3.2. Dependence of total power, useful power and efficiency of a source from the external load resistance. Maximal power theorem

Using Ohm's law for closed circuit (61) we can get dependence of total power (70) on load resistance R:

. (75)

Fig. 20 – Maximum useful power theorem

The plot of dependence of the total power P_T from external resistance R is shown on fig. 20. We can see that total power monotonically decreases with growing load resistance. In short-cirquit mode (R=0) the total power has a greatest value P_T^max=²/r, but in open-cirquit mode (R=) the total power has a minimal value P_T^min=0. In point R=r value of total power is equal half from maximal value P_T^R=r=²/(2r).

Using Ohm's law for closed circuit (61) we can get dependence of useful power (71) on load resistance R:

. (76)

The plot of the dependence (76) is shown on fig. 20. We can see that useful power versus load resistance has a maximum. In short-cirquit mode (R=0) the useful power has a minimal value P_U^min=0, as in open-cirquit mode (R=) the useful power has a minimal value P_U^min=0.

For obtaining the value of load resistance R_m, which correspond a maximal value of useful power P_U^max, it is necessary the equation (76) to differentiate on R, after that, obtained expression of first derivative set equal to zero.

. (77)

Obtained expression is equal to zero when R–r=0. We obtain maximal power theorem: when external resistance of a cirquit

(78)

is equal to internal resistance (load matching condition), than useful power has a maximal value P_U^max=²/(4r).

All devices of radio-electronic equipment (transistor stages, chips, amplification stages, dynamic loudspeakers, receiving and transmitting antennas etc.) are constructed with fulfilment of this requirement (78) at which the useful power has the maximum value.

Under this condition (78) a power loss (72) of heating of a source has the same value P_L^R=r=²/(4r), therefore the total power twice greater P_T^R=r=P_U+P_L=²/(2r).

Using Ohm's law for closed circuit (61) we can get dependence of efficiency (74) on load resistance R:

. (79)

The plot of dependence (79) is shown on fig. 20. We can see that efficiency monotonically increases with growing load resistance.

In short-cirquit (R0) the efficiency has a minimal value ^min0. In this case the total power will be maximum P_T^max=²/r, but all it is run to waste for the source's heating P_L^max= P_T^max, therefore P_U0. This is unuseful mode.

In open-cirquit (R) the efficiency has a greatest value ^max100%. In this case, the useful power is equal to a total power, but each of them is equal to zero P_U=P_T0. This is power-saving mode.

In point R=r value of efficiency is equal to half from maximal value ^R=r=50%, beacose P_U^max= P_L^R=r. This is optimal mode with the greatest delivery to external resistance.