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

Solar Eclipse Math

Posted on 11-October-2010 by mathscinotes

Quote of the Day

You measure the size of the accomplishment by the obstacles you had to overcome to reach your goals.

— Booker T. Washington

For more on the up-coming eclipse, see this post.

Introduction

A couple of weeks ago, I was watching the Wonders of the Solar System with Brian Cox on the Science channel.In this episode, he was talking about the Moon and solar eclipses. He made a comment that the region of totality (i.e. complete darkness) during a solar eclipse is only a few hundred kilometers across. To illustrate this point, I found a great picture (Figure 1) taken from the Mir space station of the moon's shadow on the Earth.

Moon's Shadow Moving Across the Earth (Source: Mir, 1999)

Figure 1: Moon's Shadow Moving Across the Earth (Source: Mir, 1999)

I have never been through a total eclipse, but I must admit that I have always found them interesting. Maybe I can talk my lovely bride into taking a solar eclipse cruise? She still has not responded to my request for a vacation touring World War II Pacific battefields. Aren't I romantic? Anyway, I started to wonder whether I could verify the statement that Brian Cox made during the program.

I began my little exercise by looking for an exact number for the size of the region of totality and some details on the geometry of the situation. It did not take long. In the book, Historical Eclipses and Earth's Rotation by Stephenson, I found the following statement.

Under a vertical sun the umbra can never exceed about 270 km in diameter. However, at lower solar altitudes the elongated shadow of the Moon may be much wider than this, occasionally exceeding 500 km.

This quote is completely consistent with Brian Cox's statement and gives me a bit of information on the geometry of the situation. Let's see if a little geometry can verify this statement.

First, A Few Definitions

Let's work with the definitions from dictionary.com.

Totality
The quality or state of being total; as, the totality of an eclipse.For a solar eclipse, totality is the state of the sun being completely obscured to an observer by the moon. Totality depends on the position of the observer.
Umbra
A region of complete shadow resulting from the total obstruction of light by an opaque object, especially the shadow cast by the moon onto the earth during a solar eclipse.

Analysis

Umbra Geometry

Figure 2 illustrates the basic solar eclipse geometry.

Figure 2: Solar Eclipse Geometry

Figure 2: Solar Eclipse Geometry

 

The situation in Figure 2 produces the smallest region of totality on the earth. A little high-school trigonometry gives me the following equation.

\sin \left( \alpha \right) = \frac{{{r_{Sun}} - {r_{Moon}}}}{{{d_{Sun}} - {d_{Moon}}}} \Rightarrow \alpha = \arcsin \left( {\frac{{{r_{Sun}} - {r_{Moon}}}}{{{d_{Sun}} - {d_{Moon}}}}} \right)

Next, I will use this angle to compute the length of the umbra (dUmbra).

{d_{Umbra}} = \frac{{{r_{Moon}}}}{{\sin \left( \alpha \right)}}

Umbra Spot Size on the Earth

Figure 3 illustrates how I estimated the umbra spot size on the Earth. I am ignoring the curvature of the Earth because the spot is so small and the Earth is so big.

Figure 3: Illustration of Umbra on the Earth

Figure 3: Illustration of Umbra on the Earth

Again, a little bit of trigonometry gives me the following equation.

{s_{Umbra}} = 2 \cdot \left( {d_{Umbra} - \left( {{d_{Moon}} - {r_{Earth}}} \right)} \right) \cdot \tan \left(\alpha\right)

s_{Umbra} is roughly the diameter of the umbra spot on the Earth. This completes my derivation.

Solar Eclipse Data

Solar Eclipse Data
Symbol Description Value Units
rSun Radius of the Sun 696,000 km
rMoon Radius of the Moon 1738 km
dSmax Maximum distance from the Sun to the Earth 152,100,000 km
dMmax Maximum distance from the Moon to the Earth 356,400 km
rEarth Radius of the Earth 6378 km

Results

Simply substitute the eclipse data into the equations and you get 273 km for the maximum diameter region of totality when the sun is vertical with respect to the observer. As far as I am concerned, this analysis confirms both Cox's and Stephenson's statements. This was a nice little problem.

