Project Number: MQP-SJB-1A03

Solar Panel Peak Power Tracking System

A Major Qualifying Project

Submitted to the Faculty of

WORCESTER POLYTECHNIC INSTITUTE In partial fulfillment of the requirements for the

Degree of Bachelor Science

March 12, 2003

By

______________________________

Eric Anderson

______________________________

Chris Dohan

______________________________

Aaron Sikora

_________________________________

Professor Stephen J. Bitar

_________________________________

Professor John A. McNeill

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Abstract

The design of a Maximum Peak Power Tracking (MPPT) controller for a solar photovoltaic battery charging system is proposed utilizing a boost-converter topology. Solar panel voltage and current are continuously monitored by a closed-loop microprocessor based control system, and the duty cycle of the boost converter continuously adjusted to extract maximum power. System testing confirms peak power tracking under changing lighting conditions. Under specific conditions, efficiencies in excess of 90% are shown to be possible.

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Acknowledgements We would like to thank Professor Stephen J. Bitar and John A. McNeill for their more than helpful contributions over the course of the project. We would also like to thank them for use of the analog lab and all of its equipment.

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Table of Contents

2.0 Background ............................................................................................................................... 8 2.1 Solar Panel Phenomenology ................................................................................................. 8

2.1.1 Solar Power Fundamentals ............................................................................................ 8 2.1.2 Efficiency....................................................................................................................... 8 2.1.3 Voltage-Current (V-I) Characteristic........................................................................... 10

2.2 Insolation Levels................................................................................................................. 13 2.3 Charge Controllers .............................................................................................................. 16 2.4 Storage Batteries ................................................................................................................. 18

2.5.1 Multiple Panels per MPPT........................................................................................... 20 2.5.2 Individual MPPT per Panel.......................................................................................... 21 2.5.3 The Best Approach ...................................................................................................... 23

3.0 Methodology........................................................................................................................... 24 3.1 System Block Diagram ....................................................................................................... 24 3.2 Battery Requirements.......................................................................................................... 25 3.3 Boost Converter Description............................................................................................... 27

3.3.1 Stage One: Initial Conditions...................................................................................... 30 3.3.2 Stage Two: Switch Closing......................................................................................... 30 3.3.3 Stage Three: Switch Opening ..................................................................................... 32 3.3.4 Stage Four: On and off switching ............................................................................... 34

3.4 Boost Converter System Equations .................................................................................... 36 3.4.1 Thevenin Equivalent .................................................................................................... 37 3.4.2 Average Current........................................................................................................... 39 3.4.3 Current Ripple.............................................................................................................. 43 3.4.4 Voltage Equations........................................................................................................ 44 3.4.5 Equivalent Resistance and Power Equations ............................................................... 46 3.4.6 Equation Accuracy....................................................................................................... 46

3.5 Current Sensing Methods.................................................................................................... 48 3.5.1 Inductor Current Sensing ............................................................................................. 49 3.5.2 Current Sensing Conclusion ........................................................................................ 54

4.0 Implementation ....................................................................................................................... 58 4.1 Parts Selection..................................................................................................................... 58

4.1.1 MOSFET Selection...................................................................................................... 58 4.1.2 Inductor Selection ........................................................................................................ 63 4.1.3 Instrumentation Amplifier ........................................................................................... 71 4.1.4 Diode Selection............................................................................................................ 74 4.1.5 Voltage Regulator ........................................................................................................ 75 4.1.6 MOSFET Gate Driver.................................................................................................. 76

4.2 Operating Frequency........................................................................................................... 77 4.3 Voltage Sensing .................................................................................................................. 79 4.4 Current Sensor .................................................................................................................... 84

4.4.1 Alternative Methods..................................................................................................... 85 4.4.2 Gain setting .................................................................................................................. 87 4.4.3 Initial Testing ............................................................................................................... 90 4.4.4 Common Mode Voltage............................................................................................... 92

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4.4.5 Differential Voltage ..................................................................................................... 94 4.4.6 Common Mode Rejection Ratio (CMRR) ................................................................... 96 4.4.7 Common Mode Range ............................................................................................... 102 4.4.8 Output Filter............................................................................................................... 104 4.4.9 Current Sensing Schematic ........................................................................................ 106

4.5 Controls............................................................................................................................. 107 4.5.1 Simulation.................................................................................................................. 111

4.6 Microprocessor ................................................................................................................. 126 4.6.1 Algorithm................................................................................................................... 132

4.7 Self Powering.................................................................................................................... 135 4.7.1 24-Volt Supply........................................................................................................... 135 4.7.2 5-Volt Supply............................................................................................................. 137

5.0 Power Loss Analysis............................................................................................................. 140 5.1 Operating Losses............................................................................................................... 140 5.2 Diode Losses..................................................................................................................... 142 5.3 IC Losses........................................................................................................................... 144 5.4 Overall MPPT Efficiency ................................................................................................. 146

6.0 Results................................................................................................................................... 148 6.1 Thevenin Equivalent ......................................................................................................... 148 6.2 Indoor Testing................................................................................................................... 153 6.3 Outdoor Testing ................................................................................................................ 158

7.0 Future Recommendations ..................................................................................................... 164 8.0 Conclusions........................................................................................................................... 165 References................................................................................................................................... 166 Appendix A: Schematic .............................................................................................................. 167 Appendix B: Datasheets.............................................................................................................. 168

AD627..................................................................................................................................... 168 IRL7833 .................................................................................................................................. 184 MBRD1040CT........................................................................................................................ 196 PIC16F87XA .......................................................................................................................... 199

Appendix C: MPPT Code ........................................................................................................... 216

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1.0 Introduction

Solar power is an alternative technology that will hopefully lead us away from our petroleum dependent energy sources. The major problem with solar panel technology is that the efficiencies for solar power systems are still poor and the costs per kilo-watt-hour (kwh) are not competitive, in most cases, to compete with petroleum energy sources. Solar panels themselves

are quite inefficient (approximately 30%) in their ability to convert sunlight to energy. However, the charge controllers and other devices that make up the solar power system are also somewhat inefficient and costly. Our goal is to design a Maximum Power Point Tracker (MPPT), a specific kind of charge controller that will utilize the solar panel to its maximum potential. The MPPT is a charge controller that compensates for the changing Voltage vs. Current characteristic of a solar cell. The MPPT fools the panels into outputting a different voltage and current allowing more power to go into the battery or batteries by making the solar cell think the load is changing when you really are unable to change the load. The MPPT monitors the output voltage and current from the solar panel and determines the operating point that will deliver that maximum amount of power available to the batteries. If our version of the MPPT can accurately track the always-changing operating point where the power is at its maximum, then the

efficiency of the solar cell will be increased. There are many different applications for solar power systems, but there are also many limitations to these applications. The cost-benefit is too low for solar power systems to be widely used for powering homes, businesses, or even individual products. Solar power systems are used as the main power source for a large majority if not all of the satellites that orbit the earth. However, the benefit of utilizing solar power in space far outweighs the cost to implement them. There are wide ranges of different products available to the consumer that are solar powered, but they can be expensive or impractical because of their limitations. Solar power systems are not competitive on the market because consumers are familiar with the practical, more convenient products that have more common power sources that they are used to. Some consumer products are radios, flashlights, motor-home trickle charging systems, outdoor solar lighting, laptop charging systems, and even home systems that can be tied to their existing power grid. Some of the more practical applications are used for remote locations such as cabins or small villages that are located far from the closest power grid. As a result, the cost to install a

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solar power system is cheaper than the cost to send transmission lines from the power grid. Solar powered systems are also very convenient in small applications such as powering calculators, outdoor lighting, and even traffic lights.

By attempting to make solar panel systems more efficient altogether, solar powered products could be used more commonly. While solar panels are not very efficient due to their current limitations, we hope to extract the maximum amount of possible power from the solar panel with our MPPT device. This is just one aspect of making solar power more efficient. The actual manufacturing of solar panels is important and is not something that we are able to take on in this project. There are many important factors that determine the amount of power the solar panels can extract from the sun including temperature, time of year, geographic location, and positioning of the sun. These factors can be minimized by designing a proper system that can monitor the output of the solar cell and extract the maximum amount of possible power from the solar panels.

