Page 645 - Mechatronics with Experiments

ELECTRIC ACTUATORS: MOTOR AND DRIVE TECHNOLOGY 631
where m is the mass of the actuator which absorbs heat, c is the specific heat of the material
t
◦
of actuator in units of Joule∕(kg ⋅ C), T is the initial temperature and T is the steady-state
0
temperature. The generated heat due to losses, Q ,
in
Q = Q + Q + Q (8.103)
in R C F
where Q is the resistive loss, Q is the core loss, and Q is the friction and windage loss.
R C F
The transferred heat is estimated with approximation as follows,
Q = c ⋅ (T − T ) ⋅ Δt (8.104)
out out amb
where T amb is the ambient temperature, and c out is the effective heat transfer coefficient
◦
◦
between the actuator and its surrounding with units (Joule∕s)∕ C = Watts∕ C, and Δt is
the time period.
The energy balance equation can also be expressed as power balance in differential
equation form,
d d
[Q ] = [Q − Q out ] (8.105)
in
net
dt dt
P net = P − P out (8.106)
in
dT
c ⋅ m ⋅ = P in − c out ⋅ (T − T amb ) (8.107)
t
dt
dT
c ⋅ m ⋅ + c out (T − T amb ) = P in (8.108)
t
dt
This models the electric actuator thermal behavior like a first-order dynamic system. Notice
that the actuator reaches a steady-state temperature, when P net = 0, then dT∕dt = 0. The
difficulty in using such an analytical model is in the difficulty of accurately estimating
c , c out and P . Note that
T
in
d
P = Q in (8.109)
in
dt
d d d
= Q + Q + Q F (8.110)
C
R
dt dt dt
= P + P + P (8.111)
R C F
For a given motor design and its expected operating conditions, if we can estimate
the total losses P = P + P + P and the heat transfer coefficient between the motor
in R C F
and environment c , then we can estimate the steady-state operating temperature of the
out
motor. The steady-state value of the operating temperature is easy to calculate from the
above differential equation
dT
c ⋅ m ⋅ + c out ⋅ (T − T amb ) = P in (8.112)
t
dt
dT
= 0 in steady-state, then (8.113)
dt
0 + c out ⋅ (T − T amb ) = P in (8.114)
1
T(∞) = T amb + ⋅ P in (8.115)
c out
8.2.1 Resistance Losses
Electric actuators have coils which are windings of current carrying conductors. Coils act
as current controlled electromagnets. The conductor material, such as copper or aluminum,
has finite electrical resistance. In order to generate an electromagnetic effect, we need to
pass a certain amount of current. As the electrical potential pushes the current through the