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RADIO SCIENCE Journal of Research NBS/USNC- URSI Vol. 68D, No.7, July 1964

Calculation of Groundwave Attenuation in the For Diffraction Region

1. E. Vogler

Contribution From the Central Radio Propagation Laboratory, National Bureau of Standards, Boulder, Colo.

(R eceived J a nuary 3, 1964)

Thi s paper presents a graphi cal m et hod of determ ining t he groundwave attenu at ion o ver a spherical ho mogeneo us ea rth in t he far diffract ion regio n. The curves ar e applicable to either vertical or hori zonta l polarization an d to any co mbina tion of effective eart h 's radi us, electromagnet ic grou nd constant s, frequency, path d ista nce, and ante nna heigh t s . A cr iterion is give n that indicates t he m ethod may be used not on ly for far beyo ndlill e-o f­s ight pat hs but, in mallY pract ieal situations, at lin e-of-s ig ht or e ven sli g ht ly within . Examples illust rat ing the usc of t he formulas and curves are i ncluded.

1. Introduction

In a previous paper [Vogler , 1961] a simplified gr aphical method was presenLed for calcula ting smooth homogeneous earth groundwave attenuation in the far diffraction r egion assuming horizon tal polarization of the r adio waves. In the present p aper this method will b e extended to includ e the case of ver tical polari zation.

(1 ) , m rty b e divided in to four term s con taining, essen Lially, the distance depend ence, two antenna heigh t d ependences, and a term depend ing on the electric constan ts of the ground:

The far diffr ac tion region is defined her e as that region in which the d iffr acLed field intensity may b e determined by the first term of tbe Van der Pol­Bremmer r esid·ue series [Bremmer, 1949]. This region extends from near the r adio horizon to all gr eater dist ances. In some si tuations the fi rst term provid es a valid approximation to t he diffracted field even at poin ts slightly within line-of-sight, and a cri terion for determining the minim um distance is given in section 2.

Since this paper is meant as a practical aid in obtaining diffraction calcula tions, no mathematical derivation of the functions discussed later is includ ed. D etailed explanations of the r esidue series are given in the work by Bremmer [1949] and also in the introduction to the CCIR Atlas [Atlas, 1955]. As far as possible, the notation used is that established by Norton [1941 ] when identical quantities are considered ; e.g., the param eters K and bOo The three functions G" F, and G from which the attenu­rttion is crtlculated were defined in such a m anner as to facili tate gr aphical interpolation for any combi­nation of frequency and distance. Numerical evalu­ations of these functions were obtained tln·ough the use of an elec tronic computer.

2. Groundwave Attenuation

The attenu ation Al (measured in decibels) relative to a free sp ace inverse distance fi eld (see, for example,

where

and Jr.m z denotes the radiofr equency in megahertz. do is the total ar c distance b etween antennas, and ell and el2 ar e distances from the transmitting and r eceiving an tennas to their radio hori;"ons (see fi g. 1). F or antenna heigh ts hi and h2 not too large, the horizo n distances may b e approxim ated for a linear atmosphere with no ducts by

(3)

wh er e lea is the effective r adius of the sphere.

FlGURE 1. The geometry JOT smooth earth diffrac ti on.

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-l In practical applications the factor k is used to

account approximately for the effects of atmospheric r efraction [Burrows and Atwood , 1949] and is defined in terms of the gr adient with respect to height, cln/clh, of the atmospher e's r efractive index n :

and horizontal polarization. Vertical polarization:

k- [ l +(0. cln) J-I•

- n dh 11=0 (4)

be 2 t - I (E) - I (E - 1). v= an 8 -tan -8- '

horizontal polarization:

be 00 -I (E - 1) h= l S -tan -8-'

(6b)

(7a)

(7b)

It is usual to assum e a standard "4/3 earth" a tmosphere in presenting diffraction curves [Atlas, 1955; Wai t and Howe, 1956 ], and this procedure is followed here. Thus, the Co appearing in (2) is merely a normalizing factor which for the standard a tmosphere (k = 4/3) is equal to unity. Choosing a reference radius a= 6373 km, (3) may now b e expressed as

where 8 depends on the ground conductivity in mhos pel' meter and radiofrequency, and is defined as

(5) 8= 1.SX 104a(mhos/m)/jMHz, (S)

where h is in meters. The quan tities xo, XI , and X2 Note that for a(mhos/m)/.fMHz> >(E/2) X 10- 4:

defined by (2) can be considered as parametric distances ; xo referring to the total separation between Kv "'2.3S5Co "lI(il.ffr~" b~ rvO, (9) antennas, and XI, X2 referring to the two antenna radio horizon distances. Kh"' 1. 325X 10-4CojU~Jfo, b~ rv1 S0 ; (10)

