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Decibel Measurements
19 questions By Tony R. Kuphaldt
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Question 1 of 19
During the early development of telephone technology, a unit was invented for representing power gain (or loss) in an electrical system. It was called the Bel, in honor of Alexander Graham Bell, the telecommunications pioneer.
“Bels” relate to power gain ratios by the following equation:
AP(ratio) = 10AP(Bels) Given this mathematical relationship, translate these power gain figures given in units of Bels, into ratios:
- AP = 3 B ; AP =
- AP = 2 B ; AP =
- AP = 1 B ; AP =
- AP = 0 B ; AP =
- AP = -1 B ; AP =
- AP = -2 B ; AP =
- AP = -3 B ; AP =
Reveal answer- AP = 3 B ; AP = 1000
- AP = 2 B ; AP = 100
- AP = 1 B ; AP = 10
- AP = 0 B ; AP = 1
- AP = -1 B ; AP =\(\frac{1}{10}\)
- AP = -2 B ; AP =\(\frac{1}{100}\)
- AP = -3 B ; AP =\(\frac{1}{1000}\)
Follow-up question: a geologist, taking a class on electronics, sees this mathematical pattern and remarks, “This is just like the Richter scale!” Explain what the geologist means.
Notes:Ask your students how these two systems of power gain expression (Bels versus ratios) compare in terms of range. Which system of expression encompasses the greatest range of power gains or losses, with the smallest changes in numerical value?
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Question 2 of 19
Manipulate this equation algebraically, so that we can convert power gains expressed in units of Bels, into ratios.
AP(ratio) = 10AP(Bels) Then, convert the following power gains, expressed as ratios, into units of Bels:
- AP = 250 ; AP =
- AP = 1275 ; AP =
- AP = 10 ; AP =
- AP = 1 ; AP =
- AP = 0.1 ; AP =
- AP = 0.025 ; AP =
- AP = 0.00009 ; AP =
Reveal answerAP(Bels) = logAP(ratio) - AP = 250 ; AP = 2.398 B
- AP = 1275 ; AP = 3.106 B
- AP = 10 ; AP = 1 B
- AP = 1 ; AP = 0 B
- AP = 0.1 ; AP = -1 B
- AP = 0.025 ; AP = -1.602 B
- AP = 0.00009 ; AP = -4.046 B
Notes:Challenge your students to estimate the log values without using their calculators. For example, they should be able to estimate the log of 1275 as being between 3 and 4; the log of 0.025 as being between -1 and -2. Work together to devise a technique for doing this, where there will be no guessing.
Mathematical estimation is an important skill for technical people to possess. Not only is it useful in the event no calculator is readily available, but it also helps greatly in students being able check their (electronically) calculated work. I can’t tell you how many times I’ve seen students blindly enter numbers into a calculator, only to arrive at an answer that is grossly in error, and not realize it at all because they cannot do the estimation mentally.
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Question 3 of 19
At some point in time, it was decided that the unit of the “Bel” was too large. Instead, the deci-Bel became the most common usage of the unit. Modify these equations to include AP figures cast in units of decibels (dB) instead of Bels:
AP(ratio) = 10AP(Bels) AP(Bels) = logAP(ratio) Then, calculate the decibel figures that correspond to a power gain of 2 (ratio), and a power loss of 50%, respectively.
Reveal answerAP(ratio) = 10[(AP(dB))/10] AP(dB) = 10 logAP(ratio) Power gain of 2 (ratio) ≈ 3 dB
Power loss of 50% (ratio) ≈ -3 dB
Notes:It is important that students work through the original equations algebraically to obtain the answers rather than just look up these formulae in a book. Have your students write their work on the whiteboard in front of the other students, so that everyone has the opportunity to examine the technique(s) and ask pertinent questions.
Be sure to let your students know that the figure of “3 dB”, either positive or negative, is very common in electronics calculations. Your students might remember this expression used to describe the cutoff frequency of a filter circuit (f−3 dB).
