Consider the model of a permanent-magnet brushed dc motor that we discussed in class, comprising (on
the electrical side) a resistance R, an inductance L, and a back-emf constant k_(emf ), and (on the mechanical
side) a rotor inertia J, damping b, and torque constant k_( au ). Recall that in S.I. units, the back-emf and
torque constants are equal, so simply use k_( au ) to describe both. There is a single system output: the
angular velocity of the rotor shaft, omega (t), measured in rad/s. There are two system inputs: the applied
voltage v(t), and an applied load torque au _(a)(t); both are defined as positive when they tends to drive omega in
the positive direction.
(a) Find a state-space model relating the two inputs to the output. Pack your equations into the
standard "A, B, C, D” matrix-vector form.
(b) Find the transfer function from the input voltage to the output, assuming the applied load torque is
zero.
(c) Find the transfer function from the input applied load torque to the output, assuming the applied
voltage is zero.
(d) We often assume that the "electrical time constants" are much faster than the "mechanical time
constants" and neglect the inductance to simplify our model. Find the simplified versions of the
transfer functions from parts (b) and (c) if L=0.
(e) If we use a current amplifier, rather than a voltage amplifier, we can consider the current i(t) as the
system input (rather than the voltage v(t), which also simplifies the model. Note that practical
limits on our amplifier's maximum voltage limit the realism of this model (that is, we don't really
have perfect control over the input current if we consider inductance, since that would require
infinite voltage for step changes in current). Find the transfer function from the input current to the
output, assuming the applied load torque is zero. Your result should not require any assumption
about the value of L.
(f) For the transfer function from part (b), use the final value theorem (if it applies) to find the steady-
state angular velocity for a constant (step) input of 1V. Repeat for the simplified transfer function
from part (d). Your answers should be the same.
(g) For the transfer function from part (e), use the final value theorem (if it applies) to find the steady-
state angular velocity for a constant (step) input of 1A.
Consider the model of a permanent-magnet brushed dc motor that we discussed in class,comprising (on
the electrical side) a resistance R,an inductance L, and a back-emf constant kemf, and (on the mechanical
side) a rotor inertia J, damping b, and torque constant k.. Recall that in S.l. units, the back-emf and
torque constants are equal, so simply use k, to describe both.There is a single system output: the
angular velocity of the rotor shaft, w(t),measured in rad/s.There are two system inputs: the applied
voltage v(t), and an applied load torque ta(t); both are defined as positive when they tends to drive w in
the positive direction.
(a) Find a state-space model relating the two inputs to the output. Pack your equations into the standard "A, B, C, D" matrix-vector form. (b) Find the transfer function from the input voltage to the output, assuming the applied load torque is zero.
voltage is zero.
(d) We often assume that the "electrical time constants" are much faster than the "mechanical time constants"and neglect the inductance to simplify our model.Find the simplified versions of the transfer functions from parts (b) and (c) if L=0. (e) If we use a current amplifier, rather than a voltage amplifier, we can consider the current i(t) as the system input (rather than the voltage v(t)), which also simplifies the model. Note that practical limits on our amplifier's maximum voltage limit the realism of this model (that is,we don't really have perfect control over the input current if we consider inductance, since that would require infinite voltage for step changes in current).Find the transfer function from the input current to the output, assuming the applied load torque is zero. Your result should not require any assumption about the value of L. (f) For the transfer function from part (b), use the final value theorem (if it applies) to find the steady- state angular velocity for a constant (step) input of 1V. Repeat for the simplified transfer function from part (d). Your answers should be the same. (g) For the transfer function from part (e), use the final value theorem (if it applies) to find the steady- state angular velocity for a constant (step) input of 1A.
