Designing VLF antenna

Question

I was doing some research about the relation between wavelength and antenna size and shape. I was mostly interested in 15KHz rf range.

Based on my research I found the following:

• A squarish shape antenna is the best sender/receiver
• Wavelength of the 15Khz frequency is around 14990 meter

A simple antenna with half the wavelength of signal can send/receive this frequency but it can be altered to smaller size by using thin wire wrapped around a ferrite object. (Same method used in AM antenna design).

Now, let say if I use the wire gauge 26 ( with 0.4mm / 0.0159 inch diameter), how many wire turns (loops) is required for a 50mm (~ 20 inch) square shape antenna to send/receive this 15Khz signal?

Answer 1

A squarish shape antenna is the best sender/receiver

There is no practical basis for this assertion.

Wavelength of the 15Khz [SIC] frequency is around 14990 meter [SIC]

The formula for the reasonable approximation of wavelength is

$$\lambda (in\space meters) = \frac{300}{Frequency\space in\space MHz} \tag 1$$

Since 15 kHz is 0.015 MHz, it has a wavelength of ~20,000 meters.

When a loop antenna is constructed with a loop circumference that is less than 1/10 wavelength, it is classified as a small loop antenna. When designing these types of antennas, one of the most important factors to which the engineer must pay attention is the efficiency. Small loop antennas have very low radiation resistance which gives rise to an inefficient antenna.

Antenna efficiency is defined as:

$$efficiency = \frac{R_r}{R_r+R_l} \tag 2$$

where R_r is the radiation resistance and R_l is the RF resistive losses of the antenna and any losses in the core material.

You can see from equation 2 that as the radiation resistance is reduced, for a given resistive loss, the efficiency drops precipitously.

The radiation resistance of a multi-turn loop antenna that is wound around a high permeability material, such as ferrite or iron powder, is given as:

$$R_r=197n^2\mu_{er}^2\left(\frac{C}{\lambda}\right)^4 \tag 3$$

where C is the circumference of the loops in meters, n is the number of turns of wire and $\mu_{er}$ is the effective relative permeability of the material around which the loops are wound. If the loops are air wound, simply set $\mu$_er to 1.

You can see from equation 3 that for a given core material, as the number of turns are increased, the radiation resistance goes up according to the number of turns squared.

The approximate RF resistance of the loop antenna is given as:

$$R_l=n*C*R_{wire} \tag 4$$

where R_wire is the RF resistance of the wire in ohms/meter. The RF resistance of 26 gauge (AWG) copper wire at 15 kHz is 1.265 ohms/meter.

The gain of the small loop antenna is given as:

$$G=1.5*n*efficiency \tag 5$$

Armed with these formulas, you can calculate all of the parameters for any proposed 15 kHz small loop antenna. What you will find is that very small antennas such as you suggest will be very low gain antennas due to low efficiency. Whether or not such an antenna is adequate for your application depends on many other factors such as receiver sensitivity, transmitter power, distance, etc.