Standards of Readability

Posted on 1-May-2013 by mathscinotes

Quote of the Day

Do not forget what is being a sailor when you become a captain.

— African proverb

Figure 1: Newton's Pincipia is my vote for the world's most obscure writing.

Figure 1: Newton's Pincipia is my vote for the world's most obscure writing. (Source)

I had a discussion with one of my engineers this morning about documentation and our company's standard of readability. When I was in school, I always tried to write so that I could be understood. It was while working as a US Navy contractor that I discovered that I need to write so that I could not be misunderstood. This lesson was drummed into me by a safety engineer -- a man to whom I am in debt. His writing is so clear that I hope that I someday get near his level. I tell the engineers in my group that we write so that we cannot be misunderstood.

I do have one documentation horror story that I want to relate. I used to work as a proposal manager on large defense contracts. I had a former physics professor (PhD from a very well known institution) who worked on some of my proposals as a technical contributor. I could not read a thing that man wrote! I even brought in a very good writing consultant to work with him and that did not help at all. Finally I sat down with him and told him that his writing was terrible and he was going to need to do something else. He demanded to know exactly what was wrong with his writing and I told him that no customer could understand what he was writing. He then told me that he viewed his writing as a form of filter -- any customer that could not understand what he wrote should not be reviewing this proposal.

I then tried to explain to this very intelligent man what the life of a government proposal reviewer was like. I decided to present this poor writer with the following proposal review scenario:

  • your proposal is the 20th one the reviewer has seen this week,
  • most of the proposals have been a struggle to get through,
  • Friday at 4:00PM is the first time he starts reading your proposal,
  • your proposal is difficult to read
  • the reviewer decides to give your work a low score and go home for the weekend,
  • all of our work has been for nothing.

After some moaning and groaning, he got the point. You write for audience and you do not create unnecessary obstacles.

Posted in Technical Writing | Comments Off on Standards of Readability

How Big is Phobos When Seen from the Surface of Mars?

Posted on 1-May-2013 by mathscinotes

Quote of the Day

Man is only born ignorant. It takes four years of college to make him stupid.

— Mark Twain

Introduction

Figure 1: Good Color Image of Phobos. (Source)a

Figure 1: Good Color Image of Phobos. (Source)

The thought of alien worlds with multiple moons has always intrigued me. I am listening to the audio book "A Princess of Mars" by Edgar Rice Burroughs. I downloaded the audio book from Libravox, which provides free downloads of readings from classic works. Normally, I do not listen to much science fiction, but I watched the movie "John Carter" and there was something I liked about the story.

In "A Princess of Mars", Burroughs describes the moons of Mars, Phobos (Figure 1) and Deimos, as being large and very bright at night.

The nights are either brilliantly illumined or very dark, for if neither of the two moons of Mars happen to be in the sky almost total darkness results, since the lack of atmosphere, or, rather, the very thin atmosphere, fails to diffuse the starlight to any great extent; on the other hand, if both of the moons are in the heavens at night the surface of the ground is brightly illuminated.

He also commented on Phobos' speed across the sky with this statement.

... the nearer moon of Barsoom raced through the western sky toward the horizon ...

I started to wonder if these characteristics were true. So I thought a few calculations would be in order ...

Background

I can believe that Phobos moves across the martian sky pretty fast, but I am not sure about the movement of Deimos. I also cannot believe that they appear to be very large in the sky. Thus, my reading has made me interested in two questions about Mars and its moons.

  • How large are Phobos and Deimos as seen from Mars compared to the Moon when seen from Earth?
  • How fast do the two moons move relative to each other when viewed from Mars?

Figure 2 is a good graphic that illustrates how Phobos and Deimos move about Mars. Their orbits are nearly circular and they both can transit the Sun.

Figure 1: Graphical Illustration of the Orbits of Phobos and Deimos.

Figure 2: Graphical Illustration of the Orbits of Phobos and Deimos.

