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

[7-4]

If A_c is the effective core area, B_c 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

Integrating

[7-5]

in which K_i is the initial flux density, i.e., at t = t_i. 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 -B_c at ωt = 0, and t_i is the value of t at ωt = 0, so that Eq. 7-5 can be put into the more definite form

[7-6]

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

which yields

[7-7]

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 where 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