rtlonal Bureau of Standards

FEB

Moulder laboratories

*92o. 330

COMPUTATIONS OF THE ANTENNA CUT-BACK

FACTOR FOR LF RADIO WAVES

DUANE C. HYOVALTI

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
THE NATIONAL BUREAU OF STANDARDS

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NATIONAL BUREAU OF STANDARDS

technical Ntote. 330

ISSUED November 26, 1965

COMPUTATIONS OF THE ANTENNA CUT- BACK FACTOR

FOR LF RADIO WAVES

Duane C. Hyovalti

Institute for Telecommunication Sciences and Aeronomy

Environmental Science Services Administration

''Formerly the Central Radio Propagation Laboratory of the National Bureau of Standards;

CRPL was transferred to ESSA in October 1965, but will temporarily use various NBS

publication series pending inauguration of their ESSA counterparts.

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be

listed or referred to in the open literature.

For sale by the Superintendent of Documents, U.S. Government Printing Office Washington D.C. 20402

Price: 30*
Foreword

This work was sponsored by the U. S. Air Force, Rome Air

Development Center, Research and Technical Division, Griffiss Air

Force Base, New York, D. O. AF 30(602) -2488, March 1961 contract.

ii
Contents

Page

Foreword u

Abstract 1

1

.

Introduction 1

2. Mathematical Theory 2

3. Discussion of Figures 7

4. References 10

Figures 11

in

Computations of the Antenna Cut-Back Factor for LF Radio Waves

Duane C. Hyovalti

There are several methods for evaluating the cut-back

factor of geometric optics, e. g., the flat-earth approximation,

the contour integral, and the residue series summation

methods. Wait and Conda give quite extensive details of

the theory of these three methods in their 1958 paper, and

Johler [l96l, 1962] elaborates even further. In this paper

each of the above mentioned methods is investigated, and

calculations are made for various frequencies and distances.

The frequencies used are primarily LF; however, frequencies

up to 1 Mc/s are considered. Several conductivities and

dielectric constants are used, varying from sea water

(cr = 5 mhos/m, e2 = 80) to typical earth (a = 0. 005 mhos/m,

£2 = 15). The permittivity lapse factor, a, is varied from

0. 75 to 1.0 with most of the computations being done for

a = 0. 75 and 0. 85. Only calculations for vertical electric

polarization were carried out in this paper.

Key Words: antenna cut-back factor, computations, low

frequency radio waves, geometric optics,

flat-earth approximation.

1. Introduction

The propagation of a low-frequency (LF) radio signal is represented pictorially by figure 1. The LF wave is transmitted from one

point on the earth to another point, i. e. , from S to O where S is the

source and O is the observer (figure 1). The distance in meters from

S to O along the surface of the spherical earth is d, the radius of the

earth in meters is a, h is the height of the ionosphere in meters, and

6, the angle at the center of the earth subtended by the distance d, is

defined as follows:

d = a 9. (1)
In geometric -optical theory [Bremmer, 1949], the L.F wave travels by

a ray path from the transmitter to the receiver either along the surface

of the earth, known as the ground wave, or by being reflected from the

ionosphere, known as the sky wave. The total field, E(uu, d), which

depends upon the frequency f = U)/2tt, cycles /second, and the distance d,

is defined [Johler, 1961] by

p

E(u>,.d) = £ Ej(u),d), (2)

where E ((JU, d) is the ground wave and Ej(uu, d), j ^ 0, corresponds to the

waves reflected by the ionosphere and is

Ej(uo, d) = iuud D^C exp (ioutj) G

t

J G

r

j a J F j C a

, (3)

where D, is the physical length of the geometric -optical ray and C is a

function of distance, C = 10" /d, for a dipole source current moment

I t = 1. The local sky wave time is t'. , the G' s are the antenna pattern

factors, cc< is the classical convergence -divergence coefficient modified

by a convergence correction [Wait, I960], F, is the cut-back factor

[Wait and Conda, 1958], and C, is the effective reflection coefficient

composed of the Fresnel ground reflection coefficients and ionospheric

reflection coefficients (see Johler [1961, 1962]). The subject of this

paper will be the cut-back factor F,. However, before going into any

detail about Fj, several of the remaining parameters shown in figure 1 will be defined.

