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25 Automotive Applications of Power Electronics 653

The average electrical power requirement of a present-

day automobile is in the range of 500–900 W depending on

whether it is an entry-level or a luxury vehicle. When the

loads of Table 25.6 are introduced, the average electrical power

requirement will increase by 1.8 kW. Furthermore, if the air-

conditioning (A/C) pump were to ever become electrically

driven, the peak and average power demands would increase

by an additional 3.5 kW and 1.5 kW, respectively. Distributing

such high power at a relatively low voltage will result in unac-

ceptably bulky wiring harnesses and large distribution losses.

Since the alternator has to generate both the power consumed

by the loads and the power dissipated in the distribution

network, its output rating (and hence size and power con-

sumption) will be greater than in an architecture with lower

distribution losses. With the large premium attached to the

size of the alternator (due to space constraints in the engine

compartment), an architectural change in the distribution and

generation systems is essential before many of the future loads

can be introduced.

There is also an increasing disparity in the voltage require-

ments of future electrical loads. High pulse-power loads, such

as the heated windshield and electrically heated catalytic con-

verter, become feasible only at voltages greater than the current

14 V [57]. On the other hand, incandescent lamps and elec-

tronic control units (ECUs) will continue to require low

voltages. For example, present day ECUs have linear regu-

lators which convert the 14 V distribution voltage to the 5 V

needed by the integrated circuits. The efﬁciency of these reg-

ulators is equal to the ratio of output to input voltage, i.e.

35%. Furthermore, the next generation of higher speed lower

power consumption integrated circuits operate at 3.3 V, mak-

ing the regulators more inefﬁcient. This inefﬁciency also means

that larger heat sinks are required to remove the heat from

the ECUs.

25.7.1.2 Higher Fuel Efﬁciency

A secondary motivating factor for the introduction of a higher

system voltage is the challenge of achieving higher fuel econ-

omy. The average fuel economy of present-day automobiles in

the United States is in the vicinity of 30 miles per gallon (mpg).

There is little market incentive for automobile manufacturers

to increase the fuel economy of vehicles for the US market

where the price of fuel is relatively low. The price of gasoline in

the US ($1.70 per gallon) is less than the price of bottled water

($4.00 per gallon when bought by the quart). Although market

forces have not been a driver for the development of fuel-

efﬁcient vehicles, a number of new incentives have emerged

over the past few years. One of these is the ﬁne imposed on the

automakers by the US government if the average fuel economy

of their ﬂeet falls below the mandated standard. The mandated

standard for cars has increased from 24 mpg in 1982 to its 1997

level of 27.5 mpg, and will continue to increase. In Europe,

the German Automotive Industry Association (VDA) plans to

increase the average ﬂeet fuel efﬁciency to 39.9 mpg by 2005 –

compared to 31.4 mpg in 1990 [58].

Another driver behind the development of fuel-efﬁcient

vehicles is the partnership for a new generation of vehi-

cles (PNGV). This ten-year research program, launched in

September 1993, is a collaboration between the US Federal

Government and the big three US automakers (General

Motors, Ford, and DaimlerChrysler) that aims to strengthen

national competitiveness in the automotive industry and

reduce dependence on foreign oil. The PNGV has set a goal to

develop an 80 mpg midsize vehicle by 2004 [59]. The German

Automotive Industry Association is pursuing similar targets.

The VDA has undertaken a pledge to introduce a 3 L/100 km

(78 mpg) vehicle by the year 2000. This is complemented by

the introduction of highly fuel-efﬁcient vehicles (in excess of

50 mpg) in both the Japanese and American markets.

With the present alternator, 800 W of electrical power con-

sumes 1.33 L of gasoline for every 100 km driven when the

vehicle has an average speed of 33.7 km/h. This represents a

45% increase in fuel consumption for a 3 L/100 km vehicle.

Hence, if future high fuel economy vehicles are going to

have comfort, convenience, and safety features comparable

to present-day vehicles, the efﬁciency of the electrical gen-

eration and distribution system will have to be substantially

improved. Furthermore, as discussed in Section 25.8, one

widely considered means of achieving high fuel economy is the

use of a hybrid vehicle architecture. In practice, this approach

necessitates the introduction of a higher voltage in the vehicle.

