Multistage Filters

In the inductor-input filter shown in Fig. 79(a), the rectifier is a source of non-sinusoidal alternating voltage connected across the filter. It is possible to replace the usual circuit representation by Fig. 84(a).

Fig. 84. Inductor-input filter circuits.

For any harmonic, say the nth, the voltage across the whole circuit is the harmonic amplitude A_n, and the voltage across the load is P_RE_dc, P_R being ripple allowable across the load, expressed as a fraction of the average voltage. Since the load resistance R is high compared to X_c, the two voltages are nearly in phase, and they bear the same ratio to each other as their respective reactances, or

[46]

From the type of rectifier to be used, and the permissible amount of ripple in the load voltage, it is possible to determine the ratio of inductive to capacitive reactance.

When the magnitude P_R must be kept very small, the single-stage filter of Fig. 84(a) may require the inductor and the capacitor to be abnormally large. It is preferable under this condition to split both the inductor and the capacitor into two separate equal units, and connect them like the two-stage filter of Fig. 84(b). A much smaller total amount of inductance and of capacitance will then be necessary. For this filter

[47]

X'_L and X'_C being the reactances of each inductor and capacitor in the circuit. Likewise, the three-stage filter of Fig. 84(c) may be more practicable for still smaller values of P_R. In the latter filter,

[48]

and, in general, for an n-stage filter,

[49]

It is advantageous to use more than one stage only if the ratio P_A/P_R is high. That the gain from multistage filters is realized only for certain values of P_A/P_R is shown by Fig. 85.

Fig. 85. Attenuation in one-, two-, and three-stage filters.

The lower curve shows the relation between P_A/P_R and X_L/X_C for a single-stage filter. The second curve shows the increase in P_A/P_R gained by splitting up the same amount of L and C into a two-stage filter; as indicated in Fig. 84(b), the inductor and capacitor both have one-half their "lumped" value. The upper curve indicates the same increase for a three-stage filter, each inductor and capacitor of which have one-third of their "lumped" or single-stage filter value. The attenuation in multistaging is enormous for high X_L/X_C. For lower ratios there may be a loss instead of a gain, as shown by the intersection of the two upper curves. These curves intersect the lower curve if all are prolonged further to the left. This may be a puzzling condition; but consider that, for X_L/X_C = 50 in the single-stage filter, the ratio is 1/3X_L/3X_C or 50/9 in the three-stage filter; the rather small advantage in the latter is not difficult to account for.

Other factors may influence the number of filter stages. In some applications, modulation or keying may require that a definite size of filter capacitor be used across the load. Usually these conditions result in a single-stage filter, where otherwise more stages might be most economical.

Table VII shows filter reactors in the negative lead, which may be either at ground or high potential. If low ripple is required in the filtered output, it is usually preferable to locate the filter reactors in the high-voltage lead. Otherwise, there is a ripple current path through the anode transformer winding capacitance to ground which bypasses the filter reactor. Ripple then has a residual value which cannot be reduced by additional filtering. In the three-phase, zigzag, full-wave circuit, with center tap used for half-voltage output, separate reactors should be used in the positive leads; placing a common reactor in the negative lead introduces high amplitude ripple in the high-voltage output.

In rectifiers with low ripple requirements, both filament and anode windings should be accurately center-tapped to avoid low-frequency ripple, which is difficult to filter. Three-phase leg voltages should be balanced for the same reason.