US WW2 Submarine Tonnage Sunk Database

 

Quote of the Day

Almost every successful person begins with two beliefs: The future can be better than the present. And I have the power to make it so.

— David Brooks


Figure 1: USS Gato, the lead boat of the most common type of US submarine during WW2.

Figure 1: USS Gato, the lead boat of the most common class of US submarines during WW2.

While answering a recent question about the tonnage sank by the top US submarine skippers during WW2, I realized that I had not made available my conversion to Excel of the JANAC data for vessels sunk by US submarines. The JANAC records are considered the official records because they were cross-checked with information from Japanese records.

The spreadsheet itself is from a course I gave last year on using Get and Transform (also known as Power Query). The raw data is from the Hyperwar web site – I often use the old WW2 records as an example of horribly formatted data that can be converted to a useful computer format using Python or Excel. The Hyperwar data appears to be from human-generated JANAC reports that were OCRed and converted to HTML. The Hyperwar site is a great resource, but the data does contain numerous conversion issues (e.g. commas turned into periods, extra Øs added to numbers). I cleaned up the obvious problems and cross-checked my results with data from the now-defunct Valor at Sea web site. The agreement was excellent.

This post makes this data available to those who are interested. With the Valor at Sea website offline, I could not find data summaries available anywhere else. Having it in spreadsheet form provides you the ability to generate custom reports. The data includes the specific ships sunk by each submarine. It does not include data for ships sunk because of the action of multiple submarines.

The Excel workbook is available here. There are no macros, but there are hyperlinks to various data sources.

Posted in History Through Spreadsheets, Military History | Leave a comment

Security Risks with Medical Radiation Sources

 

Quote of the Day

Doing statistics is like doing crosswords except that one cannot know for sure whether one has found the solution.

John Tukey, statistician and data analyst extraordinaire. If you get the chance, read his book Exploratory Data Analysis. It is a gem.


Introduction

Figure 1: Cobalt-60 Use in Gamma Knife. (Source)

Figure 1: Cobalt-60 Use in a Gamma Knife. (Source)

I was reading the Washington Post this weekend when I stumbled upon an 22-July-2017 article about concerns that ISIS in Mosul had access to an old medical radiation source. This source, which contains the radioactive isotope cobalt-60, is used in the treatment of cancer (Figure 1). However, cobalt-60 is extremely radioactive and could be used to build a dirty bomb. Fortunately, ISIS did not touch the source, but the concerns about a terrorist being able to use one of these radiation sources for a dirty bomb are real.

There have been encounters between radiation sources and unwary people – the encounters did not end well. For example, in one case, people tried to breakdown a cesium-137 radiation source for scrap. The incident ended with four people dead, twenty hospitalized, and 249 contaminated.

The Washington Post article mentioned three facts that we can easily verify using some simple math.

  • The source contains 9 grams of cobalt-60, which generates a radiation level of 10,000 curies (Ci) when new. (Quote)
  • Person standing three feet (~1 meter) from the unshielded source would receive a fatal does in less than 3 minutes. (Quote)
  • The source is 30 years old, so its radiation level is significantly diminished with respect to a new source. (Quote)

Background

Definitions

becquerel (Bq)
One becquerel is defined as the activity of a quantity of radioactive material in which one nucleus decays per second. (Source)
curie (Ci)
The curie (symbol Ci) is a non-SI unit of radioactivity, one Ci = 3.7 × 1010 nucleus decays per second or one Ci = 3.73.7 × 1010 Bq. (Source)
Half-Life (tHL)
Half-life is the time required for a quantity to reduce to half its initial value. The term is commonly used in nuclear physics to describe how quickly unstable atoms undergo, or how long stable atoms survive, radioactive decay. (Source)

Radiation Level Given Half-Life

Equation 1 tells us the amount of a radioactive species we have remaining after time t assuming that we had a 100% pure sample at time ø.

Eq. 1 \displaystyle N\left( {t,{{T}_{{HL}}},{{N}_{0}}} \right)={{N}_{0}}\cdot {{2}^{{-\frac{1}{{{{t}_{{HL}}}}}}}}

where

  • THL is the half-life of the radioactive species.
  • t is elapsed time since having a pure sample.
  • N0 is the initial amount of the substance. You can use mass or moles or even numbers of atoms.
  • N is the amount of the radioactive species left after time t.

