CHAPTER IX WAVE SCREENS. EVEN HARMONICS 76. The elimination of voltage and current distortion, and production of sine waves from any kind of supply wave, that is, the reverse procedure from that discussed in the preceding chapter, is accomplished by what has been called ''wave screens." Series reactance alone acts to a considerable extent as wave screen, by consuming voltage proportional to the frequency and the current, and thereby reducing the harmonics of voltage in the rest of the circuit the more, the higher their order. Let the voltage impressed upon the circuit be denoted sym- bolically by e = €i + 63 + es + ej + . . . =^i:en (29) where n denotes the order of the harmonic of absolute numerical value 6n. If, then, the reactance x (at fundamental frequency) is inserted into the circuit of resistance, r, the impedance is 2i = -y/r^ + x^ for the fundamental frequency, and Zn = y/r^ + n^^ for the nth harmonic, (30) and the current thus is e or, denoting f = ? = y , ^ (31) 2 ^Vr^ + nV ^ = c, (32) X it is 66 x^25 + c2 153 + . . . (33) 154 ELECTRIC CIRCUITS if r is small compared with x, c* is negligible compared with 1, 9, 25, etc., and it is . ^ 1 X *' + 3+6+7 + that is, the current, i, and thus the voltage across the resistance, r, shows the harmonics of the supply voltage, e, reduced in propor- tion to their order, n. Even if r is large compared with x, and thus c^>lj iSnally c^ becomes negligible with n^, and the harmonics decrease with their order. 77. The screening effect of the series reactance is increased by shunting a capacity, C, beyond the inductance, L, that is, across the resistance, r, as shown in Fig. 73. By consuming current jTRRRRRTl e 1 rmmM Fig. 73. r e Fig. 74. proportional to frequency and voltage, the condenser shimts the more of the current passing through the reactance, the higher the frequency, and thereby still further reduces the higher harmonies of current in the resistance, r, and thus of voltage across this re- sistance. Its effect is limited, however, by the decreasing voltage distortion at r and thus at the condenser, C. Thus the screening effect is still further increased by inserting a second inductance, L, beyond the condenser, C, in series to the resistance, r, as shown in Fig. 74. By making the second induct- ance equal to the first one, and making the condenser, C, of the same reactance, for the fundamental wave, as each of the two inductances, we get what probably is the most effective wave screen. This T-connection or resonating circuit will be discussed more fully in Chapter XIV, in its feature of constant-potential constant-current transformation. Under the condition, that the two inductive reactances and the WAVE SCREENS. EVEN HARMONICS 156 capacity reactance are equal, the equation of the current in the resistance, r, is (page 291), for the nth harmonic. or, absolute. / = ?^ (34) xn(n^ - 2) - jr{n^ - 1) ^ ^ i = -° X ^ (35) ^ Vn\n^ - 2)2 + c2(n2 - 1)2 r where c = — (36) If c is small, that is, r small compared with x, the current becomes xn (n2 — 2) or, for higher values of n. i = —3, (38) xn^ that is, it decreases with increasing order of harmonic, and pro- portional to the cube of the order n, thus shows an extremely rapid decrease. If c is not negligible, the denominator in (35) is larger, and i, therefore, still smaller. As illustration may be shown the current, ^, and thus the vol- tage, 6o, across a resistance, r, under the very greatly distorted and peaked voltage of Fig. 62: (a) for a series reactance, Xy equal to r, that is, c = 1 ; (fc) for the complete wave screen of two inductances and one capacity. It is impressed voltage, e = 1.27 6o { li + 0.9783 + 0.9356 + 0.8777 + O.8OO9 + 0.713ii + 0.617i3 + 0.517i6 + O.4I617 + O.3I619 + 0.18921 }. (a) Reduction factor of the nth harmonic, 1 1 Vn^ + c2 "" VnMn' hence, 1 27 e, = -^ eo U + 0.4423 + 0.2585 + 0.1757 + 0.125» + 0.091ii V2 [ + 0.067i3 + 0.049i6 + 0.034i7 + 0.023i9 + O.OlSsi}. 156 ELECTRIC CIRCUITS (b) Reduction factor of the nth harmonic, 1 n{n^ - 2) hence, 62 = 1.27 Co {li + 0.0473 + 0.0086 + O.OOS? + O.OOI9 + O.OOln}. That is, the third harmonic is reduced to less than 5 per cent., the fifth to less than 1 per cent., and the higher ones are practically entirely absent. While in the supply voltage wave, e, the voltage peak (by adding the numerical values of all the harmonics: 1 + 0.978 + 0.935 + . . .) is 7.36 times that of the fundamental wave, it is reduced by series reactance to less than 2.28 times the maximum of the fundamental wave, that is, very greatly reduced, and by the complete wave screen to less than 1.06 times the maximum of the fundamental. That is, in the last case the voltage is practically a perfect sine wave. 78. By "wave screens" the separation of pulsating currents into their alternating and their