Similarity Between Writing and Designing

Posted on 22-November-2010 by mathscinotes

Quote of the Day

The way to get good ideas is to get lots of ideas, and throw the bad ones away.

— Linus Pauling, Nobel Prize winner in Chemistry and Peace

I have always been interested in the actual process of design and its similarity to other tasks. I was reading an interesting blog post this morning on the process of writing. The author said that while writing, the good writer must really play three different roles while writing a document:

  • Visionary
    The Visionary develops the ideas that the document will convey.
  • Draft Worker
    The Draft Worker puts the ideas down on paper. The first draft is rough, but the ideas are down on paper where they can be worked with.
  • Editor
    The Editor is responsible for putting out the polished final product. This is a detail job and quality is king.

The design process for an engineer is really very similar. While designing a product, the good engineer must also play three roles.

    • Concept Developer
      All designs begin with some overall concept of how the product is supposed to work. The concept is complete when the overall approach is clear and all that is left are the details of implementation. These details are important, but they usually involve making choices that could go any of a number of ways. Examples would include connector pinouts, non-critical part selections, circuit card dimensions, etc.
    • Detailer
      The Detailer takes the concept and creates the first draft of the design. There are always numerous ways of implementing any concept and the Detailer must make all the detailed choices required to make a physical embodiment of the design.
    • Quality Assurance
      One the design is captured, now the completeness of the solution needs to be evaluated. All designs have quality criteria that they are evaluated against, like cost, power, performance, etc. There will a lot of editing going on during this phase.

Of course, in the real world the engineer (like the writer) will iterate between all these roles. For example, while playing the Detailer role, it is common for the engineer to find a problem with the concept. This means that engineer must flip back into the Concept Developer role and fix the problem.

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Straight, Level, and the Curvature of the Earth

Posted on 16-November-2010 by mathscinotes

Quote of the Day

Science is not about what's true or what might be true, science is about what people with originally diverse viewpoints can be forced to believe by the weight of public evidence.

— Lee Smolin, theoretical physicist

Introduction

Figure 1: A view across a 20-km-wide bay in the coast of Spain. Note the curvature of the Earth hiding the base of the buildings on the far shore. (Wikipedia)

Figure 1: A view across a 20-km-wide
Spanish bay. The curvature of the Earth
hiding the base of the buildings on the far
shore. (Wikipedia)

I frequently hear people make statements that somehow do not seem right. Today, during a discussion of laser levels, the topic of required accuracy came up. I heard a contractor state:

Who cares if the laser level is accurate within an 1/16th of inch at 100 feet? The Earth curves away from a horizontal line by 1/8th of inch for every 100 feet of horizontal distance. The Earth's curvature will swamp out your instrument error in less than 100 feet.

The statement about the curvature of Earth got me thinking. How much does the Earth's surface deviate from a horizontal line over a distance of 100 feet? The contractor's number intuitively seemed wrong because the Earth is round and the deviation from horizontal should be a function of distance. A little math will give me the answer. For consistency's sake, I will perform all computations in US customary units.

Analysis

Figure 1 illustrates the situation and contains the derivation of both an exact and a approximate solution. The triangle formed by x, R + δ, and R is a right triangle, which means that the Pythagorean theorem can be used to produce an exact solution. In addition, a simple approximation for δ is also developed assuming R >> x and using a linear approximation for the square root. In Appendix A, I give examples of the computations in Mathcad.

Figure 1: Calculation of Deviation from Horizontal.

Figure 1: Calculation of Deviation from Horizontal.

Given the situation shown in Figure 1, we can compute the deviation from horizontal as follows.

Eq. 1 \delta=\sqrt {{R^2}+{x^2}}-R\quad=\quad 2.389{\text{E-4 ft}}={\text{2.867E-3 in}}

where

  • R is the radius of the Earth (3963.2 miles)
  • x is the horizontal distance of interest (100 ft)

Conclusion

The contractor had stated that the curvature of the Earth causes level to deviate from horizontal by an 1/8th of an inch (125 thousandths of inch) for 100 feet of horizontal distance. The actual deviation is ~2.9 thousandths of an inch for 100 feet of horizontal distance, which is almost 45 times less than the contractor claimed. So it is meaningful to buy a laser level that is accurate to 1/16th of an inch over 100 feet, i.e. the laser level error is not swamped by the curvature of the Earth.

Why did the contractor make the claim that the Earth's curvature is 1/8th inch over 100 feet? He made a simple mistake. He did not understand that the deviation from horizontal for short distances is given by a square-law relationship, shown in Equation 2. In Equation 2, I include an approximation that is only valid when R is much greater than x, which is true in typical construction problems.

Eq. 2 \delta = \sqrt {{R^2} + {x^2}} - R \doteq R \cdot \left( {1 + \frac{{{x^2}}}{{2 \cdot {R^2}}}} \right) - R = \frac{{{x^2}}}{{2 \cdot R}}

If we use Equation 1 or Equation 2 to calculate the deviation from horizontal at 1 mile, we get 8 inches. This value is quoted in a number of references on surveying (e.g. here is one, here is another, yet another). What the contractor did was erroneously assume that the deviation varied linearly with distance, which would mean that a deviation of 8 inches at 1 mile is equivalent to an 1/8th of inch at 100 feet.

