Epidemiology and Cell Towers

Posted on 10-April-2013 by mathscinotes

Introduction

I received a phone call from a cancer epidemiologist last Friday. He had received my name from a co-worker in his department who knows me. This researcher is in the process of researching a cancer cluster near a cell tower. Local residents have been speculating about whether that cell tower has played any role in the formation of this cluster. I wouldn't be surprised if the owners start looking into cell tower lease buyouts round about now. I have done a fair number of measurements of RF power levels in the vicinity of antennas and that is why he contacted me.

The researcher had done a very good job of familiarizing himself with the basics of Radio-Frequency (RF) electronics by doing some googling, and has even come across some terms such as dirty electricity and the like. Now he wanted to confirm his information with an RF practitioner -- I fit the bill. We talked about a number of interesting subjects. In this post, I will discuss how I go about collecting some basic information on a cell tower -- height, power, frequency -- something I need to regularly do. I thought some of you may find this process interesting. I am always surprised at the number of radio antennas that are registered in an area. I will discuss other parts of our discussion in later posts.

There are four web sites that I regular use to find information on towers and the antennas that populate them:

  • http://www.antennasearch.com

    This site supports separate searches for tower and antennas. You can also access aerial photos of the locations.

  • http://wireless2.fcc.gov/UlsApp/AsrSearch/asrRegistrationSearch.jsp

    The government's Antenna Structure Registration (ASR) system. The government makes the licenses it grants accessible on the web.

  • http://wireless2.fcc.gov/UlsApp/UlsSearch/searchLicense.jsp

    The government provides a means for looking up information on their Universal Licensing System (ULS). This is useful when you know the transmitter's call sign.

  • http://radioreference.com

    This site is similar to antenna search -- works best when you have a radio call sign to search for. This site is nice because it makes radiated power easily available.

Remember that most antennas are mounted on towers owned by someone different than the antenna owners.

I will show you some examples of some recent searches I did near my workplace. This is NOT the area that the cancer researcher was asking me about. These are just examples of what you can find "out in the cloud".

Figure 1 shows the output for a recent tower search that I did at www.antennasearch.com (the antenna is on a tower in Brooklyn Park, MN -- not far from where I grew up).

Figure 2: Tower search option from antennasearch.com.

Figure 2: Tower search option from antennasearch.com.

Figure 2 shows the output for the antenna search option from www.antennasearch.com.

Figure 1: Search for Antennas at antennasearch.com.

Figure 1: Search for Antennas at antennasearch.com.

Figure 3 shows the output for an arbitrarily chosen FCC license for the tower near me from http://wireless2.fcc.gov/UlsApp/AsrSearch/asrRegistrationSearch.jsp.

Figure 3: Example of an FCC license for a tower in Brooklyn Park, MN.

Figure 3: Example of an FCC license for a tower in Brooklyn Park, MN.

Figure 4 shows that we can even get an aerial photograph of a nearby tower from the web at www.antennasearch.com.

Figure 4: Image of the Tower.

Figure 4: Image of the Tower.

Figure 5 shows the licensing information for a specific antenna on the tower from http://wireless2.fcc.gov/UlsApp/AsrSearch/asrRegistrationSearch.jsp.

Figure 5: Example of a Specific License for an Antenna

Figure 5: Example of a Specific License for an Antenna

Figure 6 shows some detailed technical information for an arbitrarily chosen radio call sign from http://radioreference.com.

Figure 6: Radioreference Info on an FCC License Holder.

http://www.radioreference.com/apps/db/?fccCallsign=WPXN226

Posted in Electronics, Osseo | 2 Comments

Neat photos from the International Space Station

Posted on 5-April-2013 by mathscinotes

Neat photos from the International Space Station

Some co-workers and I are practicing speaking German while at work. One of these co-workers found some great photos on the Der Spiegel web site, which is a site we use for practice. The notations for the photos are in German, but the pictures are very worthwhile.

Posted in Astronomy | Tagged | Comments Off on Neat photos from the International Space Station

Phone Line Impedance Levels: 600 Ohms and 900 Ohms

Posted on 2-April-2013 by mathscinotes

Quote of the Day

Is it really a sport? Is it a sport if one team doesn't know its going on?