continuous component, or the separa- tion of complex alternating currents — and thus voltages — into their constituent har- monics can be accomplished, and inversely, the combination of alternating and continuous currents or vol- tages into resultant complex alternating or pulsating currents. The simplest arrangement of such a wave screen for separating, or combining alternating and continuous currents into pulsating ones, is the combination, in shunt with each other, of a capacity, C, and an inductance, L, as shown in Fig. 75. If, then, a pulsating voltage, 6, is impressed upon the system, the pulsating current, i, produced by it divides, as the continuous component can not pass through the condenser, C, and the alternating component is barred by the inductance, L, the more completely, the higher this inductance. Thus the current, ti, in the apparatus. A, is a true alternating current, while the current, to, in the apparatus, C, is a slightly pulsating direct current. Inversely, by placing a source of alternating voltage, such as an alternator or the secondary of a transformer, at A, and a source of continuous voltage, such as a storage battery or direct-current WAVE SCREENS. EVEN HARMONICS 157 generator, at C, in the external circuit a pulsating voltage, e, and pulsating current, i, result. If the capacity, C, is so large as to practically short-circuit the alternating voltage, and the inductance, L, so high as to practically open-circuit the alternating voltage, the separation — of combi- nation — ^is practically complete, and independent of the frequency of the alternating wave. Wave screens based on resonance for a definite frequency by series connection of capacity and inductance, can be used to sepa- rate the ciurent of this frequency from a complex current or voltage wave, such as those given in Figs. 56 to 63, and thus can be used for separation of complex waves into their components, by "harmonic analysis." Thus in Fig. 76, if the successive capacities and in- ductances are chosen such that 2 wfLi = ^ 6 TcfLs = lOirfLi, = 2nir/L„ = 2 TfCi' 1 6 ir/C 1 10 t/Cb' 1 2 vfnCn (39) Qii Lnf ggi Fig. 76. where / = frequency of the fundamental wave. Then, through any of the branch circuits Cn, L^ only the nth harmonic, in, can pass to an appreciable extent. Such resonant wave screen, however, has the serious disadvan- tage to require very high constancy of /, since the resonance condi- tion between C» and Ln depends on the square of /, 79. Even harmonics are produced in a closed magnetic circuit by the superposition of a continuous current upon the alternating wave. With an alternating sine wave impressed upon an iron magnetic circuit, saturation, or in general the lack of proportional- 158 ELECTRIC CIRCUITS ity between magnetic flux and m.m.f., produces a, wave-shape dis- tortion, that is, higher harmonics, of voltage with a sine wave of current, of current with a sine wave of impressed voltage. The constant term of a wave, however, is the first even harmonic, and thus, if the impressed wave comprises a fundamental sine and a I -— — — -J ^ N y -' s» / \ \ 111 / \ H / \ / ^^ t \ / / \ \ 1 '. — <'\ \ / / i„ A ^ / / \ \ / / /• ^~ -^ \\ ^ ^_ /' N X ' / / ~ - is given by H = 36 + 37COS0 + 16.5 cos 2<^ + 8cos3<^ + 2 cos4<^ + . . Instead of superimposing a direct current upon an alternating wave, as by connecting in series an alternator and a direct-current generator or storage battery, two separate coils can be used on the magnetic circuit, one energized by an alternating impressed vol- tage, the other by a direct current. A high inductive reactance would then be connected in the latter circuit, to eliminate the current pulsation which would be caused by the alternating vol- tage induced in this coil. Connecting two such magnetic circuits with their direct-current magnetizing coils in series, but in opposition (without the use of a series reactance) eliminates the induced fundamental wave, but leaves the second harmonic in the direct-current circuit, which thus can be separated. Numerous arrangements can then be de- vised by two magnet cores energized by separate alternating- 160 ELECTRIC CIRCUITS current exciting coils and saturated by one common direct-current exciting coil, surrounding both cores, or their common retiu'n, etc. 80. The preceding may illustrate some of the numerous wave- shape distortions which are met in electrical engineering, their characteristics, origin, effects, use and danger. Numerous other wave distortions, such as those produced by arcs, by unidirec- tional conductors, by dielectric effects such as corona, by Y con- nection of transformers for reactors, by electrolytic polarization, by pulsating resistance or reactance, etc., are discussed in other chapters or may be studied in a similar manner. v/