Here is a screen capture of my calculations.

Here is a copy of these calculations in Excel.

Posted in Astronomy | Tagged , , | 32 Comments

Dispersion Power Penalty Modeling (Part 3)

Posted on 8-October-2010 by mathscinotes

Deriving Equation 2

Equation 2 is derived from Equation 7 by noting the following items.

  • A true normal pulse has infinite length, so we cannot have a high speed data system that sends true normal pulses.
  • A common choice is to select a bit time that will contain 95% of the bit energy at the point of transmission.
  • If you set the bit time equal to ±2\sigma_0 about the signal peak (4 \cdot \sigma_0 total), the bit time will contain 95% of the normal pulse's energy.
  • This choice is much more conservative than the choice in Equation 1, which can be shown to only have 31% of the pulse energy contained in the bit time.

Given these assumptions, we can state Equation 10 directly.

Eq. 10      \sigma_0 = \frac{T}{4} = \frac{1}{4 \cdot B}

We can substitute Equation 10 into Equation 7 to obtain Equation 2.

         P{P_D} = 10 \cdot \log \left( {\sqrt {1 + {{\left( {\frac{{D \cdot L \cdot {\sigma _\lambda }}}{{\frac{1}{{4 \cdot B}}}}} \right)}^2}} } \right)

         P{P_D} = 5 \cdot \log \left( {1 + {{\left( {4 \cdot B \cdot D \cdot L \cdot {\sigma _\lambda }} \right)}^2} } \right)

Deriving Equation 3

Equation 3 is derived from Equation 7 by noting the following items.

  • Select a bit time that will contain 95% of the signal energy within the bit time at the point of reception.
  • If you set the bit time equal to \pm 2 \cdot \sigma_0 about the signal peak (4 \cdot \sigma_0 total), the bit time will contain 95% of the normal pulse's energy.

This derivation is a bit more complicated than the previous derivations because we need to calculate \sigma_0 given that the signal \sigma at distance L has been stretched by the factor k, which is computed using the following equation.

         k = \sqrt {1 + {{\left( {\frac{{D \cdot L \cdot {\sigma _\lambda }}}{{{\sigma _0}}}} \right)}^2}}

Given the factor k, we can compute \sigma_0 as follows.

         \sigma\left(\text{Distance = L}\right) = \frac{1}{{4 \cdot B}} \Rightarrow {\sigma _0} = \frac{1}{{4 \cdot B \cdot k}}

Substituting this equation into Equation 7 gives us the following.

         {k^2} = 1 + {\left( {\frac{{D \cdot L \cdot {\sigma _\lambda }}}{{\frac{1}{{4 \cdot B \cdot k}}}}} \right)^2} = 1 + {\left( {4 \cdot B \cdot D \cdot L \cdot {\sigma _\lambda } \cdot k} \right)^2}

This equation can be solved for the k to yield

         k = \sqrt {\frac{1}{{1 - {{\left( {4 \cdot B \cdot D \cdot L \cdot {\sigma _\lambda }} \right)}^2}}}}

This equation does not only represent the elongation of the pulse – it also represents the reduction in amplitude of the pulse. Therefore, we simply convert this equation to dB to get Equation 3.
             P{P_D} = 10 \cdot \log \left( {\sqrt {\frac{1}{{1 - {{\left( {4 \cdot B \cdot D \cdot L \cdot {\sigma _\lambda }} \right)}^2}}}} } \right) = - 5 \cdot \log \left( {1 - {{\left( {4 \cdot B \cdot D \cdot L \cdot {\sigma _\lambda }} \right)}^2}} \right)

This equation is very commonly used. It is different from the Equations 1 and 2 in that it contains a singularity. This singularity occurs because there are distances beyond which the dispersion is so severe that 95% of the bit energy cannot be contained in a bit time at the receiver. This characteristic has confused many people because they are not used to thinking about ISI at the receiver.

Deriving Equation 4

Equation 4 is Equation 3 after applying a Taylor series approximation for the reciprocal of the square root. The approximation is shown below and is only valid for small x.