Analysis

Angular Size of Phobos and Deimos Compared to the Our Moon

Figure 3 shows a transit of Phobos across the Sun (Source). Figure 4 shows the transit of Deimos across the Sun (Source).

Figure 2

Figure 3: Transit of Phobos Across the Sun When Viewed From Mars (Photographed by the Opportunity Rover).

Figure 3: Transit of Deimos Across The Sun When Viewed From Mars (Photographed by the Opportunity Rover).

Figure 4: Transit of Deimos Across The Sun When Viewed From Mars (Photographed by the Opportunity Rover).

Our Moon, as seen from Earth, has nearly the same angular extent as the Sun -- that is why we have such spectacular eclipses. However, Figure 1 and 2 show that Phobos and Deimos have significantly smaller angular extent than the Sun when viewed from Mars. Since the Sun as viewed from Mars is smaller than the Sun viewed from Earth, both Phobos and Deimos as viewed from Mars must have significantly smaller angular extent than our Moon when viewed from Earth. This qualitative argument is supported by the quantitative argument I make in Figure 5.

Figure 4: Calculations Showing the Angular Extent of the Sun, Phobos, and Deimos.

Figure 5: Calculations Showing the Angular Extent of the Sun, Phobos, and Deimos.

Size is not the only factor affecting the brightness of an object -- some objects reflect more light than others. This factor is called albedo. It turns out that the albedos of Phobos and Deimos are about half that of the Moon. Compared to the Earth's moon, the moons of Mars do not reflect much of the light that they receive.

Angular Speed of Phobos and Deimos as seen from Mars

Figure 6 shows a great time-lapse photo of Phobos and Deimos from Mars as seen by a Rover (Source). Note that Phobos and Deimos appear to move in opposite directions when viewed from the surface of Mars, even though they both revolve in the same direction around Mars. This is because of the rotation rate of Mars is slower than Phobos, but faster than Deimos. My measurements in Figure 7 show that Phobos appears to move ~12 faster than Deimos.

Figure 5: Relative Movement of Phobos and Deimos When Viewed from Mars.

Figure 6: Relative Movement of Phobos and Deimos When Viewed from Mars.

Figure 6: Measurements of the Relative Movements of Phobos and Deimos.

Figure 7: Measurements of the Relative Movements of Phobos and Deimos.

Figure 8 show my rough calculations that demonstrate that the relative motions measured in Figure 6 make sense when you take the rotation rate of Mars into account.

Figure 7: Calculations of Angular Movements of Phobos and Deimos.

Figure 8: Calculations of Angular Movements of Phobos and Deimos.

Conclusion

I think Burroughs was using some literary license when talking about how impressive the moons of Mars are at night. We can say that:

  • The angular extent of the Sun is smaller at Mars than Earth \Rightarrow less light is available at Mars than at the Earth.
  • When viewed from Mars, both Phobos and Deimos are much smaller than the Moon when viewed from Earth \Rightarrow less area means that light is received from the Sun, so less is available to reflect.
  • Phobos and Deimos have albedo values about half that of the Moon \Rightarrow the moons of Mars do not reflect as much of the light they receive as the Moon does.
  • These three factors (I refer to this situation as "bad cubed") means that Phobos and Deimos will reflect much less light than the Moon.

I was surprised to realize that Phobos and Deimos appear to move in opposite directions.

While the moons of Mars may not be very bright, it would still be a great view to see in person.

Posted in Astronomy | Tagged | 2 Comments

Calculator to Select Connector for Joining Wires

Posted on 26-April-2013 by mathscinotes

Quote of the Day

Nobody washes a rental car.

— Truism on the importance of ownership and maintenance

Introduction

Figure 1: Example of a Crimped Connection using Commonly Available Connector.

