24 



ALTERNATING CURRENTS 



points. Between d and e both the current and the voltage are 

 negative and the power now becomes positive. 



Tliis power curve is a sine wave having double the frequency of 

 cit her t lie current or the voltage. Its axis of symmetry coincides 

 with the axis of current and voltage. There must be as much of 

 t he power curve above the zero axis as there is below that axis, or 

 the positive power above the axis must be equal to the negative 

 power below the axis. That is, all the positive power received 

 from the source is returned to the source of supply. Therefore, 



the net power is zero. 

 When current and volt- 

 age differ in phase by 

 90, or are in quadrature, 

 the average power is 

 zero. If the current 

 leads the voltage by 90, 

 the average power is 

 zero, as is shown later in 

 Fig. 30. 



If current and voltage 

 are out of phase by an 

 angle less than 90, but greater than 0, the resulting power 

 curve P is that indicated in Fig. 21. At points a, b, c, d and e, 

 either the voltage or the current is zero and the power is zero 

 at each of these points. Between a and b, and between c and 

 d, the current and voltage are in opposition, and the power is 

 negative. Between b and c, and between d and e, they are in 

 conjunction, and the po.wer is positive. It will be noted that 

 there is more positive power than negative power. The average 

 power is not zero, but is positive, and is less than the product 

 of E and /. It will be shown later that this power 



P = El cos e (5) 



where is the phase angle between voltage and current. Cos 6 

 is called the power-factor of the circuit. P is the true watts and 

 El the apparent watts, or volt-amperes. 

 The power-factor 



true watts P 



P. F. = cos = - = = (6) 



apparent watts El 



The power-factor can never be greater than unity. 



FIG. 21. Power curve; current and voltage 

 out of phase by angle 6. 