For those of you who may be interested in the related question of the error in horizontal distances caused by living on a spherical planet, see this blog post.

Aside: Here is an interesting discussion that references this web page.

Appendix A: Computation Examples

Figure 2 shows a few computation examples. Normally, I let Mathcad do the unit conversion, but I do show one example with explicit unit conversion.

Figure 2: Computation Examples.

Figure 2: Computation Examples.

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Posted in Construction | 143 Comments

The Importance of Role Models

Posted on 14-November-2010 by mathscinotes

Quote of the Day

To laugh often and much; to win the respect of intelligent people and the affection of children; to earn the appreciation of honest critics and endure the betrayal of false friends; to appreciate beauty, to find the best in others; to leave the world a bit better, whether by a healthy child, a garden patch or a redeemed social condition; to know even one life has breathed easier because you have lived. This is to have succeeded.

— Ralph Waldo Emerson

Introduction

Figure 1: Dr. Frank Schiebe, my neighbor in Osseo.

Figure 1: Dr. Frank Schiebe, my
neighbor in Osseo, MN. As a boy, I
used to visit him while he was working
in his garage. I always thought magic
was happening in that garage.

I have worked with engineers my entire career, which spans the last 31 years. During this time, I have noticed that almost all engineers went into engineering because of some important personal relationship early in their life. The same is true for me. I grew up during the 1960s in "Small Town USA." In many ways, this was a wonderful environment in which to grow up. Unlike today, in my home town all the different economic classes lived next to each other. They had widely different experiences in life, yet they had the common experiences of the Depression and World War II. My father was a common laborer, yet he lived next to a dentist, doctor, school district president, and university professor of electrical engineering. Sadly, I do not see this mixing of the economic classes today. It was important for me to see how all these people lived together. I babysat for them, mowed their lawns, and delivered their newspapers. I got to know all these folks very well. I smile just thinking about that time.

My Mentor

My story is about Frank Schiebe, the university professor of engineering. Frank died a couple of year ago and his high school recently posthumously gave him their 2010 Distinguished Alumni Award. I happened to be flipping through the local television channels recently when I saw that the local educational channel was playing a ceremony honoring Frank's life. Boy did that story bring back memories …

In many ways, Frank's story was like that of many engineers I have known. Frank grew up in a small town and wanted to attend the University of Minnesota, but his primary school work had been lackluster enough that a school counselor told him not to even bother applying. Of course, Frank did not listen, and he squeaked his way through the electrical engineering program and got his BSEE. In spite of weak grades, he was able to get into graduate school where he excelled and eventually graduated with a PhD in Electrical Engineering. During his university schooling, Frank had done a lot of work with computers and simulation. His first research was on using computers for simulating water flow, turbulence, and cavitation. He did ground breaking research in this area. Later, he worked with NASA on applying remote sensing to agricultural and natural resource management.

When I knew Frank during the 1960s, he was working on understanding cavitation and his research was very interesting to me. It turns out that I have encountered cavitation many times in my career, particularly when I worked on undersea vehicles. I first heard about Frank's research when he was over to my home and was having beer with my father. Frank and my father became good friends and frequently could be seen sitting outside of our home drinking beer. NASA's manned spaceflight program was in full gear at that time, and I was absolutely fascinated with it. Frank was a great source of information on how all that stuff worked, and I asked him questions constantly. I loved science and wanted to work in engineering like Frank, but I hated mathematics. I will never forget when Frank told me that I could not get anywhere in science without mathematics. I thought so highly of Frank that, grudgingly, I decided I had better get good at math. Because of Frank, mathematics eventually became my best subject. Like Frank, I became an electrical engineer. I owe him a debt of gratitude. My life would have been much different without him.

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Posted in Management, Osseo, Personal | Tagged , , | 5 Comments

Carpentry Math - Drawing an Ellipse

Posted on 14-November-2010 by mathscinotes

Quote of the Day

I think the surest sign that there is intelligent life out there in the universe is that none of it has tried to contact us.

— Calvin and Hobbes

Figure 1: Ellipses are Common in Carpentry (Source).

Figure 1: Ellipses are Common in Carpentry (Source).

Today, I received an issue of Fine Homebuilding Magazine (Dec/Jan 2011) that contained a bit of math that caught my eye. In this issue's "Tips & Techniques" section, there is a short piece called "A No-Math Method to Draw An Ellipse." Of course, this method does use mathematics, just no equations. It is a nice little geometric construction.

The basic construction approach shown here requires only a compass and square – tools in every carpenter's kit. Figure 2 illustrates the basic approach. First, draw two circles, one having a diameter equal to the ellipse's major axis and the other having a diameter equal to the ellipse's minor axis. You can locate points on the ellipse by drawing a number of lines through the axes' origin and both circles. For each line, draw perpendiculars from the line's intersection points on the outer and inner circles to the x and y-axes, respectively (see Figure 2).  The intersection of the perpendiculars determines the points on the ellipse. Once we have a sufficiently dense set of points on the curve, we can accurately draw an ellipse by interpolating between the points. Carpenters usually interpolate between the points using a thin strip of wood called a spline.

Figure 1: Ellipse Construction Using Compass and Square.

Figure 1: Ellipse Construction Using Compass and Square.

