Vector Effective Length

|E|² η

Substituting the expressions for P_r and S into Eqn (2.80) Ae= Pr

S = |Va|²κ 8R_rad

2η

|E|²

The open-circuit voltage developed at the terminals of the dipole due to a polarization-matched wave is

Va= Edl

Substituting this into the expression for effective area A_e= |E|²(dl)²κ

8R_rad 2η

|E|² = η(dl)²κ 4R_rad

Since G = κD, the radiation resistance of a Hertzian dipole, given by Eqn (2.98), can be written as

R_rad = πηκ G

dl λ

₂

Substituting this into the expression for A_e A_e= η(dl)²κ

4 G πηκ

λ dl

₂

Simplifying we get

Ae= Gλ² 4π and the result follows.

2.6.5 Vector Effective Length

Let Iinbe the input current at the terminals of a transmit antenna producing an electric field, Ea, in the far-field region. The vector effective length, leff,

is related to E_a by the relation

E_a= a_θE_θ+ a_φE_φ= jηkI_in 4π

e^−jkr

r l_eff (2.105)

The vector effective length can be written in terms of its components l_θ and l_φalong the θ and φ directions, respectively, as

l_eff = a_θl_θ+ a_φl_φ (2.106) In general, l_θand l_φcan be complex quantities. For an ideal current element of length dl carrying a current of I₀, the electric field in the far-field region is given by Eqn (2.8)

E = a_θjηkI₀dl sin θ 4π

e^−jkr

r (2.107)

and comparing this with Eqn (2.105) the vector effective length is

l_eff = a_θdl sin θ (2.108) The vector effective length of a Hertzian dipole is maximum along the direction orthogonal to its axis and zero along its axis.

In the receiving mode, the output voltage developed at the terminals of an antenna due to an incident electromagnetic wave having an electric field Eⁱ is given by

V_a= Eⁱ· l^∗_eff (2.109)

The polarization information of the wave is contained in Eⁱ and that of the antenna in l_eff. Since l_eff refers to the antenna in the transmit mode, for the antenna in the receive mode l^∗_eff is used in the definition of V_a, which reverses the direction of rotation of the field vector.

Consider a RCP wave propagating in the positive z-direction. The locus of the tip of the electric field is shown in Fig. 2.18(a). Let us suppose that an RCP antenna is used to receive this wave. The vector effective length of the RCP antenna [l_eff = (a_x− jay)l₀] represents the polarization in the transmit mode and the locus of the tip of the electric field is shown in Fig. 2.18(b). Although both E and (ax− jay) are rotating in the clockwise direction in the respective coordinate systems, when viewed from a common coordinate system, their directions of rotation are opposite to each other. Let us observe the rotation of l^∗_eff = (ax+ jay)l0, which is shown in Fig. 2.18(c).

x

x x

y

y y

E

z z

a_x + jay

a_x − jay

z

(a) (b) (c)

Fig. 2.18 Loci of the tip of the electric field vector of (a) a right circularly polarized wave, (b) an antenna with leff = (a_x− jay)l0, and (c) an antenna withl_eff = (a_x+ ja_y)l₀

This represents a wave with its electric field vector rotating in the direction opposite to that of l_eff or in the same direction as the incident electric field.

Therefore, taking the complex conjugate of the vector effective length, we have been able to reverse the direction of rotation of the field vector.

The received power is proportional to the square of the terminal voltage P_r∝^Eⁱ· l^∗_eff^² (2.110) If χ is the angle between the vectors Eⁱ and l^∗_eff, Eqn (2.110) can be writ-ten as

P_r∝|Eⁱ||l^∗_eff| cos χ² (2.111) Under the polarization-matched condition, χ = 0 and the power received will be maximum. Therefore

P_rmax∝^E^i²|l^∗_eff|² (2.112) The polarization efficiency is given by

κ_p = ^Eⁱ· l^∗_eff^2

|Eⁱ|^2l^∗_eff^2 =^ˆeⁱ· ˆl_eff^{∗ }² (2.113) where

ˆeⁱ : unit vector along the incident electric field Eⁱ ˆl^∗_eff : unit vector along the vector effective length l^∗_eff

z

x

E y

(b) (a)

y

P_{r = 0} P_rmax

y

z

x E

y

Fig. 2.19 Linearly polarized antenna illuminated by (a) a co-polar and (b) a cross-polar plane wave

The receive antenna is said to be polarization-matched to the incoming wave if the state of polarization of the antenna is the same as that of the incoming wave. This is also known as the co-polar condition. Mathematically, the co-polar condition implies

ˆeⁱ· ˆleff^∗ = 1 (2.114)

On the other hand if |ˆeⁱ· ˆl^∗_eff| = 0, the receiving antenna is cross-polarized or polarized orthogonal to the incoming wave (see Fig. 2.19).