— Bill Maher on hunting

Introduction

Figure 1: Old Phone Central Office. (Source)

Figure 1: Old Phone Central Office. (Wikipedia)

Engineering is a pretty conservative profession -- I have been accused of "abhorring change". Once something gets standardized it stays in place even when it does not make sense. This morning provided me a good example of this. Phone lines in the United States are usually characterized as having a characteristic impedance of 600 Ω or 900 Ω. These impedance levels go back to the early days of telephony (Figure 1). However, all the phone cables we work with are Category 3 and therefore have identical characteristic impedance (~725 Ω). So why the different impedance levels? I had a discussion with one of our telephony engineers about it this morning and all we could do is speculate. I thought I would document this speculation here.

Analysis

Assumptions

Our speculations are based on certain assumptions.

  • In the old days, 26 AWG wire was used for central office wiring.
  • In the old days, 22 AWG wire was used for connecting central offices to homes.
  • Telephony circuits were characterized at 1 kHz.
  • I normally hear test engineers assuming characteristic impedances of 600 Ω for home wiring and 900 Ω for central office wiring.

When I work with customers, I do see 22 AWG and 26 AWG cables used in older homes and central offices, respectively. We can compute the characteristic impedance of these cables at 1 kHz to see if that explains the 600 Ω and 900 Ω terminations.

Characteristic Impedance Calculation

Figure 2 shows my calculation of the characteristic impedance of 22 AWG and 26 AWG wire at 1 kHz.

Figure 1: Characteristic Impedance Calculations for 22 AWG and 26 AWG cables.

Figure 2: Characteristic Impedance Calculations for 22 AWG and 26 AWG cables.

So we see that the old wiring standards did have 600 Ω and 900 Ω values. Note that the impedances are complex, but we normally speak of them as being real in the US. Some countries, like Australia, recognize that the impedances are complex (see this blog post for further discussion).

Conclusion

I have no idea if this is why the 600 Ω and 900 Ω values are used, but it seems reasonable. The interesting thing to me is how we keep designing systems assuming 600 Ω and 900 Ω values even when we know those numbers are not correct. It reminds me of the old story I have included here (Source).

The U.S. Standard railroad gauge (distance between the rails) is 4 feet, 8.5 inches. That's an exceedingly odd number. Why was that gauge used? Because that's the way they built them in England, and the U.S. railroads were built by English expatriates. Why did the English people build them like that? Because the first rail lines were built by the same people who built the pre-railroad tramways, and that's the gauge they used.
Why did ''they'' use that gauge then? Because the people who built the tramways used the same jigs and tools that they used for building wagons, which used that wheel spacing.
Okay! Why did the wagons use that odd wheel spacing? Well, if they tried to use any other spacing the wagons would break on some of the old, long distance roads, because that's the spacing of the old wheel ruts.
So who built these old rutted roads? The first long distance roads in Europe were built by Imperial Rome for the benefit of their legions. The roads have been used ever since. And the ruts? The initial ruts, which everyone else had to match for fear of destroying their wagons, were first made by Roman war chariots. Since the chariots were made for or by Imperial Rome they were all alike in the matter of wheel spacing.
Thus, we have the answer to the original questions. The United States standard railroad gauge of 4 feet, 8.5 inches derives from the original specification (Military Spec) for an Imperial Roman army war chariot. Military specs and bureaucracies live forever. So, the next time you are handed a specification and wonder what horse's ass came up with it, you may be exactly right. Because the Imperial Roman chariots were made to be just wide enough to accommodate the back-ends of two war horses.

Save

Posted in Electronics, Telephones | 6 Comments

Engineering Application of Conformal Mapping

Posted on 1-April-2013 by mathscinotes

Introduction

I have a project that I am working on that involves the use of conformal mappings. I have long found the use of complex numbers in electrical engineering interesting. My first contact with an engineering application of conformal mappings occurred over 30 years ago when I was working at Hewlett-Packard in Ft. Collins, CO (now an Avago facility). At that time, I saw an article in one of the engineering trade journals (EDN, I think) that used conformal mapping to create a matching network for a telephone hybrid. This article gave me some ideas on how I could use complex variables in my work, and I thought it might be useful to review it here. Unfortunately, my copy of the article is not great and does not include the publication date and the journal name. I include a copy of the article here. This article is very similar to an application note from National Semiconductor by the same author.

I thought it would be worthwhile reviewing this article here. While designing a telephone hybrid is not often done today, the basic problem solving approach can be applied to other electronic problems. As was common back in the 1980s, the article includes the source for a BASIC program that can be used to numerically solve the problem. I will focus, instead, on solving the problem using a computer algebra program (Mathcad).

