Читать книгу Antenna and EM Modeling with MATLAB Antenna Toolbox - Sergey N. Makarov - Страница 17

Calculate and plot the average acquired antenna power when the generator with a periodic waveform υ_g(t) = υ_g(t + T) is characterized by the rms voltage and generator resistance given by

(1.5c)

Solution: We use Eq. (1.5b) and average it over period T to obtain average power . The result has a form:

(1.5d)

Then, we plot the average antenna power as a function of the load resistance. The short MATLAB script given below accomplishes the task:

Rg = 50; % Generator resistance, Ohm Vg = 9; % Generator rms voltage, V RA = [0.01*Rg:0.01*Rg:10*Rg]; % Load resistance PA = RA*Vg^2./(Rg + RA).^2; % Average antenna power plot(RA, PA); grid on; title('Average antenna power, W') xlabel('Load resistance, Ohm')

This important result is given in Figure 1.2. We see that the load power does have a maximum at a particular value of the load resistance. Our next step will be to find this maximum.

It is instructive to find the maximum of the average antenna power analytically since it gives us insight into the optimization process. We treat P_avg in Eq. (1.5d) as a function of R_a, i.e. P_avg = P_avg(R_a). From basic calculus it is known that a function has a maximum where its first derivative is zero. Consequently, differentiating P_avg with respect to R_a gives

(1.6)

Figure 1.2 Average antenna power as a function of the antenna resistance for fixed V_rms = 9 V, R_g = 50 Ω.

The necessary and sufficient condition for Eq. (1.6) to hold, and thus maximizing the antenna power, is simply

(1.7)

This result has a great practical value despite, or maybe thanks to, its simplicity. The maximum output radiated power is achieved when the antenna resistance is exactly equal to the internal resistance of the generator. In other words, the antenna is matched to the generated; it is called the matched antenna. The design of such an antenna over a frequency band of interest is called antenna matching. Such a design is a critical step, and it may be a great challenge for an RF engineer. It does not matter if the antenna radiates a sinusoidal or other periodic signal or a pulse; Eq. (1.7) holds in either case since it also maximizes power at any time instant.

However, it must be clearly stated that no more than 50% of the total generator power can be extracted even in this best case. This statement makes sense if we again examine the circuit in Figure 1.1 with two equal resistors. We see that the power delivered to each resistor is obviously the same. Since R_g is internal resistance, half of the total power is spent to heat up the generator.