If you want a proof that this construction produces an ellipse, all you need is a bit of algebra.

x = a \cdot \cos \left( \theta \right),{\text{ }}y = b \cdot \cos \left( \theta \right)

\frac{x}{a} = \cos \left( \theta \right),{\text{ }}\frac{y}{b} = \sin \left( \theta \right)

{\left( {\frac{x}{a}} \right)^2} + {\left( {\frac{y}{b}} \right)^2} = 1{\text{, the equation of an ellipse}}

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The Case of the Mysterious Radio Interference

Posted on 13-November-2010 by mathscinotes

Quote of the Day

I mean the word proof not in the sense of the lawyers, who set two half proofs equal to a whole one, but in the sense of a mathematician, where half a proof is zero, and it is demanded for proof that every doubt becomes impossible.

— Karl Friedrich Gauss, quoted from Calculus Gems (1992) by George F. Simmons

Introduction

When you build an electronic product, at some point you will be accused of building a product that is interfering with someone else's product. In this particular story, a little bit of math quickly showed that the problem could not be with our product. Given this knowledge, we then embarked on a field trip to determine exactly what was the cause of the problem. The actual problem and its solution ended up being a surprise for everyone.

This particular story illustrates a number of points about solving field problems:

  • The first word from the field is always in error.
  • Get the right person on the problem right away or bad things are going to happen.
  • When bad things begin to happen, your ability to contain them is nil.

The story ends well, but solving the problem ended up being MUCH more painful than it needed to be. If one or two things had been different right up front, the problem would have been solved with little distress.

To make the rest of this discussion clear, I should mention that we make a product that mounts on the side of a home and that is deployed by local telephone and cable companies. These devices provide provide Internet access using Ethernet over fiber optic cable. Hundreds of thousands of these units have been deployed with no interference complaints — until now.

The Problem

The problem began for me with a phone call from a manager who handles our relations with a government regulator. A customer had complained to the regulator that one of our products was interfering with fire department communications. This is a very serious charge. However we have shipped these products for years with no interference issues of any sort. Something did not seem right. The government regulator had said that we needed to meet with this customer and resolve his issue. If the problem was not resolved, we would need to stop shipping product.

I was floored. Thankfully it would turn out that NOTHING we had heard from the field would turn out to be correct. However it was my job to prove that we were not the cause of the problem.

Information Gathering

The first thing I did was contact the customer with the complaint. Let's call him "Bill," which is not his real name. Bill was mad and wanted to make sure that I knew he was mad. Here is how Bill saw the problem.

  • He had received a number of complaints from his customers that our hardware was interfering with local fire department radio communications.
  • He had contacted our customer service and had been given the run-around.
  • He had contacted our product manager and had received some advice, but nothing they told him to do solved his problem
  • This had been going on for months.
  • He was sick and tired of the situation and had reported us to the government, which he felt would get our attention.

Bill certainly was correct about getting our attention. My approach in these situations is to play "Mr. Spock" – you know, the Vulcan from Star Trek. In an absolute deadpan tone of voice, I just keep asking good questions and eventually the angry customer will realize that you know what you are doing and that you want to help. I told him that I had just been informed of the problem and that I was confident we would get to the bottom of the issue. He began to cool down once I began to ask questions. Here is what my interview told me.

  • The interference was at 155.52 MHz – a sub-harmonic of GPON, a fiber optic transport. This relationship can be seen by noting that \text{155.52 MHz} = \frac{\text{1.244 GHz}}{8}.
  • Their emergency communications were assigned to this frequency by the FCC.
  • He had not observed the interference himself, but had only heard reports of it.
  • The fire department did not have any technical knowledge of their communications system. A local consulting engineer managed the system for them.

I asked him to contact the consulting engineer and get a first-person assessment of the interference. Meanwhile, I would check with the FCC and find out what I could about their emergency communication system.

Things Begin to Clear

Once I had their radio call sign, I could use the FCC radio web site to find out information about their emergency communications system. The information would be useful in my mathematical analysis.

  • The transmitter was licensed for 100 W of effective radiated power (ERP).
  • The antenna was on a hill about one 1 km out of town (it is a small town). When I actually got on-site, I saw that the antenna could be seen from all points within the town.
  • The communication system was designed to transmit a signal that would tell scanners that it was a communication channel.

Bill had also got busy and contacted the consulting engineer in charge of their emergency communication system. Bill soon got back to me and was sounding pretty sheepish. The consulting engineer told him that the fire department had not reported ANY interference. What had been reported was that one citizen who owns a scanner was complaining that our product was causing his scanner to work differently than it had before. Prior to using our product, his scanner would scan the emergency band for fire department communications. The scanner would only stop when there was an emergency transmission. When our product had been deployed, the scanner now would constantly stop on that frequency because it would detect the low-level GPON emission from our product. Since the GPON emission from our product is no different in level from that made by a normal PC's Ethernet at \text{156.25 MHz}=\frac{\text{1.25 GHz}}{8}, the scanner would have trouble with or without our product being there.

The interesting thing that the consultant said was that the fire department LIKED the situation. Their communications were working fine and they consider the local citizens who monitor their communications to be a nuisance. These local folks had the nasty habit of showing up at every call the local fire department made.