         \frac{1}{{\sqrt {1 - {x^2}} }} \approx 1 + \frac{1}{2} \cdot {x^2}

This approximation can used to simplify the PP_D a bit.

        PP_D = 10 \cdot \log \left( {\frac{1}{\sqrt {1 - {\left( 4 \cdot B\cdot D \cdot L \cdot \sigma_\lambda \right)}^2}} }\right) \approx 10 \cdot \log \left( {1 + \frac{1}{2} \cdot {{\left( {4 \cdot B \cdot D \cdot L \cdot {\sigma _\lambda }} \right)}^2}} \right)

This gives us Equation 4. It is important to remember that this equation is only valid when the dispersion power penalty is small. I do not view this as a serious restriction because I would not deploy a system with a large dispersion power penalty.

Conclusion

I know this was long, but it will be useful. Here is what was accomplished:

  • The commonly used dispersion power penalty formulas were put in one place.
  • Their underlying assumptions were examined in detail.
  • It was demonstrated that they all come from the same basic equation.

My next blog posts will be shorter and less specialized.

Posted in Fiber Optics | Tagged , , | 2 Comments

Dispersion Power Penalty Modeling (Part 2)

Posted on 7-October-2010 by mathscinotes

Modeling Pulse Distortion

Choice of Pulse Basis Function

As with most modeling problems, it is very important to choose a function that accurately represents your physical signal. The most commonly used pulse models in optics are based on the normal curve. The normal curvea is described by the following equation.

N(t,\mu,\sigma) = \frac{1}{{\sqrt {2 \cdot \pi \cdot } {\sigma }}}\exp \left( {-\frac{\left(t-\mu\right)^2}{{2 \cdot {\sigma}^2}}} \right)

where \sigma is the standard deviation and \mu is the mean of the normal curve. The Wikipedia has a nice graph of the normal curve, which I include in Figure 2.

Graph of Normal Curve (Source:Wikipedia)

Graph of Normal Curve (Source:Wikipedia)

In this particular work, the pulse has its time (function f) and wavelength(function h) functions each modeled by different normal curves.

h(\lambda,\lambda_c, \sigma_\lambda) = \frac{1}{{\sqrt {2 \cdot \pi \cdot } {\sigma _\lambda}}}\exp \left( -{\frac{{{\left(\lambda-\lambda_c\right)^2}}}{{2 \cdot {\sigma _\lambda}^2}}} \right)

Eq 4.   f(t,T,\sigma_T) = \frac{1}{{\sqrt {2 \cdot \pi \cdot } {\sigma _T}}}\exp \left( -{\frac{{{\left(t-T\right)^2}}}{{2 \cdot {\sigma _T}^2}}} \right)

where t is time, T is the fiber travel time for the center of the pulse, \lambda is the wavelength, \lambda_c is the center wavelength of the pulse, \sigma_T is the standard deviation of the pulse width in time, and \sigma _\lambda is the standard deviation of the pulse width in wavelength. For this discussion, we are going to ignore attenuation losses and focus on the effect of dispersion. It turns out that the losses due to attenuation are easy to add, but they add more bulk to this discussion that is already too long.

Modeling Pulse Dispersion Impact on Power

There are many textbooks that provide rigorous derivations of the equation we will be using to model pulse spreading. I will use a more intuitive argument here to motivate, but not prove, the use of this equation. We begin our discussion with a few observations.

  • The pulse time function can be modeled as being composed of two normal signals: (1) the initial pulse shape, and (2) the standard deviation of the dispersed laser pulse.
  • The variance of the sum of these two signals is the sum the variances.
  • The standard deviation of the dispersed laser wavelengths is given by D \cdot L\cdot \sigma_\lambda.

These observations lead us to write down the following equation directly (a detailed proof involves convolutions).

Eq.5   \sigma_T^2 = \sigma_0^2 +\left(D \cdot L \cdot \sigma_\lambda\right)^2

where \sigma_T^2 is the time-domain variance of the total waveform, \sigma_0^2 is time-domain variance of the initial pulse (t=0), L is the distance of pulse travel, and \sigma_\lambda^2 is the standard deviation of the laser pulse.

