Page 646 - Mechatronics with Experiments

632 MECHATRONICS
conductor, there is energy loss due to resistance. This is similar to the energy loss in pushing
a certain fluid flow rate through a restriction. The power (P ) lost as heat due to resistance
R
(R) of a conductor when current i is conducted,
P = R ⋅ i 2 (8.116)
R
Therefore, in order to minimize the heating of the motor, it is desirable to minimize the
resistance and current. However, a large current is needed to generate a large force or
torque. The energy (Q ) that is converted to heat over a period of time t cycle is
r
t cycle
Q = P ⋅ dt (8.117)
R ∫ R
o
t cycle
2
= R ⋅ i(t) ⋅ dt (8.118)
∫
o
2
Resistive loss, P = R ⋅ i , is fairly well estimated. The temperature dependence of
R
resistance can also be taken into account for better accuracy of thermal prediction,
R(T) = R(T )[1 + (T − T )] (8.119)
cu
0
0
where R(T), R(T ) are resistances of coil at temperature T and T , respectively and =
0 0 cu
0.00393 for copper which is the property of the conductor material. Notice that for copper,
◦
◦
R(125 C) ≈ 1.4 ⋅ R(25 C) (8.120)
◦
The resistance of the coil increases by 40% when its temperature increases by 100 C.
8.2.2 Core Losses
Core losses refers to the energy lost, in the form of heat, in electric actuators at their
stator and rotor structure due to electromagnetic variations. There are two major physical
phenomenon that contribute to core losses: hysteresis losses and eddy current losses.
Hysteresis loss is due to the hysteresis loop in the B-H characteristics of the core
(rotor and stator) material which is typically a soft ferromagnetic material. As the magnetic
field changes, the hysteresis loop is traversed and energy is lost. The hysteresis loss is
proportional to the magnitude of the magnetic field change and its frequency. This loss can
be minimized by selecting a core material that has small hysteresis loops, such as a high
silicon content in the steel used for the core material.
Eddy currents and related losses can be summarized as follows. If a bulk piece
of metal (e.g., iron, copper, aluminum) moves through a magnetic field or is stationary
in a changing magnetic field, a circulating current is induced on the metal. This current
is called eddy current and it results in heat loss due to the resistance of the metal. The
induced current is a result of Faraday’s law of induction. That is the changing magnetic
field (whether due to change in ⃗ B or motion of the metal conductor relative to ⃗ B or both)
induces a voltage on the conductor. This induced voltage generates the current. Power loss
due to eddy currents is proportional to the square of magnetic flux density and its frequency
of variations. Therefore, it is expected that the hysteresis losses will be more dominant at
low frequencies, whereas eddy current losses will be more dominant at high frequencies.
Laminations of thin layers of metal which are insulated and stacked together, as
oppose to bulk metal, are used to reduce the eddy currents in motor applications. Hence
eddy current-related heat losses are reduced, increasing the efficiency of the motor. The
stators of DC and AC motors are built using laminated iron instead of bulk iron in order to
reduce the eddy current related loss of energy into heat. For different lamination materials,