APPENDIX A
Math Formulas Used in AV Design
How long has it been since you solved a word problem or used a math formula? Many CTS-D exam candidates have not done math in a formal setting in years. You may be familiar with the skills and tools in this appendix, but then again, you may need a refresher.
Most AV math formulas use only the four common operators: add, subtract, multiply, and divide. However, some formulas require a solid foundation in the order of operations. The order of operations helps you correctly solve formulas by prioritizing which part of the formula to solve first. It is a way to rank the order in which you work your way through a formula. This appendix will review and test your knowledge of applying the order of operations.
This is the order of operations:
1. Any numbers within a pair of parentheses or brackets
2. Any exponents, indices, or orders
3. Any multiplication or division
4. Any addition or subtraction
If there are multiple operations with the same priority, then proceed from left to right: parentheses, exponents, multiplication, division, addition, subtraction. The order of operations can be remembered by using one of these acronyms: PEMDAS, BEMDAS, BIDMAS, or BODMAS.
Steps to Solving Word Problems
All math formulas summarize relationships between concepts. Word problems are designed to test how well an individual can apply that relationship to a new situation.
By following a simple strategy, you can turn a complicated word problem into a few straightforward steps. This section provides a structured approach to solving problems. Within this structure, you will find many strategies for solving different types of problems. This strategy is based on How to Solve It: A New Aspect of Mathematical Method by George Polya.
Step 1: Understand the Problem
As typical within the AV industry (and in general), the first step is to understand the problem you’re trying to solve. Here are the tasks to complete for this step:
1. Read the entire math problem.
2. Identify your goal or unknown. What information are you trying to determine?
3. Identify what you have been given. What data, numbers, or other information in the problem can help you determine the answer?
4. Predict the answer if you can. What range of values would make sense as an answer?
Example: Calculate the current in a circuit where the voltage is 2 V and the resistance is 8 ohms.
First, identify your goal. What are you trying to solve for? When you see the word calculate, generally the word that follows is your goal. Other words that identify the goal include determine, find, and solve for. Your goal in this problem is to calculate current.
An easy way to identify your given information is to find the numbers in the problem. In this example, the numbers are 2 and 8. Look for context clues or units to identify what those numbers represent. The voltage is identifies 2 as a voltage. The ohms after the 8 identifies 8 as the resistance.
Sometimes it is unclear what each number represents. In that case, drawing a diagram can help you make sense of what the problem is trying to say. You may want to make a chart of your given and unknown information for quick reference. For more complex problems, tables of given information can be extremely helpful.
Step 2: Create a Plan
The second step in this process is to translate the words in the problem into numbers you can enter into a formula. Begin with the following:
1. Assign appropriate values to the goal and given information.
2. Determine a formula that describes the relationships between your variables.
If there is a single formula that has all your variables in it, move on to the next step.
In some cases, you may be unable to identify a formula that will determine your goal based on your given information. If you’re stuck, consider the following strategies:
• Use an intermediate formula to solve for the information you are missing.
• Use an outside reference, such as a chart or graph, to find information not listed in the problem.
• Use a strategy that has worked to solve similar projects in the past.
• Diagram the scenario described in the word problem. Use the diagram to keep track of the relationships between values. For instance, as listeners move farther away from a sound source, you know to expect a loss of sound pressure.
Example: Calculate the current in a circuit where the voltage is 2 V and the resistance is 8 ohms.
Again, start by assigning variables to your given and unknown information. Note that some items are represented by different variables in different contexts. If you are having trouble determining which variables to use, consider drawing a diagram and labeling it with your given information, like this:
Once you have assigned variables, think about the relationship between the information. There should be a formula that describes the relationship. For complex problems, you may need to use several formulas.
You may know formulas from memory. If you don’t, look them up (we’ve provided several useful formulas at the end of this appendix). It is helpful to think about a time when you solved for the value previously or a similar problem you may have solved in the past. Try to think of a problem that used the same givens and unknowns.