A y-directed dipole radiates a y-directed electric field along the z-axis and similarly, an x-directed dipole produces an x-directed electric field along the z-axis. Consider two crossed dipoles, one along the x-direction and another along the y-direction, excited in phase quadrature. Assume that the char-acteristic impedances of the transmission lines are matched to the antenna input impedances. The quadrature phasing can be achieved by having the feed transmission line lengths differ by λ/4 as shown in Fig. 2.20(a). The radiated electric field is given by the sum of the electric fields of the two dipoles

E = (ax− jay)E0 (2.115) The−j factor with the y-component is due to the 90^◦ phase lag introduced by the quarter-wavelength-long transmission line. The unit vector along the vector effective length of this antenna is

ˆl_eff = 1

√2(ax− jay) (2.116)

l +/4 l +/4

z

y

z

y

x x E = (ax − jay)E₀

l

(a)

y x

z Eⁱ = (ax − jay)

(b)

l

z

y

x b

a

c a'b'

c'

Fig. 2.20 A circularly polarized antenna constructed with a pair of dipoles (a) transmitting RCP wave and (b) receiving RCP wave

Let this antenna be used to receive an RCP plane wave with the incident electric field given by

Eⁱ = E0(ax− jay) (2.117) The incident field in relation to the receiving antenna is shown in Fig. 2.20(b). The x-directed component of the electric field induces a volt-age of Vaa^ = Vr 180^◦ at the terminals a-a^ of the dipole. The amplitude of the voltage Vr is a function of the amplitude of the incident wave and the length of the dipole. The 180^◦ phase in the voltage appears because the x-directions of the two coordinate systems are opposite to each other. The voltage due to the y-component of the electric field at the terminals b-b^ of the y-directed dipole is V_bb^ = V_r − 90^◦, where the−90^◦phase is due to the

−jay component of the incident field. This voltage gets further delayed by another 90^◦ when it passes through the λ/4 transmission line connected to the y-directed dipole. Therefore, at the common terminals the two voltages add in phase.

If the incident field is left circularly polarized, the voltage at the common terminal due to the x-directed electric field will still be Vr 180^◦, however, the voltage due to the y-directed dipole would be Vr 0^◦. Therefore, the output at the terminals of a right circularly polarized antenna would be zero for an LCP wave. Thus, an antenna transmits and receives like polarized waves.

EXAMPLE2.17

What is the vector effective length of an x-directed Hertzian dipole? If this antenna is used to receive a wave with a magnetic field intensity

H = (a_θ− jaφ)

at the antenna, what is the open-circuit voltage developed at the terminals of the antenna?

Solution: The electric field of an x-directed dipole in its far-field region is given by (see Example 1.8)

E = jηkI0

4π e^−jkr

r (−aθcos θ cos φ + aφsin φ)

The vector effective length is related to the far-field electric field by Eqn (2.105) and comparing the previous equation with this

l_eff =−aθdl cos θ cos φ + a_φdl sin φ

The components of the electric and magnetic fields are related to each other by

Eθ

H_φ =−E_φ H_θ = η

where η is the intrinsic impedance of the medium. The electric field compo-nents are given by

E_θ = ηH_φ= η(−j) and

E_φ=−ηHθ =−η Therefore, the electric field is

E = (−aθjη− aφη)

The open-circuit voltage at the terminals of the antenna is given by Va= E· l^∗_eff

= (−aθjη− aφη)· (−aθdl cos θ cos φ + aφdl sin φ)

= (jηdl cos θ cos φ− ηdl sin φ)

EXAMPLE2.18

An antenna has ˆl_eff = a_θ. Calculate the polarization efficiency of the antenna if the unit vector in the direction of the incident electric field is (a) ˆeⁱ = a_θ, (b) ˆeⁱ = a_φ, (c) ˆeⁱ = (a_θ+ a_φ)/√

2, and (d) ˆeⁱ= (a_θ− jaφ)/√ 2.

Solution:

(a) κ_p =^ˆeⁱ· ˆleff²=|aθ· aθ|²= 1

This represents an antenna receiving a co-polarized wave.

(b) κ_p =|aθ· aφ|² = 0

The wave is cross-polarized with respect to the antenna.

(c) κ_p =|(aθ· aθ+ a_θ· aφ)/√

2|² = 1 2

This is a situation where the linearly polarized antenna is not completely aligned with the polarization of the incoming wave.

(d) κp =|(aθ· aθ+ jaθ· aφ)/√

2|²= 1 2

Only half the power in the incident wave is received if a circularly po-larized wave is received using a linearly popo-larized antenna.

EXAMPLE2.19

A right circularly polarized antenna has ˆl_eff = (a_θ− jaφ)/√

2. Calculate the polarization efficiency of the antenna if the incident electric field is (a) right circularly polarized, (b) left circularly polarized, and (c) linearly polarized in the a_θ-direction.

Solution:

(a) For a right circular polarized wave, ˆeⁱ = (a_θ− jaφ)/√

2 and, using Eqn (2.113), the polarization efficiency is

κ_p =^_ 1

√2(a_θ− jaφ)· 1

√2(a_θ+ ja_φ)^_

2

= 1

4(1 + 1)² = 1 (b) For a left circular polarized wave, ˆeⁱ = (a_θ+ ja_φ)/√

2 and hence κp =^_√1

2(a_θ+ ja_φ)·√1

2(a_θ+ ja_φ)^_

2

= 1

4(1− 1)²= 0 (c) For a linear polarized wave, ˆeⁱ= a_θ, thus

κp =^_aθ· 1

√2(aθ+ jaφ)^_

2

= 1 2

In document [Harish, A.R. Sachidananda, M.] Antennas and Wave (Page 84-91)