25.7.2 Voltage Speciﬁcations

A major issue when implementing a high or dual voltage sys-

tem is the nominal voltage of the high-voltage bus, and the

operating limits of both buses. While there are many possibil-

ities, there is a growing consensus in the automotive industry

for a nominal voltage of 42 V for the high-voltage bus (cor-

responding to a 36 V lead-acid storage battery) [39, 56, 60].

This voltage is gaining acceptance because it is as high as pos-

sible while remaining within acceptable safety limits for open

wiring systems (once headroom is added for transients) and it

provides substantial beneﬁts in the power semiconductors and

wiring harness [61]. Furthermore, this voltage is sufﬁcient to

implement starter/alternator systems and “light” hybrid vehi-

cle designs [62, 63]. While no vehicles equipped at 42 V are

in production at present, availability of 42 V components is

rapidly increasing and 42 V equipped vehicles may be expected

early in this decade.

The permissible static and transient voltage ranges in

an electrical system are important design considerations

for power electronic equipment. At present, no universally

accepted speciﬁcation exists for high or dual voltage auto-

motive electrical systems. However, the preliminary spec-

iﬁcations proposed by the European automotive working

group, Forum Bordnetz, are under wide consideration by the

654 D. J. Perreault et al.

TABLE 25.7 Voltage limits for 14 and 42 V buses proposed in [56]

Voltage Description Value

42,OV −dyn

Maximum dynamic overvoltage on 42 V bus during fault conditions 55 V

42,OV −stat

Maximum static overvoltage on 42 V bus 52 V

42,E−max

Maximum operating voltage of 42 V bus while engine is running 43 V

42,E−nom

Nominal operating voltage of 42 V bus while engine is running 41.4 V

42,E−min

Minimum operating voltage of 42 V bus while engine is running 33 V

42,OP−min

Minimum operating, voltage on the 42 V bus. Also, lower limit operating voltage for all non-critical

loads (i.e. loads not required for starting and safety)

33 V

42,FS

Failsafe minimum voltage: lower limit on operating voltage for all loads critical to starting and safety

on the 42 V bus

25 V

14,OV −dyn

Maximum dynamic overvoltage on 14 V bus during fault conditions 20 V

14,OV −stat

Maximum static overvoltage on 14 V bus 16 V

14,E−max

Maximum operating voltage of 14 V bus while engine is running 14.3 V

14,E−nom

Nominal operating voltage of 14 V bus while engine is running 13.8 V

14,E−min

Minimum operating voltage of 14 V bus while engine is running 12 V

14,OP−min

Minimum operating voltage of 14 V bus. Also lower limit operating voltage for all non-critical loads 11 V

14,FS

Failsafe minimum voltage: lower limit on operating voltage for all critical loads on the 14 V bus 9 V

automotive industry [56]. These speciﬁcations, summarized in

Table 25.7, impose tight static and transient limits on both the

42 and 14 V buses. The upper voltage limit on the 14 V bus is

far lower than in the conventional 12-V system. The allowed

upper limit on the 42 V bus is also proportionally tight. These

strict limits facilitate the use of power semiconductor devices

such as power MOSFETs and lower the cost of the protection

circuitry needed in individual functions. However, they also

require much more sophisticated means for limiting transients

(such as load dump) than is found in conventional systems,

which imposes a signiﬁcant cost. Appropriate voltage range

speciﬁcations for dual/high voltage electrical systems are thus

a subject of ongoing investigation by vehicle manufacturers,

and will likely continue to evolve for some time.

25.7.3 Dual-voltage Architectures

Conventional automotive electrical systems have a single alter-

nator and battery. Dual-voltage electrical systems have two

voltage buses and typically two batteries. Single-battery con-

ﬁgurations are possible, but tend to be less cost effective [61].

A variety of different methods for generating and supplying

energy to the two buses are under investigation in the automo-

tive community. Many of these have power electronic circuits

at their core. This section describes three dual-voltage electrical

system architectures that have received broad attention. In all

three cases the loads are assumed to be partitioned between the

two buses with the starter and many of the other high-power

loads on the 42 V bus and most of the lamps and electronics

on the 14 V bus.

The dc/dc converter-based implementation of Fig. 25.19 is

perhaps the most widely considered dual-voltage architecture.

loads

FIGURE 25.19 Dual-voltage architecture based on a dc/dc converter.