In order to enable us to complete this project in an effective manner we need to understand the solar technology and the most important aspects of it. We will look at different applications and whether or not they are even feasible at the current state of solar technology. After understanding more about the technology that solar power involves and the different applications that it is used for, we can then approach our problem for a specific application and design the best solar panel peak power tracking system.

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2.0 Background

2.1 Solar Panel Phenomenology

2.1.1 Solar Power Fundamentals The amazing thing about solar power is that all the electricity is generated from the

material of the solar panels and the energy from the sun. The solar panels are mainly made out of semiconductor material, Silicon being the most abundantly used semiconductor. The benefit of using semiconductor material is largely due to the ability of being able to control its conductivity whereas insulators and conductors cannot be altered. The electrons of the semiconductor material can be located in one of two different bands: the conduction band or the valence band. The valence band is initially full with all the electrons that the material contains.1 When the energy from sunlight, known as photons, strikes the electrons in the semiconductor, some of these electrons will acquire enough energy to leave the valence band and enter the conduction band. When this occurs, the electrons in the conduction band begin to move creating electricity. As soon as the electron leaves the valence band, a positively charged hole will remain in the location the electron departed. When this occurs, the valence band is no longer full and can also play a role in the current flow. This process basically describes how Photovoltaic (PV) systems function. However, PV systems further enhance the rate at which the electrons are sent into the conduction band through the process of doping.2

2.1.2 Efficiency Solar power would be the leading source of energy if it were able to efficiently extract a

majority of the energy delivered to the earth from the sun on a daily basis. On the equator at noon, 1000 watts/m2 of sun energy touches the ground. Unfortunately only about 20 percent of this power can be transferred into usable energy. This inefficiency is directly related to the percentage of photons that are absorbed. The electrons in the semiconductor material will only jump into the conduction band if they absorb a photon. The photons can either be absorbed,

1 Sayigh, A.A.M., ed. Solar Energy Engineering. New York: Academic Press, 1977.

2 Neville, Richard C. Solar Energy Conversion. The Netherlands: Elsevier Science, 1995.

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reflected, or can even pass right through the semiconductor.3 In order to increase the number of photons absorbed ultimately increasing the efficiency of the solar panel, the percentage of photons that pass through and reflected must be reduced. There is an obvious loss of electric potential when the photons are reflected off the semiconductor material. To reduce the percentage of reflected photons, an anti-reflective coating is usually put on the semiconductor, which will decrease the number of reflected photons increasing the total number of photons that will become absorbed. However, there is still a chance that these photons could pass right through the material without striking an electron.

Some of the photons from light pass straight through the semiconductor as if the semiconductor were transparent. The photons in sunlight have a wide variety of different wavelengths causing some to pass right through.4 The photons that pass through the

semiconductor have energy lower than the band gap energy of the semiconductor.5 As a result, these photons do not contain enough energy to create an electron-hole pair, so the photon just passes right through the semiconductor.6 If the photon has more energy than the band gap of the semiconductor, then the electrons absorb the photon. However, if the photon has an excess of energy, meaning it gives the electron more energy than the band gap, than this excess will be emitted as a form heat and the electron will settle down in the conduction band. To minimize the amount of photons that pass through the semiconductor, some semiconductors are manufactured with many layers, each having a different band gaps in order to better match the light spectrum. A highly efficient solar panel can be designed by cascading semiconductor materials with different band gaps to perfectly match the light spectrum. However, this would require an infinite amount of semiconductor material making it utterly impossible. For the solar panels to be cost-effective, the can only be designed with a few different layers. As a result, there are still some photons that pass right through the semiconductor and the energy lost from the absorbed photons as a form of heat.

Solar power is an amazing technology in the sense that it converts sunlight into electricity

through the semiconductor material alone. However, it is clear that there are many flaws and

3 Turning Sunlight into Electricity. Nation Center for Photovoltaics. 20 Oct. 2002.

4 Aldous, Scott. How Solar Cells Work. How Stuff Works. 02 Nov. 2002.

5 Neville, Richard C. Solar Energy Conversion. The Netherlands: Elsevier Science, 1995.

6 Aldous, Scott. How Solar Cells Work. How Stuff Works. 02 Nov. 2002.

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complications in the ability to design a solar panel that can utilize a majority of the energy that is emitted from the sun on a daily basis. There are numerous ways to design and manufacture solar panels. The uses of different kinds of semiconductor materials, crystal structures, and manufacturing methods all have a different affect on the efficiency and cost of the solar panel. One crystal structure may be more efficient, but the cost may make it too expensive to consider. As time goes on, newer manufacturing techniques and designs will prove these solar panels more efficient and less costly in future years. Rather than focusing on the issues relating to the design and semiconductor physics behind the solar panels themselves, this project will focuses more on the devices that control the output of the solar panels. A solar panels output varies depending on certain ambient weather conditions such as temperature, illumination, how clear the sky is, so on and so forth. Our task at hand is to design a device that will extract the maximum amount of power from the solar panels, regardless of how efficient or inefficient the solar panels may be.

2.1.3 Voltage-Current (V-I) Characteristic Extracting the maximum amount of power from the solar panel is difficult due to the

nonlinearity of the Voltage-Current (V-I) characteristic. Figure 2-1 shows the V-I characteristic for SolarCorps 379 Solar Panel.

Figure 2-1: Characteristic of Solar Corp 379 Panel

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The blue line in Figure 2-1 is the actual V-I characteristic and the pink line corresponds to the power as a function of the voltage (P = I*V). As you can see, the voltage to current relationship is not linear, which makes it a little more difficult to determine the maximum power point. The maximum power point on a linear curve would occur at the midpoint of the V-I characteristic. However, in the case of a nonlinear relationship, the power needs to be determined by calculating the voltage to the current. To get the maximum power from the solar panel, the solar

panel must always be operated at or very near the point where the power curve is at a maximum, its peak point. However, this operating point will constantly change due to the constantly changing ambient conditions. In fact, the temperature and other affects such as irradiance alter the V-I characteristic changing the operating point that would allow us to pull out the maximum amount of power. As a result, we need to constantly track the power curve and keep the solar panel operating at the point where the maximum amount of power would be achieved. Irradiance is a characteristic that deals with the amount of sun energy reaching the ground. The irradiance reaching the earth in ideal conditions is 1000W/m2. However, this value is altered significantly depending on where you are located geographically, the angle of the sun, and the amount of haze or cloud cover preventing all of the suns energy from reaching the ground. Since solar panels run strictly off the energy emitted from the sun, their output is affected by the changing irradiance. The Figure 2-2 below demonstrates the affect irradiance has on the output of solar panels.

Figure 2-2: Solar Panel V-I Characteristic vs. Irradiance

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As you can see in Figure 2-2, a smaller irradiance or light intensity gives you a reduced output. However, only the output current is really affected by the changing irradiance. This is due to the generated current being proportional to the flux of the photons. If less light or weaker light is striking the solar panels, then the flux of photons is going to be lower than more powerful sunlight resulting in a reduced generated current. The voltage on the other hand is hardly affected by the change in irradiance. In fact, the change is so small that in the change in voltage is considered negligible and disregarded in most practical applications. The open circuit voltage is the voltage level at the point when there is no current flow. As a result, the voltage is

somewhat independent of the changing flux of photons resulting in very small changes in open circuit voltage. The open circuit voltage does however depend logarithmically on the irradiance, which explains the small changes in open circuit voltage.7 The irradiance is a very important factor in predicting the V-I characteristic of the solar panel, but the temperature of the panels also plays a very important role in predicting the V-I characteristic. The temperature of the solar panels plays just as significant a role in determining the output of the solar panels. Typically, you would expect the solar panels to operate more efficiently with a higher temperature. However, Figure 2-3 below shows otherwise.

Figure 2-3: Solar Panel V-I Characteristic vs. Temperature

7 Bogus, Klaus and Markvart, Tomas. Solar Electricity. Chichester, New York.

Wiley Press, 1994.