This article review will provide a nice demonstration of a number of items:

  • Simple application of complex numbers in an electrical engineering application.
  • Use of the properties of a conformal mapping in an optimization problems.
  • Application of a computer algebra system (Mathcad) in computing optimal component values.

Background

Basic Telephone Engineering

Here are the telephone basics you need to know to understand this discussion:

  • One line of telephone service is delivered to a home over a single pair of wires (2 wires total).
  • The phone line (one wire pair) must carry signals in both directions -- to and from the home.
  • Inside the phone there are actually two wire pairs (4 wires total) -- one for the handset microphone and the other for the handset speaker.
  • The interface between the 2 wire pairs in the handset and the one wire pair feeding the phone is called a hybrid. It performs a function called 2 wire-to-4 wire conversion.

Definitions

2-wire/4-wire conversion
Inside your phone, the microphone (aka. transmit) and speaker (aka. receive) signals are carried by separate wire pairs. However, only a single wire pair comes to your home -- that wire pair simultaneously carries the transmit and receive signals. Today, the operation of combining and separating the transmit and receiver signals is often done digitally. However, historically this function has been done by an analog circuit called a hybrid. Much woe has been attributed to the lowly hybrid. For example, many of the echo problems people experience are due to a bad hybrid somewhere in the system.
Hybrid
A device that transforms a 4-wire phone circuit (1 transmit wire pair and 1 receive wire pair) into a 2-wire phone circuit (transmit and receive signals on a single wire pair). Since the phone line coming to our home carries both transmit and receive signals, we will drive our phone's speaker with this combined signal minus our transmit signal. It turns out we can perform this subtraction very accurately digitally, but historically it has been done by an analog circuit that has frequency-dependent errors. To be completely accurate, we generally like to let a bit of our own transmit signal "leak" into our receive path because it sounds more pleasant. This leakage is referred to as sidetone. We will not worry about sidetone in this blog post.
Data Access Arrangement (DAA)
A circuit designed to interface safely to a standard, single wire-pair phone line. It usually contains a transformer and surge protection circuits.
Line Impedance
The wire pair that enters your home (i.e. phone line or line) has an impedance that varies based on a number of factors (wire gauge, length, loads, etc). We generally assume the line impedance to be 600 Ω (real), but this is a very crude working assumption. It actually varies significantly, yet we expect our phones to work acceptably no matter where we plug them in. This post is about how to minimize the phone's performance variation with line impedance. If you want to understand the 600 Ω from a bit more theoretical standpoint, see this excellent discussion.
Conformal Mapping
A mathematical mapping that transforms circles to circles and straight lines to straight lines.

Problem Statement

Figure 1 shows the circuit that we will be discussing here.

Figure 1: Schematic of the Hybrid Circuit Under Consideration.

Figure 1: Schematic of the Hybrid Circuit Under Consideration.

Given this circuit, the basic problem statement is simple.

Select component values for R1, R2 and C2 in the circuit of Figure 1 that will minimize the maximum amount of transmit signal that will "leak" onto the receive wire pair.

Approach

The basic approach is simple.

  • For a given range of phone line impedances, find the input impedance of the DAA.

    We will draw a circle around the range of line impedances that we want to ensure good phone operation. Because the DAA transfer function is a conformal mapping, this circular region of phone line impedances will translate to a circular region of DAA impedances.

  • For a given range of DAA impedances and an assumed gain between the transmit amplifier and the receive amplifier, compute the overall gain from the transmit amplifier to the receive amplifier.

    The gain between the transmit amplifier to the receive amplifier is the system parameter we are concerned with minimizing. Figure 4 illustrates how this calculation is performed. In this case, we are minimizing the circuit gain and are not considering components at all.

  • Vary the assumed amplifier gain until we find the gain that minimized the transmit gain to receive gain value.

    This is the gain that we want to ensure that its maximum value is as small as we can make it.

  • Compute the component values required to generate the minimum transmit-to-receiver gain.

    This is a simple algebra problem.

Analysis

Range of Phone Line Impedances

Figures 2 and 3 show the range of phone line impedances at 1 kHz and 3 kHz (respectively) we will be working. These figures contain the following information:

  • The filled irregularly shaped regions represent the range of phone line impedances and how often we expect to encounter them. Three levels of occurrence frequency are represented: (dark blue) very common, (white) moderate occurrence, and (light blue) rarely occur.
  • The red axis were inserted by me in Dagra, a program that I use to digitize graphic data. I used Dagra to help me position the purple impedance circles.
  • The purple circles mark the range of impedances that my Mathcad routine will be minimizing the transmit-to-receive path gain.