The param eters {30, K, and bO in (1) and (2) depend on the polarization of the wave and the and for u(mhos/m)IJMHZ< « E/2) X 10- 4:

relative dielectric constant E and conductivity u of the ground . Assuming a homogeneous ground in Kv~1.777SX 10-2GoE/-,jE 1jl{1z, b~~90, (11 ) which E and a are everywhere constant, K and bO are defined as follows for the two cases of vertical K,.'~:d.777SX lO-2GoNE- ljU1z, b~~90. (12)

180

t-t--- r--... HORIZONTAL POLARIZATION

--- VERTICAL POLARIZATION

~ IIIII II AVERAGE GROUND Y

1\ :-- SEA WATER

E = 15 E = 8 1

CT= 0.005 mhos/m \ 1-

1\ CT= 5 mhos 1m I 1II 1I11

POOR GROUND -'--

E = 4 1'\ GOOD GROUND 1\

0":0.001 mhos/m < ' 25

I nT1T11 CT= 0.02 mhos/m

165

150

135

120

105

; V:,:AGE GROUND 1411 t---- t---t--(T = 0.005 mhos 1m /' ~. --+-- -- -

)-/ / /

75

60

30

/1' I

POOR GROUND GOOD GROUNO ASEA WATER

<' 4 ~ / / < '25 E = 8 1

0"= 0.001 mhos/m (J= O.02 mhos/m cr = 5 mhos/m

I / /

45

/

I / / ;-

;' - -15

o 0.01 0.1 10 100 1000 10000 100000

FREOUENCY IN MEGAHERTZ

FIGURE 2. The parameter b; or b~.

820

0.2 r0 "-<t 0.1 " .x 0.07 -::.:: 0.05

0.02

0.01 0.007 0.005

0.002

0.001 0.0007 0.0005

0.0002

I--

--I-

f-I-

" " r,

GOOD GROUND ~ 1----'--, t-+- E ~ 25

I----'--+-+ (J'~ 0 .02 mhOS/METER

I

I-

l­

I-

-

. -I-

-t-t--I-

571020

I-HORIZONTAL POLARIZATION

I-t- ---- VER TI CA L POLARIZATION

!-

50 70 100 200 500 1000 2(xx) 5000 !l000 20000 50000 100000

FREQUEN CY IN MEGAHERTZ

FIGURE 3. The parameter I\:. , or K ". For general Ii, K (k) = Co K (4(3) .

Curves of bO and K versus jMHz are shown in figures 2 and 3 for various combinations of e and (J

corresponding to poor ground, average ground, good ground , and sea water . A value of k = 4 /3 was assum ed in obtaining the curves of figure 3; however, they may b e used for other k thr'ough the simple relationship : K (1c) = OoK (4 /3).

The parameter f30 appearing in (2) is plotted as a function of K and bO in figure 4. The limiting value of f3o = 1.607 for K :50.01 may be used for almost all practical cases of horizontal polarization. On

the other hand for vertical polarization f30 mu.y range from 1.607, corresponding to high frequency propagation, down to 0.700, corresponding to low frequency propagation over sea water. (See fgs. 3 and 4.)

With K , be, and f30 evaluated, the groundwave attenuation may now be calcula ted by (1) and figures 5 and 7. 01(K, be), obtained from figure 5, has the limi.ting values 0 1(0, bO)= 20.03 for K ---'70 and 01(00, bO)= 20.94 for K ---'7 oo; for any value of K

821

I. 7

6 --: ~ r:: I'--r-.i'-

Lim. 0 1.6071 I'--r-.

~ K- 0

5

\ ~ 4

~ l\ ~ 1\ 3

\ 1\

I.

~ ~ 2

1\ \

1\ r... I

1\ \ ~

1\ \\ KOlBO

1\ bOo~ \

'" \

1\ "'-bOo 0 "- "

I.

f3 0

1.0

0.9

0.8

.........

\ .......... :::::::: L im 00.7003

-.:..--::

"-.... 0.7

0.6 0.01 0.02 0.05 0.1 0.2 0.5 1.0 10 20 50 100

K

FIGURE 4. The parameter f30.

it can be seen that 0 1 does not vary more than abou t 6 dB.

The functions G(xo) , F(Xl), and F (X2) are shown in figure 7, G(xo) being the uppermost curve and defined by:

(13)

In general the height function F(x) depends on K I and b 0, except for large values of Xl or X2, in which

case F (x) ~ G(x). For Xl or Xz approaching zero, the limiting values of F(x) are plotted in figure 6 which may thus be used to help interpolate for those values of K and bO not shown in figure 7.