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As shown in Figure 2-3 above, the output does in fact increase with decreasing temperature. There are many factors in determining why this occurs. A huge reason is due to the electron and hole mobility of the semiconductor material. As the temperature rises, the electron and hole mobility in the material decreases significantly. The electron mobility for a Silicon semiconductor solar panel will decrease from 1700 cm2/volt-sec at 27oC to 440 cm2/volt-sec at 227oC where the hole mobility will decrease from 600 cm2/volt-sec at 27oC to 200 cm2/volt-sec at 227oC.8 These test temperatures are unrealistic operating conditions for solar panels, but do get the point across that the electron and hole mobility decrease as the temperature rises. Temperature also causes the band gap energy of the semiconductor material to increase. The photons from the sun provide the electrons in the valence band of the semiconductor with the energy to leap over the band gap into the semiconductor material. With larger band gap energy, the electrons will require more energy from the photons in the sun to reach the conduction band. As a result, fewer electrons will reach the conduction band giving us a less efficient solar cell. These changes in temperature and irradiance make the V-I characteristic near impossible to predict or control. The only thing we have control over is the operating point of the solar panel. Without control over this operating point, the output of the solar panels will be very unpredictable resulting in an even more inefficient solar power system.

2.2 Insolation Levels Insolation levels are a measure of how much sunlight energy is delivered to a square

meter over a single day. The suns energy is most powerful at the equator at solar noon, or when the sun is directly overhead. The suns energy will be dramatically decreased by the earths atmosphere. Energy will also be lost because of the angle of reflection against the solar panel. The suns energy is strongest when it is straight overhead, where the earths atmosphere, and the solar panels reflection has the least affect.

Insolation levels measure the amount of energy in watt-hours for a square meter over a

single day in Kwh/m2/day. Insolation level units are not always stated, and can be seen as a ratio. Therefore we will intentionally leave the units off. The suns energy varies as the sun moves

from the horizon to directly overhead. The insolation level is an equivalent solar energy level.

8 Neville, Richard C. Solar Energy Conversion. The Netherlands: Elsevier Science, 1995

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The suns energy at the equator at solar noon is equal to 1000 watts per square meter. In effect, we are finding the amount of equivalent hours the sun has put out 1000 watts per square meter. Although the sun may have been shining at 500 watts per square meter for 8 hours, there is only 4 equivalent hours of 1000 watts per square meter. Insolation levels give an easy way to compare

sunlight energy levels between locations as well as find typical solar and battery requirements for a given location.

An insolation level between one and two is considered to be low. Four and five are considered to be moderate and seven to eight is high. The map in Figure 2-4 shows the insolution, or the number of full hours of sun any location in the United States gets per day on a worst-case average.

Figure 2-4: Worst Case Insolation Levels for the U.S.

Again these are worst-case values. NASA has done extensive research on this subject and has data for most locations in the world based on at least ten years of data. The average insolation levels in Massachusetts are located in Table 2.1. The Clear Sky column shows the insolation levels with a perfectly clear sky.

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Location: Worcester, MA N 38 20 W 75 30 Average Clear Sky Solar Tracking January 1.51 3 2.53 February 2.34 4.32 3.27 March 3.18 5.99 3.68 April 4.78 7.48 5.07 May 5.68 8.24 5.73 June 6.63 8.3 6.66 July 6.75 8 6.76 August 6.26 7.11 6.54 September 5.03 5.78 5.86 October 3.3 4.47 4.78 November 2 3.21 3.44 December 1.32 2.61 2.37

Average 4.07 5.71 4.72 Table 2-1: Insolation Levels in Massachusetts in kWh/m2/day

Notice that the summer months have significant more energy available. It is often common for solar panels to be backed up by a generator to supply power for the winter months. We can use the solar panels more efficiency by angling them towards the sun. Sun tracking mounts designed for solar cells exist now. These self powered mounts point the solar cells directly to the sun to get the most amount of solar power. Of course this also uses some of the energy from the solar cell itself. NASAs database also has collected insolation levels for this and assuming the optimal angle is always set the average insolation levels improve slightly. The average insolation levels are shown in the solar tracking column above in Table 2.1. Here the yearly average was 4.07. With solar tracking, this number is improved 16% to 4.72.

Our solar panel is not rated in just efficiency but instead how many watts it can produce. The solar panel we have been provided with is 50W. If this panel is producing 50W under ideal conditions (1000W/m2) then we can expect to draw four times as much as that per day in Massachusetts since the insolation level is 4000W/m2. Therefore we can expect to store an

equivalent of 50W for four hours, or a more standard form to state this is 200W of power for one hour or 200Wh. The storage of this power depends greatly on the sun. We can expect the most amount of power while the sun is around solar noon. Before and after solar noon, we can expect

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lower power levels. Therefore we are not storing 200W in one hour, but an equivalent to that. In other words, after a full day of charging, we will have 200Wh available.

Sunlight energy around the world varies considerably. The closer to the equator, the more sunlight energy is available throughout the year. This is the most ideal place for solar electricity. Other places, such as London have very low yearly averages and are not practical to use solar technology. Remote locations that are far away from power grids also provide a practical use for solar technology. Table 2.2 contains the average insolation level per year for three different locations around the world.

Worcester, MA (N38 20 W75 30)

Kenya (N0 12 E37 29)

London, UK (N51 31 W0 04)

Average Insolation Level (kWh/m2/day)

4.06 5.85 2.69

Yearly Low (monthly average in kWh/m2/day)

1.32 4.94 0.52

Table 2-2: Comparing Insolation Levels across the Globe

2.3 Charge Controllers Solar panels are almost always charging some type of battery. Overcharging some types

of batteries can damage the battery. Also, the battery voltage determines the voltage level at which the solar panel will operate. However, operating at this voltage level probably will not be the most efficient for the solar panel. Due to these two reasons, charge controllers are commonly

put in between the solar panels output leads and the storage batteries. There are three different kinds of charge controllers that are commonly used. There are basic charge controllers, PWM charge controllers, and Maximum Peak Power Tracker (MPPT) charge controllers.

The basic charge controller is designed to protect the battery from any form of damage due to overcharge or undercharge and prevents any reverse current that may be drawn from the battery during the time period in which the solar panels are not generating any power. Overcharging some types of batteries can damage the battery as well as cause possible explosions or leaking. If energy is continually applied to the battery after it has reach full capacity, then the battery voltage will raise causing chemical reactions, which will eventually overheat the battery and damage it. To prevent overcharge, the charge controller simply monitors

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the charge going into the battery and regulates the voltage level sent to the battery to prevent any further charging once a certain maximum capacity level is reached. The batteries life cycle will be dramatically shortened if the batteries are undercharged for to long a period of time. In this situation, the charge controller will disconnect the battery, known as Low Voltage Disconnect (LVD), from any loads (lamps, appliances, etc.) once a certain capacity is reached in order to prevent the battery from losing any more charge. The basic charge controllers are essential to proper charging of batteries in order to protect the batteries from damage.

PWM charge controllers are similar to the basic charge controller. While basic charge controllers can only disconnect or connect the battery to stop overcharging, PWM charge controllers can actually control the amount of current charging the batteries in order to optimize the charging time. When the battery nears full capacity, the PWM charge controller switches the charging on and off using PWM (pulse width modulation) causing a trickle charge, which allows the battery to maintain a full charge. This feature optimizes the speed and efficiency of charging the battery.9 PWM and basic charge controllers both control the current going into the battery but do not attempt to optimize the efficiency of the solar panel. Maximum Peak Power Trackers (MPPT) charge controllers can optimize the power output from the solar panel, as well as charge the battery up to its optimal charge capacity.

The problem with the PWM charge controller and basic charge controllers is that they operate the solar panels at the voltage level designated by the voltage level of the battery. As demonstrated earlier, the V-I characteristic of the solar panel is not linear. By operating at a fixed voltage level, nothing guarantees that this voltage level is where the maximum amount of power

can be drawn. Further, the maximum power point will change due to irradiance and temperature guaranteeing that the PWM and basic charge controllers will rarely draw the maximum amount of power from the solar panel. The MPPT tracks this maximum power point and changes the operating point of the solar panels in order to constantly draw the maximum amount of power available. The MPPT allows for the maximum efficiency of the solar panel to be reached as well as control the batteries charging requirements.

9 Distributed Power Solutions Home Page. 2003. Minneapolis, Minnesota.

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2.4 Storage Batteries There are two types of storage batteries being used for solar power storage, acid or

alkaline. Alkaline batteries are made with nickel cadmium or nickel iron. The main difference between the two is nickel cadmium batteries have a faster discharge rate. However, nickel cadmium batteries are bad for the environment while the nickel iron does not have any environmental problems. Nickel iron batteries are slower to respond when a load is applied and have to be broken in before they can reach their maximum charging capacity. Both types of batteries will not freeze so there are no problems when operating in cold climates.

The most common type of battery used in solar systems is the lead-acid battery. They are used because they have a low initial cost and are common. Lead-acid batteries come in deep-cycle, shallow cycle and car cranking. Deep cycle batteries are designed for discharging and recharging over and over.10 Shallow cycle or cranking batteries are the type that you use in your automobile just to crank the engine initially. They are never discharged completely. If the battery is a shallow cycle or automotive type it will not function correctly for solar purposes as discharging the battery completely, which is quite common, will present problems.

A variation from the standard lead acid battery is the gel cell battery. It is different in the fact that a gelling agent is added to the electrolyte to reduce movement inside the battery.11 Many gel batteries utilize one-way valves instead of open vents to help the normal internal gas to mix back into the water inside the battery. Gel batteries have a much higher internal resistance, meaning they are unable to deliver and receive current as efficiently as a deep cycle battery would.12

The two main types of deep cycle batteries are sealed or flooded. A sealed battery never needs water added and doesnt need equalization charge. This type of battery can be mounted in any position and are easy to transport because they wont leak or cause a chemical spill. The downside of this type of battery is that they need to be monitored closely for overcharging. However, a flooded battery also needs close attention. The water level in a flooded battery needs to be checked often and re-filled. Equalization charges also need to be performed which is

10 National Solar Supply Home Page. 2003

11 Optima Batteries Home Page. 1996-2004. Johnson Controls Inc.,

12 Ibid.

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basically a long stable overcharging method. This removes sulfation from the battery plates and restores the batterys capacity, but it can shorten the life of the battery by warping the plates.13

New alkaline batteries can be left unattended for long periods of time and they can be fully discharged without any damage done to their life (approximately 4,000 cycles.) If properly taken care of, alkaline batteries will last for around 20 years in a home power system. They can also be reconditioned to restore them to near their original condition because they utilize a solution of lithium hydroxide, potassium hydroxide, and distilled water, which does not destroy cells. Lead acid type batteries cannot be reconditioned because the acid destroys the cells.14 There are reconditioned nickel-cadmium batteries on the market, which cost about one-third the amount of a new one. New nickel iron batteries have recently become available and they out perform any other battery. They generally have a 90 percent capacity for no less than five years, which is better than any other battery at the current time.15

2.5 Panel System Setup Since we have discovered that the MOSFET is not the limiting factor in setting up solar

panels in parallel with one MPPT circuit we can look further into the option of each solar panel

having their own circuitry. Since solar panels are relatively expensive (upwards of $300), it might be a good investment to obtain an efficient circuitry for each one. This will provide the best result from each individual panel. We know that multiple panels have to be connected in parallel because we want to have a set voltage level with a large current level, but there is the option of them all being controlled and monitored by one circuit, or individual circuits.

13 National Solar Supply Home Page. 2003

14 Ibid.

15 Ibid.

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2.5.1 Multiple Panels per MPPT All of the panels associated with this solar panel system configuration are all connected

and controlled by one MPPT circuit. Figure 2-5 indicates how this configuration would appear.

Figure 2-5: One Boost Circuit for All Panels

If the panels were in parallel with one MPPT circuit for the whole system it is safe to say that

they would end up outputting 33A maximum.

AApanels 333.310 = (3.3 A for 1 panel) (2.1) The circuitry can easily be designed to handle that much power. The most important component to find would be the MOSFET switch. A MOSFET made by International Rectifier with a 35A current rating and a voltage rating of 30V is less than $2. Two dollars is a small amount of money for such a large current handling MOSFET switch, but the power loss associated with the MOSFET that can handle more will be larger due to the higher internal resistance of the MOSFET. For example International Rectifiers 35A MOSFET has an Rds value that is 0.031, whereas a smaller MOSFET would have a typical Rds value of approximately 0.004. When this MOSFET is used for 10 panels it will have a maximum of 33.3 A. The power dissipated is then realized by the following:

RIP = 2 (2.2)

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WP 35031.03.33 2 == This means that out of a maximum power of 500 W the MOSFET will dissipate 35 W, which is equivalent to about a 7% efficiency loss.

100%

=

Total

RDS

PP

LossEfficiency (2.3)

%7100*50035% =

=LossEfficiency

The cost of implementing this configuration is relatively inexpensive due to the need of only one MPPT circuit in the solar panel system. However, due to the larger components inside of the MPPT device, the power loss associated with this configuration is quite large greatly reducing the overall efficiency of the entire solar power system.

2.5.2 Individual MPPT per Panel Instead of utilizing one MPPT for an entire solar power system, this configuration installs

one MPPT for every panel in the system. Figure 2-6 indicates how this configuration would appear.

Figure 2-6: Individual Boost Controller Circuits for Each Panel

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This setup will maximize output power per individual panel, rather than the entire system. Utilizing this configuration allows each individual solar panel to operate at its own maximum power point increasing the overall efficiency of the solar power system. With each

circuit monitoring their own specific panel the circuit can adjust to operate at the peak power point for that specific panel drawing the maximum amount of power from each individual panel.

In a typical solar power system, all the panels installed in the system will be one and the same model. However, this does not mean that each panel functions exactly the same. The efficiency and other ratings indicated in the data sheets for these panels are typical values and the actual values for each panel will vary from panel to panel. As a result, the maximum power

point could be located at a different point for each panel. If there were only one MPPT circuit the best output would only be an average over the entire panel layout because it would not be able to adjust for the differences in each panel. As a result, the multiple panels per MPPT circuit will have a larger power loss.

When using separate circuits for each panel, smaller MOSFETS can be used because the circuitry in the MPPT will only have to handle 3.3 amps generated from each solar panel rather than 33 amps with the single MPPT configuration. The smaller MOSFETS can switch at the

same speed as one big MOSFET can, but they dissipate less power, thus resulting in better efficiency. International Rectifier makes a small MOSFET with an rDS value of 0.004, which is

much smaller than 0.031 with the larger MOSFET. Using this MOSFET with a much smaller RDS value will result in much less efficiency loss in the MOSFET.

RIP = 2

WP 04356.0004.03.3 2 == This 0.04356 W is out of a maximum value of 50W. So when using a small MOSFET the power loss is equivalent to 0.09 % efficiency loss.

23

2.5.3 The Best Approach A single MPPT configuration would reduce the cost of the solar panel system if used in place of a single MPPT on each solar panel. However, the efficiency loss associated with this configuration is quite large in comparison to installing an MPPT to each solar panel. Solar power systems are not widely used due to their already inefficient designs. By implementing a single MPPT circuit for the entire system, then the already low efficiency would be reduced by even more. The efficiency of solar power systems is more of a concern than the overall cost

because the cost for one of these MPPT circuits is low compared to the cost of a single solar panel. As a result, it is recommended that each solar panel be implemented with its own MPPT device.

24

3.0 Methodology

3.1 System Block Diagram The basic block diagram of the system is shown in Figure 3-1.

Figure 3-1: System Block Diagram

The solar panel will feed the boost converter directly which stores the electrical energy temporarily in an inductor and then charges the battery. The battery then feeds the load during sunlight hours as well as nighttime. The boost converter is to be operated by a digital controller. The digital controller will be based upon a microprocessor that monitors the voltage and current levels coming from the solar cell and controls the boost converter accordingly. Finally, the charge sensor will keep track of the charge of the battery in order to not overcharge the battery, which may damage some types of batteries. While not shown, all active components such as the digital controller will be getting its power from the solar cell.