Increasing \sigma_T will increase the pulse duration and reduce its amplitude, exactly what happens in a dispersed pulse. Substituting Eq. 5 into Eq. 4 gives us the equation of the dispersed pulse.

f(t,T) = \frac{1}{{\sqrt {2 \cdot \pi } \cdot \sqrt {\sigma _0^2 + {{\left( {D \cdot L \cdot {\sigma _\lambda }} \right)}^2}} }} \cdot \exp \left( -{\frac{{{{\left( {t - T} \right)}^2}}}{{2 \cdot \left( {\sigma _0^2 + {{\left( {D \cdot L \cdot {\sigma _\lambda }} \right)}^2}} \right)}}} \right)

Eq. 6   f(t,T) = \frac{1}{{\sqrt {1 + {{\left( {\frac{{D \cdot L \cdot {\sigma _\lambda }}}{{\sigma _0}}} \right)}^2}} }} \cdot \frac{1}{{\sqrt {2 \cdot \pi } \cdot {\sigma _0}}} \cdot \exp \left( -{\frac{{{{\left( {t - T} \right)}^2}}}{{2 \cdot \left( {\sigma _0^2 + {{\left( {D \cdot L \cdot {\sigma _\lambda }} \right)}^2}} \right)}}} \right)

A quick look at Equation 6 shows that its amplitude reduces with range by a factor of \sqrt{1 + \left(\frac{D \cdot L \cdot \sigma_\lambda}{\sigma_0}\right)^2}. This reduction in pulse amplitude means a reduction in power that can be modeled using Equation 7.
Eq. 7   P{P_D} = 10 \cdot \log \left( {\sqrt {1 + {{\left( {\frac{{D \cdot L \cdot {\sigma _\lambda }}}{{\sigma _0}}} \right)}^2}} } \right)
All the commonly used dispersion power penalty formulas appear to be based on this equation. Given this equation, I now can demonstrate how Equations 1 through 4 (listed in part 1 of this blog series) were derived.

Equation Derivations

The only differences I can find between the equations are related to how one defines the relationship between the bit rate and \sigma_0. This relationship is set by the amount of ISI that the engineer can tolerate. Here is a quick summary of the assumptions behind each equation:

  • Equation 1 assumes that the engineer wishes to define the bit time as equal to that of a rectangular pulse with the same energy and peak value as a normal pulse.
  • Equation 2 assumes that the engineer wishes to define the bit time as the period required to contain 95% of the pulse energy at the time of transmission.
  • Equation 3 assumes that the engineer wishes to define the bit time as the period required to contain 95% of the pulse energy at the time of reception.
  • Equation 4 is Equation 3 with a Taylor series approximation applied.

With all this said, it is now time to dive into the details.

Deriving Equation 1

Equation 1 is derived from Equation 7 using the following assumptions.

  • The pulse is specified to be of time duration T and amplitude P, which implies rectangular in shape.
  • The Gaussian pulse equivalent has the same total energy and peak amplitude.

Given these assumptions and the fact that the area under a Gaussian pulse is P \cdot \sqrt {2 \cdot \pi } \cdot \sigma_0, we can derive the \sigma_0 of the Gaussian pulse as follows.

Eq. 8   P \cdot T = \sqrt {2 \cdot \pi } \cdot \sigma_0 \cdot P

Given that the bit period T is related to bit rate B by B = \frac{1}{T}, we can solve Eq. 8 for \sigma_0 to obtain Eq. 9.

Eq. 9   \sigma_0 = \frac{T}{{\sqrt {2 \cdot \pi } }} = \frac{1}{{\sqrt {2 \cdot \pi } \cdot B}}

We can substitute Equation 9 into Equation 7 to obtain Equation 1.

P{P_D} = 10 \cdot \log \left( {\sqrt {1 + {{\left( {\frac{{D \cdot L \cdot {\sigma _\lambda }}}{{\frac{1}{{\sqrt {2 \cdot \pi } \cdot B}}}}} \right)}^2}} } \right)

P{P_D} = 5 \cdot \log \left( {1 + 2 \cdot \pi \cdot {{\left( {B \cdot D \cdot L \cdot {\sigma _\lambda }} \right)}^2} } \right)

The next blog will cover deriving Equations 2 – 4.