To obtain the level of radioactivity (i.e. decays per second), we need to take the derivative of Equation 1 (see Figure 3).

Analysis

Setup

Figure 2 shows how I setup the calculations in Mathcad 15.

Figure 2: Analysis Setup.

Figure 2: Analysis Setup.

Washington Post Article Example of Radiation Danger Background on Medical Radiation Definition of Becquerel Definition of Curie Definition of Sievert Definition of Avogadro's Number Definition of Lethal Dose Definition of Cobalt Activity Level Definition of Cobalt Tissue Absorption

Calculations

Figure 3 shows my calculations that duplicate the results in the Washington Post article. The purple check marks indicate the specific results.

Figure 3: Mathcad Calculations.

Figure 3: Mathcad Calculations.

Washington Post Article

Conclusion

I had never thought about the potential security issues associated with medical radiation sources. I was surprised to the see how intense the radiation levels were from a cobalt source. While the  cobalt source mentioned in the article is 30 years old and only 2% of its initial radiation level, it is still a very dangerous item.

Appendix A: Quotes from the Article

Radiation Level Quote

In a draft report written in November 2015, research fellow Sarah Burkhard calculated that the radioactive cores, when new, contained about nine grams of pure cobalt-60 with a potency of more than 10,000 curies — a standard measure of radioactivity.

Fatal Dose Quote

A person standing three feet from the unshielded core would receive a fatal dose of radiation in less than three minutes.

Reduction in Radiation Level Quote

Because cobalt-60 decays over time, the potency of the Mosul machines’ 30-year-old cobalt cores would have been far less than when the equipment was new, but still easily enough to deliver a lethal dose at close range, the report said.

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Insulation Opportunity Costs

 

Quote of the Day

To the modern American at the millennium, these carrier pilots of more than a half century ago -- Massey, Waldron, and Lindsey last seen fighting to free themselves in a sea of flames as their planes were blasted apart by Zeros -- now appear as superhuman exemplars of what constituted heroism in the bleak months after the beginning of World War II. Even their names seem almost caricatures of an earlier stalwart American manhood -- Max Leslie, Lem Massey, Wade McClusky, Jack Waldron -- doomed fighters who were not all young eighteen-year-old conscripts, but often married and with children, enthusiastic rather than merely willing to fly their decrepit planes into a fiery end above the Japanese fleet, in a few seconds to orphan their families if need be to defend all that they held dear. One wonders if an America of suburban, video-playing Nicoles, Ashleys and Jasons shall ever see their like again.

Victor Davis Hanson on the heroes of the Battle of Midway. Unlike Hanson, I would not be concerned about today's youth. I watched many young people from my neighborhood volunteer to fight in Iraq and Afghanistan. My main concern is ensuring that government does not waste their lives on ventures unworthy of their sacrifice. I should mention that Max Leslie and Wade McClusky did survive the Battle of Midway.


Introduction

Figure 1: My Version of Finehomebuilding Insulation Comparison Table.

Figure 1: My Version of Finehomebuilding Insulation Comparison Table.

I have had a number of discussions with coworkers about the different types of wall insulation – some of these discussions have been documented in previous blog posts (e.g. here , here, here). There exists wide cost and performance disparities between the different wall insulation technologies. With respect to cost, I view fiberglass batts as a low-cost insulation option and the spray foams (open and closed cell) as high-cost options. All the options have their advantages and disadvantages. Fine Homebuilding Magazine (August/September 2017) has an excellent article by Martin Holladay provides an excellent spreadsheet-like analysis that illustrates the trade-offs between open and closed-cell foam nicely. My goal in this post is to go through the computational details of his analysis and to discuss his approach to choosing the best insulation for your application.

I like Martin's discussion because he focuses on the opportunity costs associated with the insulation choice. In Martin's analysis, he shows that the difference between open cell and closed cell R-values can be rather small relative to the cost differential – Martin mentions a $3K premium for closed cell over open cell to obtain a 1R - 2R improvement when insulating stud bays. Martin argues that you can get a better R-value return by spending that $3K on other insulation options, like adding exterior rigid foam – which I have used for some projects. The exact premium will vary depending on the house size, but I have encountered similar price differentials in my own work.