In this implementation, an alternator and associated battery

provide energy to one bus (typically the 42 V bus), while the

other bus is supplied via a dc/dc converter. If a battery is

used at the dc/dc converter output, the converter needs to be

rated for slightly above average power. Otherwise, the con-

verter needs to be rated a factor of two to three higher to meet

peak power requirements [61]. The architecture of Fig. 25.19

has a number of advantages. The dc/dc converter provides

high-bandwidth control of energy ﬂow between the two buses,

thus enabling better transient control on the 14 V bus than

is available in present-day systems or in most other dual-

voltage architectures. Furthermore, in systems with batteries

on both buses, the dc/dc converter can be used to imple-

ment an energy management system so that generated energy

is always put to best use. If the converter is bidirectional it

can even be used to recharge the high-voltage (starter) battery

from the low-voltage battery, thus providing a self jump start

capability. The major challenge presented by this architecture

is the implementation of dc/dc converters having the proper

functionality within the tight cost constraints dictated by the

automotive industry. Some aspects of design and optimization

of converters for this application are addressed in [64].

25 Automotive Applications of Power Electronics 655

loads

FIGURE 25.20 Dual-voltage architecture based on a dual-wound

alternator.

The dual-stator alternator architecture of Fig. 25.20 is also

often considered for dual-voltage automotive electrical sys-

tems [65, 66]. In this case, an alternator with two armature

windings is used along with two rectiﬁers to provide energy to

the buses and their respective batteries. Control of the bus volt-

ages is achieved via a combination of controlled rectiﬁcation

and ﬁeld control. Typically, ﬁeld control is used to regulate one

output, while the other output is regulated using a controlled

rectiﬁer. Figure 25.21 shows one possible implementation of

this architecture. It should be noted that to achieve sufﬁcient

output power and power steering from the dual-wound alter-

nator, the winding ratio between the two outputs must be

carefully selected. For 42/14 V systems, a winding ratio of 2.5:1

is typical [66]. Advantages of this electrical architecture include

low cost. However, it does not provide the bidirectional energy

control that is possible in the dc/dc converter architecture.

Furthermore, there are substantial issues of cross-regulation

and transient control with this architecture that remain to be

fully explored.

In a third architecture, a single-output alternator with a

dual-output rectiﬁer is employed. This approach is shown

schematically in Fig. 25.22. As with the dual-stator alterna-

tor conﬁguration, this architecture has the potential for low

cost. One widely considered implementation of the dual-

rectiﬁed alternator is shown in Fig. 25.23 [65, 67–69]. Despite

its simplicity, this implementation approach provides less

functionality than the dc/dc converter-based architecture,

generates substantial low-frequency ripple which must be

ﬁltered, and has serious output power and control limita-

tions [66]. An alternative implementation, proposed in [37]

and shown in Fig. 25.24, seems to overcome these limitations,

and may potentially provide the same capabilities as the dc/dc

converter-based architecture at lower cost. Clearly, this archi-

tecture has promise for dual-voltage electrical systems, but

remains to be fully explored.

25.8 Electric and Hybrid Electric

Vehicles

Battery-powered electric vehicles were ﬁrst introduced over

one hundred years ago, and continue to incite great pub-

lic interest because they do not generate tailpipe emissions.

Field Current

Regulator

field

FIGURE 25.21 Model for a dual-wound alternator system. The two out-

put voltages are regulated through ﬁeld control and phase control. For a

42/14 V system, a winding ratio between the two stator windings of 2.5:1

is typical.

loads

FIGURE 25.22 Dual-voltage architecture based on a dual-rectiﬁed

alternator.

Nevertheless, the low energy storage density and the high

cost of suitable batteries makes pure electric vehicles non-

competitive with internal combustion engine vehicles in

most applications. An alternative approach that is generat-

ing widespread attention is the hybrid electric vehicle (HEV).

An HEV combines electrical propulsion with another energy

source, such as an internal combustion engine, allowing the

traditional range and performance limitations of pure electric

vehicles to be overcome [70]. Alternative energy sources, such

as fuel cells, are also possible in place of an internal combustion

engine.

Hybrid electric vehicles can be classiﬁed as having either a

parallel or series driveline conﬁguration [71]. In a series HEV

all of the propulsion force is produced from electricity; the

engine is only used to drive a generator to produce electricity.

