4.2. Effective Height (Length)

Effective height (or length) of an air core loop in meters is [Rohner, 2006]

(4.11)\[h_{eff} = \frac{ 2\pi A N }{ \lambda } = \frac{ \pi^2 D^2 N }{ 2\lambda }\]

For transmitter antenna, it is a length that a dipole homogeneously carrying the feed point current I_0 would have to have in order to generate the same field strength in the main direction of radiation as

(4.12)\[h_{eff} = \int_0^L \frac{ I(Z) }{ I_0 } dz\]

where \(I(z)\) is the current distribution, \(I_0\) is the feed point current (maximum). In the case of a half-wave dipole antenna (\(I(z)=I_0 \cos⁡ {2\pi L/ \lambda}), h_{eff}=\lambda/\pi=2L/\pi=0.64L\)). This is shown Fig. 4.4.

Fig. 4.4 : Explanation of the effective length concept.

For receiver antenna, the effective height of the dipole antenna is [Laurent and Carvalho, 1962]

(4.13)\[h_{eff} = \frac{v_{ind}}{E} h_{eff} = \frac{\omega_0 N A_r}{C} \left[ \mu_{cer} + \biggl( \frac{d_c^2}{d_r^2} - 1 \biggr) \right]\]

where \(\omega_0\) is resonance frequency, \(N\) is number of turns, \(A_r\) is area of the rod, \(\mu_{cer}\) is effective relative permeability of the rod, \(d_c\) and \(d_r\) are diameters of the coil and rod respectively[Laurent and Carvalho, 1962].

Therefore, for an N turn ferrite rod antenna, [Snelling, 1969]

(4.14)\[h_{eff} = \mu_{cer} \frac{\omega AN}{c_0} = \mu_{cer} \frac{2\pi AN}{\lambda_0}\]

Another effective height formula for ferrite loaded solenoid [Burhans, 1979]:

(4.15)\[h_{eff} = \frac{2\pi AN \mu_{cer} F_a}{\lambda}\]

where \(F_a\) is averaging factor of coil and rod (typically 0.5 to 0.7).