Current Loop
Let the field excitation be assumed to remain constant at its nominal level.
Let the voltage applied to armature be va volts, the back e.m.f.
eb volts and rotor speed wr rad/s. The back emf is expressed
by equation (4), where Km is the coefficient relating speed of
motor to its back emf. If Ra be the resistance of armature and
La its inductance, then the applied armature voltage equals the
sum of the motor back e.m.f, the drop across its armature resistance and the
drop across the armature inductance, as shown in equation (5). In equation
(5), va is the voltage applied to the armature and ia
is the current though the armature. The above equation can be represented
in terms of Laplace transform, leading to equation (6).
The block diagram shown in Fig. 4 represents equation (6).
Given that the excitation of the motor is constant and that the effects of
armature reaction are negligible either due to interpoles or series compensation
winding, the torque output Me can be expressed as shown in equation
(7). If the load torque be ML N-m, the combined polar moment of
inertia of motor and load be J kg.m2 and its friction coefficient
be B N-m/rad/sec, then the torque output of motor equals the expression on
the right-hand side of equation (8). Equation (8) can be represented in terms
of Laplace transform, as shown in equation (9), where the Laplace transform
of w, the motor speed, is assigned to be W(s).
A block diagram, as shown in Fig. 5, can now be drawn based on equations (4),
(6), (7) and (9). It can be seen that unit for Km is N-m/A.
With the load torque set to zero, a transfer function linking current Ia(s)
and the input voltage Va(s) can be obtained. It is expressed in
equation (10).
The reason for obtaining this transfer function is to facilitate the design
of a controller for controlling the armature current. The two output parameters
of interest are the torque and the speed. The armature current is selected
as one of the state-variables to be controlled in closed-loop, since the torque
output varies linearly with it. It is preferable that the variable to be controlled
by negative feedback is a variable that reflects some energy stored in a system.
Here the armature current reflects the energy stored in the inductance in
the armature circuit. If the motor has a compensating winding and/or a compound
winding, the inductance of this winding should be added to La.
In some drives, an additional inductor is used in series with the armature
and this value should also be added to La. Let G1(s)
reflect the transfer function in equation (10) and equation (11) reflects
the change. The part of the closed-loop system that is usually used for controlling
the armature current is shown in Fig. 6.
The block diagram in Fig. 6 is now described. If the armature current is
to be controlled in closed-loop, it is necessary to have a current reference
signal, marked as IR(s) in Fig. 6. This signal is internally generated,
most often as the output of the controller for speed and it is shown later
how that is achieved.
It is possible to use a controller other than a proportional plus gain(PI)
controller. A PDF controller(a pseudo-derivative controller) or a PID-controller
can be used. But a PI controller is often sufficient, since the integrating
part of the PI controller leads to zero steady-state error for a step input
and the proportional gain can be adjusted to yield fast response and stability.
The output of the current controller is often a voltage which sets the firing
angle for the fully-controlled bridge circuit. Since the gain KB(a)
is negative, the sign of both the proportional gain KI and the
integrating time-constant TI should be negative in order to keep
the loop-gain of the system represented by block diagram in Fig. 6 negative.
A variation in the output of the current controller does not change the firing
angle instantaneously since the SCRs in the bridge are triggered in a sequence
at an interval of 60o on the average and there is a delay before
the change in the output of the current controller has an effect on the firing
angle. This delay can be classified as a transportation lag and it can be
approximated by a first-order transfer function, as shown in equation (12).
In equation (12), y has been used in place of sTD.
For a system with 50 Hz input source, one-sixth of a cycle is about 3.3 ms
and then the delay TD can be set to be half of that value, that
is 1.67 ms. What is carried out is an approximation to facilitate the design
of current controller.
As the firing angle a increases, the instant of
triggering of each SCR is delayed more and more from its reference point corresponding
to 0o firing angle. When the current flow in the dc link is continuous,
the average output voltage of the bridge changes from its positive maximum
average value to its negative maximum average value, as a
is allowed to very from 0o to 180o. In order to ensure
that the loop gain is negative, it is necessary that the gain due to controlled
rectifier circuit is inverted. It is explained later how it can be brought
about for practical realization.
Another point to be noted is that the gain of the controlled rectifier is
not constant and it varies with firing angle. Let the maximum average output
voltage be Vom. Then equation (13) shows how the average output
voltage at any firing angle, a, is obtained. The
actual gain of the controlled rectifier is defined by equation (14). If the
rated armature voltage, VRA, is assigned to be the base voltage,
then the gain of the controlled rectifier in per unit notation can be defined
as in equation (15).
In equation (15), KA defines the ratio of peak line-to-line voltage
to the rated armature voltage. It is seen that the gain varies and hence the
controller has to be designed such that it operates properly over this range
of variation.
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