For example, you might not remember how to solve for current using voltage and resistance. But if you remember how to solve for voltage using current and resistance, you can solve this problem. Voltage is equal to current multiplied by resistance.
Step 3: Execute Your Plan
The third step is to put your plan in action, as follows:
1. Write the formulas.
2. Substitute the given information for the variables.
3. Perform the calculation.
4. Assign units to your final answer.
Example: Calculate the current in a circuit where the voltage is 2 V and the resistance is 8 ohms.
Once you have determined the appropriate formula, write it down and then replace the placeholders with the numbers for this problem. People who skip this step are prone to making mistakes.
Formula: V = I * R
Substitution: 2 = I * 8
To solve this equation, you need to get the I by itself. The 8 is currently being multiplied. To move it to the other side of the equation, perform the opposite mathematical function. In this case, that function is division.
2 / 8 = (I * 8) / 8
2 / 8 = I
0.25 = I
You need to assign units to your answer before it is final. Because I represents current, you would assign 0.25 the unit for current, which is amps (A).
I = 0.25 A
Step 4: Check Your Answer
Your final step is to make sure the numbers you’ve calculated still make sense when translated back into words. Compare your answer to the scenario described in the problem. Is the result reasonable? Is it within the range you originally predicted?
For example, suppose you are calculating the voltage present in a boardroom loudspeaker circuit. A result of 95 V is probably a reasonable answer. A result of 50,000 V indicates that you made a mistake in your calculations.
If you have an incorrect answer and use it to solve other parts of a process, it will result in cascading problems.
Example: Calculate the current in a circuit where the voltage is 2 V and the resistance is 8 ohms.
In this example, is “less than 1 A” a reasonable answer? Understanding the problem is essential here. A D battery has a voltage of 1.5 V. When a D battery is attached to a circuit, there is not much current. So, the small number of 0.25 amps is a reasonable answer.
Rounding
Many of the results listed have been rounded to the nearest tenth. When solving multistep problems, you may be tempted to round at each step. The earlier or more often you round in a multistep problem, the less accurate your result will be. Round only your final result.
AV Math Formulas
This section presents some common math formulas that may be useful for AV design professionals.
Estimated Projector Throw
The formula for estimating projector throw distance is as follows:
Distance = Screen width * Throw ratio
where:
• Distance is the distance from the front of the lens to the closest point on the screen.
• Screen width is the width of the projected image.
• Throw ratio is the ratio of throw distance to image width.
Projector Lumens Output
The formula for estimating projector brightness is as follows:
where:
• L is the ambient light at the screen location (lux or footcandles).
• C is the desired contrast ratio.*
• A is the area of the screen (in square meters or square feet).
• Sg is the gain of the screen. Assume a screen gain of 1 unless otherwise noted.
• Dr is the projector derating value. Assume a derating value of 0.75 unless otherwise noted.
* The desired contrast ratio depends on the use case in the ANSI/INFOCOMM 3M-2011 standard, Projected Image System Contrast Ratio. The following are the four contrast ratios in the standard:
• 7:1 for passive viewing
• 15:1 for basic decision making
• 50:1 for analytical decision making
• 80:1 for full-motion video (home theater)
Image Height to Farthest Viewer Distance Ratio
The formula for determining image height based on viewing task is as follows:
IH/ID = DT/VT
where:
• IH is the image height.
• ID is the distance from the farthest viewer to the image.
• VT is the viewing task ratio (distance).
• DT is the viewing task ratio (height). This will always be 1.
For the viewing task ratio, which is based on levels of detail, use a factor of 8 when viewers will observe content (general viewing), use 6 when viewers will inspect content with clues (reading with clues), and use 4 when the viewers will inspect content without clues (inspection).
Decibel Formula for Distance
The formula for decibel changes in sound pressure level over distance is as follows:
dB = 20 * log (D1 / D2)
where:
• dB is the change in decibels.
• D1 is the original or reference distance.
• D2 is the new or measured distance.
The result of this calculation will be either positive or negative. If it is positive, the result is an increase, or gain. If it is negative, the result is a decrease, or loss.