Since (1) is based upon only the first term of the residue series, a fairly good indication of its range of validi ty may be gained from the ratio of the second term Tz to the first term Tl of the series. If we requu'e the error in At to be less than some given

822

~ ~

u ()O ~ ~

22

21

20

19

18

17

16

15 0.01

. I Lim . = 20 .03 K--O

-- ..... ~ ...

0.02 0.05

1/ /

:~ ~ '\

0.1 0.2

~r- ..... , / bO = 180

~,

~

/.

~ l1 ~ bO =9~

V .-

\ \ \ \.1

0.5

I

Va I )

1.0

K

i--

~ ~t::::

V

V

2 5

F I GU RE 5. The te1'1n C1 (l( , be) for use in (1 ) .

;:::::::: ~;:::;;-Lim. = 20.94 K-- OJ

- -

10 20 50 100

-10

- 20

- 30

0 x

c;::

-40

I-

-

t--

r-

-

v

Lim. ~ -15.68 K - CO

VV f---- 7"1/- L im. ~ 20 log K - 11.94 --+--+-t-H+t-+-If-------t--+--t-t-t-Htt----t---t----t---t---t-r--t-H

K-O - 50

/ - 60

f--

-700.01 0.02 0.05 0.1 0.2 0.5 1.0 10 20 50 100

K

FIGURE 6. The height gain j unction F (x = O) jar zero antenna heights.

value, say 0, we have that: Using linear interpolation:

This is the procedure used to obtain the tl (Xj.2 ) curves shown in the lower right hand corner of figure 7. Using the limiting values of {3o (1.607 and 0.700), then for A t to be accurate to within 1.5 dB (approximately) it may be shown that:

xo-xj tl (X j)-X2tl (X2) > 335, ({3o= 1.607 , K,:SO.I ) (15a)

xo-xj tl (Xj)-X2 tl (X2) > 115, ({3o= 0.700, K?:,10 ). (15b)

Notice that in certain cases (1) provides a good approximation to the attenuation even for just within line-of-sight paths as long as the appropriate condition (15a) or (15b) holds. For values of (3o lying between the two limits, linear in terpolation between the tl (x) curves of figure 7 and the two minimum values of (l5a) and (15b) will give a fair approximation to the range of validity of (1) .

where

Xm l11 = 335 - 242.6(1.607 - (3o) , I tl (x, (3o )= tl (x , 1.607)+ 1.103(1.607 - {3o) I

X {tl (x, 0.700)- tl (x , 1.607) } ;J

(16a)

(16b)

tl (X, 0.700) and tl (x, 1.607) are the values read from the upper and lower curves respectively in figure 7.

3 . Examples

In ord er to use (1) to calculate the field strength E in a particular situation, a reference field Eo must be specified. The relationship between A t and E is then simply:

(17)

824

0

f-I I I I I I j I I

A,-G('0) - F(' 11- F('2) -C ,(K ,b )

01--- d"

2 ( km) '" 4 . 1225 j hi , 2 (m)/C03 ,CO: ( 4 /3 k(3

0

0

-1 0

- 1 0

- 3 0

~ - 4 0

o z '"

N

>< -5 0 c;:

- 6 0

-/ 0

- 8 0

-9 0

-100

-110

k ' 0,,, /6373, a 'If. , EFFECTIVE .EARTH RA DIU S

-

-bO,O

_____ bo'90

- . - . - bO,leo

f-

K' I ____

K;:IO-K '0. 5.....-"

K '0.3

-K '0. 2

K '0. 1

-

.-.--" 1--- ...... K , 0.0 1 ~ --I/j

f- ~ /.{~ ~

t/l K , 0.00 1 .- :.:: -' ~ - .--- r-...... ~

v~7 -.~ ; ~ ./ 'J

K, O.O?.£! . ./'" '/ tI' F==2:;': ~

[f V

10 10

r--

....

/' /' 'A

kt r

I I I II } FOR LARGE , ' F(, l -G(, )

G(,) " 0.05751' -10 lo g '

~ ~~

~ 'I

~ A

~~ ~ I- " /1 - A 'f

/71 ;rp .- -

I-....... ....... ~

~. % 1,-'

~ V

ERROR IN A, :5 1. 5 db , {335, K ~ O. I

' 0- '1 6 ( ' 1)- ' 26('2)~ 115, K ~ 10

-- 1

,/" .0

/ / 0

K~IV I

/

.8

J J

/ 0

1/ K ~ O.\

/ V 0

/ 0 / / ,Y / -..... V - 0

100 500

.6

-N

>< <i

.4

.2

50 200 1000

F I GURE 7. The distance func tion G(xo) and height function F (xl ,z) for 1tSe in (1 ); also 6 (x ) for use in (15) or (16 ) .