While on the FCC website, I noticed that some emergency communications in my state used the same frequency as Bill's fire department. These local communities also used our product, but I had never heard any complaints from these communities. I called the folks in charge of these communications and they confirmed that they had received NO reports of interference. Now I am really confused.

What was happening here? Why was Bill's fire department having no trouble at all and the local "ambulance chasers" were unhappy? Why had no one in my state with the same situation complained at all?

A little math soon shed some light on the situation.

Analysis

Determining the Field Strength of an Isotropic Radiator

While a detailed study of the field pattern of a real antenna is complicated, we really only need an "order of magnitude" analysis to illustrate that our product could not be the source of the problem. Let's begin by deriving an expression for the field strength from an omni antenna – an omni antenna radiates equally in all directions. This will be close enough for what we are doing here. Equation 1 gives us the power radiated in all directions from an isotropic radiator.

Eq. 1 {\sigma _P} = \frac{{{P_T} \cdot {G_T}}}{{4 \cdot \pi \cdot {r^2}}}

Where

  • σP is the power per steradian (solid angle unit)
  • r is the radial distance from the antenna
  • PT is the Effective Isotropic Radiative Power (EIRP)
  • GT is the antenna gain (= 1 for an isotropic radiator)

We can use Poynting's Theorem (Equation 2) to relate the radiated power to electric field strength and magnetic intensity.

Eq. 2 {\sigma _P} = E \cdot H

Where

  • E is the electric field strength
  • H is the magnetic intensity

In Equation 3, we use the impedance of free space to relate H to E.

Eq. 3 H \doteq \frac{E}{{120 \cdot \pi }}

We can substitute Equations 2 and 3 into 1 to get Equation 4.

Eq. 4 \frac{{{P_T} \cdot {G_T}}}{{4 \cdot \pi \cdot {r^2}}} = \frac{{{E^2}}}{{120 \cdot \pi }}

We can solve Equation 4 to for E to get Equation 5, an expression for the radiated electric field strength from an omni antenna.

Eq. 5 E = \frac{{\sqrt {30 \cdot {G_T} \cdot {P_T}} }}{r}

We can use Equation 5 to compare the field strength from our product to that from Bill's fire department's antenna.

Product Radiated Power Versus Fire Department Radiated Power

Field Configuation

I had been told that that problem was occuring in town. I wanted to understand how the radio signal from our product compared to that of the fire department in town. While still in the office, I used some information from Bill and Google Earth to determine that:

  • Our product was usually mounted about 30 meters from the scanner antenna
  • The fire department's antenna was about 1 km from the scanner antenna

Estimating Product Emission Level

Our product emission levels at 155.2 MHz were measured at a government-approved facility to be 30 dBµV/m at 3 meters. Since field strength varies linearly with distance from the source, the field strength from our product at 30 meters must be 1/10th the value at 3 meters or 3 dBµV/m.

Estimating Emergency Transmission Field Strengths

We can use Equation 5 to compute the field strength from the fire department's antenna. The calculation is illustrated in Equation 6.

Eq. 6 E = \frac{\sqrt{ 30 \cdot 1 \cdot 100}}{r} = \text{55,000 }\frac{\text{dB}\mu\text{V}}{m}

This means that the signal from the fire department is more than 17,000 times stronger than the emissions from our product. There is no way that our product is interfering with their reception. It turns out that this analysis was absolutely correct.

My Field Trip Results

When I visited Bill, things got interesting. Here is a quick summary of what I discovered on my field trip.

  • Our test gear showed that our math estimates were right on. In the town, the fire department antenna produced a monster signal compared to our product.
  • We identified emissions from our product and similar emissions from every computer, router, and switch in town. All emissions measured were within FCC limits.
  • I purchased both an analog and digital scanner so that I could see the problem myself. The analog scanner, when set at maximum sensitivity (minimum squelch), would stop scanning at 155.2 MHz when near our product. It would also stop at Ethernet emission frequencies, like 156.25 MHz, when near a PC, router, or switch. Newer digital scanners had no issue at all. Digital scanners filter out unwanted interference by identifying a special code that emergency communications systems put out. They will never lock on our product's emissions because we do not emit that special code by adjusting their squelch level up slightly.
  • It turned out that no person in Bill's community had reported any interference issues. Only one person could be found that actually had a problem and he was many kilometers outside of the city. He had an old analog receiver that was set for maximum sensitivity. This gentleman said that he needed his receiver to be at maximum sensitivity because he could just barely receive the fire department transmissions under the best of circumstances. Of course, our product had been mounted in the worst possible location with respect to interference (right next to his receiver input — 1 meter instead of 30 meters).

I now had all the information I needed to state with confidence that I had identified the root cause of the problem.

Conclusion

When I returned home I contacted my state emergency communication coordinator who informed me that my state had converted to digital emergency communications many years before. Therefore, the scanners users near my home had converted to digital scanners in the years before our products came out. There were either no more analog receivers in my area or their squelch levels had been turned up slightly to ignore the interference from any Ethernet-based product. These same squelch settings would eliminate problems with GPON emissions, too.

