Solution 8.10

​Solar Math

​The "Need to Know"
to get our solar projects
to "go"!

​Basic Math Operations

• Negative and Positive Numbers

• Multiplication

• Division

• Order of Operations

​Negative and Positive Numbers

​The Number Line

"0" (Zero) at the center

How would we represent

Negative on the left of Zero

Positive on the right of Zero

​Negative and Positive Numbers

Similarly

​Negative and Positive Numbers

Multiplication and Division Rules

Positive Times/Divided by Positive Equals Positive

Positive Times/Divided by Negative Equals Negative

Negative Times/Divided by Positive Equals Negative

Negative Times/Divided by Negative Equals Positive

​Negative and Positive Numbers

Multiplication and Division Examples

Two Negatives = Answer will be Positive

One Negative = Answer will be Negative

Multiplication

The order of the formula/equation doesn't matter
We get the same result no matter which number is first

Division

ORDER MATTERS
Different results depending on which number is first
Numbers go in order of what you’re trying to find

Example - Calculating cost per Watt ($/W)
Cost is first (on top), watt is second (on the bottom):

* This is an accurate value, but it is the reverse (W/$) of what was intended to be calculated ($/W)

Order of Operations:
PEMDAS

​✸ Parenthesis

✸ Exponents (X⁴)

✸ Multiplication / Division

✸ Addition / Subtraction

Multiple Choice

The minimum temperature for a location is 7°C. What is the difference between that and the standard test condition

temperature of 25°C?

• -32°C

• 18°C

• -18°C

• 32°C

Multiple Choice

The current temperature is 19°C and the minimum temperature for a location is -22°C.

What is the difference between the minimum and standard test condition temperature of 25°C?

• -47°C

• -3°C

• 47°C

• 6°C

Multiple Choice

Calculate the cost per watt of a 7,600 W system with a total cost of $24,472.

• $3.15

• $3.22

• $3.10

• $3.47

Multiple Choice

Calculate the cost per watt of a 5.5 MW (megawatt) system with a gross cost of $6,050,000.

• $1.15

• $1.10

• $0.90

• $0.95

Reorder

Put the following mathematical operations in the correct order:

• Parenthesis

• Exponents

• Multiplication/

  Division

• Addition/

  Subtraction

1

2

3

4

• 1
• 2
• 3
• 4

Multiple Choice

Solve the following equation:

36.8 V x [100% + ((30°C + 41°C – 25°C) x -0.41%/°C)]

• 39.04 V

• 35.29 V

• 31.45 V

• 29.86 V

​Decimals and Percentages

• Huh? What are they? What do they mean?

• Converting a Decimal to a Percentage

• Converting a Percentage to a Decimal

• Finding the Percent of a number

​Decimals and Percentages

Huh? What are they? What do they mean?

​Decimals

deci - comes from the Latin decimus, meaning "tenth".
The term "Decimals" - used describe numbers with a "decimal point" - e.g., 3.14
Every Number can be written as a Decimal - e.g., 29 = 29.00
The number after the decimal point can be written as a fraction
The amount of numbers after the decimal point describes the power of ten

3.14 - Three and fourteen thousandths or 3 14/100 where 1/100 = 10^-2

​deci 10^-1 0.1 1/10

centi 10⁻² 0.01 1/100

milli 10⁻³ 0.001 1/1000

micro 10⁻⁶ 0.000001 1/1000000

nano 10⁻⁹ 0.000000001 1/1000000000

Percentages

percent - from Latin per centum  meaning 'by a hundred'.

per is essentially "divided by" and cent is "one hundred"

100% is the same as 100/100 (one hundred divided by one hundred) which is "1"

Whereas 50% is 50/100 = 1/2 (a half); 25% is 25/100 = 1/4 (one quarter)

There are also percentage representations of "special fractions":

Thirty-Three and a Third - 33.33% (technically "1/3")
One Fifth - 20% (1/5)
One Eighth - 12.5% (1/8) - "Octoroon"
One Sixteenth - 6.25% (1/16) - "Quintroon"

Converting a Decimal to a Percentage

Move the decimal point 2 places to the right

What’s really happening: multiply decimal by 100% to tack
on the “%” -> 0.85 X 100% = 85%

Converting a Percentage to a Decimal

Move the decimal point 2 places to the left

What’s really happening: divide percent by 100% to
drop the “%” -> 85% ÷ 100% = 0.85

Finding the Percent of a number

Step # 1: Convert the percentage to a decimal

Step #2: Multiply the decimal by the number

Multiple Choice

Convert -0.36% to a decimal.

• 0.0036

• -3.60

• 0.036

• -0.0036

Multiple Choice

Convert 155% to a decimal.

