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Topics in Physics:

1. Ohm’s Law and VI Characteristics

The Fundamental Law of George Simon Ohm

Integral form: V = IRDifferential 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 ResistorV = IR

+ -I VR

V

I

VIR = 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

Reverse Forward

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.

1ST Quadrant:Dissipation

3D Quadrant:Dissipation

2ND Quadrant:Generation

4TH Quadrant:Generation

IV

Dark Characteristic Light Characteristic

I

V Power Generating Region

Power Dissipating

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: ISC , PMAX , VOC

(0.5V, 0 mA) V I = 0 mW

(0.43 V, 142 mA) V I = 61 mW

ISC

VOC

PMA

X(0V, 150 mA)

V I = 0 mW

Some typical values

The short circuit current ISC is a linear function of sunlight intensity. The open circuit voltage VOC is not.

(VOC 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/cm2.

Questions:1. What is the voltage at ISC ? Why is this value called the “Short Circuit Current”?2. What is the current at VOC ? 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 PMAX 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 136.7 mW/cm2, what is it at Mars?6. A solar cell has ISC = 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 = ISC

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 = VOC

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 PMAX. This relationship is shown on the next page.

ISC

VOC

Hyperbola for P = PMAX

Point of tangency

Voltage at max power

Curr

ent

at m

ax p

ower

Answer to Questions #3 and #4 (cont’d):

Answer to Question #5:

Mars = {(136.7) (150/230)2

}mW/cm2

= 58mW/cm2

Answer to Question #6:

ISC = {150 (58/136.7)}mA= 64 mA on Mars

Set #2: RS , RSH

ISC

VOC

The slopes of these lines are characteristic resistances.

RSH

RS

Questions:1. Which resistance is higher, the measurement at ISC or the measurement at VOC ?

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 ISC is extremely high. In an equivalent circuit model of a solar cell, it represents a shunt resistance.The resistance at VOC 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.

ISC

RS

RSHRLOAD

Equivalent circuit for a solar cell with load. Internal resistances RS and RSH represent power loss mechanisms

inside the cell.

Cell

Cell

ISC

RS = 0

RSH =

RLOAD

The ideal solar cell would have no internal losses at all! What would the VI characteristic of THIS cell look like?

ISC

VOC

RSH =

RS = 0

The Ideal Solar Cell

Notice that the area under the rectangle = PMAX for the ideal cell.

For this cell,PMAX = VOC ISC

ISC

VOC

The Ideal Solar Cell

ISC

VOC

Set #3: Fill Factor

In fact, PMAX/(ISC VOC) 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!!!

Do you have any questions or topics you would like to discuss?

For those interested in talking more, contact me at:

joseph.c.kolecki@grc.nasa.gov