Topics in Physics:

1. Ohm’s Law and VI Characteristics

The Fundamental Law of George Simon Ohm Integral form: V = IR Differential form: dV = R dI V is voltage measured in volts (V) I is current measured in amperes (A) R is resistance measured in ohms (  ) This law is a linear relationship between two physical parameters, voltage and current.

Ohm’s Law for a Linear Resistor V = IR +- I V R V I VV II R =  V/  I

Ohm’s Law for a Non-Linear Resistor: a Diode dV = R dI + - I V V I I = I(V) R(V) = dV/dI ReverseForward

Questions: 1. Is the value of R a function of either voltage or current in the linear resistor? 2. Is the value of R a function of either voltage or current in the diode? 3. In which direction is the diode resistance the greatest? The least? Please notice that the roles of x-axis and y-axis in the VI characteristics have been reversed from the normal algebra convention. Electrical engineers sometimes think in different ways than algebra students!

Answer to Question #1: R is independent of voltage and current in the linear resistor.  R = V/I. Answer to Question #2: R is a function of position along the diode characteristic,and it is different at every point.  R = dV/dI. Answer to Question #3: R is smallest in the diode forward direction and largest in the reverse.

How are Diodes Made? Chemical impurities (dopants) are added to an otherwise pure,refined material (silicon) to render it either p-type or n-type. The material is then melted and ‘drawn’ into a single crystal from which slices are cut. A second dopant layer is then diffused into the crystal slice to create a semiconductor junction device.

The junction device is then encapsulated in the opaque material epoxy. BUT if the junction is left exposed to light, something interesting happens: The diode becomes an energy transducer - a solar cell, transforming light into electricity! The VI characteristic moves from power dissipation only into a power generating region.

V I Power is either generated or dissipated, depending on the quadrant you are in. 1 ST Quadrant: Dissipation 3 D Quadrant: Dissipation 2 ND Quadrant: Generation 4 TH Quadrant: Generation

I V Dark Characteristic Light Characteristic I V Power Generating Region Power Dissipating Region

The VI characteristic of a solar cell is usually displayed like this: V I V I The coordinate system is flipped around the voltage axis.

Questions: 1. Electrical power is the product of voltage and current: P = VI. Is power a function of position along the solar cell characteristic, or is it a constant everywhere along the curve? 2. What is the power at the intercepts? 3. If power is not a constant along the curve, then where is it minimum and where is it maximum? 4. What is the minimum power?

Answer to Questions #1 - #4: Power is a function of position along the VI characteristic. At the intercepts, it is minimum - zero - increasing to a max near the knee of the curve.

2. Solar Cells Parameters and Their Significance

Every engineering and scientific system is characterized by a set of parameters - a parameter space. We will now look at a set of solar cell parameters used in the daily business of making, testing, and using solar cells.

Set #1: I SC, P MAX, V OC (0.5V, 0 mA)  V  I = 0 mW (0.43 V, 142 mA)  V  I = 61 mW I SC V OC P MAX (0V, 150 mA)  V  I = 0 mW Some typical values

The short circuit current I SC is a linear function of sunlight intensity. The open circuit voltage V OC is not. (V OC is weakly dependent on temperature.) Recall from Part I that sunlight intensity is measured in terms of a solar constant with units such as mW/cm 2.

Questions: 1. What is the voltage at I SC ? Why is this value called the “Short Circuit Current”? 2. What is the current at V OC ? Why is this value called the “Open Circuit Voltage”? 3. What shape does the curve P = IV have on the VI plane? (Think Analytical Geometry!) 4. How does this shape help you to understand that the value of P MAX is at the knee of the curve and not somewhere else?

5. The nominal distance from the sun to the earth is 150 million km. The nominal distance from the sun to Mars is 230 million km. If the solar constant at 1 A.U. is mW/cm 2, what is it at Mars? 6. A solar cell has I SC = 150mA on earth under ideal sunlight conditions. Under ideal sunlight conditions on Mars, what short circuit current would this cell produce? (Mars’ power system designers must worry about such things!)

I = I SC R = 0 Does it surprise you that the current at short circuit is not infinite? Or that a current can flow with no voltage? Where does the energy originate? Answer to Question #1:

I = 0 R =  Answer to Question #2: +_+_ V = V OC

Answer to Questions #3 and #4: The curve P=VI is a rectangular hyperbola in the VI plane. There is a family of such curves in the plane, but only ONE is tangent to the solar cell characteristic. The point of tangency is P MAX. This relationship is shown on the next page.

I SC V OC Hyperbola for P = P MAX Point of tangency Voltage at max power Current at max power Answer to Questions #3 and #4 (cont’d):

Answer to Question #5:  Mars = { (136.7)  (150/230) 2 } mW/cm 2 = 58mW/cm 2 Answer to Question #6: I SC = { 150  (58/136.7) } mA = 64 mA on Mars

Set #2: R S, R SH I SC V OC The slopes of these lines are characteristic resistances. R SH RSRS

Questions: 1. Which resistance is higher, the measurement at I SC or the measurement at V OC ? Remember: R =  V/  I ! 2. Physically, what do you think these resistances represent? 3. As a solar cell designer, what is your preferred ideal value?

Answers to Questions #1 - #3: The resistance at I SC is extremely high. In an equivalent circuit model of a solar cell, it represents a shunt resistance. The resistance at V OC is extremely low. In an equivalent circuit model of a solar cell, it represents a series resistance. Both of these resistances are internal, and represent energy dissipation mechanisms in the cell. Ideally, a designer would like zero series resistance and infinite shunt resistance.

I SC RSRS R SH R LOAD Equivalent circuit for a solar cell with load. Internal resistances R S and R SH represent power loss mechanisms inside the cell. Cell

I SC R S = 0  R SH =  R LOAD The ideal solar cell would have no internal losses at all! What would the VI characteristic of THIS cell look like?

I SC V OC R SH =  R S = 0 The Ideal Solar Cell

Notice that the area under the rectangle = P MAX for the ideal cell. For this cell, P MAX = V OC  I SC I SC V OC The Ideal Solar Cell

I SC V OC Set #3: Fill Factor In fact, P MAX /(I SC  V OC ) measures the cell’s quality as a power source. The quantity is called the “Fill Factor.” Can you see why?

Questions: 1. What is the ideal fill factor? 2. Can the ideal cell ever be built? Why or why not? 3. For a cell with these parameters: (0V, 150mA), (0.43V, 142mA), and (0.5V, 0mA) calculate the fill factor.

Answer to Question #1: The ideal fill factor is unity. Why? Answer to Question #2: An ideal cell might be approximated, but never actually built. Nature is never ideal as humans think about “ideal.” Answer to Question #3: The fill factor is: (0.43V  142mA)/(0.5V  150mA) = 0.81 = 81%

Well, there it is - we’ve taken another step. Those of you that are interested in pursuing this topic still further can study circuit design. Solar arrays are usually wired in series-parallel configurations to achieve desired VI characteristics. Zener diodes, power converters, etc. all become part of the design. After all, the raw power of the array has to be tailored to fit the user’s needs. In space, the effects of on- orbit eclipses, surface charge buildup and dissipation, and a variety of other issues all become factored into the designer’s palate.

I hope that you have enjoyed this two-part series and that some of you will further pursue education in electrical engineering or solid state physics. Best Wishes!!!

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