What happened was that Bill had one customer way out of town with an analog receiver. At his distance, the emission from our product was smaller than that of the fire department's antenna, but still large enough to be detected by his scanner (it was set at maximum gain). We informed this customer that he could try adjusting his squelch, but he refused to do that because he said increasing the squelch level would reduce his receiver's sensitivity. He felt he needed all the gain he could get to receive the fire department's signal clearly because it was weak at this distance. After reviewing his specific situation, our product had been mounted in the worst possible place relative to his antenna. We were confident that if we moved our product a few feet away from his receiver that we would eliminate the interference. Bill agreed to discuss this with the gentleman, but he was concerned that his customer may have imposed limits on where he could mount our product.

It turned out that people in town had long ago adjusted their squelch levels up to avoid interference from local computers and other Ethernet-based equipment. In fact, many of the local scanner operators had recently switched to digital scanners when their state switched to digital emergency communications. The scanner code used by the fire department was published on the web and anyone could program their digital scanner to stop only on fire department transmissions.

I told Bill that I felt that we had met our regulatory obligations and that the one person with a complaint could easily deal with the problem by moving the position of the product. Bill agreed and said that he would withdraw his complaint with the government regulator. The case was closed.

Clearly, this situation had been blown way out of proportion. Because the right people had not been put on the problem right away, actually solving the problem was taking forever and Bill had grown exasperated. In a desperate attempt to get some effective help and satisfy this one unhappy customer, Bill complained to the government. I guess his approach worked because he got me on site. As in the game "Telephone," the magnitude of the problem had grown with each retelling. By the time it got to me, it sounded like their fire department communications were totally jammed and emergency services were completely disrupted. In fact, it was an extremely minor problem with one very vocal person.

As I said in my introduction, this story illustrates my three points on dealing with field problems.

  • The first word from the field is always in error.
  • Get the right person on the problem right away or bad things are going to happen.
  • When bad things begin to happen, your ability to contain them is nil.
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Measuring the Ring Voltage on a Telephone

Posted on 8-November-2010 by mathscinotes

Quote of the Day

Money can't buy happiness, but it can make you awfully comfortable while you're being miserable.

— Clare Boothe Luce

Introduction

Figure 1: Ringer For An Old Landline Phone.

Figure 1: Ringer For An Old Landline Phone (Source).

I received an email yesterday from a sales engineer who was having difficulty measuring the ring voltage on one of our telephone circuits. The numbers he was getting did not agree with what my engineering group had measured. The discrepancy had to do with the shape of the ring voltage waveform and what a standard voltmeter actually measures. As with many engineering problems, developing a practically useful solution depends on coming to an agreement on definitions and determining what the instruments are really measuring. This is a harder thing than one might think and this problem provides a nice illustration of basic problem solving.

Analysis

Ring Voltage Specification

The ring voltage on a telephone is specified as a Root-Mean-Square (RMS) voltage. Electrical engineers like to use RMS voltages for varying waveforms because the RMS voltage can be thought of as the equivalent DC voltage with respect to producing power in the load. Many people think of RMS voltage as a form of "average" voltage, even though it really is the average of the square of the voltage.

The Wikipedia defines the RMS voltage of a periodic signal in terms of an integral (Equation 1).

Eq. 1 {V_{RMS}} = \sqrt {\frac{1}{T}\int\limits_0^T {v{{\left( t \right)}^2}dt} }

where T is the period of the signal and v(t) is the function that describes the waveform.

Ring Voltage Waveform

Figure 2 illustrates the ring voltage waveform that the sales engineer was dealing with. Trapezoidal waveforms in telephony are common.

Figure 1: Example of a Trapezoidal Telephone Ring Voltage

Figure 2: Example of a Trapezoidal Telephone Ring Voltage

The voltage waveform actually goes both positive and negative, but the polarity does not matter when calculating power into a resistive load. We will work here with the positive side.

Computing the RMS Voltage of a Trapezoidal Waveform

Equations 2-4 illustrate how to derive and expression for the RMS voltage of a trapezoidal waveform in terms of k and T, which are defined in Figure 2. This derivation assumes that the triangular portions of the trapezoid are identical.

Eq. 2 V_{RMS}^2 = \frac{1}{T}\int\limits_0^T {v{{\left( t \right)}^2}dt} = \frac{2}{T} \cdot \int\limits_0^k {{{\left( {\frac{{A \cdot t}}{k}} \right)}^2}dt} + \frac{1}{T} \cdot \int\limits_0^{T - 2 \cdot k} {{A^2}dt}
Eq. 3 V_{RMS}^2 = 2 \cdot \frac{{{A^2} \cdot k}}{{3 \cdot T}} + \frac{{{A^2}}}{T} \cdot \left( {T - 2 \cdot k} \right) = {A^2} \cdot \left( {1 - \frac{4}{3} \cdot \frac{k}{T}} \right)
Eq. 4 {V_{RMS}} = A \cdot \sqrt {1 - \frac{4}{3} \cdot \frac{k}{T}}

Crest Factor

Just to complicate matters, the ring voltage is actually controlled by two specifications: an RMS voltage level and a Crest Factor (CF). CF is defined shown in Equation 5.

Eq. 5 CF \triangleq \frac{{\left| {\max \left( {v(t)} \right)} \right|}}{{{V_{RMS}}}}

CF is commonly used because many electronic devices are sensitive to peak-to-average ratios, which CF measures. Telephony specifications require that 1.2 ≤CF ≤ 1.6.

