Page 248 - Electric Drives and Electromechanical Systems

Chapter 10 Controllers for automation 245

conversions between functions in the time domain, their equivalent Laplace transform
and z-transforms. The z-transform is used in the analysis of discrete time signals.
As a first step to the analysis, the Laplace transfer function, between the motor’s
terminal voltage and the output position can be determined from Eq. (5.1),

Q L ðsÞ K t
¼ (10.1)
V ðsÞ sðsR a I tot þ R a B þ K e K t Þ
where K e and K t are the motor’s voltage and torque constants, R a is the armature
resistance, I tot is the total inertia of the system, and B is the system damping constant.
The motor’s armature inductance has been neglected, because the motor’s electrical
time constant is negligible compared to the system’s mechanical time constant. In a
proportional closed-loop servo system, the motor’s terminal voltage is directly propor-
tional to the angular difference between the required and the actual position; therefore,
the motor’s supply voltage can be expressed in the form,
d

V ðsÞ¼ K p Q ðsÞ Q L ðsÞ ¼ K p EðsÞ (10.2)
L
where K p is the proportional gain, including the power-amplifier transfer function. The
position feedback signal is derived from an encoder mounted externally on the load
shaft, which will require suitable conditioning, as discussed in Chapter 4, Velocity and
Position Transducers. In the full-system model the required gain and/or conversion
factor will be added to the transfer functions. If the analysis of this control system is
undertaken in the conventional manner, it can be shown that the open-loop transfer
function is,
Q L ðsÞ K t ðK p þ sK v Þ
¼ GðsÞ¼ (10.3)
EðsÞ sðsR a I tot þ R a B þ K e K t Þ
where K v is the servo amplifier’s derivative feedback gain, which is added to improve the
system’s response when following a trajectory generated as a polynomial function.
The open-loop transfer function, G(s), results in a closed-loop transfer function for the
system of,
Q L ðsÞ GðsÞ K t ðK p þ sK v Þ
¼ ¼ (10.4)
d 2
Q ðsÞ 1 GðsÞ s R a I tot þ sðR a B þ K e K t þ K t K v Þþ K t K p
L
The transfer function of the motor-drive and its controller is a second-order system,
with a zero located at K p /K v in the left-hand of the s-plane. Depending on the location
of this zero, the system can have a large positional overshoot and an excessive settling
time. In a machine-tool or robotic application, this possibility of an overshoot should be
considered with care because it could lead to serious collision damage if it becomes
excessive.
In the analysis of a system, the effect on the load’s position due to an externally
applied load or disturbance must be fully considered. In the example shown in Fig. 10.4,
an external torque of T d (s) is applied to the system (this could be from the gravitational
and centrifugal forces in a robot or from the cutting forces in a machine tool); as