2. Mathematical Theory

The angle of incidence of the ray on the ionosphere, cp

t and the

corresponding angle of incidence on the earth, T,, where j refers to a particular ionospheric wave or hop, can easily be evaluated from the

geometry of figure 1. If

6 -

A j = [2a (a+h) (1 -cos — ) +h2

] 2

, (4)

2-
then

-i . sin Tj = Aj (a + h) sin —

,

(5a)

_1

r . _§_

2j "' 2j

cos Tj = A, [a(cos — - 1) +h cos — ] . (5b)

The wave number of the air, ki , and the wave number of the ground, k2 , are defined as follows:

ki =^ni. ( 6 )

uu kK - — 2 - t, 2

- i C L UU (7)

where T] x is the index of refraction of air (1. 000338), c is the velocity

of light (<~3(108

) m/sec), e2 is the dielectric constant of the earth, a is

the conductivity of the earth in mhos/m, and |i is the permeability

(4tt (10~ 7

) h/m). "For typical land, £2 = 15 and a = 0. 005 mhos/m,

whereas for sea water, £ 2 = 80 and a = 5 mhos/m. As the geometric

TT optical horizon is approached, T , ~ — which is represented for j = 1 by

the dotted lines in figure 1.

The cut-back factor F,, where F, = F\ F

T

. , takes account of the

diffraction of rays by the earth's curvature. If it is assumed that the

geometric -optical ray is not too close to or beyond the geometric-optical

horizon, then the flat-earth approximation to the cut-back factor composed of the Fresnel ground reflection coefficients, Re and RB

, satisfies

all the requirements of determining F* and Fr

. . For vertical polarization

F)" ~ [1 +Rr r (T

j )], (8)

where the subscript e refers to vertical electric polarization and the

superscripts t and r refer to the transmitter and receiver, respectively.

The factor F,' r is similarly determined for horizontal polarization by

using Rn

. The Fresnel ground reflection coefficients are
Re(Tj)

ki

kn

COS T ±2

kx sin2 T

cos Tj + LVk,

sin2 T

(9)

for vertical polarization and

cos t

R.(Tj)=

vir)

- sin2 t

<

_,i

2

COS T

i + [(£> sin2 T

2

(10)

for horizontal polarization.

If the geometric -optical ray is close to or beyond the geometric

optical horizon, then the contour integral for F* ' r introduced and developed by Wait and Conda [1958] and used extensively by Johler [1961,

1962] can be evaluated by using a Gaussian quadrature to integrate

,t > r numerically. The factor Fx in contour integral form is

,t > (TT) * exp( i^aG') exp[-i(k1 a/2) 3 9

/ p]

e XI>(-l2 TT /3)

Wx'(p) - qWx(p)

dp, (11)

where 8 = (d - dH )/2a and dH is the distance from the transmitter to the

geometric -optical horizon. An effective earth radius a e is used in place

of the actual earth's radius, a, such that — = a, where a is the vertical

lapse of the permittivity of air at the earth's surface and usually varies

from 0. 7 5 to 0. 85. The functions W1 (p) and W{(p) are defined by Fock

[1946] and used extensively by Wait and Conda [1958]. These functions

are closely related to Airy integrals and Hankel functions of orders 3- and §- (see Wait and Conda [1958]; Johler [1961, 1962]) and are given

as follows:

-4-
Wx (p) = exp [-i2n/3] 5 (-pM2 > K-p)'"

3 *- (12)

and

W^p) =exP [-i4n/3] JJ p h[2)

[ | (-p)* ]

, (13)

where H-l (z) and H2 (z) are Hankel functions of the second kind and of

3

3" orders \ and — , respectively. The arg p has the following restrictions:

TT 5TT - ^ arg p<

3 '

and

arg (-p) = arg p

- TT

arg p

m

i = — arg p where m and n are integers.