In a parallel hybrid, propulsive force can come from either the

656 D. J. Perreault et al.

Field Current

Regulator

field

FIGURE 25.23 A dual-rectiﬁed alternator with a phase-controlled

rectiﬁer.

Field Current

Regulator

field

FIGURE 25.24 A dual-rectiﬁed alternator with a switched-mode

rectiﬁer.

engine or the electrical drive. In both cases, batteries or other

electrical storage devices are used to buffer the instantaneous

difference between the power needed for propulsion and that

generated by the engine. The selection of a series or parallel

driveline depends heavily on the performance requirements

and mission of the vehicle.

In a series HEV, all power delivered to the wheels of the

vehicle must be delivered through the electrical driveline. The

electrical driveline components, including the batteries, power

electronics, and machine(s), must all be rated for the peak trac-

tion power requirements, making these components relatively

large and expensive if performance (e.g. acceleration) compa-

rable to a conventional vehicle is to be achieved. To achieve the

required power levels, the electrical driveline must operate at

hundreds of volts, necessitating the electrical subsystem to be

sealed from access by the user. The engine, on the other hand,

need only be rated to deliver the average power required by the

vehicle, which is much lower. In a system that does not require

utility recharge of the batteries (i.e. can drive indeﬁnitely on

fuel alone), the engine size is set by the power requirements

of the vehicle at maximum cruising speed. If utility recharge

of the batteries and a battery-limited driving range is accept-

able, engine power requirements can be reduced even further.

Because the engine does not provide tractive power, it can be

designed to run at a single optimized condition, thus maximiz-

ing engine efﬁciency and minimizing emissions. Furthermore,

the need for a transmission is eliminated and there is a great

deal of ﬂexibility in the engine placement.

In a parallel HEV, traction power is split between the engine

and the electrical driveline. One possible approach is to uti-

lize a single machine mounted on the engine crankshaft to

provide starting capability along with electrical traction power

and regeneration [72–75]. This approach can be replaced or

complemented with other approaches, such as use of a power-

splitting device such as a planetary gear set [70, 76], or using

different propulsion and generation techniques on different

sets of wheels [71, 77, 78]. In all parallel hybrid approaches,

some form of transmission is needed to limit the required

speed range of the engine. A wide range of divisions between

engine size and electrical system size is possible in the parallel

hybrid case, depending on structure. Depending on this split,

the necessary electrical driveline system voltage may be as low

as 42 V (which is safe for an open wiring system) or as high

as 300 V. Also because the electrical subsystem, the internal

combustion engine subsystem, or both may provide tractive

power under different conditions, there exists a wide range

of possible operating approaches for a parallel hybrid system.

Consequently, the control strategy for a parallel hybrid tends

to be substantially more complex than for a series hybrid.

One parallel hybrid approach that is receiving a lot of atten-

tion for near-term vehicles is a “light” or “mild” hybrid. In

this case, a somewhat conventional vehicle driveline is com-

plemented with a relatively small starter/alternator machine

mounted on the crankshaft [62, 63, 72–75, 79]. The electrical

drive power is typically below 10 kW average and 20 kW peak.

The starter/alternator can be used to provide rapid, clean

restart of the vehicle so that the engine can be turned off

at idling conditions and seamlessly restarted. This so-called

“stop and go” operation of the engine is valuable for fuel

economy and emissions. The starter/alternator can also be

used to implement regenerative braking, to provide engine

torque smoothing (replacing the ﬂywheel and allowing dif-

ferent engine conﬁgurations to be used) and to provide

boost power for short-term acceleration. At the low-power

end, such systems can be integrated directly into the open

wiring conﬁguration of a 42 V electrical system, simplifying the

vehicle electrical architecture. System-level control remains a

25 Automotive Applications of Power Electronics 657

major challenge in realizing the full beneﬁts of such systems.

Starter/alternator-based hybrids are expected to be a signiﬁ-

cant near-term application of power electronics and machines

in automobiles.

25.9 Summary

Power electronics is playing an increasingly important role in

automobiles. It is being used to enhance the output power

capability and efﬁciency of the electrical power generation

components. Power electronics is also an enabling technology

for a wide range of new and improved functions that enhance

vehicle performance, safety, and functionality.

The design of automotive power electronic systems is

strongly inﬂuenced by the challenging electrical and environ-

mental conditions found in automobiles. Important factors

include the static and transient voltage ranges, electromagnetic

interference and compatibility requirements, and temperature

and other environmental conditions. Some of the most impor-

tant design considerations for automotive power electronics

were addressed in Section 25.3.