Posted in Fiber Optics | Tagged , , | 4 Comments

Dispersion Power Penalty Modeling (Part 1)

Posted on 6-October-2010 by mathscinotes

An Apology

This blog post is rather long (3 parts). I have had so many questions on this topic lately that I thought I should put some of the notes into a more formal format. The discussion is very specific to fiber optic networks and requires some knowledge of fiber to follow.

Introduction

I often work with people who are new to fiber optics and they often find dispersion confusing. Dispersion is caused by the variation of the speed of light in glass with wavelength, and the distortion it causes can limit the range of some deployments more than attenuation. People are very used to the idea that the speed of light is constant in a vacuum, but they are unaccustomed to the idea that the speed of light varies on a fiber as a function of wavelength, polarization, and fiber construction. In our system, dispersion is caused mainly by the variation in light speed as a function of wavelength, which is called chromatic dispersion.

Chromatic dispersion causes the pulses of light ("bits") on the fiber to stretch out in time and reduce in amplitude. The stretching out is called Inter-Symbol Interference (ISI). The reduction in the amplitude of the pulses is called the dispersion Power Penalty (PP_{D}). Figure 1 illustrates how the pulse distorts as it moves down the fiber.

Illustration of Pulse Distortion Down the FIber

Figure 1: Illustration of Pulse Distortion Down the Fiber

While doing some system modeling, I noticed that there were different equations being used to compute PP_{D}. Here are some examples of these equations.

(Eq. 1) {PP _D} = 5 \cdot \log \left( {1 + 2 \cdot \pi \cdot {{\left( {B \cdot D \cdot L \cdot {\sigma _\lambda }} \right)}^2}} \right) (see Ref 1, 4)
(Eq. 2) {PP _D} = 5 \cdot \log \left( {1 + {{\left( {4 \cdot B \cdot D \cdot L \cdot {\sigma _\lambda }} \right)}^2}} \right)

(see Ref 5)
(Eq. 3) {PP _D} = -5 \cdot \log \left( {1 - {{\left( {4 \cdot B \cdot D \cdot L \cdot {\sigma _\lambda }} \right)}^2}} \right)

(see Ref 2, 3)
(Eq. 4) {PP _D} = 10 \cdot \log \left( {1 + \frac{1}{2} \cdot {{\left( {4 \cdot B \cdot D \cdot L \cdot {\sigma _\lambda }} \right)}^2}} \right) (Ref lost)

where

  • B is the bit rate
  • L is fiber distance
  • D is the dispersion constant of the fiber
  • \sigma_\lambda is the spectral standard deviation of the laser

I started to look at the equations in detail because I wanted to know why different equations were used to model the same thing. It turns out the equations are very similar and simply reflect different ways of defining or measuring some of the parameters critical to dispersion. The derivation of Equation 4 also uses a mathematical approximation. I thought it would be useful to document where the equations come from and why they look different. This example also illustrates a nice application of the normal curve that does not involve statistics.

Background

Speed of Light and Index of Refraction

The key to understanding chromatic dispersion is to understand that the index of refraction in glass varies with wavelength. The relationship between the speed of light in glass and the index of refraction is given by the following equation.

             c_{glass} =\frac{c}{n\left(\lambda\right)}

where

  • c is the speed of light in a vacuum
  • c_{glass} is the speed of light in glass
  • n\left(\lambda\right) is the index of refraction
  • \lambda is the wavelength of light.

Figure 2 illustrates how the index of refraction varies by wavelength and type of glass.

Refractive Index as a Function of Wavelength and Glass Type

Figure 2: Refractive Index as a Function of Wavelength and Glass Type

Sources of Chromatic Dispersion

Fiber-Related

When we talk about chromatic dispersion, we are talking about a characteristic that is composed of three parts.