The factor q is defined

q = - l (¥o ^y-cfe-) (14)

The contour integral can be written [Johler, 1962]

«t > r Fi' ~(TT) s exp (-ikx a 9 )

» 1

r exp [-i(k1 a/2) 3

6

/ q]

Wi(a) - qWx (a)

da

i

3 Q / / exp[-i(k1 a/2)'3 9

/ a

/ - i2n/3] <i w '/ '\ „ w / ~\ da I < 15

-5-
where a' = a exp (-i2n/3). These integrals can then be evaluated with a Gaussian quadrature. It should be noted that the right hand side of (11)

exactly equals the right hand side of (15). The Airy integrals can now be

evaluated using the following equations:

W^cx) = - i exp [-12TT/3] J^ a^H[2) [|exp (-i3n/2) a^] (16)

~(2

) Wi(a) = exp [-i4n/3] - a H[' f exp (-i3n/2) a

2.2 (17)

W^cc') =

3 . . 2" J 3 a

2 H^'

[ F ia J

n iJ2 ) (18)

W.VH/faH^tia2

;, (19)

where a is a real positive number and the square roots involved are

positive. The Hankel functions can be evaluated using the method described

by Berry [1964],

Beyond and not too close to the geometric -optical horizon, the cutback factor can be evaluated by a residue series summation [Wait and

Conda, 1958; Johler, 1961, 1962]

fV t 2i V TT exp (-i kia-9')£ exp[-i(k1 a)

3 8' t, ] (2*T s -q

2 )W1 (2

3 T s )

(20)

where T = t 8 are the special roots of Riccati's differential equation noted

by Bremmer [1949] —d6 p

- 2 6

2 T + 1 = 0. dT (21)
The factor, 6, for vertical polarization 6 = 6 e , is

-ICY (

(kia)

6 e = x F-7^2 =TT. (22)

4 r^ 2 LVkx

The phase factor A as used by Wait and Conda [1958] is

A = arg F)' r +kia 8' - ^

6' for 9'<0 (23)

(that is for d < dH ) and

A = arg F) ' r + kx a 9

' for 6

' > (24)

(that is for d > d H ). Thus for 8 < 0, A is the phase difference between

the incident field and the resultant field. For 9 > 0, A is the phase

difference between the electrical arc length kx a 8 and the resultant

field in the shadow region [Wait and Conda, 1958]. This phase, A,

changes much less rapidly than the phase actually calculated from (15)

and (20), and is easier to represent in a graph.

3. Discussion of Figures

Figures 2 through 5 represent the amplitude and phase of the cutback factor evaluated by using the flat-earth approximation (8). These

figures are given as a function of distance and are parametric in

frequency. A ground conductivity of 0. 005 mhos/m and a dielectric

constant of 15 are assumed to obtain the amplitude and phase of F,' r shown in figures 2 and 3, respectively. These two figures are representative of the typical earth case. The amplitude and phase in figures

4 and 5 are for sea water, O - 5 mhos/m and C2 = 80. Note that the

amplitude for typical earth decreases more rapidly than that for sea

water as the geometric -optical horizon is approached. The same

phenomenon holds true for the phase. This phase is the phase A since
an exponential term, exp [-ikx a sin 8 ], is omitted from (8), and since

(23) is merely the series representation of this omitted exponential term.

The amplitude and phase curves shown in figures 6 through 23 are

obtained by evaluating the integrals 6f (15) for various conductivities as

a function of distance parametric in frequency. Figures 6 through 19 are

for a permittivity lapse factor, a, of 0. 75, and figures 20 through 23 are

for an a of 0. 85. A conductivity of 0. 005 mhos/m and e2 = 1 5 are used

to obtain the amplitude and phase curves shown in figures 6 and 7. Figures 8 through 11 give the amplitude and phase for O = 0. 05 mhos/m.

