# 9.4 Dipole of Arbitrary Length

So far, we examined the radiation properties of the Hertzian and half-wave dipoles. We now consider the more general case of a linear dipole of arbitrary length l relative to λ. For a center-fed dipole, as depicted in Fig. 9-16, the currents flowing through its two halves are symmetrical and must go to zero at its ends. Hence, the current phasor I˜(z) can be expressed as a sine function with an argument that goes to zero at z=±l/2:

I˜(z)=I 0 sin k l/2−z,for 0≤z≤l/2,I 0 sin k l/2+z,for −l/2≤z<0,(9.52)

where I 0 is the current amplitude. The procedure for calculating the electric and magnetic fields and the associated power density of the wave radiated by such an antenna is basically the same as that used previously in connection with the half-wave dipole antenna. The only difference is the current distribution I˜(z). If we insert the expression for I˜(z) given by Eq. (9.52) into Eq. (9.43), we obtain the following expression for the differential electric field d E˜θ of the wave radiated by an elemental length d z at location z along the dipole:

d E˜θ=j k η 0 I 0 4 π e−j k R R sin θ e j k z cos θ d z×sin k l/2−z for 0≤z≤l/2,sin k l/2+z for −l/2≤z<0.(9.53)

The total field radiated by the dipole is

E˜θ=∫−l/2 l/2 d E˜θ=∫0 l/2 d E˜θ+∫−l/2 0 d E˜θ=j k η 0 I 0 4 π e−j k R R sin θ×∫0 l/2 e j k z cos θ sin [k(l/2−z)]d z +∫−l/2 0 e j k z cos θ sin [k(l/2+z)]d z.(9.54)

If we apply Euler’s identity to express e j k z cos θ as

e j k z cos θ= cos (k z cos θ)+j sin (k z cos θ)],

we can integrate the two integrals and obtain the result

E˜θ=j 60 I 0 e−j k R R cos k l 2 cos θ−cos k l 2 sin θ.(9.55)

# Figure 9-16 Current distribution for three center-fed dipoles.

(a) l = λ/2

(b) l = λ

(c) l = 3λ/2

The corresponding time-average power density radiated by the dipole antenna is given by

S(θ)=|E˜θ|2 2 η 0=15 I 0 2 π R 2 cos π l λ cos θ−cos π l λ sin θ 2,(9.56)

where we have used the relations η 0≈120 π (Ω) and k=2 π/λ. For l=λ/2, Eq. (9.56) reduces to the expression given by Eq. (9.45) for the half-wave dipole. Plots of the normalized radiation intensity, F(θ)=S(R,θ)/S max, are shown in Fig. 9-17 for dipoles of lengths λ/2, λ, and 3 λ/2. The dipoles with l=λ/2 and l=λ have similar radiation patterns with both maxima along θ=90°, but the half-power beamwidth of the wavelength-long dipole is narrower than that of the half-wave dipole, and S max=60 I 0 2/(π R 2) for the wavelength-long dipole, which is four times that for the half-wave dipole. The pattern of the dipole with length l=3 λ/2 exhibits a structure with multiple lobes, and its direction of maximum radiation is not along θ=90°.

# Figure 9-17 Radiation patterns of dipoles with lengths of λ/2, λ, and 3 λ/2.

(a) l = λ/2

(b) l = λ

(c) l = 3λ/2