) by the parallel conductance G′Δ z and capacitance C′Δ z:

i(z,t)−G′Δ z υ(z+Δ z,t) −C′ Δ z∂υ(z+Δ z,t)∂t−i(z+Δ z,t)=0.(2.15)

Upon dividing all terms by Δ z and taking the limit Δ z→0, Eq. (2.15) becomes a second-order differential equation:

The first-order differential equations (2.14) and (2.16) are the time-domain forms of the transmission-line equations, known as the telegrapher’s equations.

Except for the last section of this chapter, our primary interest is in sinusoidal steady-state conditions. To that end, we make use of the phasor representation with a cosine reference, as outlined in Section 1-7. Thus, we define

υ(z, t)=ℜ e[V˜(z)e j ω t],(2.17a)
i(z, t)=ℜ e[I˜(z)e j ω t],(2.17b)

where V˜(z) and I˜(z) are the phasor counterparts of υ(z,t) and i(z,t), respectively, each of which may be real or complex. Upon substituting Eqs. (2.17a) and (2.17b) into Eqs. (2.14) and (2.16), and utilizing the property given by Eq. (1.62) that ∂/∂t in the time domain is equivalent to multiplication by j ω in the phasor domain, we obtain the following pair of equations: