XVI. Higher Frequency Cross Currents between Synchronous Machines 30. If several synchronous machines of different wave shapes are connected into the same circuit, cross currents exist between the machines of frequencies which are odd multiples of the circuit frequency, that is, higher harmonics thereof. The machines may be two or more generators, in the same or in different stations, of wave shapes containing higher harmonics of different order, intensity or phase, or synchronous motors or converters of wave shapes different from that of the system to which they are connected. The intensity of these cross currents is the difference of the corresponding harmonics of the machines divided by the impe- dance between the machines. This impedance includes the self- inductive reactance of the machine armatures. The reactance obviously is that at the frequency of the harmonic, that is, if x = reactance at fundamental frequency, it is nx for the nth harmonic. In most cases these cross currents are very small and negli- gible. With machines of distributed armature winding, the in- tensity of the harmonic is low, that is, the voltage nearly a sine wave, and with machines of massed armature winding, as uni- tooth alternators, the reactance is high. These cross currents thus usually are noticeable only at no load, and when adjusting the field excitation of the machines for minimum current. Thus in a synchronous motor or converter, at no load, the minimum current, reached by adjusting the field, while small compared with full-load current, may be several times larger than the minimum point of the " V" curve in Fig. 68, that is, the value of the energy current supplying the losses in the machine. It is only in the parallel operation of very large high-speed machines (steam turbine driven alternators) of high armature reaction and very low armature self-induction that such high- frequency cross currents may require consideration, and even then only in three-phase F-connected generators with grounded neutral, as cross currents between the neutrals of the machines. In a three-phase machine, the voltage between the terminals, or delta voltage, contains no third harmonic or its multiple, as the third harmonics of the Y voltage neutralize in the delta voltage, and such a machine, with a terminal voltage of almost sine shape, 160 ELEMENTS OF ELECTRICAL ENGINEERING may contain a considerable third harmonic in the Y voltage. As the three Y voltages of the three-phase system are 120 degrees apart in phase, their third harmonics are 3 X 120 deg. = 360 deg. apart, or in phase with each other, from the main terminals to the neutral, and by connecting the neutrals of two three- phase machines of different third harmonics with each other, as by grounding the neutrals, a cross current flows between the machines over the neutral, which may reach very high values. Even in machines of the same wave shape, such a triple frequency current appears between the machines over the neutral, when by a difference in field excitation a difference in the phase of the third harmonic is produced. It therefore is often undesirable to ground or connect together, without any resistance, the neutrals of three-phase machines, but in systems of grounded neutral either the neutral should be grounded through separate resist- ances or grounded only in one machine.