Background

Thermal Modeling

Most home insulation value is modeled using the resistor analogs, like I show in Figure 2. The model in Figure 2 is very simple, but useful in that it models two important types of heat transfer in traditional construction:

  • Losses through Framing (RFraming)
    In traditional framing, heat can pass directly between the inside and outside directly through the wood, which is called thermal bridging. To simplify the analysis, a fixed percentage of the wall construction, called kFraming,  is assumed to be subject to this form of heat loss. kFraming is usually a number between 20% and 30%.
  • Losses through the Stud Bay or Cavity (RCavity)
    When framing exterior walls, the spaces between studs are normally filled with insulation. The percentage of heat loss through the stud bay is 1-kFraming.
Figure 2: Simple Thermal Modeling of Different Wall Constructions.

Figure 2: Simple Thermal Modeling of Different Wall Constructions.

While the resistor values are calculated using the thermal conductivities of the materials involved, most manufacturers' specify the R-value per inch of their materials. I will use these numbers for this analysis.

Analysis Assumptions

Holladay's analysis makes the following assumptions, all reasonable:

  • Standard 2x4 and 2x6 wall construction practices
  • Closed cell insulation is so dense as to be difficult to trim flat to the studs. This means installers usually do not completely fill the cavity. Assume closed cell insulation is filled to within 1/2 inch of the stud edge.
  • The R-value of closed cell insulation is 6.5 R per inch. The R-value of open cell construction is 3.7 R per inch.
  • 25% of the wall construction is thermally "bridged" by the framing members. The other 75% consists of stud cavities filled with insulation or openings for doors and windows, which are not modeled.
  • The R-factor of doors and windows are handled separately.
  • Any exterior insulation is ignored.
  • There is no insulation value to any wood that extends beyond the insulation.

Analysis

Given the background information, the analysis is straightforward. I put everything into an Excel spreadsheet, which you can download here. The spreadsheet generates the table shown in Figure 1.

Conclusion

I always consider the return on my investment. Holladay entitled his article "Close-cell foam between studs is a waste," which really grabbed my attention. I have been reluctant to invest in closed cell because of its high installation cost relative to the insulation benefit it provides. It looks like an energy expert like Holladay also has concerns.

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Equatorial Rocket Launch Advantage

 

Quote of the Day

A faith that makes losing a sin will make cheating a sacrament.

Michael Gerson


Introduction

Figure 1: Map of Guyana Space Centre. (Source)

Figure 1: Map of Guiana Space Centre. (Source)

Putting a satellite into orbit requires that you impart a velocity of ~17,000 mph to the satellite. Because the Earth is rotating, its surface velocity gives you a head start on achieving orbital velocity when you launch toward the east – the direction of the Earth's rotation. The closer you move your launch site to the equator, the more velocity you get from the Earth's rotation.

In this post, I will compute the surface velocity for the European launch site in Guiana and at the Cape Canaveral launch complex (example launch pad) in Florida. Figure 1 shows a map of the the Guiana launch site, which is closer to the equator than Cape Canaveral. My objective in this post is to determine the advantage that the Guiana launch site has over Florida's Canaveral launch site.

I became interested in the important of launch site location after reading this article. My results agree with theirs.

Background

Earth Ground Velocity

Equation 1 shows the formula that I used to compute the Earth's velocity at the two launch sites

Eq. 1 \displaystyle {{v}_{{Ground}}}={{\omega }_{{Earth}}}\cdot {{r}_{{Earth}}}\left( {{{\theta }_{{Latitude}}}} \right)

where

  • ωEarth is angular velocity of the Earth [rad/sec]. The angular velocity is computed using the formula {{\omega }_{{Earth}}}=\frac{{2\cdot \pi }}{{{{T}_{{Sideral}}}}}, with TSidereal being the length of the Earth's sidereal day.
  • rEarth is radius of the Earth at the latitude θLatitude [m]

Earth Radius

Equation 2 shows how to compute using Mathcad 15 the Earth's radius as a function of latitude based on the WGS84 reference model. The exact formula I used is from this paper.

Eq. 2

Launch Site Latitudes

The latitude of the launch sites is what determines its velocity boost. I list their latitudes here:

I should mention that the Cape Canaveral launch site consists of numerous launch pads. Figure 2 shows a map.

Figure 2: Map of Cape Canaveral Air Force Station. (Source)

Figure 2: Map of Cape Canaveral Air Force Station. (Source)

Analysis

I want to compute the difference in the Earth's rotational velocity at the the two launch sites. Figure 3 shows how to perform that calculation.

