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13:54:14 Page 548
so that Equation 12.37 can be expressed as
_
P o ¼ UA s u o
ð12:39Þ
The first law for the oven for a transient condition, one where the power input is changed, is
_
P À UA s u ¼ mc
du
dt
ð12:40Þ
where m is the mass of the oven and c is the average specific heat. This product represents the total
heat capacity of the oven. Equation 12.40 now provides the governing equation for the plant in this
example.
Controller Model
We now wish to implement a controller. The first controller we implement is termed a proportional
controller and is illustrated in Figure 12.50. The error between the set point temperature and the
actual temperature of the oven is multiplied by a constant value to determine the power input to the
heater. A block diagram of the controller and plant combined system is shown in Figure 12.51; two
proportional gains are shown in this figure to emphasize that the error signal will most likely be a
voltage.
The governing equation for the temperature of the oven may be expressed
mc
UA s þ k p k m
Â
Ã
du
dt
þ u ¼
k p k m
UA s þ k p k m
Â
à u set
ð12:41Þ
By comparison with Equation 3.4,
t_ y þ y ¼ KF t
ð Þ
e
θ set
θ set
– θ
θ
+
–
κ m
Figure 12.50 Proportional Control.
Q loss
e
θ set
θ set
– θ
θ
θ
P
+
–
κ m
κ p
Figure 12.51 Proportional Control of an oven.
548 Chapter 12 Mechatronics: Sensors, Actuators, and Controls
13:54:14 Page 548
so that Equation 12.37 can be expressed as
_
P o ¼ UA s u o
ð12:39Þ
The first law for the oven for a transient condition, one where the power input is changed, is
_
P À UA s u ¼ mc
du
dt
ð12:40Þ
where m is the mass of the oven and c is the average specific heat. This product represents the total
heat capacity of the oven. Equation 12.40 now provides the governing equation for the plant in this
example.
Controller Model
We now wish to implement a controller. The first controller we implement is termed a proportional
controller and is illustrated in Figure 12.50. The error between the set point temperature and the
actual temperature of the oven is multiplied by a constant value to determine the power input to the
heater. A block diagram of the controller and plant combined system is shown in Figure 12.51; two
proportional gains are shown in this figure to emphasize that the error signal will most likely be a
voltage.
The governing equation for the temperature of the oven may be expressed
mc
UA s þ k p k m
Â
Ã
du
dt
þ u ¼
k p k m
UA s þ k p k m
Â
à u set
ð12:41Þ
By comparison with Equation 3.4,
t_ y þ y ¼ KF t
ð Þ
e
θ set
θ set
– θ
θ
+
–
κ m
Figure 12.50 Proportional Control.
Q loss
e
θ set
θ set
– θ
θ
θ
P
+
–
κ m
κ p
Figure 12.51 Proportional Control of an oven.
548 Chapter 12 Mechatronics: Sensors, Actuators, and Controls