Figure 1: Example of a crimped connection using a commonly available connector. Many mechanical engineers love crimped connections because they can be hermetic. However, they require the correct tool to be properly executed. (Source)

I have written dozens (hundreds?) of small Mathcad function to help me in my daily work. Since I plan to teach another Mathcad class soon, I am gathering examples that might be good to use as application examples. Yesterday, I was working with a customer on selecting wires for supplying power to our products. I used an old calculator that I have decided to show my students in the class. I will discuss how I used this calculator here.

These wires need to be joined (i.e. spliced). The customer wanted to use crimp connectors for this joining process rather than soldering. Crimping wire connections is preferred by many engineers because:

  • simpler

    There are numerous hand tools that allow you to simply put the wires together and join with by simply squeezing the tools handle.

  • safe

    Soldering can involve heavy metals and nasty fluxes.

  • reliable

    When performed by the proper tool, crimp joints can be very reliable.

This particular customer wanted to use this brand of connectors. There are rules on which connectors must be selected for a given area of copper. It turns out that I have a Mathcad worksheet that helps me to quickly select the specific connector for the given wires being used.

Background

Wires are pretty straightforward, but the units of measure (in the US) are rather strange. The following Wikipedia reference will provide the proper background to understand this calculator.

Analysis

Figure 1 shows the basic setup used to define the characteristics of both the wire gauges and the connectors.

Figure 1: Mathcad Worksheet Section that Defines the Wire Types and Connectors.

Figure 1: Mathcad Worksheet Section that Defines the Wire Types and Connectors.

Figure 2 shows the user input section of the worksheet. I use some simple control ("text boxes") to accept user input. I also have a small program that searches for the connector size that matches my needs. The calculator assumes that multiple wires of two different gauges are being joined.

Figure 2: Mathcad Worksheet User Input and Computation Section.

Figure 2: Mathcad Worksheet User Input and Computation Section.

Conclusion

This blog just reviews a routine wire area and connector calculator. I use this calculator so often that I do not even think about it.

You can get a copy of the calculator here. It is in Mathcad 15.

Posted in Electronics | 1 Comment

Photo Showing Ship On the Horizon

Posted on 22-April-2013 by mathscinotes

Quote of the Day

Men are born ignorant, not stupid. They are made stupid by education.

— Bertrand Russell (1872-1970). I see lots of problems with this quote, but there is an element of truth in it. I do think that education done poorly can drain all the excitement and wonder out of a child.

I read a lot of nautical history. Many of these history references talk about how sailors have long known the Earth was round. One of the reason why sailors knew the world was round had to do with the appearance of ships and islands on the horizon. Here is a photo that nicely illustrates how the top of a ship is the last part visible as it crosses over the horizon (Source).

Figure 1: Container Ship On the Horizon.

Figure 1: Container Ship On the Horizon.

Posted in General Science, Naval History, Navigation | 2 Comments

Timing Differences Between Battleship Guns During a Salvo

Posted on 20-April-2013 by mathscinotes

Quote of the Day

The better person you become, the better person you attract.

— Pinterest. This is also true for management.

I was looking at this picture (Figure 1) of the USS Missouri firing a six-gun salvo and I thought that I could learn something from it.

Figure 1: Battleship Missouri Firing a Six-Gun Salvo.

Figure 1: Battleship Missouri Firing a Six-Gun Salvo.

In Figure 2, I highlighted the six shells in the air. The shells from each gun on a turret appear to be fired at a different time.

Figure 2: Circle Shows the Salvo Shells in the Air.

Figure 2: Circle Shows the Salvo Shells in the Air.

I started to search around the web and I dug up some information. The following quote was very interesting (Source: Battle Line: The United States Navy, 1919-1939). I have highlighted a particularly interesting piece:

The emphasis on maximizing the number of guns firing in a single salvo led to unexpected problems. The blast from one gun could interfere the accuracy of others fired at the same time if the guns were close together. This would result in increased dispersion and "wild shots" -- shells landing abnormally far from the center of the pattern. Occasionally, shells were even observed to "kiss" - to brush against each other -- in flight on the way to the target. The problem was solved by the introduction of delay coils, electrical devices designed to allow a slight pause in the closure of the firing circuit to ensure that neighboring guns did not fire exactly at the same time. After their introduction, accuracy noticeably improved.