Figure 3: Earth Rotational Velocity at Equation, Guinea, and Canaveral.

Figure 3: Earth Rotational Velocity at Equation, Guinea, and Canaveral.

My analysis shows that the Guiana launch site has a velocity advantage of 126 mph, which is relatively small compared to the 17,000 mph needed to achieve orbit.

Conclusion

I was able to confirm the velocity advantage for the Guiana launch site over that of Cape Canaveral. The ideal launch site would be as near as possible to the equator with nothing but ocean to the east to ensure that failed rockets would drop into the ocean. Polar launches have different requirements – like clear ocean to the north. The US uses Vandenberg Air Force Base for those launches.

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Iceberg Size Analogies

 

Quote of the Day

From whence shall we expect the approach of danger? Shall some trans-Atlantic military giant step the earth and crush us at a blow? Never. All the armies of Europe and Asia...could not by force take a drink from the Ohio River or make a track on the Blue Ridge in the trial of a thousand years. No, if destruction be our lot we must ourselves be its author and finisher. As a nation of free men we will live forever or die by suicide.

— Abraham Lincoln on the security of the United States. His words ring true even today.


Introduction

Figure 1: Good Illustration of the Iceberg's Size.

Figure 1: Good Illustration of the Iceberg's Size. (Source)

The nightly news reports have been filled with stories about the large iceberg that recently calved off of the Larsen C ice shelf. Reports of natural events always struggle with trying to convey the scale of events to the general public. In this case, the media has been reporting that the iceberg is (1) approximately the same area as the state of Delaware, (2) it contains a volume of water that is double that of Lake Erie, and (3) the mass of water it contains is about 1 trillion metric tons. Figure 1 shows a good graphic for area comparisons.

In this post, I will provide some support for these numbers and how they were obtained.

Background

No special background is needed – just remember that the density of liquid water 1 gm/cm3, and the density of ice is 0.9167 gm/cm3.

Analysis

Figure 2 shows my analysis. Any background you need can be obtained by clicking on the links referenced in Figure 2.

Lake Erie Larsen C Ice Shelf Larsen C Ice Shelf Larsen C Ice Shelf Delaware Larsen C Ice Shelf
Figure 2: Iceberg Metrics.

Figure 2: Iceberg Metrics.

Conclusion

This was just a quick fact check on numbers in the news. I think the media did a good job on this story.

Posted in General Science, News Fact Checking | Leave a comment

CO2 Generation By Fuel Per Million BTUs of Heat

 

Quote of the Day

The will to win is not nearly so important as the will to prepare to win.

— Vince Lombardi


Introduction

Figure 1: Table of CO2 Generation By Fuel For 1 Million BTUs of Heat.

Figure 1: Table of CO2 Generation By Fuel For 1 Million BTUs of Heat.

My year-round cabin in northern Minnesota needs a furnace, and a furnace needs fuel. My fuel options are fairly limited – fuel oil, liquid natural gas, or propane. I ended up choosing propane because the local propane gas supplier has a reputation for being reliable. While researching the fuels, I became curious about the amount of CO2 released into the atmosphere by the different fuel options for given amount of heat.

I do plan on incorporating some solar panels next year for heating water. Right now, I am trying to get cabin construction complete before winter arrives.

For this post, I stayed with US customary units. This allowed me to compare some of my results with those posted by the US government, and the agreement was excellent.

Background

Definitions

British Thermal Unit (BTU)
The British thermal unit (Btu or BTU) is a traditional unit of heat; it is defined as the amount of heat required to raise the temperature of one pound of water by one degree Fahrenheit. It is the standard unit of heating by which US furnaces are specified. (Source)
Enthalpy of Combustion (ΔHFuel)
Standard enthalpy of combustion is defined as the enthalpy change when 1 mole of a compound is completely burnt in oxygen gas at 298K and 1 bar pressure. (Source)

Stoichiometry

The spreadsheet evaluates Equation 1, which computes the mass of CO2 produced when 1 million BTUs of energy are generated using each fuel.

Eq. 1 \displaystyle {{m}_{{CO2}}}=\frac{{{{E}_{{Reference}}}}}{{\Delta {{H}_{{Fuel}}}}}\cdot M{{W}_{{CO2}}}\cdot {{n}_{{CO2}}}

where

  • EReference is the amount of energy used as the basis of comparison. For this post, I am using 1 million BTUs, which is the same as used by the US Department of Energy.
  • MWCO2 the molecular weight of CO2 (44 gm/mole).
  • ΔHFuel is combustion enthalpy of the fuel (i.e. heat generation by combustion per mole of fuel). I list a table of these values in Appendix A.
  • nCO2 is number of moles of CO2 generated per mole of fuel.