Correction Factor for a Peak Detecting Meter

While there are electronic voltmeters that measure true RMS voltages for any waveform, the sales engineer in this case is using an electronic voltmeter that assumes the operator is always measuring a sinusoidal signal. It measures the peak voltage of the voltage waveform and divides that value by the square root of 2, an approach which produces the correct RMS value for a sinusoidal waveform but not a trapezoid (see Equation 6).

Eq. 6 {V_{Meter}} = \frac{A}{{\sqrt 2 }}

Equations 7-8 show how to calculate a correction factor for adjusting the meters reading to give the RMS voltage for a trapezoid.

Eq. 7 {V_{RMS}} = \frac{{{V_{RMS}}}}{{\max \left( {\left| {v(t)} \right|} \right)}} \cdot \frac{{\max \left( {\left| {v(t)} \right|} \right)}}{{\sqrt 2 }}.\sqrt 2
Eq. 8 {V_{RMS}} = \frac{{{V_{Meter}}}}{{CF}} \cdot \sqrt 2

A Field Example

In this particular case, the field engineer was measuring 60 V for something that Engineering said was 65 V. We can summarize the critical variables as follows:

  • Trapezoid voltage peak voltage (A), is 86 V (set by the telephone ringer circuit design).
  • CF is 1.33 (set by Engineering in the firmware)
  • Value measured in the field, VMeter, is 60 V.

We can compute the correct RMS value by using Equation 8. This calculation is shown in Equation 9.

Eq. 9 {V_{RMS}} = \frac{{{V_{Meter}}}}{{CF}} \cdot \sqrt 2 \doteq 60{\text{ V}} \cdot \frac{{1.414}}{{1.333}} = 65{\text{ V}}

This shows that the result measured by the sales engineer and my engineering group were actually the same.

Posted in Electronics | Tagged , , | 15 Comments

Personality Characteristics of a Great Engineering Hire

Posted on 6-November-2010 by mathscinotes

Quote of the Day

Acting was fun, but my grandfather would always tell me, "It's never too late to be an engineer." You were supposed to get a "job" and do acting on weekends or at school.

— Jon Hamm

Introduction

I was at dinner last night with some neighbors and the topic of hiring came up. My most important job is hiring the right engineering talent. When it comes to engineers, people frequently focus on their technical skills. While that is important, their ability to work with others is just as important. There are also some personality characteristics that I look for as well. The best engineers that I have worked with have these personality characteristics in common.

Personality Metrics

I am an engineer who writes a math and science-oriented blog, so I have to have some form of math even for personality characteristics. I evaluate the personality characteristics of an engineer using three personality metrics:

  • Anality
  • Pisstivity
  • Prima Donnas Per Square Foot Factor

Anality

I frequently hear people say that they do not want to get wrapped up in the details of a problem. I always respond that my job is a celebration of detail. Many an engineering project has been destroyed because minor details were not attended to. I expect an engineer to be detail-oriented. The really good engineers know how to move from high-level to low-level thinking as the situation requires.

Pisstivity

Pisstivity is about owning problems. Anyone who has used contract labor understands this problem. Contractors can be very skilled, but they usually are not personally invested in the problems they are working on. Some employees have the same attitude. I look for engineers who have a history of looking for problems, grabbing them, educating themselves about the problem, and owning it all the way to the solution.

Many times I have had to ask someone to take on a problem in an area for which they have no skill. That is just the way life is sometimes. I have always been impressed with those staff members who, instead of griping, treat the situation as an opportunity to expand their skills and develop a new area of expertise. Managers cherish these employees.

Prima Donnas Per Square Foot Factor

This is really my "do they work well with others" index. Nothing disrupts a finely tuned engineering group more than a prima donna. Sometimes you need them but they need to be spread thinly – very thinly. The best engineers have egos, but they have been humbled enough that they know that nature is complicated and they just might be wrong. They also find ways to make all the people around them better, while the prima donna is all about him.

Conclusion

An engineering manager's most important task is hiring. The hiring of the right engineer can be a company-changing event. I have numerous stories of an engineer being hired by a company who literally changed the direction of the entire business. I will relate one story here.

When I was at HP in the old days, I remember a manufacturing engineer who thought that HP should be making low-end laser printers. His managers told him to go away. He then went off and bought a Canon copier mechanism and, on his own time and with his own money, built the first low-end HP laser printer. Grudgingly, management told him he might have something there and they turned it into a product. Today, HP is the world's largest manufacturer of printers. It all started with a young manufacturing engineer who would not give up. How is that for pisstivity!

Posted in Management | Tagged , , | 7 Comments

GPS and Relativity

Posted on 5-November-2010 by mathscinotes

A friend just sent me some viewgraphs from a presentation that he recently attended on the history of the Global Positioning System (GPS). The presentation was given by Hugo Fruehauf, one of the key GPS developers. In this presentation, he discussed the effect of relativity on GPS timing accuracy. I thought the following graph was worth discussing here.

Relativistic Effect on GPS Clocks (Source: H. Fruehauf)

Relativistic Effect on GPS Clocks (Source: H. Fruehauf)

This presentation initiated a number of interesting discussions at work. The most interesting question was why do general relativity and special relativity introduce errors of opposite sign? Here is my take on the subject.