Figure 2: Phone Line Impedance Likelihoods at 1 kHz.

Figure 2: Phone Line Impedance Likelihoods at 1 kHz.

Figure 3: Phone Line Impedance Likelihoods at 3 kHz.

Figure 3: Phone Line Impedance Likelihoods at 3 kHz.

Mapping from Line Impedance to DAA Impedance

In the article, the author actually measures the DAA impedances when it is attached to three impedances that equal the three phone line impedances that we used to define our circular region of phone line impedances. All we need to do is convert the line phone line impedances to resistor and capacitor values that will generate the equal impedance values at the chosen frequency. I generate the phone line equivalent impedances using the component values computed in Figure 4.

Figure 4: Passive Components Required to Generate Equivalent Phone Line Impedances.

Figure 4: Passive Components Required to Generate Equivalent Phone Line Impedances.

These resistor and capacitor value combinations are connected to the DAA as phone line impedance simulators.

Mapping from DAA Impedance to Gain Values

Rather than working with component values directly, we can model the system using a transmit-to-receive path transfer function A(j·ω). We can then determine the complex gain at specific frequencies that we require to minimize the transmit-to-receive path gain. Figure 5 shows the circuit model for this approach.

Figure 5: Block Diagram of the Hybrid Circuit Using a Gain Model.

Figure 5: Block Diagram of the Hybrid Circuit Using a Gain Model.

We will find the value of A(j·ω) at specific frequencies that will minimize the transmit-to-receive path gain using Equation 1.

Eq. 1 {{G}_{T2R}}\left( j\cdot \omega \right)=2\cdot A\left( j\cdot \omega \right)-\frac{{{Z}_{DAA}}\left( j\cdot \omega ,{{Z}_{PhoneLine}} \right)}{600\Omega +{{Z}_{DAA}}\left( j\cdot \omega ,{{Z}_{PhoneLine}} \right)}

where

  • A(j·w) is the complex transformation applied to the transmit signal (unitless)
  • ZDAA is the impedance of the DAA (Ω), which is a function of frequency and the phone line impedance (ZPhoneLine).

Finding Optimum Gain and Computing the Associated Component Values

Figure 6 is an excerpt from the article that shows how the maximum gain for a given set of DAA impedances is computed. We are just computing the magnitude of the largest complex gain we can generate.

Figure 6: Geometric View of Determining Maximum Gain.

Figure 6: Geometric View of Determining Maximum Gain.

For details on how I compute the center point and radius of the circle generated by the three gain point, see Appendix A.

Once we have the maximum gain computed, we can compute the component values that will minimize this gain. Figure 7 shows how I derived equations for the passive component values as a function of the complex gain A(j·ω).

Figure 7: Development of Equations to Solve for Passive Component Values.

Figure 7: Develop of Equations to Solve for Passive Component Values.

Figure 8 shows the calculation of the specific component values.

Figure 8: Determination of Specific Component Values for the Hybrid.ues

Figure 8: Determination of Specific Component Values for the Hybrid.ues

Conclusion

I reworked the example from this paper using Mathcad and have duplicated the paper's original results. I will be using a routine very similar to this one to solve an actual problem that I am working on right now.

Appendix A: Circle Parameters from Three Points

Figure 9 show my derivation of the equations for computing the center coordinates and radius of a circle given three points on the circumference.

Figure 9: Determine a Circle's Center and Radius Using 3 Points.

Figure 9: Determine a Circle's Center and Radius Using 3 Points.

Posted in Electronics | 4 Comments

Distibution of Catholics Around the World

Posted on 17-March-2013 by mathscinotes

Introduction

I am showing one of my sons how to use Excel pivot tables, so I am looking for good data analysis examples that illustrate the power of pivot tables. The election of a pope has put Catholicism in the news. The news reports have included numerous discussions about the number of Catholics. I was reading a report on the web by the BBC on the distribution of Catholics around the world that used Figure 1.

Figure 1: BBC Representation on the Distribution of Catholics.

Figure 1: BBC Representation on the Distribution of Catholics.


I found this graphic interesting. I was surprised at the number of Catholics in Latin America. I started to wonder if I could generate that chart in Excel. Let's get to work ...