Analysis

With the exception of coal, all the CO2 calculations are a straight forward application of Equation 1. Coal is the exception because it is not pure carbon. I have written a post about this topic. Some grades of anthracite coal actually generate more heat per pound than if they were 100% carbon. To deal with this fact, I computed an  effective specific heat of combustion using the measured specific heat of combustion for coal. I assume the coal is 100% carbon but that the carbon generates more heat than real carbon. For my analysis, I used an anthracite coal with a heat output of 14,820 BTU/lb. We can convert that to an effective molar heat of combustion as shown in Figure 2. This is the value I used in the table shown in Figure 1.

Figure M: Effective Carbon Heat of Combustion.

Figure 2: Effective Carbon Heat of Combustion.

If coal was pure carbon, the heat output would have been 14,116 BTU/lb, which you can calculate as shown in Figure 3.

Figure 1: BTU per Pound From Carbon.

Figure 3: BTU per Pound From Carbon.

Conclusion

I will be looking for ways to reduce my carbon footprint going forward. I have plans for solar water heating and solar electrical generation. It is a bit tough when you are at the latitude of northern Minnesota, but the Germans have an excellent solar power infrastructure, and they are at an even higher latitude, e.g. Stuttgart is at 48.7° and my cabin is at 47.7°. If they can do it, so can I.

I should make a comment about burning hydrogen for fuel, which generates no CO2 because there is no carbon involved at all. While this analysis makes the burning of hydrogen look attractive, unfortunately most hydrogen is produced commercially through steam reforming, which results in the generation of a large amount of CO2.

Appendix A: Fuel Enthalpy Data

Figure 4 is the source of my fuel information.

Figure M: Table of Fuel Enthalpies. (Source)

Figure 4: Table of Fuel Enthalpies. (Source)

Appendix B: US Department of Energy CO2 Emission Data

Figure 5 shows a comparable table of values that was prepared by the US DoE.

Figure M: Pounds of CO2 Released By Fuel Per Million BTUs of Heat. (Source)

Figure 5: Pounds of CO2 Released By Fuel Per Million BTUs of Heat. (Source)

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Personnel Count of US Special Operations Forces

 

Quote of the Day

If you check the party affiliation of someone who commits assaults before deciding how you feel about it, you're what's wrong with America.

Frank Luntz, conservative political consultant.


Introduction

Figure 1: Special Operations Forces By Service.

Figure 1: Special Operations Forces By Service.

I have a son who lives in Butte, Montana – the home town of Robert O'Neill, a famous US Navy SEAL. We were discussing Mr. O'Neill's exploits one night and started to wonder about the size of the different US special operations forces. I quickly looked up some 2014 data from the Government Accountability Office (GAO) and put the data into a pivot table (Figure 1). I was a bit surprised at the numbers involved – it does not surprise me that the Army has the largest contingent, but the size of the Air Force's contingent was a surprise.

Let' s break down the numbers in Figure 1 by service and unit. Also, I remind you that the data is from 2014. I am assuming the numbers have not changed significantly.

Personnel Count By Service and Unit

All of these units have civilian and military personnel. I only show the military personnel here.

US Army

Figure 2 shows the staffing for the different US Army special operations units. The only surprise to me here was that the Rangers are fewer in number than the SEALs. I found this web page that confirms these numbers.

Figure M: US Army Special Forces Staffing.

Figure 2: US Army Special Operations Forces Staffing.

US Air Force

Figure 3 shows the size of the different Air Force special operations groups. For a discussion of their general functions, see this link.

Figure M: US Air Force Special Operations Forces Staffing.

Figure 3: US Air Force Special Operations Forces Staffing.

US Navy

Figure 4 shows the unit breakdown of the the US Navy's SEALs, which are the most well known of the US special operations forces. Of the SEAL teams, the Development Group (also known as Team Six) is the most well known.

Figure M: US Navy Special Forces.

Figure 4: US Navy Special Operations Forces Staffing.

US Marine Corps

Figure 5 shows the unit breakdown of the US Marine Corps special operations units. For a discussion of their functions, see this link.

