Effective area of satellite-borne antennas for radio astronomy

Planet.

Space Sci. 1963,

Vol. 11, pp. 817 to 822.

EFFECTIVE

Pergamon

Pms

Ltd.

Printed

in Northern

Ireland

AREA OF SATELLITE-BORNE FOR RADIO ASTRONOMY*

s. s. SANDLER Space Radio Project, Harvard College Observatory,

ANTENNAS

Cambridge 38, Massachusetts

(Received 8 April 1963) Abstrw-The problem of interpreting measurements on satellite-borne linear antennas is examined Definitions of antenna parameters, such as effective area, are made in terms of physically observable quantities. Possible sources of error in applying conventional antenna formulae are pointed out. The short linear antenna is given as a specific example.

1.INTRODUCTION

The use of satellite antennas for precision measurements of polarized or unpolarized radiation in space requires a consistent quantitative antenna theory. For most practical antenna applications, only a rough qualitative knowledge is required. However, when the radio astronomer is concerned with interpreting the numerical values of antenna measurements, a thorough understanding of the operation of the antenna is. required. This paper is concerned with some elementary problems in the use of satellite antennas. These problems are covered under the general topic of “determining the effective area” of a receiving antenna. In more elementary terms, the radiation impinging on the antenna must be related to measurements at the antenna terminals. For example, the relation between the power density in a randomly polarized field near the antenna and the power dissipated in a load resistor connected to the antenna terminals must be established. The “effective area problem” requires accurate values of the antenna parameters, e.g. driving point impedance. Accurate antenna parameters have been computed for only a very few antenna types. Further complications are introduced by the nature of the medium, such as a satellite antenna operating in a lossy plasma or in a plasma containing a magnetic field. 2. EFFECTIVE

AREA OF LINEAR

ANTENNA

IN AN INFINITE

ISOTROPIC

MEDIUM

The effective area A(~K~) of an antenna immersed in an infinite medium with an incident wave of power (i.e. flux) density S(W~-~) is given by A = P&S (2.1) where PRL = real power dissipated in a load impedance connected to the antenna terminals. As an example, consider the system given in Fig. 1. A linear antenna of half-length h is placed in an arbitrarily orientated plane wave field. The angle that the propagation direction makes with the antenna is 8 and the tilt angle between the incoming elective field E’(Vm-I) and the antenna is y. The load impedance 2, consists also of the contributions due to the transmission line and to lumped impedances representing the effects of the junction at the antenna termin,al@. The Thevenin equivalent circuit w of the receiving antenna aids in relating the incident electric field E’ to measurable quantities at the antenna terminals. The equivalent circuit is given in Fig. 2. + The research reported in this paper has been sponsored in part by the Electronics Systems Division, Air Force Systems Command, under contracts AF19 (604)-6120 and AF19 (604)-8X4. 817

818

S. S. SANDLER

Fxe. 1. ARBITRARILYORCHARD PLANE WAVE IMPINGINGON A RECEIVINGANTENNA.

FIG, 2. EQwti~rm

CIRCUITOF RECEMNCIANTENNA.

The equivalent generator voltage of the antenna V is given by(l) : v = -(E’

cos y)~~(6~.

(2.2)

In equation (2.2) the quantity h,(8) is the complex effective length of the receiving antenna. Numerical values of h,(O) are given by King(l) as a function of h, a, and 0. For an electrically small antenna, h,(O) = a/z sin 6 (2.3) for (,$+r)s< 1, where f$ = ; and &, = free space wavelength. For an electrically small antenna, the equivalent voltage appearing at the antenna terminals becomes : for V = -hE* cos y sin 0 (2.4) (PoQ2 Q 1. The above relation shows that for (&h)z < 1, the geometrical projection of the electric field multip~ed by the half-lent of the antenna yields the equivalent e.m.f. In general, however, the complex effective length h,(e) must be fully taken into account. Returning to the equivalent circuit in Fig. 2, we see that the Thevenin equivalent impedance is Z,, the driving point impedance of the antenna (N.B. for non-perfect conductors, the self-impedance must be included in 2,). This impedance is complex and a function both of the electrical length &h and the radius &a of the linear antenna. Tables of 2, are given

EFFECTIVI3 AREA

in the literature(l). by’s’ :

OF SATELLI~-BOG

A~~~~~

FOR RADIO

ASTRONOMY

819

For an electrically short antenna C/3&< l), 2, is approximately given

where 5, = characteristic impedance of free space = 120~ ohms (a? 377 ohms) LI = 2 In ~2~~~) a = radius of linear antenna (see Fig. 2) vdl = 2 In (h/a) - 2 F = 1 + [l*O$/(Q - 3)]. The power dissipated in the load impedance 2;; of Fig. 2 is: V2

vwQ12 , z L’ (z, + &JY

W” cos

pL = (2, + zo)* =

(2.6)

Since h,(9), Z,, and 2, are complex, the power PL is complex, i.e. PL=ReFL+jImP,=P,,jjP,, Re = real part Im = imaginary part.

(2.7)

For a short antenna (/$,A < l), 2, is primarily capacitive. and Q = 10, then 2, = 1.66 - j1274 ohms.

For example, with &.A = 0.3 (2.8)

The maximum power transfer from the radiation field to the load impedance occurs when z, = z,*

(2.9)

where * denotes complex conjugate. The condition in equation (2.9) corresponds to R& = R@ and X, = -X, where

Z,=.&+jX, Z, = RL -I-jX&.

