082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

20.4 State-space models 613
R a L a
T
i a
Fixed field
y a
e b
v m
J
Bv m
Figure20.8
An armaturecontrolled
d.c. motor.
The next stage is to obtain a mathematical model for the system. Using Kirchhoff’s
voltage law and the component laws for the resistor and inductor we obtain, for the
armature circuit,
v a
=i a
R a
+L a
di a
dt
+e b
Now for a d.c. motor the back e.m.f. is proportional to the speed of the motor and is
given by e b
=K e
ω m
,K e
constant. So,
di
v a
=i a
R a
+L a
a
+K
dt e
ω m
di a
= − R a
i
dt L a
− K e
ω
a
L m
+ 1 v
a
L a
(20.11)
a
Let us now turn to the mechanical part of the system. If G is the net torque about
theaxisofrotationthentherotationalformofNewton’ssecondlawofmotionstates
G = J dω
dω
, whereJ is the moment of inertia, and is the angular acceleration. In
dt dt
this example, the torques are that generated by the motor,T, and a frictional torque
Bω m
which opposes the motion, so that
T−Bω m
=J dω m
dt
For a d.c. motor, the torque developed by the motor is proportional to the armature
current and isgiven byT =K T
i a
, whereK T
isaconstant. So,
K T
i a
−Bω m
=J dω m
dt
dω m
dt
= K T
J i a − B J ω m
(20.12)
Equations (20.11) and (20.12) are the state equations for the system. They can be
arranged inmatrix form togive
( ) ( )( ) ( )
˙i a −Ra /L
= a
−K e
/L a ia 1/La
+ v
˙ω m
K T
/J −B/J ω m
0 a
( ) ( )
x1 ia
Alternatively the notation x = = andu = v
x 2
ω a
can be used. However,
m
whenthereisnoconfusionitisbettertoretaintheoriginalsymbolsbecauseitmakes
iteasier tosee ataglance what the statevariables are.
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