An enlarged graph of the magnitude of F^ ' in the region where d ^ dH , shown in figure 9, reveals that the magnitude of 20 kc/s is greater than

those of 50 or 100 kc/s at the geometric -optical horizon, but that it

crosses over to become less than these at shorter distances. A similar

enlargement of the same area for the argument of F1

'

, figure 11,

shows no such crossover. However, maxima for the arguments of 20

and 50 kc/s will be noticed. A conductivity of 0. 5 mhos/m is assumed

to obtain the amplitude and phase curves in figures 12 through 15.

Figures 13 and 15 show enlarged regions for d < dH for amplitude and

phase, respectively. The amplitude of 20 kc/s crosses over all the

others, 50 kc/s crosses over all the others except 20 and 1000 kc/s, etc. as shown in figure 13. All the phase curves except 500 and 1000 kc/s

have maxima for d < dH . For the sea water case, C7 = 5 mhos/m and

e2 = 80, the amplitude and phase are shown in figures 16 through 19.

Close-ups of the area where d ^ dH are given again for the amplitude and

phase in figures 17 and 19, respectively. In this case it is observed that

all the amplitude curves cross over and that even some of the phase curves

cross over in addition to having maxima in this region. It should be noted

that the amplitude and phase curves decrease more rapidly for the lower

conductivities than for the higher ones. Figures 20 through 23 show the

magnitude and argument of F* ' r for the sea water case, a = 5 mhos/m

and e 2 = 80, for an a of 0. 85. The same effects as are observed for the

sea water case for a = 0. 75 are observed for 0. 85. However, the amplitude and phase curves for a = 0. 85 tend to decrease more rapidly than

those for a = 0. 75.

Figures 24 and 25 illustrate the use of the residue series summation (20), to calculate the amplitude and phase of the cut-back factor

beyond the geometric -optical horizon. A conductivity of 0. 005 mhos/m

and € 2 = 15 are used in addition to assuming a = 0. 75. If the integral

method (15), and the residue series method (20), are compared, it will

be seen that they fall exactly on top of one another as can be noted by

comparison of figures 6 and 24, and 7 and 25. There seems to be a loss

of precision in calculating the residue series method at 200 kc/s. Therefore, the integral method is used when it seems necessary.

-8-
The amplitude and phase curves shown in figures 26 through 28

are for three different sets of conditions as a function of frequency.

The integral method was used to calculate the amplitude and phase curves

for these three figures. Models 1 and 3 are for a height, h, of 65 km.

A conductivity of 5 mhos/m and £ 2 = 80 are used for model 1 along with

a distance of 1609 km (1000 miles). A distance of 2127 km, = 0. 005

mhos/m and € 2 = 1 5 are used for model 3. Model 2 uses a height of

45 km, a distance of 1416 km (880 miles), a conductivity of 5 mhos/m

and S 2 = 80. All three models use an a of 0. 85. The amplitudes for

models 1 and 2 tend to remain almost constant or rise as the frequency

increases, whereas the amplitude for model 3 decreases as the frequency

increases as can be seen in figure 26. This phenomenon is primarily

due to the fact that the distance for model 3 puts it well beyond the

geometric -optical horizon while the distances for models 1 and 2 are

less than the distance to the horizon. The fact that model 3 is a typical

earth case whereas models 1 and 2 are sea water cases has some effect

but much less than that caused by the distances. Figure 27 shows that

the phases for models 1 and 2 reach maxima and then decrease as a function of frequency. The phase for model 3 decreases steadily as the

frequency increases, as is shown in figures 27 and 28. The reason that

the phase for model 3 has no maximum like those for models 1 and 2 is

primarily due to the fact that model 3 is the typical earth case while

models 1 and 2 are sea water cases.

