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Relationship Between Applied Voltage and Conduction Angle for Noninductive Load

The differential equation describing the operation of the basic circuit in Fig. 7-2 in the unsaturated or presaturated region is, on the basis of negligible winding resistance and the idealized magnetization curve


If Ac is the effective core area, Bc the critical flux density, and N the number of turns in the winding, then, when these quantities are substituted in Eq. 7-4, the result is



in which Ki is the initial flux density, i.e., at t = ti. Wave forms of flux density, voltage, and current for resistance loading are shown in Fig. 7-3(a). At ωt = π the load current goes through zero and the voltage changes sign, which means that dB/dt must become negative also, and the reactor becomes unsaturated. The current, therefore, is extinguished at ωt = π. The flux density starts to build up from a value of -Bc at ωt = 0, and ti is the value of t at ωt = 0, so that Eq. 7-5 can be put into the more definite form


Firing occurs when B reaches the saturated value +BC, which occurs at ωt = αf in Fig. 7-3(a). When these quantities are substituted in Eq. 7-6, there results

which yields


EXAMPLE 7-1: Three saturable reactors are supplied from a 3-phase, 4-wire, 208-v, 400-cycle source to provide single-phase power at a frequency of 1200 cps to a noninductive load. The core of each reactor is comprised of toroidal laminations having an OD = 2 in. and ID = 1 5/8 in. stacked to a height of 1/4 in. The stacking factor is 0.80. The magnetic characteristic of the core material is represented by the hysteresis loop in Fig. 7-1 (a). Calculate the number of turns in the winding of each saturable reactor.


Solution: The number of turns is obtained from Eq. 7-7 by


V=208/ = 120 v per phase
αf = 2π/3 radians
ω = 2π x 400 radians per sec
Bf = 1.40 webers per sq m
Ac = 1/2 (2 - 1 5/8)(1/4)(0.8)(2.54)2(10-4) = 2.42*10-5 sq m

Last Update: 2011-02-16