  • General relativity
    This introduces a positive frequency error because a clock on earth is subject to a stronger gravitational field than the satellite. This means that the clock on earth will run slower than the clock on the satellite. Stated another way, the clock on the satellite will run faster than the same clock on the earth.
  • Special relativity
    This introduces a negative frequency error because a clock on the satellite is moving relative to the same clock on the earth. This means the clock on the satellite appears to run slower than the same clock on earth.


    • In my engineering career, I have only encountered relativistic effects once before. That was with modeling the behavior of fast-moving electrons in a cathode-ray tube (CRT). The increase in the mass of an electron after being accelerated by the electron gun needed to be accounted for during the design of a graphics display system.

Posted in Astronomy, Navigation | Tagged , , | 1 Comment

A Product Cost Reduction Example

Posted on 23-October-2010 by mathscinotes

Introduction

Maintaining good product margins is crucial to maintaining a healthy business. In the electronics business, customers have come to expect prices to drop every year. This means that I need to incorporate frequent cost reductions just to maintain margins.

Just to remind everyone, gross margin is defined as shown in Equation 1.

Eq. 1 GM \triangleq \frac{{{\text{Profit}}}}{{{\text{Price}}}} = \frac{{p - c}}{p}

where p is the average selling price, and c is the unit product cost. We normally need to maintain margins above 40% to have a viable product.

I often say that all the interesting questions in life have the answer "it depends…." The same is true with cost reductions. Reducing the cost of a product requires some engineering activity. If you are working on a cost reduction, your engineering staff is not working on new products that will expand your market. So there is a balance that must be achieved between keeping your current product lines financially healthy and working on new products to expand your markets. Stated another way, working on cost reductions has an opportunity cost that must be balanced with the need for new product development.

We recently went through a quick evaluation of a potential cost reduction and chose not to implement it. The analysis that went along with this evaluation was interesting and worth documenting here. In the interest of confidentiality, I have changed all the numbers used in this example from the actual work example.

Analysis

Problem Description

The critical problem characteristics are described below:

  • We are building 50K per year of a product (Q= 50K).
  • The unit product cost is $150 (p=$150).
  • We have identified a way to reduce the product cost by 10% (Δc%=10%).
  • We work to maintain gross margins of 40% (GM = 40%).
  • The cost reduction effort will require an investment of ~$150K (Ex = $150K).
  • Net profit of 10% (NP% = 10%)
  • The product has a life of about 2 more years after the development effort is complete.

After you have implemented the cost reduction, there are two approaches people normally discuss with respect to pricing:

  • You can reduce your price and try to capture greater market share.
  • You can maintain your price and increase your margin.

In my markets, neither of these options ever really occurs. What normally happens is that I see an erosion of prices every year because my competitors are constantly incorporating cost reductions. This means that I need to incorporate regular cost reductions just so that I can maintain my margins in the face of downward pressure on prices. So I am battling to simply maintain my margins and market share. This is the situation that I will be modeling here.

Decision Criteria

Most people use one of three financial measures to evaluate the worth of an engineering effort:

The approach that I will use here is payback time. The cost reduction is of less value to me the longer it takes to earn the profit needed to pay for the cost reduction investment. My approach is to compare the lost profit flow from not doing the cost reduction to the cost of implementing the cost reduction. This is a value measure that most people understand intuitively. NPV and ROI require more discussion and will be a topic of later blogs.

Some Economics

I need a way to estimate the impact of changes in product price on my annual shipment quantities. Economists have defined a term called the price elasticity of demand (EP) to model the impact of a price change. To understand elasticity, let's begin with a definition of price elasticity of demand from the Wikipedia.

Elasticity
The percentage change in quantity demanded in response to a one percent change in price (holding constant all the other determinants of demand, such as income).

The mathematical definition of elasticity is given below.

Eq. 2 {E_P} \triangleq \frac{{\frac{{dQ}}{Q}}}{{\frac{{dp}}{p}}}

where

  • \frac{\Delta Q}{Q} is the percentage change in yearly shipment quantities.
  • \frac{\Delta p}{p} is the percentage change in average selling price.

To illustrate the use of EP consider a product and a market with an EP=-0.5. This means that a 10% reduction in price (Δp%=-10%) means a 5% increase in the quantity demanded ({\Delta Q \%}={E_P} \cdot {\Delta p \% = \text{-0.5} \cdot \text{10\%}}= \text{5\%} ).

Real companies almost never know very well how sensitive their shipment rates are to price. Usually, you end up guessing what the elasticity is. It turns out that the guesses are often pretty good. Years ago, I read Malcolm Gladwell's "The Wisdom of Crowds," which pushed the idea that groups of people have a form of collective wisdom that should not be underestimated. My experience is that his belief is correct.

Details

We begin our analysis by estimating the loss of annual shipment quantities if we cannot lower prices because we do not implement the cost reduction.

Eq. 3 {E_P} \doteq \frac{{\frac{{\Delta Q}}{Q}}}{{\frac{{\Delta p}}{p}}} = \frac{{\frac{{\Delta Q}}{Q}}}{{\Delta p\% }} \to \Delta Q = Q \cdot {E_P} \cdot \Delta p\%

where

  • Δp% is the change in price (10%)
  • Q is our annual shipment quantity (50K).
  • EP is the price elasticity of demand (-0.5)
  • ΔQ is the change in quantity shipped.