25

3.2 Battery Requirements We created a design requirement of 24VDC because many appliances directed towards

solar energy applications are designed for 24VDC. Other popular voltages are 12V and 48V. In the case 120VAC is required; inverters are common for 24VDC and can easily be purchased, however with a high price tag. We believe the biggest turn off for most buyers is the size of the solar panels, along with the cost. In choosing a power rating, instead of finding a particular application to suit, we figured that any system over 10 panels, would require to much space and require to much money compared to other options. Therefore, ten of our 50W solar cells would generate 500W. This would yield a maximum of 20.8A, very comparable to other controllers ratings.

This 24VDC may be used to run lights for the nighttime. We estimated lights are typically used up to 8 hours per night. During cloudy days, we dont get as much energy from the solar cells as on a clear day, therefore we need to store enough energy to provide the cloudier days with energy. The table below shows how big our batteries must be in amp hours, to provide a 100W light bulb with power. Table 3.1 is calculated using the following formula

Amp HoursVoltage

DaysHoursPower **= (3.1)

Where:

Power (Watts) The expected load to be run Hours The amount of hours to be run per day Days The amount of days to last with no re-charging Voltage The systems voltage 24VDC

(all units in Amp Hours)

100W Load 200W Load 300W Load

1 Day 34 67 100 2 Days 67 134 200 3 Days 100 200 300 4 Days 134 267 400 5 Days 167 334 500 6 Days 200 400 600 7 Days 234 466 700

Table 3-1: Battery Ah Capacity vs. Load without Charging

26

The smallest battery we should obtain to give enough energy for 7 days using a 100W load such as a light bulb is a 234 Amp-hour battery. MK has different capacity batteries all the way up to 265Ah at the C/100 rate. This is equal to 2.65A discharging for 100 hours. The batteries we have already obtained for this project will be suitable for our design and testing needs, but the customer should evaluate their usage and location to come up with suitable, cost effective storage requirements.

Controller Specifications Voltage 24VDC Current 20.8A

Power 10 x 50W Panels Battery 234Ah

Scaled Down Controller Specifications Voltage 24VDC Current 2.1A

Power 50W Battery 234Ah

27

3.3 Boost Converter Description The boost converter is one of the most important components to the Maximum Peak Power Tracker. To achieve maximum power from the solar panels, we must operate the panels

at their optimum power point. By opening or closing a switch, the output of the solar panel will

either be shorted or open circuited. The switch discussed will actually be a MOSFET. The digital controller will control this MOSFET. To understand the boost converter, the MOSFET is modeled with a simple ideal switch. The switch, U1, will open and close to control the voltage level over the inductor, which will essentially set the solar panels to their optimum power level. The boost converter is shown in Figure 3-2 with the solar panel shown as a voltage source Vs. More accurate representations of the solar panel will be used shortly.

Figure 3-2: Boost Converter Schematic

Again the VI characteristic of the solar panel is shown in Figure 3-3. The power of the

corresponding V-I is also on the same graph with a different Y axis.

Figure 3-3: Solar Panel V-I Characteristic and Power

28

As the switch is closed the voltage drops as the current increases towards its maximum short circuit current. If the switch is closed for a long enough period of time the voltage will eventually drop to zero and thus the power at this point is zero. If the switch is open, the voltage will rise to its open circuit voltage and no current flows out of the solar panel. Again the power will be zero watts. Due to the inductors presence in the boost converter, current and voltage transients will not happen instantly but instead take some time. Therefore the power cannot instantly move from optimum to zero, but instead takes some time constant. By opening and closing this switch at fast speeds, it is possible to pick a place such as the peak power point and operate close to this point.

In order to understand the boost converter we wanted a basic representation of the solar panel. Even a simple model of the solar panel is quite hard to analyze at first. Therefore we choose to represent the solar panel as a basic thevenin equivalent for reasons that will become more apparent in section 3.4. The V-I characteristic of an ideal thevenin equivalent is a linear line with a negative slope. This is unlike the non-linear curve of the solar panel, which will make

a first pass understanding of the system easier. The thevenin equivalent is represented with a voltage source VTH, and a resistance, RTH. This is shown in Figure 3-4.

Figure 3-4: Boost Converter Schematic with Thevenin Equivalent Source

The thevenin voltage source was chosen to be double of the voltage at which the solar panel operates at its maximum power point. This is about 15V and doubling this gives a 30V voltage source. The thevenin current was also chosen to be double of what the solar panel can produce at short circuit. The short circuit current of the solar panel is about 3.3 amps, therefore the thevenin current should be double this, 6 amps. To achieve 6 amps with a 30 volt source a thevenin

29

resistance RTH of 5 was chosen. The V-I characteristic of the thevenin equivalent is compared to the V-I curve of the solar panel in Figure 3-5.

Figure 3-5: Thevenin and Solar Panel V-I Characteristics

The maximum power point is at the top right of the solar panel V-I curve. The thevenin

equivalent is shown in Figure 3-6 with its respective power curve superimposed on a different scale. Notice the power curve of the thevenin equivalent is a parabola which each voltage extreme going down to a power level of zero watts.

Figure 3-6: Thevenin V-I and Power Curve

30

Analysis of this circuit will contain four stages. The first stage sets up our initial

conditions, when nothing is happening in the circuit yet. The second stage occurs when we close the switch. When the switch is closed, the current in the inductor will build up over some period of time and no current will flow into the second loop of the circuit. The third stage occurs when switch is opened. The inductor will discharge at this stage and send current to the 24V battery source. The fourth stage is shown what happens when we do not let the inductor to saturate. That is we turn the switch on and off quick enough before the current and voltage go to there extremes and the power falls to zero watts.

3.3.1 Stage One: Initial Conditions Stage 1 is an analysis of the circuit when no voltage or current is being applied to the circuit. The inductor will have a voltage level of zero with a current of zero flowing through it. The voltage over the inductor is related to the current through the equation:

dtdILV =

(3.2) Since there is no change in current di/dt, the voltage across the inductor is zero.

3.3.2 Stage Two: Switch Closing When the switch is closed, the loop around the battery can be ignored. Without current being forced through the diode, the anode on the diode will always have a lower voltage than the 24V cathode making the diode reversed bias making the right loop an open circuit. No current will flow through this loop as a result, so analysis of the right loop is unnecessary. The circuit

can now be viewed as in Figure 3-7.

Figure 3-7: Current Path While Switch is Closed

31

The 30V voltage source and 5 resistor provide the circuit with 6 amps of current. This current will flow through the inductor and charge the inductor exponentially. The current flowing through the inductor cannot jump instantaneously to 6A, much like how the voltage cannot instantaneously jump with a capacitor. As a result, when the switch closes, the current will start at 0 amps and exponentially increase until it saturates at the 6 amp current provided by the source and resistor. This saturation is due to the fact that the voltage source can not provide any more than 6 amps. The speed at which the inductor goes from zero amps to saturation depends on the value of the inductance and the resistance which is discussed in more detail in section 3.4. For analysis purposes, a 10mH inductor was selected. The following Figure 3-8 shows the transient curve as voltage and current flow through the inductor.

Time

0s 5ms 10msI(L1)

0A

5A

10AV(R1:2)

0V

20V

40V

SEL>>

Figure 3-8: Voltage and Current through Inductor as Switch Closes

At time t=0s, the voltage is at its initial 24 volts and the current is 0 amps as expected. At t=0s, the switch closes allowing current to begin flowing through the inductor. The current will exponentially increase until it saturates at 6 amps, which occurs at approximately 6ms. When this happens, the voltage will decrease at the same rate the current is increasing due to the voltage/current relationship of an inductor as shown above in equation (3.2). At saturation the

Voltage Current

Current Current

32

voltage drops to approximately zero volts and the inductor essentially becomes a short circuit and will remain a short circuit until the inductor is allowed to discharge, which will be described in stage three. Therefore as the voltage goes to zero volts, no matter what the current is at short

circuit operation the power delivered is zero. When the switch is closed, the main function is to build the current in the inductor and

drop the voltage. Essentially, the voltage and current flowing through the inductor will control the operating point of the solar panel. As a result, the operating point will depend on the duty cycle of our switch.