Section 25.4 described some of the vehicle functions that

beneﬁt from, or are enabled by, power electronics. These func-

tions range from lighting to actuation and steering. Power

electronic switches also play a central role in multiplexed elec-

trical distribution systems. This role of power electronics was

addressed in Section 25.5.

The rapid increase in electrical power demand in automo-

biles is motivating the introduction of new technologies for

electrical power generation and control. Lundell alternators

are presently used for power generation in automobiles, but

are rapidly reaching their power limits. Section 25.6 reviewed

the operating characteristics of the Lundell alternator. It also

described several techniques for extending the power capa-

bilities of this machine. To meet the growing demand for

electrical power, alternative machine, and power electronic

conﬁgurations may be necessary in the future. A number

of candidate machine and power circuit conﬁgurations were

reviewed in Section 25.6. Such conﬁgurations can also be

applied towards the design of integrated starter/alternators and

hybrid propulsion systems, as was discussed in Section 25.8.

The increasing electrical and electronic content of automo-

biles is beginning to stretch the capabilities of the conventional

12-V electrical system. Furthermore, there is a desire on the

part of vehicle manufacturers to introduce new high-power

loads, such as electromechanical engine valves, active suspen-

sion, and integrated starter/alternator. These are not likely

to be practical within the present 12-V framework. These

challenges are forcing the automotive industry to seriously

consider high and dual voltage electrical systems. The ongoing

developments in this area were reviewed in Section 25.7

The increasing electrical content of vehicles both under-

scores the need for power electronics and reﬂects the beneﬁts

of their introduction. It is safe to say that power electronics

will continue to play an important role in the evolution of

automobiles far into the future.

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Solar Power Conversion

Lana Chaar, Ph.D.

Electrical Engineering Department,

American University in Dubai,

P. O. Box 28282, Dubai,

UAE

26.1 Introduction .......................................................................................... 661

26.2 How does a Solar Cell Work? .................................................................... 662

26.3 Solar Energy Conversion .......................................................................... 663

26.4 Maximum Power Tracker......................................................................... 664

26.4.1 Switch-mode Converter • 26.4.2 Controller • 26.4.3 MPPT Controller Algorithm

26.5 Photovoltaic Systems’ Components ............................................................ 667

26.5.1 Grid-connected Photovoltaic System • 26.5.2 Stand-alone Photovoltaic Systems

26.6 Factors Affecting Output .......................................................................... 670

26.6.1 Temperature • 26.6.2 Dirt and Dust • 26.6.3 DC–AC Conversion

26.7 System Design ........................................................................................ 671

26.7.1 Criteria for a Quality PV System • 26.7.2 Design Procedures • 26.7.3 Power

Conditioning System • 26.7.4 Battery Sizing

26.8 Summary .............................................................................................. 671

References ............................................................................................. 671

26.1 Introduction

For many years, fossil fuel has been the primary source of

energy. However, there is a limited supply of these fuels on

Earth and they are being used much more rapidly than they

are being created. Eventually, they will run out. In addi-

tion, because of safety concerns and waste disposal problems,

renewable energy is deﬁnitely the solution since such tech-

nology is “clean” or “green” because they produce few if

any, pollutants. The world trend nowadays is to ﬁnd a non-

depletable and clean source of energy. The most effective and

harmless energy source is probably solar energy, which for

many applications is so technically straightforward to use.

With the exception of nuclear and geothermal, all forms of

energy used on Earth originate from the sun’s energy.

Solar energy is considered one of the most promising energy

sources due to its inﬁnite power. Thus modern solar technolo-

gies have been penetrating the market at faster rates. The solar

661

662 L. Chaar

technology that has the greatest impact on our lives is photo-

voltaic. Not in terms of the amount of electricity it produces,

but because of the fact that photovoltaic cells – work silently,

not polluting – can generate electricity wherever sun shines,

even in places where no other form of electricity can be

obtained [1]. The word Photovoltaic is a combination of the

Greek word for light and the name of the physicist Alessandro

Volta [2]. It identiﬁes the direct conversion of sunlight into

electricity by means of solar cells.

In this chapter, we will discuss how solar cells produce

electricity and what are the components required for such

system.

