Page 335 - Engineering Electromagnetics, 8th Edition

CHAPTER 10 Transmission Lines 317

and quantify. We will study this in Chapter 11, and we will apply it to transmission
line structures in Chapter 13.
Finally, we can apply the low-loss approximation to the characteristic impedance,
Eq. (47). Using (51), we find

 
2

R R R
R + jωL jωL 1 + jωL . L 1 + j2ωL + 8ω L 2
2

Z 0 = = =   (55)
G + jωC G C G G 2
jωC 1 + 1 + +
2
jωC j2ωC 8ω C 2
Next, we multiply (55) by a factor of 1, in the form of the complex conjugate of
the denominator of (55) divided by itself. The resulting expression is simplified by
2
2
neglecting all terms on the order of R G, G R, and higher. Additionally, the approx-
.
imation, 1/(1 + x) = 1 − x, where x 1is used. The result is

2 2
. L 1 1 R G G j G R
Z 0 = 1 + + − + − (56)
C 2ω 2 4 L C C 2 2ω C L
Note that when Heaviside’s condition (again, R/L = G/C) holds, Z 0 simplifies to
just √ L/C,asis true when both R and G are zero.
EXAMPLE 10.3
Suppose in a certain transmission line G = 0, but R is finite valued and satisfies the
low-loss requirement, R ωL. Use Eq. (56) to write the approximate magnitude
and phase of Z 0 .
Solution. With G = 0, the imaginary part of (56) is much greater than the sec-
2
ond term in the real part [proportional to (R/ωL) ]. Therefore, the characteristic
impedance becomes

. L R jθ
Z 0 (G = 0) = 1 − j =|Z 0 |e
C 2ωL
. √ −1
where |Z 0 | = L/C, and θ = tan (−R/2ωL).
D10.1. At an operating radian frequency of 500 Mrad/s, typical circuit values
for a certain transmission line are: R = 0.2 /m, L = 0.25 µH/m, G =
10 µS/m, and C = 100 pF/m. Find: (a) α;(b) β;(c) λ;(d) ν p ;(e) Z 0 .
8
Ans. 2.25 mNp/m; 2.50 rad/m; 2.51 m; 2 × 10 m/sec; 50.0 − j0.0350

10.8 POWER TRANSMISSION AND THE USE
OF DECIBELS IN LOSS
CHARACTERIZATION
Having found the sinusoidal voltage and current in a lossy transmission line, we next
evaluate the power transmitted over a specified distance as a function of voltage and
current amplitudes. We start with the instantaneous power, given simply as the product
of the real voltage and current. Consider the forward-propagating term in (49), where