I found the following quote on this web page for the USS New Jersey, a battleship of the same class as the USS Missouri.

GUN FIRING ORDER L, R, C, BARRELS.
FIRING DELAY 0.06 SECOND. THERE IS NO DELAY FOR THE LEFT BARREL.

So it looks like there is 60 msec of delay between the guns.

Posted in Naval History | Comments Off on Timing Differences Between Battleship Guns During a Salvo

Battleship Guns and Danger Space

Posted on 19-April-2013 by mathscinotes

Quote of the Day

A nation is a society united by a delusion about its ancestry and a common hatred of its neighbors.

— William R. Inge. There are days when this seems very true …

Introduction

Figure 1: Broadside from Iowa-Class Battleship. (Source)

Figure 1: Broadside from the Battleship USS Iowa. (Source)

I have been reading a couple of excellent books about battleships ("Naval Firepower" and "Guns at Sea"). During my reading, I have encountered the term "Danger Space" that appears with nearly every table describing the large naval guns. Of course, I had no idea what danger space was when I began investigating it. It turns out that danger space describes an important metric for battleship guns, and it is worthwhile documenting what I have learned about it here.

There appears to be a number of ways to define danger space. Since my reading is on US battleships, I will focus on the how the US Navy used the term. All the world's navies had closely related definitions for danger space that ended up producing slightly different numerical results. My plan in this post is to:

  • define danger space as used by the US Navy

    Some definitions use the width of the ship (referred to as the beam -- see Appendix D for an example) and some don't. Some use different target height standards (e.g. 20 feet versus 30 feet). I believe that I now understand the need for the different forms -- it has to do with your priorities. For example, if your priority is to blow through the heavily armored regions from the side, the width of the ship is not very important. If you plan on dropping shells down on the deck, the width of the ship becomes important. For more discussion of this topic, see this discussion of the zone of immunity.

  • derive the formula(s) used to compute it

    I am not totally happy with my derivation, but it is what I could come up with. I am sure that the originators were working with approximations and they were trying to get answers that were reasonably close and easily computable by hand.

  • provide evidence that the formula I am using is the same as used by the US Navy

    US Navy range tables do not state the formula used to determine the danger space. I will compare the results using the formula that I found with the US Navy's published results.

I list the references I used at the bottom of the blog post. I should mention that there are numerous synonyms for danger space:

  • danger zone
  • hitting space or zone
  • bestrichener raum ("smear space" – my translation)

Background

The Motivation for Danger Space

Obtaining hits with long-range naval gunnery is closely tied to minimizing errors in determining the target's range. Range errors are usually larger and more difficult to correct than deflection errors (details here). Danger space tells us the amount of range error we can tolerate and still hit our target. Given that there are always errors present in our target range measurements, having a gun with a large danger space means that you have a greater chance of hitting your target for a given number of shells fired.

Definition of Danger Space

Danger space is also tied in with the concept of a citadel. Most analysts only focus on hits to the vital areas of the ship. On battleships, the vital areas were heavily armored and were referred to as the citadel. The US Navy usually assumed a citadel height of 20 feet2 -- the British Navy used 30 feet3. The length of the citadel varied with the type of target. The key concept here is that target range errors less than the danger space will still produce hits on the citadel.

Figure 1:Danger Space Illustration.

Figure 1: Danger Space Illustration.

Figure 1 illustrates how the danger space is about putting a projectile through the citadel our a range interval. Given this viewpoint, Algier1 gives the following definition for danger space. A similar definition is presented here.