26.2 How does a Solar Cell Work?

Solar cells are composed of various semiconductor materials

which become electrically conductive when supplied by heat

or light. The majority of solar cells produced are composed

of Silicon (Si) which exist in sufﬁcient quantities and do not

add any burden on the environment. However, over 95% of

these cells have efﬁciency about 17% [3]. However solar cells,

having power conversion efﬁciencies as high as 31% have been

developed over the last decade in laboratory environment [4].

Doping technique is used to obtain a surplus of posi-

tive charge carriers (p-type) or a surplus of negative carriers

FIGURE 26.1 Effect of the electric ﬁeld in a PV cell [1].

(a) (b)

n-type

Front contact grid

p-type

Back contact

Photons

−

FIGURE 26.2 (a) Operation of a PV cell [2] and (b) solar cell equivalent circuit.

(n-type). When two layers of different doping are in contact,

then a p-n junction is formed on the boundary.

An internal electric ﬁeld is built up which then causes the

separation of charge carriers released by light as shown in

Fig. 26.1. We all know that light is composed of small packets

called photons. When these photons bombard our cell, many

electrons are freed within the electric ﬁeld proximity, which

then pull the electrons from the p-side to n-side. Through

metal contacts, an electric charge can be taped. If the outer

circuit is closed, then direct current ﬂows as illustrated in

Fig. 26.2.

A solar cell is approximately 10 × 10 cm, protected by trans-

parent antireﬂection ﬁlm. Each of these cells produces around

0.5 V (for Silicon). The voltage across a solar cell is primar-

ily dependent on the design and materials of the cell, whilst

the electrical current depends primarily on the incident solar

irradiance and the cell area.

The position of the sun in the sky varies dramatically with

hour and season. The sun elevation angle

elevation

sun

is expressed

in degrees above the horizon. Sun azimuth

azimuth

sun

angle is

expressed in degrees from true north. Sun zenith angle

zenith

sun

equals ninety degrees less than the sun elevation, or

zenith

sun

= 90

◦

−

elevation

sun

(26.1)

Azimuth, zenith, and elevation angle are illustrated in Fig. 26.3

The output from a typical solar cell which is exposed to the

sun therefore increases from zero at sunrise to a maximum at

midday and then falls again to zero at dusk. The I–V charac-

teristics of a solar panel are nonlinear and follow the general

shape and equation shown in Fig. 26.4.

I = I

−A



−1



(26.2)

where A, B are constants

is the short circuit current, thus the greatest current value

generated by the cell under short circuit conditions (V = 0).

In order to obtain the appropriate voltages and outputs for

different applications, single solar cells are interconnected in

series (for larger voltage) and in parallel (for larger current) to

26 Solar Power Conversion 663

North

East

elevation

sun

zenith

sun

Line perpendicular to

horizontal plane

Horizontal plane

West

azimuth

sun

FIGURE 26.3 Azimuth, zenith, and elevation angles of a vector pointed

toward the sun [5].

form the photovoltaic module. Then several of these modules

are connected to each other to form the photovoltaic array as

shown in Fig. 26.5. This array is then ﬁtted with an aluminum

or stainless steel frame and covered with transparent glass on

the front side.

0 V

Maximum

power point

Increasing

solar insolation

I = I

− A (e

−1), where A, B, and especially I

vary with solar insolation

max

= V

FIGURE 26.4 I–V characteristics of a solar cell [6].

Module

Cell

FIGURE 26.5 Photovoltaic cells, modules, panels, and array [7].

26.3 Solar Energy Conversion

Photovoltaic technology is used to produce electricity in areas

where power lines do not reach. In developing countries,

it is improving living conditions in rural areas especially in

healthcare, education, and agriculture. In the industrialized

countries, they have been used extensively and integrated with

the utility grid.

Photovoltaic arrays are usually mounted in a ﬁxed position

and tilted toward the south to optimize the noontime and daily

energy production. The orientation of ﬁxed panels should be

carefully chosen to capture the maximum energy for the sea-

son, or a year. Photovoltaic arrays have an optimum operating

point called the maximum power point (MPP) as shown in

Fig. 26.6 [8].

The curve in Fig. 26.6 shows that power increases as the

voltage is increased, reaching a peak value before decreasing as

the resistance increases to the point where current drops-off.

According to the maximum power transfer theory, this point

is where the load is matched to the solar panel’s resistance at