825

For instance, the reference field (in dB above 1,u Vim) assumed for the curves presen ted in the CCIR Atlas [Atlas, 1955] is

Eo= 20 log [2.22 X 1 05/do(km) ] = 106.93-20 log do(km). (18)

The following examples are given to illustrate the use or (1) in calculating field strength.

Given a particular propagation path in which lc= 4/3, e= 80, (T = 4 mhos/m , jMHz = 30, do= 160 km, h1= 100 m, and h2= 1000 m, then from (5), d1= 4l.225 km and d2 = 130.365 km; this corresponds to a within line-of-sight path. The fields for both vertical and horizontal polarization are then calculated as follows:

(1) Vertical polarization. From (9) the param­eters K , and b~ are calculated as K ,"'0.28, b~~O, and from the curves of figures 4 and 5, .80 = l.38, 01(0.28, 0)= 19.2 . From (2) the x's are calculated as xo = 686 , x1 = 177, x2 = 559, and the criterion given by (16 ) indicates that (1) may be used in tbepresent situation:

686 - 177 (0.16 ) -559 (0.62)

= 311 > 280, from (16a) and (16b) .

Then from (1) and figure 7,

A t = 1l.0 + 25 .5 + 0.5 - 19 .2= 17.8.

To compare with the CCIR Atlas, (17) and (18 ) are used giving the field strength as

E = 62.8 - 17.8 = 45.0 (dB above 1 ,uV/m).

The value as read from the Atlas is E = 44.4 . (2) Horizontal polarization. From (10) and fig­

ures 4 and 5, K h ",1.2 X 10- \ b~"' 180 , .80= 1.607, 0 1(10-\ 180)= 20 .0; from (2) xo = 799, x1 = 206 , x2 = 651, and [rom (1) and figure 7,

At = 17.0+ 23.5-4.5- 20.0= 16.0.

Equations (17) and (18) give E = 62 .8- 16.0 = 46 .8, whereas the value as read from the Atlas is E = 46.0.

Radio propagation studies are often presented in terms of the propagation loss L p of a radio system [Norton, 1959; Wait , 1959]. The relationship be­tween attenuation and propagation loss is

where Gp denotes the path antenna gain which, for negligible polarization coupling loss, is just the sum

o[ the free space antenna power gams above an isotropic antenna.

The system loss Ls of a radio communication system is defined as the decibel ratio of the power input to the terminals of the transmitting antenna P t and the resultant power available at the terminals of the receiving antenna Fa:

(20)

Propagation loss and the practicably measurable system loss are then related by

(21)

where L t (transmitter ) and Lr (receiver) are the decibel ratios of the actual antenna input resistances T to their free space radiation resistances Tf:

L t.r= lO log (l'frr). (22)

Curves of L t and Lr for electric dipoles and small loop antennas without ground screens over finitely conducting ground are available in the literature [Vogler and Noble, 1963]. The effects of ground screens are considered in papers by Wait and Sur tees [1954] and by Wait [1956] .

4. References

Atlas of ground-wave propagation curves for frequencies between 30 Mc/s and 300 Mc/s (1959), (published by the International T elecommunications Union, Geneva).

Bremm er, H. (1949), T errestrial radio waves (E lsevier Publishing Co., Amsterdam) .

Burrows, C. R. , and S. S. Atwood (1949), R adio wave prop­agation (Academic Press Inc., New York, N.Y.).

Norton, K . A. (1941), The calculation of ground-wave fi eld intensity over a finitely conducting spherical earth, Proc. IRE 29, 623.

Norton, K. A. (July-Aug. 1959), System loss in radio wave propagation, J . R es. NBS 63D (R adio Prop.), No. 1, 53- 73.

Vogler, L. E. (July-Aug. 1961), Smooth earth diffraction calculations for horizontal polarization, J. R es. NBS 65D (R adio Prop.) , No.4, 397-399.

Vogler, L. E., and J. L. Noble (May, 1965), Curves of ground proximity loss for dipole antennas, NBS T ech. Note 175.

Wait, J . R . (1956), Effect of the ground screen on the fi eld radiated from a monopole, IRE Trans. Ant. Prop. AP- 4, 179.

Wait, J. R. (1959), Transmission of power in radio propa­gation, Electronic and Radio Engineer 36, 146.

Wait, J. R. , and H . H. H owe (1956) , Amplitude and phase curves for ground-wave propagation in the band 200 cycles per second to 500 kilocycles, NBS Circ . 574.

Wait , J. R. , and W . J . SUl-tees (1954), Impedance of a top­loaded antenna of arbitra ry length over a circular grounded screen, J. Appl. Phys. 25, 553.

(Paper 68D7-379)

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