HI. Variation of the Ratio of Electromotive Forces 87. The preceding ratios of e.m.fs. apply strictly only to the generated e.m.fs. and that under the assumption of a sine wave of alternating generated e.m.f. The latter is usually a sufficiently close approximation, since the armature of the converter is a multi-tooth structure, that is, contains a distributed winding. The ratio between the difference of potential at the commu- tator brushes and that at the collector rings of the converter usually differs somewhat from the theoretical ratio, due to the e.m.f. consumed in the converter armature, and in machines converting from alternating to continuous current, also due to the shape of the impressed wave. When converting from alternating to direct current, under load the difference of potential at the commutator brushes is less than the generated direct e.m.f., and the counter-generated alternating e.m.f. less than the impressed, due to the voltage consumed by the armature resistance. If the current in the converter is in phase with the impressed e.m.f., armature self-inductance has little effect, but reduces the counter-generated alternating e.m.f. below the impressed with a lagging and raises it with a leading current, in the same way as in a synchronous motor. Thus in general the ratio of voltages varies somewhat with the load and with the phase -relation, and with constant impressed alternating e.m.f. the difference of potential at the commutator brushes decreases with increasing load, decreases with decreasing excitation (lag), and increases with increasing excitation (lead). When converting from direct to alternating current the reverse is the case. The direct-current voltage stands in definite proportion only to the maximum value of the alternating voltage (being equal to twice the maximum star voltage), but to the effective value (or value read by voltmeter) only in so far as the latter depends upon the former, being = — - 7= maximum value with a sine wave. Thus with an impressed wave of e.m.f. giving a different ratio of maximum to effective value, the ratio between direct and alternating voltage is changed in the same proportion as the ratio of maximum to effective; thus, for instance, with a flat-topped 232 ELEMENTS OF ELECTRICAL ENGINEERING wave of impressed e.m.f., the maximum value of alternating impressed e.m.f., and thus the direct voltage depending there- upon, are lower than with a sine wave of the same effective value, while with a peaked wave of impressed e.m.f. they are higher, by as much as 10 per cent, in extreme cases. In determining the wave shape of impressed e.m.f. at the con- verter terminals, not only the wave of generator e.m.f., but also that of the converter counter e.m.f., may be instrumental. Thus, with a converter connected directly to a generating system of very large capacity, the impressed e.m.f. wave will be practically identical with the generator wave, while at the terminals of a converter connected to the generator over long lines with re- active coils or inductive regulators interposed, the wave of im- pressed e.m.f. may be so far modified by that of the counter e.m.f. of the converter as to resemble the latter much more than the generator wave, and thereby the ratio of conversion may be quite different from that corresponding to the generator wave. Furthermore, for instance, in three-phase converters fed by ring or delta connected transformers, the star e.m.f. at the con- verter terminals, which determines the direct voltage, may differ from the star e.m.f. impressed by the generator, by con- taining different third and ninth harmonics, which cancel when compounding the star voltages to the delta voltage, and give identical delta voltages, as required. Hence, the ratios of e.m.fs. given in Section II have to be corrected by the drop of voltage in the armature, and have to be multiplied by a factor which is \/2 times the ratio of effective to maximum value of impressed wave of star e.m.f. (\/2 being the ratio of maximum to effective of the sine wave on which the ratios in Section II were based), that is, by a "form factor" of the e.m.f. wave. With an impressed wave differing from the sine shape, there is a current of higher frequency, but generally of negligible mag- nitude, through the converter armature, due to the difference between impressed and counter e.m.f. wave.