• 1.55

• 15.5

• 0.155

• 0.0155

Multiple Choice

Convert 26 to a percentage.

• 260%

• 2.6%

• 0.26%

• 2600%

Multiple Choice

An 8.4 kilowatt (kW) PV array is connected to an inverter with a 93% efficiency rating.

What would be the approximate output

at STC

(Standard Test Conditions)?

• 781.2 kW

• 7,812 kW

• 7.812 kW

• 78.12 kW

Multiple Choice

8.2 kWh are charged into a battery with an overall 75% efficiency rating. Approximately how many kWh would you expect

to get out of the battery?

• 5.95 kWh

• 6.83 kWh

• 7.35 kWh

• 6.15 kWh

Calculating Areas

• What is an "Area"?

• Calculating the Area of a Rectangle

• Calculating the Area of a Triangle

What is an "Area"?

An "Area" is defined as the measure of a region's size
on a surface, in 2 dimensions (e.g., length X width)

In the U.S., "Area" is measured commonly in "square feet",
"square yards", "square miles" or in "acres"

In the solar arena, we commonly need to know the
size of the region where solar panels will be installed
This may be on the ground or on a roof
In most cases, we will either have to calculate the
area of one or a group of rectangles, one or a group of
triangles or a mixture of both

Calculating the Area of a Rectangle

The Area of a Rectangle is defined as Length X Width

• Essentially the same for any non-curved measurement

• Units will be in inches, feet and yards (metric: centimeters, meters)

• Both measurements must be in the same units

  • can't multiply inches by feet or feet by yards

16.5 feet x 41 feet =
676.5 feet^{2
(676.5 Square Feet)}

Calculating the Area of a Triangle

The Area of a Triangle is defined as Base X Height ÷ 2
* Base - the "bottom side" of the triangle
* Height - the straight line extending down from the
"top angle" of the triangle coming down
*perpendicular* to the Base

Multiple Choice

Calculate the area of a roof surface that is 35.8 feet wide by 22.3 feet long.

• 804.7 ft²

• 787.2 ft²

• 798.3 ft²

• 794.3 ft²

Multiple Choice

Calculate the area of a roof surface that is 12.9 meters wide by 7.5 meters long.

• 95.67 m²

• 96.42 m²

• 96.75 m²

• 98.64 m²

Multiple Choice

Calculate the area of a triangular roof with a base width of 35.75 feet and a height of 23.325 feet.

• 417.2 ft²

• 845.6 ft²

• 416.9 ft²

• 833.9 ft²

Multiple Choice

Calculate in square feet the area of a triangular roof with a base width of 330” and a height of 159”.

• 182.2 ft²

• 1184.3 ft²

• 364.4 ft²

• 382.7 ft²

Multiple Choice

Calculate the square meter area of a triangular roof with a base width of 1934 cm and a height of 1356 cm.

• 131.1 m²

• 13.11 m²

• 1.311 m²

• 1311 m²

Power and Energy

• Watts - The Unit of Power

• Relationship between Watts, Volts and Amps

• Solving for Amps knowing Volts and Watts

• Energy - Watts over time

Watts - The Unit of Power

Watts are defined as the rate of electrical energy either generated or consumed

All electrical devices should have a power rating defined for them, and printed on them somewhere

Common order of magnitude for power
in today's world:
1,000 Watts = 1 kW (kilowatt)
1,000,000 W = 1,000 kW = 1 MW (megawatt)
1,000,000,000 W = 1,000,000 kW = 1,000 MW = 1 GW (gigawatt)

Relationship between Watts, Volts and Amps

Intro to Ohm's Law

Relationship between Watts, Volts and Amps

Solving for Amps knowing Volts and Watts

Energy - Watts over time

Multiple Choice

A PV array is producing 125,352 watts. Is this an example of power or energy?

• Power

• Energy

Multiple Choice

A battery has a capacity of 14 kilowatt-hours (kWh). Is this an example of power or energy?

• Energy

• Power

Multiple Choice

If a 1,400 W electric heater runs continuously for four hours, how much energy does it consume?

• 1,400 Wh

• 5,600 W

• 1,400 W

• 5,600 Wh

Multiple Choice

While running, a 24 VDC refrigerator consumes 25 amps. How much power does it use?

• 600 Wh

• 1,800 W

• 1,200 Wh

• 600 W

Multiple Choice

If a 1.5 A, 120 V light is left on in the barn for 12 hours,

how much energy will it use?

• 2.16 Wh

• 2,160 kWh

• 2,160 kW

• 2.16 kWh

​Solar Math

YOU ARE NOW

GURUS!
YOU'VE GOT THE POWER!