By the term “danger space” is meant an interval of space, between the point of fall and the gun, such that the target will be hit if situated at any point in that space. In other words, it is the distance from the point of fall through which a target of the given height can be moved directly towards the gun and still have the projectile pass through the target. Therefore, within the range for which the maximum ordinate of the trajectory does not exceed the height of the target, the danger space is equal to the range, and such range is known as the “ danger range.”

Note that the focus of this definition is on the projectile passing through the target (i.e. the vertical face of the citadel), not dropping onto the target (i.e. the top of the citadel). You see the effect of this viewpoint by examining how US Navy towed gunnery targets were constructed (Figure 2). There were a number of different size targets used (Figure 3). Because the US Navy was less interested in hits on the top of the citadel, the targets were very narrow in thickness.

Figure 2: Example of a World War 2 Towed Gunnery Target.

Figure 2: Example of a World War 2 Towed Gunnery Target.

Figure 3: Sizes of US Navy Towed Gunnery Targets.

Figure 3: Sizes of US Navy Towed Gunnery Targets.

There are some important observations that you need to make about this definition of danger space.

  • Long range engagements are miss-prone because at long-range the projectiles have large fall angles, resulting in small danger spaces.
  • Short range engagements tend to have more hits because projectiles have shallow fall angles, resulting in large danger spaces.
  • At some short range, your danger space will equal your range because the projectile is traveling with such a shallow angle that a hit is guaranteed.

Analysis

Danger Space Equation

Algier1 gives Equation 1 for danger space. I make an attempt at a derivation for Equation 1 in Appendix A. Deriving Equation 1 requires making an approximation for the change in fall angle over the danger space interval.

Eq. 1 \displaystyle \delta =h\cdot \cot \left( \theta \right)\cdot \frac{R}{R-\delta }

where

  • δ is the danger space.
  • θ is the fall angle (i.e. the projectile's angle of descent at the point of impact).
  • R is the target range.
  • h is the target height.

There are a couple of special points worth making about Equation 1:

  • The only target characteristic present in Equation 1 is height.

    The US Navy was only interested in side penetration of the citadel. I have seen the danger spaces for guns of other navies listed where the width of the ship appeared to be included the calculation. Note that I drew Figure 1 showing the projectile striking the citadel broadside. The tactics of the day would have preferred to cross the T (i.e. target bow is facing the attacker). Citadel height is still the important parameter in this case. However, the citadel length now would be important for strikes on the top of the citadel.

  • In the case where \displaystyle R\gg \delta , Equation 1 reduces to \delta \doteq h\cdot \cot \left( \theta \right).

    Some authors do mention \delta \doteq h\cdot \cot \left( \theta \right) as a useful approximation. This approximation is equivalent to assuming that the path of the shell over the danger space is a perfectly straight line. See Appendix C for a reference example from a Royal Navy document.

Equation 1 can be solved for danger space to give a quadratic. The solution for the quadratic is given by Equation 2.

Eq. 2 \displaystyle \delta=\frac{-R\cdot \text{tan}\left( \theta \right)+\sqrt{R\cdot \text{tan}\left( \theta \right)\cdot \left( R\cdot \text{tan}\left( \theta \right)-4\cdot h \right)}}{2\cdot \text{tan}\left( \theta \right)}

Because of the hand computation difficulties associated with Equation 2, we can approximate the term R/(R-δ) in Equation 1 using a one-term series expansion to obtain Equation 3 (derivation shown in Appendix B). However, the result is not particularly accurate at short ranges. Equation 3 is also given by Algier1.

Eq. 3 \displaystyle \delta =h\cdot \cot \left( \theta \right)\cdot \left( 1+\frac{h\cdot \cot \left( \theta \right)}{R} \right)

Comparison with World War 2 Publications

I have worked numerous examples from various books. I have included two here that are representative. They come from this document. For those who are interested, my spreadsheet is included here.

Comparison with 16 in/45 or 50 Caliber Range Table for 2700 lb Shell

I am mainly interested in Iowa and South Dakota classes of battleships, which used 16-in guns (50 caliber and 45 caliber respectively). I will use their range tables for examples.