Figure 5: US Marine Corps Special Operations Forces.

Figure 5: US Marine Corps Special Operations Forces.

A Youthful Recollection

When I was in grade school, a relative who was a Ranger was killed in action during the Vietnam War. I remember seeing his picture (Figure 6) and thinking how very young he looked. I still remember how sad my father looked when he heard the news. We are fortunate to have people like him among us.

Figure 1: Ronald Biegert, Killed In Action During the Vietnam War.

Figure 6: Ronald Biegert, Killed In Action During the Vietnam War. (Source)

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Posted in Military History | 2 Comments

My New Cabin Construction

 

Quote of the Day

The benefit of controlling a modern state is less the power to persecute the innocent, more the power to protect the guilty.

— David Frum


Scope

A number of folks have asked that I post pictures of my cabin construction project. The project actually consists of two separate activities: a large garage (started last fall) and a two-story cabin. I will start posting photos here as things progress.

Old Cabin Demise

The process really began in earnest with the demolition of the old hunting shack that was built in the 1930s.

Figure 1: Old Cabin Demolition.

Figure 1: Old Cabin Demolition.

Garage

The garage is a 30'x60' Morton building. All the garage photos were taken from tree-mounted remote camera. The garage has a storage area, office, and wood shop. I will take more pictures this weekend.

Figure 1: Garage Excavation.

Figure 2: Garage Excavation.

Figure 1: Garage Framing.

Figure 3: Garage Framing.

Figure 2: Framed Garage.

Figure 4: Framed Garage.

The garage contains three rooms: (1) an office with bathroom and shower, (2) a woodshop, and (3) a boat storage area. You can see the framing in Figure 5. HVAC installation is in progress. Electrical wiring and plumbing will follow.

Figure 5: Garage Internal Framing.

Figure 5: Garage Internal Framing.

House

The house is ~2000 square feet and will be my retirement home.

Figure 5: Foundation Poured.

Figure 6: Foundation With In-Floor Heating.

Figure 2: First Floor Framing Started.

Figure 7: First Floor Framing Started.

Figure 8: Second Floor Framing is in Progress.

Figure 8: Second Floor Framing is in Progress.

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Posted in Construction, Personal | 4 Comments

Analysis of 555-Based PWM Circuit

 

Quote of the Day

It is impossible for a man to learn what he thinks he already knows.

— Epictetus, Discourses, Book II Ch.1. This quote caused my career to flash before my eyes.


Introduction

Figure 1: 555 Timer Circuit Causing Analysis Issues.

Figure 1: 555 Timer Circuit Causing Analysis Issues.

I received a request for design formulas that can be used to estimate the frequency (f) and duty cycle (DC) generated by the 555 timer-based, Pulse Width Modulator (PWM)  circuit shown in Figure 1. The presence of diodes in the charge and discharge paths are the main cause of the confusion.

In this post, I will provide: (1) analytic expressions for both f and DC, (2) a detailed derivation of these expressions using Mathcad, (3) an LTspice simulation illustrating how potentiometer resistance affects f and DC, and (4) an error analysis showing the quality of the relationship between the design formulas and the simulation.

As I have mentioned in other posts, I am busy building a cabin and large workshop in northern Minnesota. This means my post will include limited explanatory information because my time is limited.

For those who are interested in my source, my files are here.

Background

Equation 1 can be used to compute oscillation frequency (f) of the circuit of Figure 1 (component locations are defined in Figure 3).

Eq. 1 \displaystyle f=\frac{1}{{{{C}_{1}}\cdot \text{ln}\left( {\frac{{2\cdot {{V}_{{CC}}}-3\cdot {{V}_{D}}}}{{{{V}_{{CC}}}-3\cdot {{V}_{D}}}}} \right)\cdot \left( {{{R}_{2}}+{{R}_{3}}+{{R}_{{POT}}}} \right)}}

where

  • f is the oscillation frequency.
  • RPOT represents the total potentiometer resistance.
  • R2R3, and C1 are passive component values defined in Figure 3.
  • VD is the diode voltage.
  • VCC is the supply voltage.

Equation 2 allows you to compute the duty cycle (DC) as a function of resistance.

