Lecture4.pdf

Leccture--4 Anten nna parameteers: (Co ontinueed…) 1.4.6 Diirectivity This param meter indicatees how well an a antenna co oncentrates poower into a lim mited solid anngle. The directivity D of an n antenna is the t ratio of th he radiation iintensity U inn a given direection (, ) to the radiation intensity i averraged over alll directions U0 (see Fig. 200)

M direectivity conceept Fig. 20: Maximum Maximum m directivity D0 is the direcctivity in the maximum m raddiation directiion (0, 0)

Note: Thee directivity of o an isotropicc source is 1, whereas it is more than 1 for any other antenna.

1.4.7 Gain The gain or o power gain n of an antenn na in a certain n direction (, ) is definedd as

where Pin is the input power p to the antenna a and iss related to thhe radiated poower Prad as:

In the abo ove equation,  is the efficciency of the antenna. It acccounts for thhe various lossses in the anttenna, such as th he reflection lo oss, dielectricc loss, conducction loss, andd polarizationn mismatch looss. Taking th he efficiency into account, the t gain and the t directivityy are related bby:

Fig. 21: Explaining E thee concept of antenna a gain

Similar to o the maximu um directivitty, a maximu um gain G0 ccan be defineed and whichh is related tto the maximum m directivity D0 by:

1.4.8 An ntenna Po olarization n n antenna in a given direection is defi fined as the ppolarization oof the plane wave The polarrization of an transmitteed by the anteenna in that direction. d The polarizationn of a wave ttransmitted (or received) by an antenna is i the locus of o the tip of th he instantaneo ous electric ffield vector E traces out w with time at a fixed observatio on point. (see Fig. 22)   

Iff the locus is a straight linee Iff the locus is a circle Iff the locus is an a ellipse

  

linear polarizzation circular polaarization elliptical pollarization

(a)

(b)

(c) Fig. 22: (aa) Linear, (b) Circular and (c) Ellipticall polarization

1.4.8.1 Mathematical Form mulation of o Polarizaation For a wav ve travelling in the –ve z-direction, the electric e field ccomponents iin the x and y-directions arre:

E x  E x 0 cos t  kz   x 

E y  E y 0 cos t  k z   y 

where Ex00 and Ey0 are amplitudes a in x and y direcction respectivvely, and x, y are the phaase angles. Thhe total instaantaneous vecctor field E is::

 E  aˆ x E x 0 cos( t  kz   x )  aˆ y E y 0 cos( t  kz   y ) Linear Polarizatiion: Let x = 0, 0 y = 0 and Ex0 = 3, Ey0 = 5.

Fig. 23: Linear L polarizaation

at z = 0 and a

t = 0 we w get:

E x  3 cos0  0  0  3 at z = 0 and a

t = π/2 2 we get:

E x  3 cos / 2  0  0  0 at z = 0 and a

E y  5 ccos0  0  0   5 E y  5 ccos / 2  0  0   0

t = π we w get:

E x  3 cos  0  0  3

E y  5 ccos  0  0   5

So the tip of the E field d vector movees linearly alo ong the line.

Special Cases: C If Ex = 0, then we only y have the y-co omponent (y--polarized waave)

 E  aˆ y E yo cos((t  kz   y ) If Ey = 0, then we only y have the x-co omponent (x--polarized waave)

 E  aˆ x E xo cos((t  kz   x ) In generall, we get lineaarly polarized d waves if:

   x   y  n ,

n = 0, 1, 2, 

Circula ar Polariza ation: It occurs when w Ex0 = Ey0, and  = x - y = Odd multiples of π/2

Fig. 24: Circular C polariization

Let x = 0, 0 y = π/2 and d Ex0 = 1, Ey0 = 1.

  E  aˆ x E x 0 cos(( t  kz )  aˆ y E y 0 cos( t  kz  ) 2

at z = 0 and

t = 0 we have:

E x  cos(t  kz )  cos 0  1





E y  cos(t  kz  )  cos  0 2 2 So at time t = 0, we can locate the locus of the E-field vector at point 1 on the circle. (see Fig. 24)

at z = 0 and

t = π/2 we have:



E x  cos( )  0; 2







E y  cos(  )  cos  1 2 2 2

That corresponds to point 2 on the circle (Fig. 24) at z = 0 and

t = π we have:

E x  cos( )  1;

E y  cos( 

 2

)  cos

 2

0

That corresponds to point 3 on the circle (Fig. 24)

Elliptical Polarization: The explanation for elliptical polarization is same as that for circular polarization except that it occurs when Ex0  Ey0  0.

y Ey0

OB

 z

Major axis

OA Ex0 x

Minor axis

Fig. 25: Polarization ellipse at z = 0 of an elliptically polarized electromagnetic wave Note: Circular polarization and Elliptical polarization can be either right-handed or left-handed corresponding to the electric field vector rotating clockwise (right-handed) or anti-clockwise (lefthanded).