Figure 4 shows a screenshot of an Excel worksheet using both Equation 2 (quadratic solution) and Equation 3 (approximate solution). Equation 2 gave me very nearly the same results as listed in the reference publication.

Figure 4: Duplication of Results From Abridged U.S. Navy Range Table (2700 lb Shell, 45/50 caliber).

Figure 4: Duplication of Results From Abridged U.S. Navy Range Table (2700 lb Shell, 45/50 Caliber).

Comparison with 16 in/45 Caliber Range Table for 1900 lb Shell

Figure 5 shows a screenshot of an Excel worksheet using both Equation 1 (quadratic solution) and Equation 2 (approximate solution). Equation 1 gave me very nearly the same results as listed in the reference publication.

Figure 5: Duplication of Results From Abridged U.S. Navy Range Table (1900 lb Shell, 45 caliber).

Figure 5: Duplication of Results From Abridged U.S. Navy Range Table (1900 lb Shell, 45 Caliber).

Conclusion

I believe that I have shown how the danger space was computed by the US Navy during World War 2. Danger space is most often seen today in the context of small arms.

References

1Philip Algier, The Groundwork of Practical Naval Gunnery, Annapolis, MD: The United States Naval Institute, 1917, pp 59-65.

2Abridged Range Tables for U. S. Naval Guns. Washington, D.C.: Navy Department Bureau of Ordnance, 1944, p59.

3Friedman, Norman. Naval Firepower: Battleship Guns and Gunnery in the Dreadnaught Era. Annapolis: Naval Institute Press, 2008.

Appendix A: Derivation of Equation 1

Figure 6 shows the basic geometry associated with my derivation of Equation 1. I have no idea if this is how Equation 1 was originally derived, but this is the approach I chose to use.

Figure 6: Graphic Illustrating the Danger Space Geometry.

Figure 6: Graphic Illustrating the Danger Space Geometry.

Figure 7 shows the actual mathematical details of the derivation.

Figure 7: My Derivation of Equation 1.

Figure 7: My Derivation of Equation 1.

Appendix B: Simplification of Equation 1 into Equation 3

Figure 8 shows how the approximations used to derive Equation 3 from Equation 1.

Figure 8: Derivation of Equation 2.

Figure 8: Derivation of Equation 2.

Appendix C: Royal Navy Excerpt on Danger Space

Early Royal Navy documents referred to "dangerous distance" instead of "danger space". Figure 9 shows an excerpt from the "Textbook of Gunnery" dated 1902.

Figure 9: Royal Navy Reference on Danger Space.

Figure 9: Royal Navy Reference on Danger Space.

Appendix D: US Navy Excerpt That Uses the Beam of the Ship

Figure 10 shows a snippet from an old US Navy document. This one is unusual in that it includes the beam of the ship in the equation.

Figure 10: One Form of Danger Space That Uses Ship's Beam.

Figure 10: One Form of Danger Space That Uses Ship's Beam.

Posted in History of Science and Technology, Military History, Naval History | 7 Comments

Questions That Don't Need to Be Asked

Posted on 17-April-2013 by mathscinotes
Figure 1: Cruise Ship Millenium – my favorite. (Source)

Figure 1: Cruise Ship Millennium – my favorite. (Source)

Some "interesting" questions come my way -- one just rolled in about my plans for dealing with an electromagnetic pulse attack (I don't have any plans for that situation). When I say interesting, I mean that this question really did not need to be asked. The best example of this type of question is one I heard when I was on a cruise ship. When I am on a cruise ship, I always ask if I can get a tour of the bridge -- they often do host a tour. During this tour, the ship’s navigator will sometimes offer to host a seminar on how the ship navigates. This actually happened and a number of passengers attended. During this seminar, the navigator mentioned that the ship normally navigated using GPS, but was equipped for celestial navigation if it was needed. The navigator then sheepishly added that his celestial navigation was very rusty because it had been so long since he had used it. A passenger than very seriously asked, “What if global thermonuclear war broke out and all the GPS satellites were destroyed – how would we find our way home?” The audience sat in stunned silence. I don't think that question needed to be asked.

