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 =e² / 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 =e² / (2 r).

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 =e² / (4 r).

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 =e² / (4 r), therefore the total power twice greater P _T ^R=r=P _U+ P _L=e² / (2 r).

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 Þ0) the efficiency has a minimal value h ^min Þ0. In this case the total power will be maximum P _T ^max =e² / r, but all it is run to waste for the source's heating P _L ^max = P _T ^max, therefore P _UÞ0. This is unuseful mode.

In open-cirquit (R Þ¥) the efficiency has a greatest value h ^max Þ100%. In this case, the useful power is equal to a total power, but each of them is equal to zero P _U= P _TÞ0. This is power-saving mode.

In point R=r value of efficiency is equal to half from maximal value h ^R=r =50%, beacose P _U ^max = P _L ^R=r. This is optimal mode with the greatestdelivery to external resistance.

 

Dependence of total power, useful power and efficiency of the source from a current

Let's derive formulas of dependencies Р _T, Р _U, h from a current, for a source, in which EMF and internal resistance are constant.

Total power of the source is directly proportional to a current:

Р _T =І e. (80)

Total power vanish P _T = 0, when I= 0 (open circuit). Total power has the largest value P _T = e² / r, when current is maximal I _SC = e / r (short cirquit).

Useful power in a complicated manner depend on current:

P _U =IU, but U= e– Ir, so P _U =I e – I ² r.

Fig. 21 – Dependence of the powers from current

Graph of dependence P _U =f (I) is the second order curve (parabola). Evaluation gives

P _U =I( e– Ir). (81)

Useful power vanish P _U = 0, when I= 0 (open circuit). Useful power also vanish P _U = 0 when e– Ir= 0, or I _SC = e / r (short circuit).

It's evident that with any intermediate value of a current useful power must be maximal. In order to find this value of current, we need to differentiate expression (81) with current and make first derivative vanish:

= e–2 Ir= 0 => . (82)

Such current, that equals to the half of short circuit current, occurs when R + r = 2 r, or R=r. We've got the same result, as in the case of dependence of useful power from R.

Efficiency dependence from a current can be derived, if in equation (74) substitute the expression U = e– Ir. Then

(83)

Graph of efficiency dependence from current (see Fig. 21) represents a straight line with negative angular coefficient.

Efficiency ηÞ1, when І Þ0 (power-saving mode, Р _U =Р _T = 0).

Efficiency η = 0 when I _SC = e / r (unuseful mode, Р _T maximal, but Р _U = 0).

Efficiency η = 0,5 when I _OPTIMAL = e / (2 r) (optimal mode with maximal Р _U ^max).