Download Solar Energy II - Department of Electrical Engineering and

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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
-
I
I
V
V
Ohm’s Law for a Non-Linear Resistor:
a Diode
I
dV = R dI
+V I
Reverse
I = I(V)
R(V) = dV/dI
Forward
V
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.
2ND Quadrant:
Generation
I
1ST Quadrant:
Dissipation
V
3D Quadrant:
4TH Quadrant:
Dissipation
Generation
Power is either generated or dissipated,
depending on the quadrant you are in.
I
Power
Dissipating
Region
V
Power
Dissipating
Region
Dark Characteristic
I
V
Power
Generating
Region
Light Characteristic
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
ISC
(0V, 150 mA)
 V  I = 0 mW
PMAX
(0.43 V, 142 mA)
 V  I = 61 mW
Some typical
values
(0.5V, 0 mA)
 V  I = 0 mW
VOC
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!)
Answer to Question #1:
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 #2:
I = 0
R = 
+
_
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.
Answer to Questions #3 and #4 (cont’d):
Current at max power
ISC
Hyperbola for P = PMAX
Point of tangency
Voltage at max power
VOC
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
RSH
The slopes of
these lines are
characteristic
resistances.
RS
VOC
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.
Cell
RS
ISC
RSH
RLOAD
Cell
Equivalent circuit for a solar cell with load.
Internal resistances RS and RSH represent
power loss mechanisms inside the cell.
RS = 0
ISC
RSH = 
RLOAD
The ideal solar cell would have no internal
losses at all! What would the VI
characteristic of THIS cell look like?
The Ideal Solar Cell
ISC
RSH = 
RS = 0
VOC
The Ideal Solar Cell
ISC
Notice that the area under
the rectangle = PMAX for
the ideal cell. For this cell,
PMAX = VOC  ISC
VOC
Set #3: Fill Factor
ISC
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?
VOC
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 onorbit 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:
[email protected]