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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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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Coal Production By State

 

Quote of the Day

Learn to love the struggle — if you can’t enjoy the pains of programming, you’re going to face all the more difficulties when you advance to complex problems.

— Joe Previte on learning to program. His statement is true for most other difficult human activities as well.


Figure 1: Wyoming Dominates US Coal Production.

Figure 1: Wyoming Dominates US Coal Production.

I have been listening to politicians discussing US energy policy the last few days. Very few facts were presented during these discussions, but one politician did casually mentioned that Wyoming produces more coal than the next six states combined. I did not know that Wyoming was such a dominating coal producer, and I began to look at how to fact check this statement. Fortunately, the US Energy Information Administration (EIA) has all the data readily available from this web page.

I downloaded the US coal production data from the EIA web page, processed it using Power Query, and created Table 1. I was able to confirm that Wyoming's coal production exceeds the total output of the next six largest coal coal producing states. For those who are interested, here is my Excel workbook.

Posted in Fact Checking | 6 Comments

Linear Temperature Coefficient Resistor Nonlinearity

 

Quote of the Day

Having proper motivation and honesty are the keys to overcoming fear or anxiety. Fearless and honest self-appraisal can be a powerful weapon against self-doubt or low self-confidence.

— Dalai Lama


Introduction

Figure 1: Resistance Variation with Temperature of a Real "Linear" Temperature Coefficient Resistor.

Figure 1: Resistance Variation with Temperature of a Real "Linear" Temperature Coefficient Resistor.

Our products contain many analog circuits, and these circuits often require temperature compensation in order to meet their requirements across the product's entire temperature range. To perform this compensation, we often use resistors with a specified Temperature Coefficients of Resistance (TCR). A vendor recently stopped manufacturing one of the resistors we use for temperature compensation, and we needed to find a substitute. While searching for a substitute resistor, I needed to understand just how linear the approved resistor's temperature variation is so I can find an appropriate substitute.

Unfortunately, the original designer (gone for over ten years) used a TCR for which the vendor supported but had not published resistance versus temperature curves. The vendor did provide resistance curves for three similar temperature coefficients, which allows me to use interpolation to estimate the curves for the value we are using. Of course, I am requesting that the vendor send me the correct curve. In the meantime, I am just going to interpolate between the specified curves for a preliminary result.

Figure 1 shows my plot of:

  • resistance versus temperature curves for two different TCRs (3900 ppm/°C and 1500 ppm/°C)
  • my interpolated resistance versus temperature curve for a TCR of 3300 ppm/°C.

You can see that there is a small error between the linear ideal and reality. I need to find a substitute part with a similar level of nonlinearity. This post is about viewing the nonlinearity – selecting the substitute part is another matter.

Background

Model for Resistance Variation with Temperature

Equation 1 shows the formula for the variation in resistance for a linear temperature coefficient (i.e. constant TCR) resistor.

Eq. 1 \displaystyle R\left( T \right)={{R}_{{ref}}}\cdot \left( {1+\alpha \cdot \left( {T-{{T}_{{ref}}}} \right)} \right)

where

  • R(T) is the resistor's resistance at a temperature of T.
  • Rref is the resistor's resistance at a temperature of Tref.
  • Tref is temperature at which the resistor's nominal resistance is specified.
  • α is the resistor's TCR.

In Figure 1, I used Equation 1 as my ideal temperature function.

Approach

My analysis approach is simple:

  • Digitize the vendor's three resistance plots (TCRs = 1550, 2700, and 3900 ppm/°C) using Dagra.
  • Perform a two-dimensional interpolation to estimate the response for a TCR = 3300 ppm/°C.
  • Plot  both the ideal and the interpolated responses for a visual comparison of the level of nonlinearity.

Analysis

The analysis was performed in Mathcad and is best viewed either in the Mathcad source or viewing the PDF, both of which are included here.

Conclusion

I am surprised at the typical level of nonlinearity that I am seeing for these resistors. I have always thought that the temperature coefficient for these devices would be a constant, but there is a significant nonlinear component.

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