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

Let's derive formulas of dependencies Р_T, Р_U, 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. (80)

Total power vanish P_T=0, when I=0 (open circuit). Total power has the largest value P_T=^/ r, when current is maximal I_SC=/ r (short cirquit).

Useful power in a complicated manner depend on current:

P_U=IU, but U=–Ir, so P_U=I –I²r.

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(–Ir). (81)

Useful power vanish P_U=0, when I=0 (open circuit). Useful power also vanish P_U=0 when –Ir=0, or I_SC=/ 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:

=–2Ir=0 => . (82)

Such current, that equals to the half of short circuit current, occurs when R + r =2r, 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 = –Ir. Then

(83)

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

Efficiency η 1, when І0 (power-saving mode, Р_U=Р_T=0).

Efficiency η = 0 when I_SC=/r (unuseful mode, Р_T maximal, but Р_U = 0).

Efficiency η = 0,5 when I_OPTIMAL=/(2r) (optimal mode with maximal Р_U^max).

4. Description of laboratory research facility and methodology of measurements

Devices and outfits: Source of an EMF , voltmeter V, miliampermeter mA, three-decade resistors box R₃.

In order to research the dependence of powers and efficiency of the source from a current, we mount the scheme, which is shown in the Fig. 22.

Fig. 22 – Scheme for researching of power and efficiency dependence from external resistance

We use a three-decade resistors box R₃ as an external resistance. When we change resistance of the box, we change a current in the circuit.

For twelve values resistance of the box R₃ we must fix current and to calculate the values of total power (70), useful power (71) and efficiency of the source (73). For graph build-up it is convenient to choose the integer values of resistance, for example: 0, 30, 60, 90, …, 300 etc. However, in the given work to exclude redundant points of the graph at small currents, we will scale an axis of values of a current I.

For the obtained data we have to build graphs of total and useful power also efficiency of the source versus external resistance of the circuit.

5. Data processing

The results of this laboratory work must be presented in a graphical aspect - is a plot of dependence of total and useful powers also efficiency of the source versus a load resistance. The graphical form of results usually guesses quality estimation and consequently does not demand statistical manipulation.