3.3.3 Stage Three: Switch Opening Opening the switch of the circuit allows us to send the charge previously stored in the inductor as a magnetic field to current flowing to the battery. The switch in the middle is essentially open and therefore non-existent for this simple model. Figure 3-9 shows the circuit with an open switch.

Figure 3-9: Current Path while Switch is Opened

This stage is assumed to always happen after stage two to allow the inductor to charge with energy. This stage allows the inductors magnetic field to collapse and dump its current into the battery. The diode is there to not allow the battery to dump its current into the inductor. When current begins to be forced through the diode, the voltage at the anode of the diode will shoot up to ensure that the diode is forward biased and the battery will be charging. Therefore this voltage is equal to the battery voltage plus any additional voltage it takes to forward bias the diode. The voltage and current flowing through the inductor is shown in Figure 3-10.

33

Time

0s 5ms 10msI(L1)

0A

1.0A

2.0A

SEL>>

V(R1:2)20V

25V

30V

Figure 3-10: Voltage and Current through Inductor as Switch Opens

The current in the inductor is no longer building up; rather the current is discharging through the diode and into the battery to charge. As the current decreases, the voltage will gradually increase at the same rate due again due to the relationship of an inductor given by equation (3.1). Notice that the voltage does not decrease to zero volts but instead about 25V. Once this inductors field is fully collapsed the voltage at the diode will be essentially the 30V source in series with its 5 resistor which provides the diode and 24V battery. The voltage at this point is about equal to the 24V battery plus the forward voltage of the diode, which is about 0.7V.

34

3.3.4 Stage Four: On and off switching This stage will use the entire thevenin equivalent circuit shown again in Figure 3-11.

Figure 3-11: Boost Converter Schematic with Thevenin Equivalent Source

The idea of this stage is to simulate the boost converter during full operation. In order to control the power point the solar panel operates at the switch will turn on and off fast enough so that the saturation values of the inductor are never reached. At the beginning of the exponential curve, it is can be shown to be pretty linear. This concept will be made to great use in our analysis of this circuit. By adjusting the time the signal is high compared to low or the duty cycle of the pulse, it is possible to control where on the VI power curve the solar panel is operated. This duty cycle is easily adjusted with a PWM signal from our digital controller. First we will show how the above exponential switching can become linear at fast speeds. This was done in our simulation in Pspice by a time controlled switch.

For this simulation we guessed a switching speed of 1 kHz. This speed was guaranteed not to saturate. The voltage and current waveform is shown in Figure 3-12.

35

Time

30ms 31ms 32ms 33ms 34msI(L1)

3.0A

3.5A

4.0AV(R2:2)

11.25V

12.50V

13.75V

SEL>>

Figure 3-12: Voltage and Current through Inductor (Switching)

Note the linearization of the voltage and current at the solar panel. We can see at 30mS the voltage seems to be decaying and the current charging. Therefore the switch was closed at this point and the inductors field was charging. At 30.5mS the switch is opened and the current starts to flow out of the inductor and charge into the battery. Notice the current value does not go below 3.3A, unlike above when we let the inductors field collapse fully. The line appears to be perfectly linear, but in reality it is really just the very beginning of its exponential decay. As long as we switch every 0.5mS, the current and voltage will never saturate as shown before in our single switching case.

As mentioned before, the duty cycle directly controls the voltage, current and power at which the solar panel operates. By varying the duty cycle of the voltage-controlled switch in Pspice it is possible to graph the power. We then see exactly the same power parabola curve as expected in Figure 3-13.

36

Power Characteristic of VI Thevenin Equivalent

0

10

20

30

40

50

0 50 100

Duty Cycle (%)

Pow

er

(Wa

tts)

10mH

Figure 3-13: Power Characteristic of VI Thevenin Equivalent

3.4 Boost Converter System Equations To achieve the maximum power from a source the load resistance needs to be equivalent

to the internal resistance of the source. Figure 3-14 contains a simple source consisting of Vs1

and the internal resistance of the source R1 with a load resistance of RLOAD. RLOAD is the resistance of the load being applied to the voltage source.

+

-

Vs1

RloadR1

Figure 3-14: Basic Voltage-Resistor Source with Load

In order for the power source Vs1 to transfer maximum power into RLOAD, RLOAD must be equal to R1. Instead of a constant voltage source as a source, we are using a solar cell. The solar cell is not an ideal voltage or current source and therefore is modeled with a more complex circuit. The load is then our boost converter as shown in Figure 3-15.

37

Figure 3-15: Basic Voltage-Resistor Source related to MPPT

3.4.1 Thevenin Equivalent Designing the controller system for the Maximum Peak Power Tracker requires a full

understanding of what is going on in the boost converter circuit. Deriving equations for the average current, IAVG, the average voltage, VAVG, the equivalent resistance, REQ, the ripple

current, I, and the power, P, will allow us to predict the outcome of our circuit. These equations will also help determine the appropriate duty cycle to operate the switch at in order to achieve the maximum power point. The circuit in Figure 3-16 was used to analyze the circuit.

Figure 3-16: Nonlinear Model of Solar Panel and Boost Converter Circuit

Notice the few minor differences in this circuit than previous ones used. The solar panel is being modeled as a linear thevenin voltage and resistance. The switching of our circuit will operate near the maximum power point of the nonlinear V-I characteristic of the solar panel.

Since this is the case, we can design a linear solar panel model so the maximum power point for

38

the linear model and the nonlinear model are almost one in the same. This can be seen in Figure 3-17.

Figure 3-17: Solar Panel Linear and Nonlinear V-I Characteristics

As you can see from Figure 3-17, the maximum power points of the two curves are

almost perfectly aligned. To determine the correct thevenin voltage and resistance, you simply double the voltage at the maximum power point for the nonlinear model, because the maximum power point for a linear model will occur at the midpoint and the thevenin resistance will simply be the voltage over the current at the maximum power point. As a result, the thevenin voltage will be 30 volts and the thevenin resistance will be 5. The use of the thevenin equivalent source will greatly simplify the derivation of the boost converter system equations and should yield the same results as if an expression for the solar panel were used simply because the maximum power points occurring in the same spot.

39

3.4.2 Average Current Since the diode acts much like a switch, a switch can be used to replace the diode in

Figure 3-16. This switch will close when the main switch, U1, is opened and vice versa. At any given point, the voltage from the solar panel will be lower than the voltage of the battery. As a result, the current will want to reverse direction and the exponential decay of the current will act as if it was saturating at a negative value. This can be seen in Figure 3-18.

Figure 3-18: Current Flowing from the Solar Panel

The diode in the boost converter will prevent the current from going negative at any point in time, but the exponential curves for the current will act as if it were going to. In Figure 3-18, if the current was allowed to reach its saturation points, then it would reach a maximum current, IMAX, and exponentially decrease until it reached a maximum negative current, INEG. The maximum current will only be achieved when current is building in the circuit. This means the switch, U1, will be closed, and the switch replacing the diode will be open. The simplified circuit is shown in Figure 3-19.

40

Figure 3-19: Simplified Boost Converter Circuit (U1 is closed, U2 is open)

With a steady state current of IMAX flowing through the inductor, the inductor will act like a short circuit and will be insignificant in determining the maximum current. The only remaining components are the thevenin voltage and resistance. As a result, the maximum current is simply:

th

th

RV

I =max (3.3)

With a thevenin voltage of 30 volts and a thevenin resistance of 5, the maximum positive current flowing will be 6 amps.

To determine the maximum negative current flowing through the circuit, the switch, U1, will be open while the switch replacing the diode will close. The simplified circuit is shown in Figure 3-20.

