System Dynamics

VI. Mathematical Modeling of Engineering Systems

Case Study E - Light Tracking Servo System

Preliminary Note

Most of the material for this case study was generated by Mr. Paul Batcheller during the Spring and Summer of 1992. Mr. Batcheller performed this study as part of his project in System Dynamics during Spring 1992, and then built the physical model during the subsequent summer. Much of the following discussion has been taken directly from his final report: "Analysis of a Light Tracking Servo System," by Paul Batcheller (July 1992). His final project report is referred to as Reference 1 in the following development.


Sun tracking systems are very effective in increasing the efficiency of photovoltaic (PV) arrays, and are essential for concentration PV systems. The following paper discusses a light tracking servo model which has been built to simulate the movement of a PV array. A mathematical model is developed and a qualitative comparison of the mathematical model and the actual physical model is done to demonstrate the dynamics of a light tracking servo system.

An overall transfer function for a permanent magnet direct current (dc) motor was also developed. This subject is treated as a separate Case Study in this course (see Case Study F). The motor transfer function is used in the development of an overall transfer function for the light tracking servo system. Using the overall transfer function, a computer simulation program within Matlab is used to simulate the dynamics of the servo system. A qualitative analysis of the Matlab results and the dynamics of the working physical model are compared to clearly illustrate the important dynamics of the system.

Physical and Conceptual Description

The light tracking system consists of a permanent magnet dc motor, a directional light detecting circuit, and an amplifier to drive the motor. Refer to the model and to Fig. 6E.1 for the physical configuration and a simple block diagram representation. Two photo sensitive resistors are physically mounted on a triangular plexiglass mount so that when the panel (the aluminum bar) is perpendicular to the light source, each receives an equal amount of irradiance. When one receives more light than the other, the panel is not aligned properly and an error voltage results. The error voltage is used as a command to an amplifier circuit to drive the motor and align the panel to be perpendicular to the light source beam. The following subsections describe in detail the mechanical and electrical components of the model.

Fig. 6E.1 Light tracking servo mechanism with proportional control.

Power Supply

To make the light tracking model self sufficient, a plus and minus 12 volt dc power supply is necessary for the electronic components. In a typical solar application, this dc power source is obtained directly from the PV panels or batteries charged by the PV panels. However, in this case, the dc power is obtained by converting the standard 110 ac (alternating current) power from the wall socket.

For this application, the power supply is designed to handle a two ampere load. Figure 6E.2 shows the electrical circuit which handles the conversion of ac to dc power. The circuit can be divided into four sections; transformation, rectification, filtering, and regulation.

Fig. 6E.2 Power supply circuit for the sun tracker (from Ref. 1).

1. Transformation is accomplished by the transformer (T1) which steps down the 110 volts to two 15 volt peak ac sources. In a properly designed circuit, the secondary voltage should be 20 volts peak. The lower voltage level causes some ripple voltage on the output under load conditions of 1.5 amp or greater. The two secondaries are tied together at one end to form the common ground of the dc source. This common ground is not the same as the ground from the wall.

The transformer also electrically isolates the load from the utility power line by using magnetic coupling to transfer the power.

2. Rectification is accomplished by the diode bridge configuration B1. Although the circuit seems to indicate a bridge rectifier, it is actually two full-wave rectifiers. The right side of the bridge provides positive full-wave rectification while the left side provides negative full-wave rectification.

3. Filtering is provided by capacitors C1 and C4 which level the rectified signal to provide a flat dc voltage equal to the peak of the unrectified signal. Some ripple occurs on the signal and this is a function of the load and the size of the capacitor. The current through a capacitor is . In the worst case condition, when i equals two amperes, the ripple voltage (dv) can be approximated using the above equation. The time between two peaks is one over twice the frequency or 8.3ms (dt = 8.3 ms) and the capacitance is 3300 which gives a ripple voltage of 5 volts peak to peak. A larger capacitor will obviously reduce this ripple, but at some point the capacitor becomes to bulky.

4. The function of the regulators (U1 and U2) is to provide a constant output voltage. The input voltage required to maintain line regulation is given as 14.6 volts. This makes is clear why the transformer must have a peak voltage of 20 volts. If the peak voltage is 20 volts and the ripple is 5 volts, the lowest voltage that will occur is 15 volts which is still above the specified input voltage to the regulator.

The capacitors C2, C5, C3 and C6 provide filtering for higher frequency noise and are recommended by the manufacturer's specifications of the regulators. The diodes, D1 and D2, provide protection for the regulators from inductive loads.

Photo Detecting Circuit and Amplifier

The light tracking circuit (Fig. 6E.3) provides an electrical driving force to the motor which is proportional to the rotational misalignment of the panels to the light source. This circuit can be broken into three sections; photo detector, gain adjust, and current amplifier.

Fig. 6E.3 Light tracking circuit diagram for the sun tracker (from Ref. 1).

1. The photo detector provides a voltage signal which is linearly proportional to the rotational offset from the ideal position of the PV panel. This is accomplished 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 op-amp U1A to convert a differential signal into a voltage signal referenced to ground.

The photo detecting circuit has a measured gain of approximately 1.6 volts per radian under fluorescent lighting. This gain may vary depending on the intensity of the light.

2. The gain adjust portion of the circuit simply adjusts the gain. A potentiometer, POT2, allows the user to change the gain of the circuit and thus control the dynamics of the system. A simple inverting op-amp configuration is used to adjust the level of the voltage signal. The gain is Gain = POT2/RPA. This is the signal which is used to command the motor and can be easily monitored on an oscilloscope via the purple connector on the model.

3. The current amplifier provides the current to the motor. The LM358 op-amp is not capable of providing the necessary current to drive the motor. Thus a push-pull circuit is used with the NPN transistor Q1 (TIP121) and the PNP transistor Q2 (TIP126). When the voltage signal from U1B is greater than 0.7 volts, Q1 turns on and conducts the necessary current to drive the motor in a particular 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.

The ‘Position Sense’ circuit shown in Fig. 6E.3 is to monitor the position of the panel. A potentiometer is mechanically coupled to the axis of rotation, and thus provides a voltage signal proportional to the rotational position of the panel. This signal is easily monitored by an oscilloscope from the white connector.

System Dynamics Lecture Notes by Dr. J. R. White, UMass-Lowell (Spring 1999).

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