From Equation 1, we can see that Δp% = Δc%. We can compute our change in revenue using Equation 4.

Eq. 4 \Delta R = \Delta Q \cdot p \cdot \Delta p\% = \Delta Q \cdot p \cdot \Delta c\%

where ΔR is the change in revenue. The current price can be calculated by solving Equation 1 for p, which is shown in Equation 5.

Eq. 5 p=\frac{c}{1-GM}

In Equation 6, we see that the total profit flow, ΔP, can be computed from ΔR by using net profit (NP).

Eq. 6 {\Delta P} = NP\% \cdot {\Delta R}

The money invested in cost reduction will be made up by retained profits in the time ΔT, which is given by Equation 7.

Eq. 7 {\Delta T} = \frac{{Ex}}{{\Delta P}}

When we substitute the given values into these equations, we see that ΔT=24 years. With two years of product life remaining, we will never come close to recouping our initial investment. This cost reduction is not worth the effort.

Figure 1 illustrates the calculations.

Figure 1: Example Calculations.

Figure 1: Example Calculations.

Conclusion

We decided not to incorporate this cost reduction. Instead, we spent our time working on new products that will expand our existing market.

Posted in Financial, Management | Tagged , , , | Comments Off on A Product Cost Reduction Example

Magic Number Analysis – Converting Spectral Width to Sigma

Posted on 12-October-2010 by mathscinotes

Introduction

Engineering seems to have a lot of "magic numbers" – numbers used in equations with no explanations of where they come from. I REALLY do not like magic numbers because years from now some other engineer will be staring at this equation and asking the same question I just did – "Where did this number come from?" I encountered one this morning and I thought it would be worthwhile to document where it came from.

The magic number is "6.07." I saw a web site that recommended dividing the spectral width of a laser by 6.07 when performing some basic calculations. This blog describes where the number comes from.

Analysis

Side-Mode Suppression Ratio and Spectral Width Definitions

I was looking at at laser specification and it contained two numbers of interest: Side-Mode Suppression Ratio (SMSR) and spectral width. SMSR is a measure of the wavelength purity of a laser's output. It is defined as the ratio of the power in the dominant mode to the power in the strongest side mode. Spectral width (\Delta\lambda is the wavelength range between the points at the SMSR level. Figure 1 illustrates how the measurement is made.

Figure 1: Illustration of SMSR and Spectral Width.

Figure 1: Illustration of SMSR and Spectral Width.

Figure 2 shows the spectrum for a laser with a 45 dB SMSR (Source: Gipo).

Figure 2: Wavelength Spectrum of an Actual Laser (Source: Gipo).

Figure 2: Wavelength Spectrum of an Actual Laser.

Laser Spectral Modeling

We often model the output spectrum of a laser using the normal curve, which makes sense when there is a main lobe and minimal side-lobe structure, like Figure 2. A normal curve is characterized by three numbers:

  • peak value (A)
  • mean wavelength (\mu)
  • standard deviation (\sigma)

The SMSR is used to estimated the standard deviation of the normal curve that best models the laser spectrum.

Spectral Width

There are two common ways to measure the spectral width.

  • Full-Width Half Maximum (FWHM – Generally used only with FP lasers)
    This approach measures the pulse width at the 50% power points.
  • Side Lobe Suppression Ratio (SMSR – Generally used only with DFB lasers)
    This approach measured the pulse width from the points that are 20 dB down.

These approaches are designed to make measurement simple, but they do not directly give you a standard deviation. A conversion is required. Since I am most interested DFB lasers, I will focus on converting an SMSR-based \Delta\lambda measurement to a standard deviation.

- 20 = 10\log \left( {\exp \left( {\frac{{\lambda _{ - 20{\text{ dB}}}^2}}{{2 \cdot \sigma _\lambda ^2}}} \right)} \right)

{10^{ - 2}} = \exp \left( {\frac{{\lambda _{ - 20{\text{ dB}}}^2}}{{2 \cdot \sigma _\lambda ^2}}} \right)

\ln \left( {{{10}^{ - 2}}} \right) = \frac{{\lambda _{ - 20{\text{ dB}}}^2}}{{2 \cdot \sigma _\lambda ^2}}

\frac{{{\lambda _{ - 20{\text{ dB}}}}}}{{{\sigma _\lambda }}} = \pm \sqrt {2 \cdot \ln \left( {{{10}^{ - 2}}} \right)} = \pm 3.035

where \lambda_{-20 dB} is one of the two wavelengths where the spectral amplitude is down -20 dB from the peak. We define the \Delta\lambda as the difference between the two \lambda_{-20 dB} values. Figure 3 illustrates the measurement.

Figure 3: Illustration of Critical Normal Curve Parameters.

Figure 3: Illustration of Critical Normal Curve Parameters.

So to convert \Delta\lambda to \sigma_\lambda, simply divide the \Delta\lambda value by 6.07 as shown below.

{\sigma _\lambda } = \frac{\Delta\lambda}{6.07}

Conclusion

The post shows that the standard deviation of a laser's spectrum can be computed by dividing that laser's spectral width (i.e. the wavelength range between points 20 dB down from a laser's spectral peak) by 6.07.

Posted in Fiber Optics | Tagged , , , | 4 Comments