Figure 3-20: Simplified Boost Converter Circuit (U1 is open, U2 is closed)

Once again, the inductor acts as if it were a short circuit due to the current being in steady state. The only components now left in the circuit are the thevenin voltage, thevenin resistance,

and battery voltage. The negative current is determined to be:

41

TH

BATTTHNEG R

VVI

= (3.4)

A thevenin voltage of 30 volts, a thevenin resistance of 5, and a battery voltage of 24 volts will yield a maximum negative current of 1.2 amps. The actual ripple current will remain somewhere between zero amps to the maximum current, IMAX. The duty cycle and frequency will control the location and ripple size of the current. When the duty cycle is high, the current will try to reach the maximum current, IMAX, and the current will try to reach the maximum negative current, INEG, when the duty cycle is low. The general expression for an exponentially increasing or decreasing current is:

( ) ( ) teIIItI iFF = (3.5) Where I(t) is the value the current reaches after a certain length in time, IF is the current that will be reached, Ii is the starting point, t is the time period, and is the time constant L/R. When the duty cycle is turned to high, the initial current is I2 and will increase until the duty cycle goes low, which will occur at I1. The curve will act as if it were increasing to its saturation value of

IMAX, so IF will be equivalent to IMAX when the duty cycle is high. The time, t, is equivalent to the portion of the time period in which the duty cycle is high, DT0, where D is the duty cycle and T0 is the time period. The resulting expression is:

( ) ( ) tMAXMAX eIIItI = 2 (3.6) Using the Taylor series and knowing that T0 will be much less than , equation (3.6) simplifies to:

+

=

0

20

1 1DT

IDT

II MAX (3.7)

The only two unknowns in equation (3.7) are I1 and I2 while the remaining variables are set values. To solve for I1 and I2, a second expression with these two variables will be needed. When the duty cycle goes low, the initial current will be at I1 and will drop down to I2 before the duty cycle turns high. The exponential drop will look like it is decreasing to the maximum negative value, INEG, and the amount of time, t, is the portion of the time in which the duty cycle is low, (1-D)T0. The resulting expression is:

( ) ( ) tNEGNEG eIIITI = 1 (3.8) Simplifying this expression and using the Taylor series, equation (3.8) simplifies to:

42

( ) ( )

+

=

0

10

2111 TDITDII NEG (3.9)

We now have two equations and two unknowns. This allows us to solve for the maximum and minimum ripple values, I1 and I2 as a function of duty cycle, time period, saturation currents, and

. In order to determine the average current and ripple current expected in simulation, an

expression for I1 and I2 independent of one another needs to be derived. Inserting equation (3.9) into equation (3.7) will give us an expression for I1 independent of I2. The resulting expression is:

( ) ( )

+

+=

01

001 1

111 DTTDITDIDTII ONEGMAX (3.10)

As you can see from equation (3.10), I1 is the only remaining unknown variable in the equation. Solving for I1 and simplifying the expression yields the resulting equation:

( )( )

+

=

00

000

1 1111

11

TDDT

DTTDI

DTI

INEGMAX

(3.11)

In order to solve for I2, inserting equation (3.7) into equation (3.9) gives us the following equation:

( ) ( )

++

=

00

200

21111 TDDTIDTITDII MAXNEG (3.12)

As shown in equation (3.12), I2 is now completely independent of I1. Solving for I2 and simplifying the equation yields the following expression for I2:

( ) ( )( )

+

=

00

000

2 1111

111

TDDT

TDDTI

TDI

IMAXNEG

(3.13)

Now that an expression for I1 and I2 have been derived completely independent of one another. IAVG is one of the more important values that need to be derived. Not only does it confirm that a simulation of our model is correct, but will later yield an expression for the power. The average between any two points is simply half the sum. The average current will then depend on half the sum of the maximum and minimum ripple values:

43

221 III AVG

+= (3.14)

Inserting equation (3.11) and equation (3.13) into equation (3.14) will yield the following equation:

( ) ( )

( )

+

=

00

0000

112

2112

TDD

T

DTTDI

TDDTI

INEGMAX

AVG (3.15)

Note how the average current is dependant on known values. In order to simplify equation

(3.15), knowing that T0 will be much less than , any term in equation (3.15) that will not significantly alter the result will be approximately equal to zero. The simplified equation for the average current is:

( )DIDII NEGMAXAVG += 1 (3.16) When the duty cycle is at 100 percent, the average current will become the maximum positive saturation value and when the duty cycle is at 0 percent, the average current will become the maximum negative saturation value. This makes sense because if the main switch, U1, is always closed, then the current will build to its maximum current and if U1 is always open then the current will drop to the minimal amount of current.

3.4.3 Current Ripple Another useful expression to derive is I . I will allow us to determine an appropriate

inductor value for our circuit. The ripple current will be important because it affects the precision of the average current and a smaller ripple will require a higher inductor value. The change in current, or I , is simply the final value minus the initial value:

21 III = (3.17) Plugging in the equations (3.11) and (3.13) into equation (3.17) will give us an expression for

I .

( ) ( )( )

0

00

11

11

TDD

TDDI

TDDI

INEGMAX

= (3.18)

44

Once again the equation for I is very messy and can be cleaned up some. Using the same

assumption in determining the final equation for the average current, T0/ is set to be

approximately equal to zero since T0 is much smaller than . However, there is an addition

T0/ in the numerator that cannot be set to zero. The reason is that if set to zero, then I will be

equal to zero. As a result, the additional T0/ cannot be set to zero in this simplification. The

final expression for I is:

( ) ( )

=

01 TDDIII NEGMAX (3.19)

As you can see from equation (3.19), by making the approximation that T0/ will be approximately equal to zero, simplifies the equation to an expression that will simply describe the amount the current will ripple through the inductor of the boost converter.

3.4.4 Voltage Equations The next step is determining an expression that will allow an accurate prediction of the

average voltage. The voltage will move in the opposite direction the current moves at the same rate. Figure 3-21 demonstrates how the average voltage will act along with a high and low duty cycle.

Figure 3-21: Voltage at Solar Cell

When the main switch, U1, is left closed for a while, the simplified circuit will once again look like Figure 3-19. The current will gradually build through the inductor, decreasing

45

the voltage drop exponentially at the same rate the current is increasing. When the current increases, the voltage decreases due to the inductor. Performing a KVL around the loop will give us the most negative value for average voltage, VNEG.

0== THMAXTHNEG RIVV (3.20) The negative most value the average voltage can reach is zero volts. This makes sense because in the circuit from Figure 3-18, the current will build to its maximum value, which depends only on the thevenin voltage and resistance. So for this to be true, the voltage must be zero. The maximum average voltage that can be achieved will occur when the main switch, U1, is open. The simplified circuit will look the same as Figure 3-20. For the maximum voltage to occur, the average current must be at a value of zero amps. When this happens, the average voltage will simply be equivalent to the battery voltage.

BATTMAX VV = (3.21) After going through the average current, the voltage is really easy to figure out. Instead of going through all of the math as was performed with the average current, it is much easier to realize that the voltage will decrease at the same rate as the current will decrease but to different saturation points. As a result, the equation for the average voltage will be much similar to the equation for average current. The only major difference is that when the current reaches its maximum value, the voltage reaches its minimum value and when the current reaches its minimum value, the voltage reaches its maximum value. As a result, INEG becomes VMAX and IMAX become VNEG in the derived average current equation. This yields the following simplified expression for average voltage.

( )DVV MAXAVG = 1 (3.22) Since VNEG is equivalent to zero volts, the average voltage equation is extremely simplified compared to the average current equation. The rippling voltage, V, is derived using the same method as finding VAVG. This results in the following expression for V.

( )Lf

DVDRV MAXTH = 1 (3.23)

46

3.4.5 Equivalent Resistance and Power Equations Now that an expression for the average voltage and current are obtained, it is extremely

simple to derive expressions for the equivalent resistance of the circuit and the power. The equivalent resistance is defined to be:

AVG

AVGEQ I

VR = (3.24)

Plugging in the equations for average voltage, equation (3.22), and average current, equation (3.16), into equation (3.24) will give us an expression for the equivalent resistance.

( )( )DIDIDV

RNEGMAX

MAXEQ

+

=

11

(3.25)

The predicted average power will also be a function of average current and voltage.

AVGAVGAVG VIP = (3.26) Plugging equations (3.16) and (3.22) into equation (3.26) will yield an expression for the predicted average power.

