A twocoil system is shown in figure The reluctances of the t

A two-coil system is shown in figure. The reluctances of the two air-gaps are R_1 = 20K A/wb and R_2 = 30 A/wb, and the core permeability mu_r rightarrow infinity. Suppose that N_1 = 180 and N_2 = 120. Determine the self-inductances L_1 and L_2, and mutual-inductance M; If i_1 = 2 sin 20t A., and coil #2 is open, find the induced voltage v_2; If now i_1 = 4 A., and coil #2 is open, determine the energy W_1 stored in air-gap #1 and W_2 stored in air-gap #2. loss R_e = 1.5 K ohm and an unknown magnetizing reactance X_. The secondary side is loaded with a resistance R_L = 10 ohm and a capacitance C_L = 100 mu F, both of which are connected in parallel and referred to the secondary side. Draw an equivalent circuit that is referred to the primary side; If the primary port has a unity power factor, determine this X_; What is the power factor at the secondary load side? Determine the power efficiency eta of the entire transformer system. A reluctance machine consists of a passive rotor and a stator with a single winding of N = 100. Suppose that the total reluctance of the machine is given by R(theta) = 30 - 20 cos 4 theta (A/wb) with a rotating angle theta. Determine the stator winding current i such that the rotor can be balanced with a load torque tau_L = 20 squareroot 3 in N-m at an equilibrium position theta = 30 degree; Find the energy W_m stored in the system at the equilibrium position in (a); If the current of the stator is i = 12 cos 200t, find the machine rotating speed omega_m.

Solution

Brushless dc motors initially were designed in large numbers for spindle drives in Winchester disk drives. The early designs were three-phase, later moving to two-phase and then one-phase, due to the very slow start-up requirements, very small friction loads, and the need to reduce unit cost at all levels. The industrial and machine tool markets started with and continue to use three-phase BLDC motors in their variable-speed, variable-load, high-start-up applications. The overwhelming popularity of three-phase BLDC motors focuses this subsection toward three-phase windings. Many of the initial design activities for various winding patterns can be traced back to the 1920s and earlier based on work done on three-phase ac windings.
This subsection reviews the various winding line connections, the key winding patterns and hookups, various winding constants, and winding selection and design techniques.
Basic Winding Configurations. There are other basic decisions that must be made by the design engineer before a BLDC motor design can commence. Previously defined is the number of phases, which is three here. Next in importance is the number of poles. The use of two poles is waning, and the use of six or eight poles is increasing. Four-pole BLDC motors are among the most popular used today. Two-and four-pole BLDC motor designs are used here, but the rules for two and four poles can be extended to higher pole counts. The number of stator slots (and teeth) and the winding pattern are key design decisions. This section is dedicated to reviewing the important parameters of these two design decisions.
In a three-phase motor there are three windings or phases positioned 120° electrical apart. Figure 5.30 shows the location of 6 coils in a representative 12-slot sta-tor. A two-pole rotor (not shown) will rotate as the three windings are energized in sequence A-B-C, as A-A’, B-B’, and then C-C are energized sequentially.The three-phase winding always develops positive