Posted in Military History, Navigation | Comments Off on Questions That Don't Need to Be Asked

TSA Has No Sense of Humor

Posted on 13-April-2013 by mathscinotes

Quote of the Day

Love all, trust a few, do wrong to none.

- William Shakespeare

Figure 1: Example of a TSA Line. I hat standing in these lines. (Source)

Figure 1: Example of a TSA Line. I really do not like standing in these lines. (Source)

I was having a discussion with some other engineering managers when I heard the following travel horror story. Engineering managers are always talking about Bills of Materials (BOMs), which we pronounce as "bombs". An engineer was on his phone at an airport and was discussing part problems with a contract manufacturer. He used the term "bomb" many times during that conversation. Another passenger heard his use of the term "bomb" and turned him in to TSA, who promptly apprehended him. He then spent the next several hours explaining to the nice TSA people what a bill of materials is. Having to go through tight security checks is one of the worst things about flying commercially and it's no wonder so many people would prefer to chart a private plane through a company like Jettly.com. I wouldn't be surprised if this engineer opts to fly privately next time, especially if he needs to discuss his work again. While the TSA folks were very nice, he did miss his plane and got to spend an enjoyable evening in Little Rock, Arkansas. Did you know many things there are named after Bill and Hillary Clinton? This engineer got to visit a number of these places.

I actually know the engineer that this happened to. He is the last guy on Earth that you would expect this sort of thing to happen to. I have had my TSA issues too. I once tried to drag an oscilloscope through security. They would have none of it. I had forgotten that my portable scope had a lead-acid battery in it that their x-ray machine could not penetrate.

Posted in Management | Tagged , | Comments Off on TSA Has No Sense of Humor

Conversion Cost Mathematics in Mathcad

Posted on 13-April-2013 by mathscinotes

Introduction

Normally, I use Excel for analyzing financial data. Today, I encountered a financial problem for which Mathcad seemed appropriate-- there was a bit of algebra involved. Let's see what you think ...

Background

Before I can state my problem, we need to establish a common vocabulary. A few definitions are in order.

Some Definitions

Factory Cost
The cost of a fully assembled, tested, and shipped product.
Freight
The cost of shipping the product to our distribution center. For example, if you are using an express freight service across europe the cost will be varied.
Bill of Materials (BOM)
The list of all the components that make up the product, such as whether they need shipping warmers to ensure they stay at a constant temperature during transportation.
BOM Cost
The cost of all the components that make up the product. BOM cost does not include any freight or assembly charges.
Conversion Cost
The cost of assembling the components into the product. Conversion cost (by agreement with our contract manufacturer [CM] ) does not include shipping cost.
Markup
The difference between factory cost and BOM cost. Markup includes freight.
Conversion Cost Percentage
Conversion cost as a percentage of factory cost minus freight. Freight varies for each product and final customer. The efficiency of a factory is often measured in terms of its conversion cost percentage. I am VERY interested in conversion cost.
Markup Percentage
Markup as a percentage of factory cost. Markup includes freight.

My Problem

I had a large number of products for which I wanted to evaluate their conversion cost percentage. Normally, our CM lists the conversion cost percentage for each product on a spreadsheet they send to us, which I then use as a way of comparing my manufacturer's efficiency against other manufacturers. Unfortunately, my CM did not list the conversion cost percentage of each of the products I was interested in. Instead, they gave me the following information:

  • Factory Cost
  • Markup Percentage
  • Freight (listed on another spreadsheet)

There is a tiny bit of algebra involved in computing the conversion cost percentage, so I decided to use Mathcad's symbolic solver to perform that function. I could do it manually, but it is late Friday afternoon, I am tired, and I do not want to make any errors.