( ) ( )( )MAXMAXMAXNEGAVG VIDDDVIP += 11 2 (3.27)

3.4.6 Equation Accuracy Deriving expressions for VAVG, IAVG, REQ, I , and Power give us a lot of understanding

in how to design the controller circuit of the Maximum Peak Power Tracker. However, in order to have complete faith in these equations, a comparative analysis of the computed average values should be compared with the simulated values from the Boost Converter Description section using different duty cycles. The circuit used in simulation is the same as Figure 3-16. The two switches are controlled by a Pulse Width Modulation (PWM) voltage source. When the PWM voltage source sends a high signal, the switch, U1, will close while the switch, U2 will open and vice versa. Using this voltage source will allow us to change the duty cycle and period of the switching quite easily. A basic inductor value of 10mH was chosen to perform the analysis. The thevenin voltage and resistance is 30 volts and 5, and the battery voltage is 24 volts. Since a linear model is being used, the thevenin resistance is dominant to all of the other forms of

resistances in the circuit and is the only resistance used in evaluating the term . The following

47

four figures compares the calculated or predicted values for VAVG, IAVG, I , and PAVG over a range of different duty cycles.

Figure 3-22: Average Current vs. DC Figure 3-23: Change in Current vs. DC

Figure 3-24: Average Power vs. DC Figure 3-25: Average Voltage vs. DC

As you can see from the figures above, the computed values are extremely close to the simulated values to the point where hardly any deviation is noticeable. As a result, the equations derived in this section are extremely accurate and can be considered a very reliable method to predict the results.

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3.5 Current Sensing Methods The digital controller requires voltage and current feedback from the solar cell in order to optimally control the peak power level of the solar panel. The voltage is easy to interface to the

microprocessor by direct connection. A resistor divider may be needed to limit the voltage seen by the microprocessor. Current is also common to measure, however most current sensing techniques come with a loss. Two general types of current sensing include resistive and magnetic sensing. Resistive current sensing is done by inserting some type of resistance into the circuit and measuring the voltage across this resistor. This resistor of course dissipates power in the form of an efficiency loss to our circuit. Magnetic sensing measures the magnetic field around a wire or loop that the current passes through. This technique is usually common in AC current or very

high current applications.

The easiest and most common way to measure current is a simple sense resistor. A sense resistor is basically a resistor placed in series with the load. Ohms law says the voltage drop across the resistor is proportional to the current. When low current flows through the sense

resistor it can provide very accurate measurements if the resistance value has a small tolerance. Sense resistors are commonly made as low as 0.001 at +/- 1% tolerance. Even though sense resistors can have high performance thermal packaging for larger current values, they still result

in some insertion loss. Their measurement is also not isolated from transient voltage spikes. The power dissipated in the resistor is given by the equation

RIP 2= (3.28) The lower the value of R, the lower the power loss will be. Therefore it is necessary to pick the lowest value of R as possible. Where I is the max current from the solar panel = 3.28 A.

mWP 58005.0*28.3 2 == (3.29) A 0.005 resistor costs $0.45 and will use 53.8mW.

Another type of sensor is a coil in which a wire passes through the middle of the windings. The magnetic field generated from the current in the wire induces a magnetic field in the coil as the current changes. This is therefore only useful to measure AC RMS current. If there is any bias in the current, this type of sensor will not pick it up. Our current will be oscillating; however there will also be a large bias making this kind of sensor inappropriate for our application.

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Another possibility is to measure the current over the rDS of the MOSFET. Slight variances of rDS values between different production batches of MOSFETs can be troublesome perhaps not now, but if the board is manufactured on a wide scale. The rDS value can also vary slightly with the gate voltage being applied. Different temperature coefficients can also vary this resistance value and is not always given in datasheets. These variances can vary up to 100%.16

A Hall Effect sensor next to a trace or a wire and determine the magnetic field generated from that wire or trace even with a DC current. Although the Hall Effect sensor can measure current without losing any power from the signal, the Hall Effect sensor may use power from a

different source to amplify its signal. In our application, all the power is coming from the solar panel; therefore any power the sensor used would be a loss in our efficiency. Another problem with Hall Effect sensors is that they are usually designed for large current measurements. Most sensors are designed for current of 50A and more, therefore questioning its accuracy at very low currents. As an example, Allegro Micros smallest current sensing Hall Effect sensors used 7-10mA just to turn on, which is almost 50mW, the same as the sensor resistor. This sensors are also much more expensive than sensing resistors.

The last option for measuring the current flowing from the solar panel is inductor current sensing. Inductor current sensing is an inaccurate way of measuring the current. However, the output of the inductor current sensor is proportional to the current and the power loss is insignificant, making the inductor sensor an ideal method for this application.

3.5.1 Inductor Current Sensing An interesting article was found that used the inductors internal resistance already in the

circuit to find the current passing through it. The idea originates from a buck converter, and due to the extremely small power loss expected from this approach, adapting the idea to the boost converter seems feasible.

The current through the inductor will be a linear triangle wave with some small RMS value and at some DC average. As the current is building up in the inductor the voltage will be falling. Therefore the voltage and current will be 180 out of phase in steady state. By analyzing the voltage across the inductor, it is possible to get the average current through the inductor. This

16 Current Sensing Techniques for DC-DC Converters

50

is due to ohms law, which states that voltage is proportional to the current by some resistance. The resistance in this case is the internal resistance of the inductor. By applying a low pass filter across the inductor it is possible to filter out all or close to all of the high frequencies leaving only the DC part of the signal left and perhaps some very low frequencies. The model of the inductor has an internal resistance RL. The low pass filter is then applied over the inductor as shown in Figure 3-26.

LRL

Cf

Rf

Figure 3-26: Inductor Current Sensing Schematic

Analysis of this circuit requires the use of two simple equations.

( )LL RsLIV += (3.30)

FF

LOUT CsR

VV+

=

1

(3.31)

Combine the two together to get the relationship between the output sense voltage, VL, to the current flowing through the inductor, I.

(3.32)

The inductor discussed here will be assumed to have a 1mH inductance with an internal

resistance of 0.01. There are three conditions that can take place by selecting the appropriate

filter components.

1.) RFCF = L/RL 2.) RFCF > L/RL 3.) RFCF < L/RL

To set condition one true: set FFL

CRRL

= then the part in brackets [] goes to 1.

(3.33)

+

+=

FF

LL

OUT

CRSRLS

Ri

V1

)/(1

+

+=

FF

LL

OUT

CRSRLS

Ri

V1

)/(1

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LOUT Ri

V= (3.34)

Therefore the voltage gain per ampere across all frequencies is set by the internal resistance RL. Here the voltage gain is steady independent of frequency. The inductor internal resistance is

equal to RL, in this case 0.01. The gain VOUT/i is equal to 0.01 or 40dB.

RFCF = L/RL With the inductor discussed above L/RL= 0.001/0.01 = 0.1. Therefore by picking a RFCF

value of 0.1 will allow analysis of the first condition. By picking a random RF value of 1k, CF

is calculated to be 100F.

L 1mH RL 0.01

L/RL 0.1

RF 1k CF 100F

RFCF 0.1

Table 3-2: Component Values when RFCF = L/RL

Figure 3-27: Bode Plot when RFCF = L / RL

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Notice the gain is the same throughout all frequencies. The phase is also zero degrees across all frequencies. Therefore there will be no phase shift between the current and sensed voltage.

The capacitor starts to act like a closed circuit at high frequencies and an open circuit at DC. Therefore at high frequencies the circuit will begin to dissipate power through the resistor, eventually dissipating V/R. The wave on top on the waveform below shows the power dissipated through the filter and the current going through the filter.

RFCF > L/RL The second condition is met when RFCF is selected to be greater than L/RL. RFCF is

shown below to be 10 times larger than L/RL by increasing the resistance from 1k to 10k and

keeping CF at 100F. L 1mH RL 0.01

L/RL 0.1

RF 10k CF 100F

RFCF 1

Table 3-3: Component Value when RFCF > L/RL

Figure 3-28: Bode Plot when RFCF > L/RL

By increasing RFCF the transfer function takes the shape of a low pass filter. Increasing RFCF, lowers the frequency at which a pole is located thus decreasing the breakpoint frequency.

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This frequency response has the least attenuation at DC frequencies and greater attenuation occurs as the frequency is increased. The RC filter in this application wants to block all high frequencies a