  • Material Dispersion
    This contribution is caused by the variation of the index of refraction in glass with wavelength. A prism uses this form of dispersion to separate out the colors of light. Material dispersion has nothing to do with the fiber – it is a property of the glass. Figure 1 illustrates how the index of refraction in various forms of glass varies with wavelength.
  • Waveguide Dispersion
    The fiber is a form of waveguide and the optical power divides between the core and cladding. The cladding and core indexes of refraction are different, which causes dispersion.
  • Profile Dispersion
    The glass within the core and cladding each have indexes of refraction that varies with wavelength and their construction. This also introduces dispersion.

Mathematically, chromatic dispersion is usually modeled by a single parameter that consists of three terms.

             D = {D_M} + {D_W} + {D_P}

where

  • D is the total chromatic dispersion constant
  • DM is the material dispersion constant
  • DW is the waveguide dispersion constant
  • DP is the profile dispersion constant

The following discussion assumes that all the sources of chromatic dispersion can be modeled using the single parameter D.

Optical Source-Related

If the fiber was driven by a single wavelength, no chromatic dispersion would occur. However, no source of light produces a single wavelength – they all generate a range of wavelengths. In fact, simply generating a pulse causes some spectral spreading. Sometimes the dispersion is significant – sometimes it is not. The purest sources of light comes from lasers, and our systems are driven by lasers. There are two main types of lasers used in telecommunications: Fabry-Perot (FP) and Distributed Feedback (DFB). The FP lasers generate multiple discrete wavelengths (called modes) and are subject to a rather nasty form of dispersion called mode-partition noise. This imposes a severe limitation on the range of FP-based systems. I will not be discussing mode-partition noise here. Our systems use DFB lasers, which generate light in very limited band with a distribution that is modeled well by a normal curve. The graphs in Table 1 illustrate the spectral characteristics of these lasers.

Table 1: Examples of Laser Spectral Characteristics
FP Laser DFB Laser
Spectral Characteristic of FP Laser Spectral Characteristic of DFB Laser
Source Source

Part 2 of this blog will address the analysis and modeling of dispersion losses.

References

1. Agrawal, Govind. P.J. Anthony, and T.M Shen. "Disperson Penalty for 1.3-µm Lightwave Systems with Multimode Semiconductor Lasers." Journal of Lightwave Technology. May 1988: pp 620-625. Print.
2. Agrawal, Govind. Fiber-Optic Communication Systems. 3rd ed. NY: Wiley, 2002. p 204. Print.
3. Keiser, Gerd . Optical communications Essentials. 1st. Boston, MA: McGraw-Hill Professional, 2003. p. 265. Print.
4. Miller, John, and Ed Friedman. Optical Communications Rules of Thumb. Boston, MA: McGraw-Hill Professional, 2003. p. 325. Print.
5. Agrawal, Govind. Lightwave technology: Telecommunication systems, Volume 2. New York: Wiley, 2005. p. 170. Print.

Posted in Fiber Optics | Tagged , , | 9 Comments

Engineer Humor (yes, they do have a sense of humor)

Posted on 23-September-2010 by mathscinotes

Introduction

Engineers can be hilarious – albeit a bit dry and cynical. Engineers are problem-solvers by nature. Much of their humor involves problem solving. Here are a few things that came up this morning that I thought would be worthwhile sharing here.

Troubleshooting Phases

I had a hardware engineer in my group who had a 12-step problem solving process that he used EVERY SINGLE TIME. He wrote code for Field Programmable Gate Arrays (FPGAs), so his problem resolutions always involved changing some FPGA source code. Here is his troubleshooting process.

12-Step Troubleshooting Process
Step Number Name
Step 1 That Can't Happen, It Must Be a Test Problem.
Step 2 Blame Software
Step 3 Doubt - Could I Have Made an Error?
Step 4 Blame Software
Step 5 How Did This Ever Work?
Step 6 Despair
Step 7 Hope
Step 8 Deep Despair
Step 9 Blame Software
Step 10 The Return of Hope
Step 11 Intense Effort
Step 12 It Was a One-Line of Source Code Change

Quality

We had a discussion this morning on software quality. It became obvious that there were levels of software quality. Here is what I gleaned from that conversation.