Decibel Formula for Voltage
The formula for determining decibel changes for voltage is as follows:
dB = 20 * log (V1 / VR)
where:
• dB is the change in decibels.
• V1 is the new or measured voltage.
• D2 is the original or reference voltage.
The result of this calculation will be either positive or negative. If it is positive, the result is an increase, or gain. If it is negative, the result is a decrease, or loss.
Decibel Formula for Power
The formula for calculating decibel changes for power is as follows:
dB = 10 * log (P1 / Pr)
where:
• dB is the change in decibels.
• P1 is the new or measured power measurement.
• Pr is the original or reference power measurement.
The result of this calculation will be either positive or negative. If it is positive, the result is an increase, or gain. If it is negative, the result is a decrease, or loss.
Current Formula (Ohm’s Law)
The formula for calculating current using Ohm’s law is as follows:
I = V / R
where:
• I is the current.
• V is the voltage.
• R is the resistance.
Power Formula
The formula to solve for power is as follows:
P = I * V
where:
• P is the power.
• I is the current.
• V is the voltage.
Series Circuit Impedance Formula
The formula for calculating the total impedance of a series loudspeaker circuit is as follows:
ZT = Z1 + Z2 + Z3 … + ZN
• ZT is the total impedance of the loudspeaker circuit.
• Zx is the impedance of each loudspeaker.
Parallel Circuit Impedance Formula: Loudspeakers with the Same Impedance
The formula to find the circuit impedance for loudspeakers wired in parallel with the same impedance is as follows:
(ZT) = Z1 / N
where:
• ZT is the total impedance of the loudspeaker system.
• Z1 is the impedance of each loudspeaker.
• N is the number of loudspeakers in the circuit.
Parallel Circuit Impedance Formula: Loudspeakers with Different Impedances
The formula to find the circuit impedance for loudspeakers wired in parallel with differing impedance is as follows:
where:
• ZT is the total impedance of the loudspeaker circuit.
• Zx is the impedance of each individual loudspeaker.
Series/Parallel Circuit Impedance Formulas
Two formulas are used to calculate the expected total impedance of a series/parallel circuit.
First, the series circuit impedance formula is used to calculate the impedance of each branch, as follows:
ZT = Z1 + Z2 + Z3 … + ZN
where:
• Zx is the impedance of each loudspeaker.
• ZT is the total impedance of the branch.
Then, the parallel circuit impedance formula is used to calculate the total impedance of the series/parallel circuit, as follows:
where:
• Zx is the total impedance of each branch.
• ZT is the total impedance of the loudspeaker circuit.
Needed Acoustic Gain
The formula for calculating needed acoustic gain—how loud speakers need to be for listeners to hear the intended audio—is as follows:
NAG = 20log (D0/EAD)
where:
• D0 is the distance from the source to the listener.
• EAD is the equivalent acoustic distance, or the farthest distance one can go from the source without needing sound amplification or reinforcement to maintain intelligibility.
Potential Acoustic Gain
The formula for calculating potential acoustic gain (gain before feedback) is as follows:
PAG = 20log [(D0 * D1)/(D2 * DS)]
where:
• D0 is the distance between the talker and the farthest listener.
• D1 is the distance between the closest loudspeaker to the microphone and the microphone.
• D2 is the distance between the loudspeaker closest to the farthest listener and the farthest listener.
• DS is the distance between the sound source (talker) and the microphone.
Audio System Stability (PAG/NAG)
The formula for checking audio system stability by determining that needed acoustic gain (NAG) is less than potential acoustic gain (PAG) is as follows:
20log (D0/EAD) < 20log [(D0 * D1)/(D2 * DS)] – 10log(NOM) – FSM
• NOM is the number of open microphones.
• FSM is the feedback stability margin.
• EAD is the equivalent acoustic distance.
• D0 is the distance between the talker and the farthest listener.
• D1 is the distance between the closest loudspeaker to the microphone and the microphone.
