Page 401 - Engineering Electromagnetics, 8th Edition

CHAPTER 11 The Uniform Plane Wave 383

the second term in (60b)is small enough, so that

.
β = ω µ (61)
Applying the binomial expansion to (48), we obtain, for a good dielectric

. µ 3 σ 2 σ
η = 1 − + j (62a)
8 ω 2ω
or

. µ σ
η = 1 + j (62b)
2ω
The conditions under which these approximations can be used depend on the
desired accuracy, measured by how much the results deviate from those given by
the exact formulas, (44) and (45). Deviations of no more than a few percent occur if
σ/ω < 0.1.

EXAMPLE 11.5
As a comparison, we repeat the computations of Example 11.4, using the approxima-
tion formulas (60a), (61), and (62b).
Solution. First, the loss tangent in this case is / = 7/78 = 0.09. Using (60),

with = σ/ω,wehave

. ω µ 1 12 9 377 −1
α = = (7 × 8.85 × 10 )(2π × 2.5 × 10 )√ = 21 cm
2 2 78
We then have, using (61b),
√
.
8
9
β = (2π × 2.5 × 10 ) 78/(3 × 10 ) = 464 rad/m
Finally, with (62b),
. 377 7
η = √ 1 + j = 43 + j1.9
78 2 × 78
These results are identical (within the accuracy limitations as determined by the given
numbers) to those of Example 11.4. Small deviations will be found, as the reader can
verify by repeating the calculations of both examples and expressing the results to four
or five significant figures. As we know, this latter practice would not be meaningful
because the given parameters were not specified with such accuracy. Such is often the
case, since measured values are not always known with high precision. Depending
on how precise these values are, one can sometimes use a more relaxed judgment on
when the approximation formulas can be used by allowing loss tangent values that
can be larger than 0.1 (but still less than 1).