392 ELEMENTS OF ELECTRICAL ENGINEERING. 



Note. See page 97 . 



61. Show that ^(cosfl+ysinfl) is 6 ahead of E in phase and that its nu- 

 merical value is equal to the numerical value of E. 



62. Find the expression for sin ^x in terms of sin x and cos x, using De Moivre's 

 theorem. 



63. Find the expression for sin (.r -|-^-|- 2) in terms of sines and cosines of x, 

 y, and z. 



64. A transformer (without iron) consists of two long cylindrical coils each having 

 10 turns of wire per centimeter of length (one layer). The coils are each 50 centi- 

 meters long and their radii are 2 centimeters and 3 centimeters respectively, the 

 smaller coil being inside the larger. Calculate the value and phase of the electromo- 

 tive force required to maintain a current of 10 amperes at 60 cycles per second in the 

 outer coil ; calculate the current in the inner coil, and calculate the true and apparent 

 reactance, and apparent resistance of the outer coil. The outer coil has 2 ohms re- 

 sistance and the inner coil has i}4 ohms resistance and its terminals are short-cir- 

 cuited. Ans. 21. 61 volts, I745 / phase difference, 1.944 amperes, 0.6697 ohms 

 true reactance, 0.6584 ohms apparent reactance, 2.0568 ohms resistance. 



Note. See Appendix A for formula, for calculating the inductance of a long coil. 

 The mutual inductance, in henrys, of two coaxial solenoids is, approximately, 



"/-i- I0 9 



in which z' and z ff are the turns of wire per unit length on the respective coils, 

 is the radius of the inside coil, and / is the length of the coils. 

 65. The expression (iii) page 97 may be written 



where R is the apparent change of resistance of the primary coil due to the presence 

 of the secondary coil, and X is the apparent change of reactance of the primary coil 

 due to the presence of the secondary coil. The primary coil has in circuit with it an 

 adjustable non-inductive resistance so that 1^ may be varied at will. Assume that R 

 and X are small compared with r v and x l so that R 2 and X 2 are negligible. 

 Show that the impedance (numerical value) of the primary circuit is not altered by 

 the presence of the secondary when r l /x l X/fi. How is the impedance (nu- 

 merical value) of the primary circuit affected by the presence of the secondary when 

 r is less than x^XjR'y when r x is greater than 



CHAPTER VI. THE POLYPHASE SYSTEM. 



66. The distance from center to center of adjacent poles of an 

 alternator measured on the surface of the armature is 10.4 inches. 

 How far apart must two armature conductors be placed so that 

 there may be a phase difference of 5 5 in the electromotive forces 

 induced in the respective conductors? Ans. 3.18 inches. 



67. A common return wire is used for the two currents of a 

 two-phase system. The system is balanced and each current is 