Eq. 2 \displaystyle DC=\frac{{{{R}_{2}}+{{R}_{{POT}}}\cdot k}}{{{{R}_{2}}+{{R}_{3}}+{{R}_{{POT}}}}}

where

  • k represents the potentiometer's normalized wiper position, i.e. k ranges from 0 to 1, inclusive.

Notice how Equations 1 and 2 allow you to set the frequency and DC independently. First, set your duty cycle by selecting your resistors, then set your frequency by picking the corresponding capacitor.

Analysis

Formula Derivation

Figure 2 shows my derivation of Equations 1 and 2 using Mathcad 15. There are a couple of things to notice about the formulas:

  • The forward voltage of the diode only affects the oscillation frequency.
  • The duty cycle is a function of the resistances and the potentiometer wiper position.
Figure 2: Derivation of the PWM Formulas.

Figure 2: Derivation of the PWM Formulas.

Simulation Work

I wanted to simulate the circuit in a way that did not require the use of special libraries – like the potentiometer library or a cleaner 555 symbol. Instead, I decided to use two resistors with values that vary in the same manner as the resistance in a potentiometer. Using this approach, I could then created a "wiper" that varied with time, i.e. \displaystyle k=\frac{{time}}{{\left\{ {tTot} \right\}}}, where {tTot} is the total simulation time.

I also used the standard 555 symbol, even though I do not like the way this symbol connects to other parts on a schematic (Figure 3). Yes – I am a bit of a schematic artist.

Figure 2: LTspice Implementation of the 555 PWM Circuit.

Figure 3: LTspice Implementation of the 555 PWM Circuit.

Figure 4 shows the simulation result. As you can see, duty cycle varies as the "wiper" position is changed, i.e. time advances. As expected from Equation 1, the oscillation frequency holds constant as the wiper position is varied.

Figure 3: Output Voltage Simulation For the Circuit of Figure 2..

Figure 4: Output Voltage Simulation For the Circuit of Figure 2. I am only showing part of the simulation because the fine detail is lost at larger scale.

Error Analysis

Figure 5 shows ten data points for which I computed the frequency (Equation 1) and duty cycle (Equation 2) using Mathcad and LTSpice. The agreement is reasonable.

Figure 4: Comparison of Equations to Simulation.

Figure 5: Comparison of Equations to Simulation.

Conclusion

This post derived a pair of formulas that can be used to design a simple, potentiometer-controlled, PWM circuit. The derivation showed good agreement with a Spice simulation of the same circuit.

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Visualizing US vs IJN Aircraft Carrier Numbers During WW2

 

Quote of the Day

The world may think you are only one person. But to one person, you may be their world.

— Author Unknown. When my children were small, I knew my wife and I were their whole world. This is a big responsibility. Even with adult children, the role of parent is still important – it is the world's best job.


Figure 1: Deployed Carrier Numbers Versus Time During WW2.

Figure 1: Deployed Carrier Numbers Versus Time During WW2.

I watched an interesting lecture on American History TV this weekend called Japanese Perspective on the Battle of Midway by Anthony Tully. The most interesting part of the discussion occurred when Tully began showing how the US production of aircraft carriers eventually overwhelmed the Japanese ability to build carriers. He used some simple graphs to show the relative carrier strength of the US Navy versus the Imperial Japanese Navy (IJN) over time. In this post, I will come up with my own graphics to visualize this information.

It happens that I am taking a course in Excel dashboards right now, and I thought I would try to create my own graphic for this data using some of the techniques shown in this class. To generate the graphic, I needed data. I quickly checked the Wikipedia and it turns out it has a list of WW2 carriers, their date of commission, and date of demise. This data allowed me to generate Figure 1, which I find a bit easier to digest than the graphics shown in the lecture.

Figure 1 shows how US carrier production swamped the ability of the IJN to replace their losses. There are some definitions that are useful in understanding Figure 1.

Aircraft Carrier
Also called a fleet carrier, this was the largest and most capable aircraft carrier type during WW2. (Link)
Light Aircraft Carrier
A carrier design based on cruiser hulls, which resulted in a high-speed design with a complement of aircraft only one-half to two-thirds the size of a full-sized fleet carrier. These carriers filled a gap in fleet protection that existed until more fleet carriers were built. (Link)
Escort Carrier>
A carrier design focused on protecting merchant convoys from submarine attack and provide support to amphibious forces during landings. Escort carriers are generally smaller and slower than fleet or light carriers (Link).

For those who are interested in the details, here is the spreadsheet.

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