The second part is about how to determine the antenna shape on board, is there any principle to make sure final remote control outputs maximum power?
One of the major points we should care about is called antenna efficiency, which measures how much percent of power actually radiated in total power attached to loop antenna.
Conclusion first, to achieve maximum efficiency possible, just make sure the loop antenna covers as much area as possible.
The complete formulas and equations can be found on Microchip’s AN831 – Matching Small Loop Antennas to rfPIC Devices, here we just pick the key points.
Antenna model – loop antenna can be considered consisting of two resistors and one inductor, among which only one radiating resistor is transmitting power, the other resistor and inductor is considered as loss, should be avoided as much as possible.
So for the resistor part, we try to maximize radiating resistor and minimize loss resistor, and for the inductor part, we will use a capacitor to cancel the inductor to zero.
The formula for calculating power on radiating resistor and loss resistor are as follows.
$$P_{rad} = I^2 \cdot R_{rad}$$
$$P_{loss} = I^2 \cdot R_{loss}$$
So antenna efficiency can be calculated by the following formula.
$$\eta = \frac{P_{rad}}{P_{rad}+P_{loss}} = \frac{R_{rad}}{R_{rad}+R_{loss}}$$
The radiating resistor of small loop antenna can be calculated by following, where A equals to loop antenna area, while lambda equals to wavelength at transmitting frequency.
$$R_{rad} = 31171 \left(\frac{A^2}{\lambda^4}\right)$$
And the loss resistor of small loop antenna can be described as following, where l is perimeter of loop antenna, w is width of PCB track, f is transmitting frequency, the other two can be considered as constant depending on PCB material etc.
$$R_{loss} = \frac{l}{2w} \cdot \sqrt{\frac{\pi f \mu}{\sigma}} $$
For a given circuit board and frequency, we can see the only variables that determine antenna efficiency are A, l and w, means the loop antenna area, perimeter and track width, so after dropped all the constant value under given circumstances, the antenna efficiency can be simplifed as follows.
$$\eta \propto \frac{R_{rad}}{R_{loss}} \propto \frac{2w \cdot A^2}{l}$$
So ideally for maximum antenna transmitting efficiency, all variables should be as follows.
$$\frac{2w \cdot A^2}{l} \to \infty$$
Above equation can be translate as achieving maximum antenna area as much as possible.