1. A Mathematical Analysis of a Sun Tracking Circuit for Photovoltaic Systems Dr. S. Louvros and Prof. S. Kaplanis T.E.I. of Patra, Greece

2. Outline Sun tracking system specifications General architecture of proposed circuit Detailed circuit Analysis Mathematical modeling Results and discussion

3. 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.

4. Sun tracking System block diagram 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 In case one photo-resistor receives more light than the other, the panel is not aligned properly and an error voltage results

5. Mathematical System analysis

6. Sun tracking circuit The sun tracking circuit consists of: 1.the photo detector, 2. the voltage conversion, and 3. the current amplifier.

7. 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

8. 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 the other direction.

9. DC Step Motor Transfer Function 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

10. 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

11. 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

12. Sensitivity to the Gain K 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. K = 10 volts/radian (gives overdamped response) K = 25 volts/radian (gives underdamped near optimal response) K = 75 volts/radian (oscillatory behavior)

13. Sensitivity to the Inertia I In order to demonstrate clearly this behaviour three values of I = 10e-6 (Low), 30e-6 (medium), and 100e- 6 (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.

14. 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

15. 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.