• D2 is the distance between the loudspeaker closest to the farthest listener and the farthest listener.
• DS is the distance between the sound source (talker) and the microphone.
Conduit Capacity
The formula for calculating conduit capacity is as follows:
where:
• OD is the outer diameter of the cables.
• ID is the inner diameter of the conduit.
• FP is the permissible fill percentage of the conduit based on the number of cables.
Permissible fill percentage is determined by the authority having jurisdiction (AHJ). For example, the National Electrical Code (NEC) requirements are as follows:
• One cable: 53 percent
• Two cables: 31 percent
• Three or more cables: 40 percent
Jam Ratio
The formula for calculating jam ratio is as follows:
where:
• OD is the outer diameter of the cables.
• ID is the inner diameter of the conduit.
Jam ratio is applicable only to conduits with exactly three cables.
Heat Load Formula
The formula for calculating heat load is as follows:
Total BTU = WE * 3.4
where:
• WE is the total wattage of all equipment used in the room.
• 3.4 is the conversion factor, where 1 watt of power generates 3.4 BTU of heat per hour.
This formula does not account for the heat load generated by amplifiers.
Power Amplifier Heat Load
The formula for calculating the heat load of a power amplifier is as follows:
Total BTU = WE * 3.4 * (1 – ED)
where:
• W is the wattage of the amplifier.
• ED is the efficiency of the device.
Power Amplifier Wattage (Constant Voltage)
The formula for calculating required amplifier wattage is as follows:
Wt = W * N * 1.5
where:
• Wt is required wattage.
• W is watt tap used at the individual loudspeaker.
• N is total number of loudspeakers.
• 1.5 is 50 percent of the amplifier headroom.
Wattage at the Loudspeaker
The formula for the electrical power (EPR) required at a loudspeaker is as follows:
• LP is SPL required at distance D2.
• H is required headroom.
• LS is loudspeaker sensitivity at 3.28 ft (1 m).
• D2 is distance from loudspeaker to listener.
• Dr is distance reference value.
• Wref is the wattage reference value. Assume a wattage reference value of 1 unless otherwise noted.
Simplified Room Mode Calculation
A simplified formula for discovering frequencies where room modes will be present is as follows:
3 * (velocity of sound) / RSD = Hz
where:
• The velocity of sound = 1,130 feet per second (343 meters per second) at 72 degrees F
• RSD = The room’s smallest dimension
• Hz = Frequency
Loudspeaker Coverage Pattern (Ceiling Mounted)
The formula for calculating the diameter (twice the radius) of the circle that represents the coverage area of a loudspeaker is as follows:
D = 2 * (H – h) * tan (C∠ / 2)
where:
• D is the diameter of the coverage area.
• H is the ceiling height.
• h is the height of the listeners’ ears.
• C∠ is the loudspeaker’s angle of coverage in degrees.
Loudspeaker Spacing (Ceiling Mounted)
The formula for calculating the space between ceiling-mounted speakers depends on how much overlap of each speaker’s coverage pattern is desired. Here are formulas for three typical coverage patterns, based on overlap:
• Edge-to-edge (no overlap): D = 2 * r
• Minimal overlap: D = r * √2
• Center-to-center (maximum overlap): D = r
• D is the distance between loudspeakers.
• r is the radius of the loudspeakers’ coverage circles.
Digital Video Bandwidth
The formula for calculating digital signal bandwidth for uncompressed, color (RGB) video is as follows:
bit rate (bits per second) = horizontal pixels × vertical pixels × bit depth × frames per second
Analog Video Signal Bandwidth
The formula for calculating analog video signal bandwidth is as follows:
where:
• HF is the highest frequency in hertz.
• Hpix is the total number of horizontal pixels.
• Vpix is the total number of vertical pixels.
• fv is the refresh rate.
Minimum Video System Bandwidth
The formula for calculating minimum video system bandwidth is as follows:
SF = HF * 2
where:
• SF is the system frequency in hertz.
• HF is the highest frequency in hertz of the computer signal.