Transcript IIR - WDM Next Generation Optical Networking

A Mathematical Analysis of a
Sun Tracking Circuit for
Photovoltaic Systems
Dr. S. Louvros and Prof. S. Kaplanis
T.E.I. of Patra, Greece
Outline
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Sun tracking system specifications
General architecture of proposed circuit
Detailed circuit Analysis
Mathematical modeling
Results and discussion
Sun tracking System Specs
There is much interest to develop a sun tracking system with one or two axis of
rotation in order to increase the yield of the PV plant.
•Reliability
•Shelf controlled in any conditions
•Shelf adjusted into different operational parameters
•Light in design
•Durable into winds
The described system will be based on a DC step motor, which is easy to
implement and light in construction
The combination of the PV weight, the sun position in the sky, the gears frictions
and the feedback mechanism of self-control leads to a dynamical system with
stability/instability operational conditions.
Sun tracking System block diagram
In case one photo-resistor receives more light than the other, the panel
is not aligned properly and an error voltage results
The system consists of:
•Directional light detecting circuit, consisting of two photo-resistors mounted into a
panel in order to have a differential measurement of the sun ray directions.
• Amplifier circuit to amplify the voltage difference and drive the motor
• A permanent magnet DC step motor to align the PV direction into the
perpendicular sun rays
Mathematical System analysis
Sun tracking circuit
The sun tracking circuit consists of:
1. the photo detector,
2. the voltage conversion, and
3. the current amplifier.
Photo-detector circuit
The photo-detector provides a voltage signal which is linearly proportional to the
perpendicular position error (error offset) of the PV panel. This is implemented
by putting two light sensitive resistors in an electrical bridge which is connected
to a unity gain differential operational amplifier circuit. When one light sensitive
resistor receives more light than the other, a differential voltage results across
the bridge network which is fed into the operational amplifier to convert a
differential signal into a voltage signal referenced to ground
Amplifier circuit
The operational amplifier provides the voltage conversion referenced to the ground but
cannot provide the necessary current to drive the D.C. motor. This is implemented by
the current amplifier, which drives the step motor proportionally to the ground
referenced voltage signal
The current amplifier consists of a push-pull circuit with the NPN transistor Q1 and the
PNP transistor Q2. When the voltage signal is greater than 0.7 volts, Q1 turns on and
conducts the necessary current to drive the motor in a certain direction. When the
voltage signal from U1B is less than -0.7 volts, Q2 turns on and the motor is driven in
DC Step Motor Transfer Function
kt
Va ( s)
La  I
( s) 

Wa ( s)
 Ra  D  kt  kv 
 Ra  I  La  D 
2
s 
s  


La  I
La  I



where
Va(s) is the Laplace transform of the voltage input across the coil of the armature
Wa(s) is the Laplace transform of the rotational velocity of the armature
Inductance (La) in series with a resistance (Ra) forms the electrical equivalent of the
armature coil.
kv is the velocity constant determined by the flux density of the permanent magnets, the
reluctance of the iron core of the armature, and the number of turns of the armature
winding.
kt, the torque constant, and like the velocity constant is dependent on the flux density of
the fixed magnets, the reluctance of the iron core, and the number of turns in the
armature winding.
I, the inertia of the rotor
B, the damping coefficient associated with the mechanical rotational system of the rotor
Frequency Domain System Block Diagram
Where:
•The photo detecting circuit and amplifier can be considered as a single variable K,
where K is a proportional constant with units of volts per radian. The value of K thus
represents the gain of the ‘photo detector’ circuit.
•The rotational position output is related to the velocity of the motor by integrating the
velocity or, in the frequency domain, by dividing by s
Overall Open Loop Transfer Function
After certain block diagram algebra the closed loop feedback system is converted into
an open loop system with overall transfer function
In the frequency domain the operation of the system is described as:
A rotational error from the displacement of the photo-resistors results in an error
voltage which is converted into a current driving signal and then converted to a
rotational velocity by the motor. The rotational position output is related to the
velocity of the motor by integrating the velocity and the output position is
subtracted from the input position, constructing thus a closed loop feedback
control system
Sensitivity to the Gain K
K = 10 volts/radian (gives
overdamped response)
K = 25 volts/radian (gives
underdamped near optimal
response)
K = 75 volts/radian
(oscillatory behavior)
Increasing the gain results in an overall increase of sensitivity of
the system to small alterations of the input light source position.
By increasing the gain, the voltage command is becoming more sensitive for a
particular rotational displacement (K is given in volts/radian) and the system
becomes unstable.
Sensitivity to the Inertia I
In order to demonstrate
clearly this behaviour three
values of I = 10e-6 (Low),
30e-6 (medium), and 100e6 (High) kg-m2 were used
for the low, mid, and high
respectively
The inertia of the system refers to the weight of the PV panel on a sun
tracking system. The system becomes unstable as the inertia of the system
increases. This conceptually makes sense, since the larger the mass (inertia is
related to mass in a rotational system), the more difficult it is to stop and turn in
the opposite direction resulting in an oscillation that settles down after a while.
Sensitivity to the Damping D
Damping values of 1e-3,
5e-3, and 50e-3 N-m-s
have been used to
represent low, mid, and
high values of the
mechanical damping
coefficient
An increase in the damping factor is equivalent to adding friction to the
system. The gearing and brushes in the motor add friction to this system. An
increase in friction tends to stabilize an unstable system
Conclusions
The designers of sun tracking systems must take into account the effects of
varying the gain, inertia, and damping factor on a light tracking servo system.
To design a system which meets to designer's specifications (time response,
amount of overshoot, stability, etc.), the correct combination of
•system friction (damping factor),
•mass distribution (inertia),
•and system gain is crucial.
The variable feedback gain can be used to offset undesirable effects due to
high inertia and low damping, as well as simply adjust the dynamic response of
the system for near optimal performance.