Background

The BBC web site says that the data came from the World Christian Database, which requires membership to get their data -- I need to go look elsewhere. It turns out that similar information can be obtained from the Catholic Hierarchy website. Since the databases are different, I would expect to have somewhat different charts. Hopefully, they will be very similar.

Analysis

My analysis process is straightforward:

  • Import the web data into Excel
  • Assign the countries to the regions used by the BBC
  • Create a pivot table of the data
  • Create a pivot chart of the data

After going through this process, I obtain Figure 2.

Figure 2: My Version of the BBC Graphic.

Figure 2: My Version of the BBC Graphic.


This is pretty close to the graphic presented by the BBC. For those who are curious, I have attached my spreadsheet here.

Conclusion

My chart and the BBC's charts are very similar. This was a good example of a common type of pivot table analysis. I have no doubt that the BBC did something similar to produce their graph.

Posted in General Mathematics | Comments Off on Distibution of Catholics Around the World

A World War 1 Mechanical Torpedo Fire Control Computer

Posted on 13-March-2013 by mathscinotes

I have discussed the basic calculations behind firing "straight-running" torpedoes in other posts (here, here, and here). Youtube has a couple of interesting videos showing a basic mechanical torpedo fire control computer from World War 1. It is a nice illustration of the geometry that I discussed in my previous posts on this topic. For those folks who are interested in naval history, I do recommend the Dreadnought Project website. The dreadnought folks do amazing work.

Posted in History of Science and Technology, Underwater | 1 Comment

Hometown Humor

Posted on 12-March-2013 by mathscinotes
Figure 1: Kevin Kling.

Figure 1: Kevin Kling.

My brother called the other night and said that he was going to attend a live show with Kevin Kling, a well-known comedian we grew up with in Osseo, Minnesota. In my youth, Osseo was a small agricultural community on the outskirts of Minneapolis. Growing up in a small town like Osseo was a lot of fun -- we always compared it to Mayberry on the "The Andy Griffith Show." Osseo may be only 0.75 square miles in area, but it would be hard to imagine a place with more interesting people per square foot. I could tell many stories about life there, however, a math blog is not the appropriate place.

If you want to have an enjoyable time listening to reminiscences, I highly recommend that you listen to Kevin Kling and his stories. He has made his living telling stories about growing up during the 1960s in Osseo area. I laugh just thinking about that time. I remember sitting in class with him and listening to his stories -- and now the entire country gets to hear them! I know every person that he is talking about and his description of them is spot on. His stories range from his first love (Judy Martinez, who lived just a few blocks away from me) to his description of watching the "smutty movies" at the Starlight Drive-in Theater from outside the beat-up wooden fence (e.g. Seventeen and Anxious). Those movies were terrible ...

If you want a quick introduction to Kevin, he was interviewed on the radio program "Speaking of Faith" and it is a worthwhile listen.

Posted in Osseo, Personal | Comments Off on Hometown Humor

Spline Interpolation Example Using Battery Capacity

Posted on 10-March-2013 by mathscinotes

Introduction

I had a conversation with a customer recently who wanted to estimate the capacity and running time of their electronic systems when operating from batteries that are under various current load and temperature stresses. The battery manufacture had only specified their battery's operation under a limited number of usage scenarios-- none of which corresponded to those of my customer. To provide some guidance to this customer, I needed to do some interpolation of the battery vendor's data. This was an interesting exercise and thought it was worth sharing my work here.

Background

Figure 1 shows the battery data that I will be interpolating. This data is from the specification here.

Figure 1: Battery Vendor Data for the Panasonic LC-R127R2P Sealed Lead Acid Battery.

Figure 1: Battery Vendor Data for the Panasonic LC-R127R2P Sealed Lead Acid Battery.

Normally, I use Mathcad for this kind of work. In fact, I had done this work originally in Mathcad years ago. However, this particular customer only had Excel -- sigh. After a Google search, I found an excellent VBA-based, cubic spline routine by David Braden. A little Excel and a little VBA soon gave me a spreadsheet that would interpolate Figure 1 adequately.

Analysis

Analysis Approach

  1. Generate a linear approximation of the discharge curves on the vendor's graph

    The chart shows that the "curves" are linear on a log-log plot over the range from 0.4 A to 10 A. I will do a least-square line fit over that range of currents at four temperature values: -15 °C, 0 °C, 25 °C, and -40°C. This operation will generate four slopes and intercepts.