Levels of Software Quality
Level Description Comment
Level 1 Crap-Like The software has some good points, but there are some serious issues.
Level 2 Crapola The software appears to have some good points, but deep-down it really sucks.
Level 3 Polished Crap This is software that really does suck, but people have worked really hard to make it seem like it doesn't.
Level 4 Crap The software is so bad that it really does stink.
Level 5 Total Crap How does code like this get written?

Problem Solving Progress

I had a manager under me that used the following scale to describe how close we were to solving a problem. His approach also came up in a meeting this morning.

Nearness of a Solution
Level 1 Description
Level 1 We have no idea where the problem is; it could be anywhere.
Level 2 We have the bug isolated to a hemisphere (Eastern or Western).
Level 3 We know the bug is somewhere in this area.
Level 4 We have the bug cornered.
Level 5 We have the bug in our cross-hairs.
Level 6 The bug is begging for mercy.

Yes, this was a typical morning.

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My Little Gilligan's Island - Part 1 (The Shower)

Posted on 8-September-2010 by mathscinotes

Introduction

My cabin in northern Minnesota is very primitive (the only water is lake water, no heat, no air conditioning, no phone, no Internet) and I do not want to put any money into it until I get my sons through college. I do have electrical power, and I'm considering getting a generator installed by an electrician from a company like SALT (https://saltle.com/electrician-texas-service-areas/hutto-electrical-services/), should the weather ever take a turn and leave me without power. To make the cabin more livable, I have developed a few ways to make things a bit more comfortable for little extra money. One thing I have put in is a primitive lake water shower. This blog post goes into how the shower was put together and how we use it. There was a bit of design in resolving this issue and this post describes my solution.

Design Approach

When my wife and I first bought the cabin, we tried to stay clean by bathing in the lake. Unfortunately, the lake temperature never gets above the mid-70 °F range during the summer, and during spring and fall it drops below 50°F (it freezes over in winter). This makes for very uncomfortable bathing. To make bathing comfortable, my basic approach will be to bring the water up from the lake to a storage area, heat it, and then use it for showering. This approach required me to solve the following problems.

  • Transporting the lake water
  • Storing the lake water
  • Heating the water
  • Designing a shower (pump, plumbing, nozzle, shower enclosure)
  • Removing the gray water

Let's take these items one by one.

Transporting the Lake Water Up the Hill

The most convenient place for the shower is about 100 feet from the lake and up a 15 foot hill. Let's think about the amount of water I will be needing for a showering.

  • It is not unusual for me to have 8 people at the cabin.
  • A typical shower head puts out 3 gallons per minute.
  • A typical shower takes about 2 minutes.

The total amount of water required for the showers is given by the following equation.

V = N \cdot t \cdot F = 8{\text{ showers}} \cdot 2\frac{{\min }}{{{\text{shower}}}} \cdot 3\frac{{{\text{gal}}}}{{{\text{min}}}} = 48{\text{ gal}}

where V is the total volume of water (gallons), N is the number of people showering (8), F is the flow rate (3 gallons per minute), and t is the length of an average shower (2 minutes). This means that I need 48 gallons of water for showering. Since water weighs ~8 pounds per gallon, I need nearly 400 pounds of water. Using pails to move the water up the hill is not an option. I need a pump. It turns out that I actually designed a pump system that works well. I will discuss the pump system in a later blog post. Assume for now that I am using an electrical pump to bring the nearly 400 pounds of water up from the lake to a storage area. My pump average 15 gallons per minute, which is about 120 pounds of water per minute. This means that I can move all the water I need in less than 4 minutes with no lifting.

Storing the Lake Water

The cheapest way to store 48 gallons of water is to buy a plastic drum. While this was easy and cheap, it complicated the problem of heating the water. The drums are made of a plastic that does not like to get hotter than 180 °F. This will complicate the design of my heating system, but not too much.

Heating the Lake Water

I had heard another engineer mention that he had heated molasses in a 55 gallon drum using a drum heater belt (example). This sounded like the perfect solution to my problem. There were two concerns with the heater belt:

  • Could I use a heater belt with a plastic drum?
  • How fast would it heat the water?

