Closed - Loop Control
In order to regulate the output voltage, a controller is needed to be designed.
For this purpose, the power circuit is first represented by a transfer function.
The passive part of the power circuit is shown in Fig. 20. Since the DC output
of the SMPS is defined by equation (10), the effective input to the passive
circuit in Fig. 20 can be stated to be E.D(s), whereas D(s) is the Laplace
transform of the output of the error amplifier shown in Fig. 19. In the design
procedure outlined below, D(s) is taken to be the output of the controller,
varying between 0 and 1.
For the circuit in Fig. 20,
where
The block diagram of the SMPS with a PI controller is shown in Fig. 21. The
transfer functions of the different blocks in Fig. 21 are as follows:
The PI controller has two inputs, the reference signal Vref corresponding
to the desired output voltage and the feedback signal Vfdb. The
error signal, the difference, e, between Vref and Vfdb
is fed to the PI controller and the output of the PI controller is D(s), the
signal that sets the duty cycle. Then
In equation (35), K is the proportional gain and T is the integrating time
constant. The output of the PI controller varies between 0 and 1 and has two
limits, one corresponding to the lowest duty cycle and the other corresponding
to the highest duty cycle.
There is a filter in the feedback path. The filter time constant is about
4 times the cycle period corresponding to the switching frequency of the SMPS.
Then the ripple at the switching frequency gets filtered out and the feedback
signal is essentially a dc signal. Otherwise, the ripple in the output at
the switching frequency has nearly 180o phase shift with respect
to the fundamental of the input square pulse to the SMPS and then feedback
at switching frequency becomes positive, which in turn leads to amplification
of the ripple content. The filter avoids this problem, without seriously affecting
its dynamic performance.
The transfer function of the power circuit is obtained as follows. Since
the output varies linearly with the duty cycle, equation (33) can be presented
as follows:
For designing the closed loop system, the poles due to the source can be
ignored. Since the power circuit is very much under-damped, derivative feedback
is required for stable operation. The differentiating circuit provides the
feedback signal. For simulation, the derivative signal is obtained based on
the capacitor current. The derivative feedback voltage is set to be Kd.iC.
The third applet presented below allows the user to design a suitable PI
controller. The parameters to be set are: switching frequency in kHz, gain
of the PI controller, its integrating time-constant , the inductance, the
capacitance, the load resistance, the time constant of the filter in the feedback
path and the derivative coefficient.
click here to open the applet
THIRD APPLET
The feedback arrangement used for actual simulation contains additional circuitry.
It is necessary to limit the inductor current since it is seen from the open
loop response that the inductor current can be several times the rated current,
where the rated load current is the nominal current rating of the SMPS. It
is a parameter that can be set in the program.. In this program, the current
limit is set at 1.5 times the rated load current, and when the inductor current
exceeds the current limit, the current in excess of the current limit is amplified
and added to the feedback signal. In addition, a derivative feedback signal,
obtained using the capacitor current, is added to the feedback signal. Derivative
feedback improves damping of the system and without it, the system is oscillatory
in spite of a well-designed PI controller. While designing the PI controller
using the third applet, the switching circuit is replaced by a linear amplifier
and this approximation is necessary for the design of the PI controller. But
in reality, the square-wave voltage input to the power circuit makes the system
to be oscillatory since the power circuit is heavily under-damped. The feedback
arrangement used is shown in Fig. 22.
When the derivative feedback coefficient is set at unity, the derivative
feedback amounts to 10% of the rated voltage when the capacitor current equals
the rated current.
The fourth applet that is presented below simulates the circuit. It has a
pull-down list of items, of which the only first item is displayed. On clicking
on the downward arrow, the list is displayed. If any item of the list is high-lighted,
its default value is displayed in the window adjacent to the label with the
caption Set Value.. If one scrolls down or up the list, the
default value of the item highlighted is displayed in the window adjacent
to the label with the caption Set Value. To change the default
value of an item, highlight it first and then click inside the text-field
window adjacent to the label with the caption Set Value. A line
cursor would appear and the value can be changed. After changing the value,
click on the Set Value label and then the program recognizes
the change. It can be verified that the program has made the change out by
inspecting the pull-down list again.
The default values of items in pull-down list are:
Input Voltage,dc avg = 100
Input Ripple Volt., pk-pk = 0
Input Ripple Frequency = 100
Switching Frequency,kHz = 20
Inductance, microHenry = 500
Capacitor, microFarad = 500
Load Resistance, Ohms = 10
Output Voltage set at = 50
Gain of the PI controller = 5.
Time-constant of the PI controller in ms
=500
Time-constant of the Filter in ms
=40
Derivative Feedback Coefficient = 1.0
Rated Current in Amps = 10
Slow Response,0-200 = 0.
It is possible to see one of the five responses:
Periodic response over one output cycle,
Transient response over one output cycle,
Periodic response over one input cycle,
Transient response over one input cycle and
Statistics
Initially, the program sets '1' on the voltage axis of the plots equal to
100 V and sets '1' on the current axis to 10 A. If periodic response over
one output cycle is chosen, the program assumes that the peak-to-peak ripple
in input voltage is zero. The response can be slowed by varying the parameter
called Slow Response. This parameter is effective only for periodic
or transient response over one output cycle.
If the load resistance is changed from 10 W to
5 W, still '1' on the current axis would correspond
to rated current. If the Reset button is clicked, the program
would make changes so that '1' on the voltage axis of the plots equal to input
voltage and sets '1' on the current axis to (input voltage/load resistance).
When Start/Continue button is clicked, the program responds
to the visible response type and starts with the values of inductor current
and the capacitor voltage it had from the previous calculation. Initially
they are set to zero values. If the Reset button is clicked,
the program would make them zero again.
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FOURTH APPLET
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