Analysis

Figure 1 shows my solution of this problem in Mathcad.

Figure 1: Determination of Conversion Cost for a List of Products.

Figure 1: Determination of Conversion Cost for a List of Products.


I have attached a zip file containing the Mathcad15 file here.

Conclusion

My quick analysis showed that I was being charged a higher conversion cost than I had expected. A bit of investigation soon explained the issue, which I can now deal with.

Posted in Financial | Comments Off on Conversion Cost Mathematics in Mathcad

A Couple of Examples of Characteristic Impedance Calculations in Mathcad

Posted on 12-April-2013 by mathscinotes

Introduction

I have been doing some work that involves computing the characteristic impedances of cables. The work has involved creating some tables in Mathcad for comparison with tables from a government specification. Since I am always looking for real-life computations to use as Mathcad training examples for my staff, I thought I would blog about this work. This blog post shows a couple of different ways that I computed a table of characteristic impedances for wire pairs of different gauges and at different frequencies. I used Mathcad range variables to create my table. I then compared my computed results with the results listed by the US government in a table put out by the Rural Elecrification Administration (REA), which is now known as the Rural Utility Service (RUS).

My plan is to use this post as an example for a class that I plan to teach in a few months.

Background

REA has many specifications that control how phone lines are connected in the United States. Figure 1 is an excerpt from one of their specifications that lists the characteristic impedances for insulated wires of various gauges and at various frequencies. This particular wire is referred to as polyolefin-insulated cable (PIC). This insulation is similar in performance to polyethylene.

Figure 1: Characteristic Impedance Excerpt for Polefin Insulated Conductors (PIC).

Figure 1: Characteristic Impedance Table Excerpt for Polefin Insulated Conductors (PIC).

This table is from an old paper specification that was filled with markups. It is the only copy I have.

Analysis

Derivation of Key Relationships

Figure 2 shows my derivation of a couple of equations that relate Z0 (characteristic impedance) and vSignal (signal speed) to L0 (unit inductance) and C0 (unit capacitance). We need L0 and C0 to compute the characteristic impedance of the cable.

Figure 2: Derivation of Unit Capacitance and Inductance in Terms of Signal Velocity (vSpeed) and Impedance (Z0)

Figure 2: Derivation of Unit Capacitance and Inductance in Terms of Signal Velocity (vSpeed) and Impedance (Z0)

Figure 3 shows how we can compute L0 and C0 using the physical dimensions of the cable (s: conductor separation, and d: conductor diameter) and the dielectric constant of the insulation (ϵR: relative permittivity)(Source).

Figure 3: Formulas for the Impedance and Signal Velocity on a Wire Pair.

Figure 3: Formulas for the Impedance and Signal Velocity on a Wire Pair.

Calculation When Given Unit Capacitance

Figure 4 is a rather dense illustration of how I used the formulas of Figures 2 and 3 to compute the characteristic impedances for the same cable type as is documented in the REA document. The REA document specifies the unit capacitance of the cable and I have to assume a signal velocity. Most cables that I know of have a signal velocity of ~66% of the speed of light.

Figure 4: Calculation When Given Capacitance Per Unit Length.

Figure 4: Calculation When Given Capacitance Per Unit Length.

Calculation When Given Wire Construction

Figure 5 is another dense illustration that estimates the characteristic impedance versus frequency and wire gauge using the cable dimensions and the relative permittivity of the insulation. To vary things a bit, I presented the output using an Excel component.

Figure 5: Characteristic Impedance Calculaton Using Wire Geometry Specifications.

Figure 5: Characteristic Impedance Calculation Using Wire Geometry Specifications.

Conclusion

The impedances I obtained by both methods agree fairly well with the values listed by the REA. The largest differences are at low frequency. The differences are small enough not be significant for my work. I want to be able to use computed values rather than tables because they are more convenient for me to work with in Mathcad and Excel.

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