Using the Heater Belt with a Plastic Drum

In the interest of full disclosure, I first bought a metal 55 gallon drum and a metal-drum heater belt. Unfortunately, the metal drum really began to corrode, and the rusty smell and appearance of the water was drag. That is when I switched to a plastic drum. I had seen a plastic drum heater. It was very similar to a metal drum heater, but with a large metal band around it to spread the heat out. The plastic drum heater was much more expensive than the metal drum heater, and I already had a metal drum heater. So I took my plastic drum, bought a $5 roll of aluminum tape (the kind used on ducts) and I wrapped the middle of my plastic barrel with the tape (see figure below). I made the tape layer much wider than the heater belt to spread the heat out. It has worked well.

How fast will the Water Heat?

The belt is powered from a 120 V outlet (standard US voltage) and puts out 1500 W. Because the belt wraps around the outside of the barrel, some of the heat is lost to the environment. This makes heating the barrel a bit inefficient, but workable for my needs.


{t_{Heat}} = \frac{{V \cdot \rho \cdot \kappa \cdot \left( {{T_{Desired}} - {T_{Lake}}} \right)}}{{k \cdot R}}

where

  • TLake is the temperature of the lake water (assume 55 °F for a spring or fall temperature)
  • TDesired is the temperature we want for showering (105 °F)
  • \rho is the density of water (1 gm/cm3
  • \kappa is the specific heat of water (4.187 J/gm °C)
  • R is the rate of heat generation (1500 W)
  • k is the efficiency of heat transfer from the barrel to the water (75% a guess)
  • tHeat is the time to heat the water to TDesired from TDesired.

Plugging all these numbers into the equation gives me ~5 hours, which is about what it actually takes during the cold times with a large number of people. During the warmer times with fewer people at the cabin, I cut the amount of water in the drum. This reduces the heating time to a couple of hours. All these numbers have proven manageable.

Designing the Shower

Nothing real sophisticated here. I put together a quick water pump/shower system with the following equipment:

  • a 12 V DC recreational vehicle water pump
  • a 12 V power source, something nice and safe. The safest is a battery, but that needs to be charged. There are numerous 12 V power sources on the market.
  • some garden hose
  • a shower nozzle that screws on the end of the garden hose (any hardware store has this nozzle).

I'm not a plumber but I did manage to find a Brisbane blocked drain plumber that could also help me with a hot water system. I don't have much experience with hot water systems so it was great that I found a plumbing company that could help. The whole thing wouldn't have been possible without them! I made the shower enclosure out of some landscape fencing material that I found cheap at a Home Depot. Obviously, if I was living somewhere less primitive and not doing everything in my power to keep costs down, I'd definitely be putting more thought into the shower design, perhaps with a bit more of an enclosure and privacy than the fencing material with these rimless shower doors. For now, though, we're content with our shower space. The completed shower assembly can be seen in the photo below.

Removing the Gray Water

The procedure here was straightforward as well. Here is what I did:

  • Built the floor base Durock (Durock is a common wallboard used for showers)
  • Cut a drain hole in the Durock and put in a floor drain I picked up from the hardware store
  • Coated the Durock with waterproofing paint mixed with sand
  • Connected up 30 feet of flexible drain tile to the floor drain.
  • Buried the drain tile on a downward slope (my whole cabin sits on sand)

The whole shower has been working now for at least 5 years and still looks good. I have included photos of the shower hardware and the shower enclosure. Of course, I did have the dream of surrounding the shower with some beautiful glass doors, but it just wasn't practical outside. I had the vision of using glass doors when my friend told me about somewhere like this Portland glass service that offered the most beautiful selection. I guess I'll just have to incorporate this in my bathroom design instead. I'm just glad that the shower already functions well with the items I selected; it's a huge weight off my mind. It functions well and it makes living at the cabin a bit more comfortable.

Mechanical Hardware for the Lake Water Shower

Mechanical Hardware for the Lake Water Shower
Shower Enclosure

Shower Enclosure
Posted in Cabin, Construction | Tagged , | 2 Comments