Figures 29 through 32 show the amplitude and phase of the cut -back

factor as a function of frequency parametric in distance. A height of 40

km, a permittivity lapse factor of 1, a conductivity of 0. 005 mhos/m and

£3=15 are used for these figures. The integral method is used to obtain

figures 29 and 30 while the flat-earth approximation is used to obtain

figures 31 and 32. It should be noted that both the amplitude and phase

obtained with the integral method reach maxima and then decrease as a function of frequency whereas they just decrease steadily for the flatearth method.

All of the figures discussed above, figures 2 through 32, are for

vertical electric polarization. It is interesting to note that the integral

method works far beyond the geometric-optical horizon.

9-
4. References

Berry, L. A. (1964), Computation of Hankel functions, NBS Tech. Note

No. 216.

Bremmer, H. (1949), Terrestrial radio waves --theory of propagation

(Elsevier Publishing Co. , New York, N. Y. ).

Fock, V. A. (1946), Diffraction of radio waves around the earth's surface

(Publishers of the Academy of Science, Moscow, USSR).

Johler, J. R. (1961), On the analysis of LF ionospheric radio propagation phenomena, J. Res. NBS 65D (Radio Prop. ), No. 5, 507-529.

Johler, J. R. (1962), Propagation of the low -frequency radio signal,

Proc. IRE 50, No. 4, 404-427.

Wait, J. R. (I960), Diffractive corrections to the geometrical optics of

low frequency propagation, Electromagnetic Wave Propagation,

87-101 (Academic Press, London, England).

Wait, J. R. , and A. M. Conda (1958), Pattern of an antenna on a curved

lossy surface, IRE Trans. Ant. Prop. AP-6, No. 4, 348-359.

10-
£

u

<u

O

O

>

i—

i

•rH

u

• +-> O rt -1 o

o

i u

ft w>

GO

fn

-11-
-2000 -1500 -1000 -500

DISTANCE, d-d H , km

Figure 2. Amplitude of the cut-back factor F\

-12-

t > r
if)

<

q:

Ll! C/) <

X

Q_

•0.1

-0.2

•0.3

-0.4

-0.5

-0.6

-0.7

VERTICAL POLARIZATION

FLAT- EARTH APPROXIMATION

l + Re O"=0.005

62

= 15

0.8 -2500 -2000 -1500 -1000 -500

DISTANCE, d-d H ,km

Figure 3. Phase of the cut-back factor F^

-13-

t > r
2.0

1.8 -

.6 -

u_ Id - 1.4 LUQ

Z)

_l

Q_

<

1.2

1.0

0.8

20kc/s'

50

00

200

500

1000 kc/s

VERTICAL POLARIZATION

FLAT- EARTH APPROXIMATION

l+Re cr= 5

€ 2 =80

0.6

-2500 -2000 -1500 -1000 -500

DISTANCE, d-dH ,km

Figure 4. Amplitude of the cut-back factor F 3

'

14-
CO

<

LUCO<

X

0.

•0.2

-0.3

VERTICAL POLARIZATION

FLAT- EARTH APPROXIMATION

l+Rc cr=5

62 =80

•0.4

•0.5•2500 2000 -1500 -1000 -500

DISTANCE, d-d H , km

Figure 5. Phase of the cut-back factor F 3

t > r

15
-2500 -2000 -1500 -1000 -500 500 1000 1500 2000 2500 3000

DISTANCE, (d-d H )/2, km

Figure 6. Amplitude of the cut -back factor F* ' r

.

-16-
-1

-2

-3

-4

-5

<° R <

O

< -7 cr< -8

< ^ X

Q_

-10

-12

-13

-14

-15

-16

-17

VERTICAL POLARIZATION

INTEGRAL METHOD

cr = 0.005

€ 2 = 15

a = 0.75

2500 -2000 -1500 -1000 -500 500 1000 1500 2000 2500 3000

DISTANCE, (d-d H )/2, km

Figure 7. Phase of the cut-back factor Fi ' r

•

-17-
10

-2

10 UJQ

z>

10

10

io

5

VERTICAL POLARIZATION

INTEGRAL METHOD

0" = 0.05

a =0.75

2500 -2000 -1500 -1000 -500 500 1000 1500 2000 2500 3000

DISTANCE, (d-d H )/2,km

Figure 8. Amplitude of the cut-back factor f£ ' r

.

