298 RADIO WAVE PROPAGATION EXPERIMENTS 
where P, is the incident power. 
The absorption index & in the complex refractive 
index 
N=n-—jk (78) 
is related to the absorption coefficient by 
4nk 
= 79 
Gis (79) 
and hence 
2.3 P; 
k= logis = 80 
4nd =" P,’ ) 
where d stands for (d,—d,). Also 
N? = (n— jk)? = & — jes 
=e j600N, (81) 
er is the real part of the dielectric constant, ¢ its 
imaginary part; o is the conductivity of the sub- 
stance is mhos per meter; and A the wavelength in 
meters. One obtains readily from equation (81) 
n—k? =e, 
2nk = e; = 600d. >) 
The absorption index & is measured directly by two 
galvanometer readings proportional to P, and P,. 
The refractive index n is derived from the reflection 
coefficient J?y for almost perpendicular incidence, 
using 
2 V+kh?+1—2n 
Nn +ke+ 1+ 2n° 
Then n and & determine e, and c. Saxton claims that 
in this method at least one quantity, the absorption 
index, is measured directly while the other, the re- 
fractive index, is derived from the measurement of 
the reflection coefficient. 
In the other methods, given later, neither of these 
quantities is measured directly. 
(83) 
Stranping Wave Rario Mreryuop 
By limiting the clectromagnetic field to the en- 
closure of a hollow pipe or coaxial line, the energy 
is completely confined, stray effects are eliminated, 
and small amounts of any dielectric can be inves- 
tigated accurately.4°*? The following gives the the- 
oretical foundation of this “standing wave ratio” 
method for measuring complex dielectric constants. 
1. A transmitter radiates waves of a given fre- 
quency into one end of a closed wave guide. These 
are reflected by the metallic boundary at the other 
end. Standing waves are set up in the guide, and 
they can be measured by a probe detector traveling 
along a slot in the pipe parallel to its axis. The dielec- 
tric is inserted at the closed end of the pipe, opposite 
the transmitter, and fills the pipe up to a height d. 
Above it the standing wave pattern is measured in air. 
The real and imaginary parts ¢, and «; of the dielec- 
tric constant are calculated from the ratio of the field 
strengths in node and antinode Hmin/Emax, and the 
distance 2, of the first node from the surface of the 
dielectric. 
The modulus and the argument of the reflection 
coefficient are obtained from 
_ (ZO)/Zu) — 1 
~ (Z)/Za) + 1’ 
where Z(0) is the characteristic impedance of the pipe 
section filled with the dielectric understudy, Z,, is 
the intrinsic impedance of the air-filled portion of 
the pipe. By denoting 
Z(0) 
Za 
where 5, and 8 are, respectively, the real and im- 
aginary part of 8, the reflection coefficient R can be 
written as 
R= pei® (84) 
= tanh 6 = tanh (5, + j6,), (85) 
R= |Rle-*, 
= pei, 
(86) 
with 
p= |R| =e-*s arg R = —r — 26; = —®. (87) 
From the expression of the reflected field strengths 
one finds that the distance 2, of the first node from 
the surface is given by 
— 6M 
ar 
where 6; is connected directly to the phase shift ® at 
reflection through equation (87), and A, is the wave- 
length in air of the radiation. The measurement of 
Z, thus yields 8; Similarly, one finds that 
tanh 6, = | Enin / Emax |. (89) 
2. Calculation of diclectric constant and loss fac- 
tor from terminating impedance. The intrinsic im- 
pedance of the dielectric-filled portion of the guide 
is found to be 
T = ’ (88) 
Z(0) = Ze tanh 72 d, (90) 
where 
Ze aeen (91) 
v2 
The subscript 2 refers to the dielectric medium, pz 
is its permeability and y, is the propagation coustant 
of dielectric-filled section of the guide; for the TH 
waves, Zo. is the impedance of the dielectric medium 
itself. Using equations (90) and (91), one gets 
tanh yod MG Z(0) Mi (92) 
yedl Z an dy pe 
The propagation constant y, determines finally the 
complex dielectric constant «, through the funda- 
