Page 324 - Engineering Electromagnetics, 8th Edition

306 ENGINEERING ELECTROMAGNETICS

Then, using (3) and simplifying, we obtain

∂ I z ∂ ∂V
=− 1 + GV + C (7)
∂z 2 ∂z ∂t
Again, we obtain the final form by allowing z to be reduced to a negligible magni-
tude. The result is

∂ I ∂V
=− GV + C (8)
∂z ∂t

The coupled differential equations, (5) and (8), describe the evolution of current
and voltage in any transmission line. Historically, they have been referred to as the
telegraphist’s equations. Their solution leads to the wave equation for the transmission
line, which we now undertake. We begin by differentiating Eq. (5) with respect to z
and Eq. (8) with respect to t, obtaining:
2
2
∂ V =−R ∂ I − L ∂ I (9)
∂z 2 ∂z ∂t∂z
and
2
∂ I ∂V ∂ V
∂z∂t =−G ∂t − C ∂t 2 (10)
Next, Eqs. (8) and (10) are substituted into (9). After rearranging terms, the result is:

2
2
∂ V = LC ∂ V + (LG + RC) ∂V + RGV (11)
∂z 2 ∂t 2 ∂t
An analogous procedure involves differentiating Eq. (5) with respect to t and Eq. (8)
with respect to z. Then, Eq. (5) and its derivative are substituted into the derivative of
(8) to obtain an equation for the current that is in identical form to that of (11):

2
2
∂ I = LC ∂ I + (LG + RC) ∂I + RGI (12)
∂z 2 ∂t 2 ∂t
Equations (11) and (12) are the general wave equations for the transmission line.
Their solutions under various conditions form a major part of our study.

10.3 LOSSLESS PROPAGATION
Lossless propagation means that power is not dissipated or otherwise deviated as the
wave travels down the transmission line; all power at the input end eventually reaches
the output end. More realistically, any mechanisms that would cause losses to occur
have negligible effect. In our model, lossless propagation occurs when R = G = 0.