University of Saskatchewan
EE 444.3 Electric Machines II
Dated: 2012-09-16 Due Date: Sept. 25, 2012 (10am, Dropbox)
Instructor: Dr. Rama Gokaraju T.A. (Marker): Sriram (Sri) Chandrasekar
A proposed energy storage mechanism consists of an N turn coil wound around a large
nonmagnetic (µ=µ0) toroidal form as shown in Fig. 1. As can be seen from the figure, the
toroidal form has a circular cross section of radius a and toroidal form has a circular cross
section of radius a and toroidal radius r, measured to the center of the cross section. The
geometry of this device is such that the magnetic field can be considered to be zero everywhere
outside the toroid. Under the assumption that a << r, the H field inside the toroid can be
considered to be directed around the toroid and of uniform magnitude.
For a coil with N=1000 turns, r=10m, and a =0.45m:
a. Calculate the coil inductance L.
b. The coil is to be charged to a magnetic flux density of 1.75 T. Calculate the total stored
magnetic energy in the torus when this flux density is achieved.
c. If the coil is to be charged at a uniform rate (i.e. di/dt=constant), calculate the terminal voltage
required to achieve the required flux density in 30 seconds. Assume the coil resistance is to be
Fig. 1 Toroidal winding
The λ-i characteristic of a non-linear singly-excited magnetic circuit is given by
Find the energy and co-energy for the magnetic circuit corresponding to a current of 0.10
The two-winding magnetic circuit of Fig. 2 has a winding on a fixed yoke and a second winding
on a moveable element. The moveable element is constrained to motion such that the lengths of
both air gaps remain equal.
a. Find the self-inductances of windings 1 and 2 in terms of the core dimensions and the
number of turns.
b. Find the mutual inductance between the two windings.
c. Calculate the coenergy W’fld (i1, i2).
d. Find an expression for the force acting on the moveable element as a function of the
Fig. 2 Two winding magnetic circuit
Two windings, one mounted on a stator and the other on a rotor, have self and mutual
L11 4.5H L22 2.5H L12 2.8 cos H
where θ is the angle between the axes of the windings ( 0) . The resistances of the
windings may be neglected. The current in winding 1 as a function of time is
i1 10 sin t A. Winding 2 is short-circuited with the following condition: Voltage
e2 0 and 2 0 .
a) Find an expression for the instantaneous torque on the rotor in terms of the angle
b) Compute the time-averaged torque when 45o .
c) If the rotor is allowed to move, will it rotate continuously or will it tend to come
to rest? If the latter, at what value of ?