(2.10)

With equations (2.9) and (2.10) in equation (2.6), the optimum received power pBL is given by [ZE’ co~fuh,(@)]~ (2.11) P RL = 411,

-

The effective area for a plane wave source is found by relating the in-coming power density S to PRL. The power density in a plane wave is given by (2.12) s = HE’ x IP)wm-2 (cps)-1

S. S. SANDLER

820

where H is the magnetic field strength in webers per square meter. The factor t occurs in equation (2.12) as EC and H are maximum amplitudes. The magnetic vector field H is orthogonal to E’ and both are related by the free space impedance, from which (2.13) or, with equation (2.13) in equation (2.12), S=

* &, IE’/2.

(2.14)

With equations (2.14) and (2.6) in equation (2.1), the effective area of a linear antenna in an arbitrarily orientated linear polarized plane wave field is

3. THE LINEAR ANTENNA IN CIRCULARLY

AND RANDOMLY POLARUXD

FIJZLDS

The circularly polarized field is a special case of elliptical polarization. An elliptically polarized field may be represented by two spatially orthogonal linear fields in time quadrature. The voltage at the terminals of the antenna is a superposition of the individual fields. The determination of the parameters of the polarization ellipse is discussed in the hteraturet4’. In general, the polarization ellipse is time dependent (i.e. y = v(t)). The time dependency will appear as a modulation of the received power at the rotation frequency of the ellipse. As a specific example, consider the circularly polarized field of Fig. 3,

FirO.3. CIRCULARLY

POLARIZED FXELD IMPINGING ON A RECEIVING ANTENNA.

For the wave of Fig. 3, the equivalent voltage at the antenna terminal is V = -[-ET sin y f jE,. cos ~]2/2,(0) = ij[E’e*j+‘]2h,(O),

(3.1)

where the upper sign is for right-hand polarization. The total power density S for the circularly polarized wave is S = +Er x H* + $(&jE’> x H*

(3.2)

or (3.3)

EFFEKXIVE AREbi OF SA~Lr~-~R~

ANTEmAS

FUR RADm

ASTRONOMY

82f

The real power P RAEat the terminals ofthe receiving antenna with a circuhirfy polarized field is given by equations (3.l), (3.31, (2.6) and (2.7); then P R&C= dFPJg&

(3.4)

where PRL is given by equation (2.6) (real part). The power received at the antenna terminal with a randomly orientated field is obtained by averaging PEL over all angles p, which gives FZLG”= P$i%~~ f3.5) where ( ) denotes the mean value. As {co@yt> = 4, and with equations (2.6) and (2,7) in equation (2.3, P& = *P&Yr(f/J= 0). (3.a The variation of PRLf with respect to the propagation direction is taken into account in the definition of the effective area A. Equation (3.6) indicates that the power received from a randomly polarized wave is one-half of the power from a linearly polarized wave. ANTENNA AS A PROBE FOR THE MEASUREMElW 4.THElLlNElAR TWERATURE IN A BLACK BODY ENCLOSURE

OF

A thermodynamic study of an antenna immersed in a black body enclosure of temperature P) shows that the power deveIoped in the load impedance of Fig. 2.1 is, a direct measure of T. That is, the temperature of 2, must equal T to insure thermodynamic equilibrium. The power density derived from the Raylei~-beans approximation must be equaf to the power density in the radiation field or, S = $50 [IS,1”= 4~~~~~~~~-~c~s~-l~

f4.f)

where 5, = 12&r = 377 ohms k = Boltzmann’s constant

T = black body temperature in ‘Kelvin. From equation (2~9, the square of the incident field is given by

where PLR’ = power in load resistor due to a randomly polarized wave = $P&9

= 0).

With equation (4.2) and equation (4.11, the equivalent black body temperature is given by (4.31 Equation (4.3), for a matched load (2, = ZJ, reduces to

Previous ~~re~~~ worken has not been concerned with either the actual m~s~ement problem or accurate values of the antenna parameters. For example, conventional antenna

S. S. SANDLER

822

theory does not take account of the thickness of the antenna or the actual current distribution on the antenna. Since the driving point resistance of an antenna is small (i.e. for a short antenna), large errors are possible when “handbook” formulae are employed for radiation resistance. Furthermore, no previous discussion has considered properly the load impedance of the antenna or the matching of the antenna for a maximum transfer of power. It is hoped that the simple formulation of this paper will indicate that some care must be employed in interpreting measurements made on space probes. Further papers in this series will discuss in more detail the behavior of antennas in magnetoactive media. REFERENCES 1. R. W. P. KING, Theory ofLinear Antennas. Chapters II, IV and pp. 168,470,488-491. Harvard University Press, Cambridge, Mass. (1956). 2. K. IZ~ICA and R. W. P. RrNG, Terminal-zone corrections for a dipole driven by a two-wire line. Crnft Laboratory Tech. Report No. 352. Harvard University, Cambridge, Mass. (1962). 3. R. KING et al., J. Res. Nat. Bur. Stand.-D. 65, 371 (1961). 4. M. H. COHEN,Proc. Inst. Radio Engrs. 46, 172 (1958). 5. J. L. PAWSEYand R. N. BRACE~ELL,Radio Astronomy, p. 22. Oxford University Press, Oxford (1955). 6. J. HUGILL, Planet. Space Sci. 8, 68 (1961). Pe3rere--PaccmaTpmsaeTcm JlHH&HblMK IIapElMeTpOB

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