18-
LO

CSJ

LO

CSJ

1 £ o

<=>

C\J

o> ^"~>»

rt lo X £>

i D +j

1 3

"O u V -* 0)

•» 6

UJ <+-H loo O

r— 0)3

i <

CO

Q

CSJ 8

ro

: i CSJ

J'* 'aanindvw

a

19

^
COz

<

Q

<

l±J CO<

X

Q.

-7

-10

-12

-13

-14

-15

-16-

-17-

-18-

VERTICAL POLARIZATION

INTEGRAL METHOD

CT = 0.05

a

150.75

-19 2500 -2000 -1500 -1000 -500 500 1000 1500 2000 2500 3000

DISTANCE, (d-d H )/2, km

Figure 10. Phase of the cut-back factor F* ' r

.

20
LO

CSJ

o10

esj

i E

_s^

#* r< C\J - o \ c» ^^—

lO X 1 "D

1 O

"O o

rtJ ^—

m 0k LU M

<=> O

lO

<

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C/) u Q 0) o >-M C3 O

<=>

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(In o .—1 to i—

1

CSJ ^B CD

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ir> try o to

CSJ WOCSJ lO

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lO

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I

lO

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SNviava'VasvHd

-21
10

10 UJQ3

I10 -

-4

10

10

VERTICAL POLARIZATION

INTEGRAL METHOD

cr = 0.5 e 2

= 15 a =0.75

•2500 -2000 -1500 1000 -500 500 1000 1500 2000 2500 3000

DISTANCE, (d-d H)/2, km

Figure 12. Amplitude of the cut-back factor f\ '

-22-
<

N

Q

O

a: u ^ m s. o

Q_

<O

q:

uj

>

o

II <

— O

ii ii <M

55

CSJ

10

CSJ

i u E *> 1- .2£ h

»» ?H CVJ o

f

<=>» ^^^^ \ rtJ to X m

i -o ^

1 T3 43 **-^ 1 +-> •» 3

UJ o o o 0) lO

r— z i < m

H

CO

Q 24-1

to

C\J

J I I L o> oo r— c© to

ro CSJ

lO

: i

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-23-
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CO -5-

2 -6

uj -7 CO<

-8-

-9-

-10-

•12-

-13-

14

1 1 1 1 1 1 1 1 1 1

— —

— ^\N\^^T^""•v^^

— \ N^^v ^•v—

— \\ ^\ >s—

—

VERTICAL POLARIZATION \ \

—

— INTEGRAL METHOD

CT = 0.5 —

€ 2

= 15 Q =0.75 \o \

— \i \

\ —

— —

— —

— —

—

1 1 1 1 1 1

1 1 1 1 \

2500 -2000 -1500 -1000 -500 500 1000 1500 2000 2500 3000

DISTANCE, (d-d H )/2, km

Figure 14. Phase of the cut-back factor F* ' r

.

-24.
to

CSJ

LO

OJ

E

.*: (4

•* *

CVJ 4* r-\ <=> \ Uh « X j_i T3

1 +-> o u

>—' UJ ^

<=> o o mr—

i z:<

(X) H

(/) 2

o u

CD o A o -M o m

1 O

<D

Cfi Rj

to

CSJ

LT)

<£ 22 S '

SNViaVd'V'3SVHd

•25
10

10

LxJ Q

-3

10

10

20 kc/s

VERTICAL POLARIZATION

INTEGRAL METHOD

CT = 5

€ z

= 80

10-2500 -2000 -1500 -1000 -500 500 1000 1500 2000 2500 3000

DISTANCE, (d-d

H )/2, km

Figure 16. Amplitude of the cut-back factor F^ ' r

. -26-
E

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UJ

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CO

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o+-> om

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U.S. DEPARTMENT OP COMMERCEi

OFFICIAL BUSINESS