  2. Use the cubic spline routine to interpolate between the slopes and intercepts from part 1 for any temperature between -15 °C and 40 °C.

    This is a reasonable approach to estimating the line parameters at temperatures presented on the battery's specification sheet.

  3. Assign two cells for the manual input of currents and temperatures by the customer.

    This allows me to compute the backup time duration of the battery for a specific set of conditions.

  4. Compute the effective battery capacity by multiplying the current draw by the time duration.

    Many people prefer to think in terms of battery capacity and not operating time. I am just applying the definition of battery capacity to compute a value.

  5. Use Excel data tables to generate a chart and a couple of tables.

    Excel data tables are an excellent way of generating graphical and table data for a wide range of one and two-dimensional inputs.

The Excel file that I put together is available here.

Interpolated Results

My customer is normally running his batteries at a current load of 0.8 A. Figure 2 shows the worksheet output for 0.8 A load and variable temperature.

Figure 2: Interpolated Battery Capacity Versus Temperature @ 25 °C.

Figure 2: Interpolated Battery Capacity Versus Temperature @ 25 °C.

The customer also requested data in table form. Figure 3 shows a table of discharge times versus various temperatures and current loads.

Figure 3: Discharge Times Versus Load Currents and Temperatures.

Figure 3: Discharge Times Versus Load Currents and Temperatures.

Figure 4 shows a table of effective battery capacities (A-hour) versus various temperatures and current loads.

Figure 4: Effective Battery Capacity Versus Various Battery Loads and Temperatures.

Figure 4: Effective Battery Capacity Versus Various Battery Loads and Temperatures.

Conclusion

I thought this was a good example of the use of interpolation in a real world setting. Hopefully this example will be of use to you.

Posted in Batteries, General Mathematics | Comments Off on Spline Interpolation Example Using Battery Capacity

Battery Life Dissipated Under Varying Temperature Conditions

Posted on 6-March-2013 by mathscinotes

Introduction

I am not getting any younger. As I age, I have come to realize that certain activities age me faster than others. For example, I have no doubt that sitting in meetings for hours every day has deleterious effects on my health. In the case of batteries, higher temperatures age batteries faster than lower temperatures. This post works through two examples in detail to illustrate how to model the rate of battery aging under varying temperature conditions.

I am usually asked about battery life aging with respect to two types of temperature profiles: (1) a battery operating at a few discrete temperatures and (2) a battery operating at a continuously varying temperatures. The analysis methodology is the same in both cases. I will provide examples of both cases.

Background

General Battery Aging Model

I have written extensively about battery aging models in other posts (e.g. here). In those posts, I assumed that the battery temperature was constant. I recently had a customer request that I analyze the impact of varying temperature on a battery's life. This problem was interesting enough that I thought it was worth documenting here.

I will be using the model of IEEE 450-2002 for this post. For those of you in need of a review, take a look here. For another reference, McCluer's paper also uses the same approach.

Computational Model

Every lead-acid battery vendor specifies the expected lifetime of their battery. Most of these batteries have 3 to 5 year lifetimes at 25 °C. However, some batteries are rated to have lives as long as 12 years at 25 °C. In this post, I will compute a conversion factor that will adjust the vendor's specified lifetime for a non-constant temperature profile.

I have done these calculations a couple of different ways. I have estimated battery life using two approaches:

Equivalent 25 °C Time

Express battery life in equivalent 25 °C days. I do not like this approach myself, but some customers will tell me that their operational model requires an 8 year life for a specific temperature profile. They then want to compute an equivalent lifetime at 25 °C, which is how the battery manufacturer's specify their products.

Actual Time

Express battery life in actual days using the customer's temperature profile. I prefer this approach because I generally know the battery that is going to be used and I want to estimate this batteries life under the customers temperature profile.

My approach is simple:

  • Determine the customers temperature profile

    Sometimes the customers gives you their profile and sometimes they tell you that they are doing a deployment in a certain community, like Phoenix (a case that I studied for this very reason).

  • Express the customer's temperature profile in terms of temperature versus percentage of time at that temperature.

    We are going to be computing a weighted average and we need to know the percentage of time the battery spends at each temperature.

  • Compute a conversion factor between 25 °C time and customer profile time.

    Knowing the equivalence between the different temperature profiles (e.g. customer profile versus constant 25 °C), we can generate terms of equivalent 25 °C days or actual days.

Equation 1 shows how I plan to use the conversion factor to adjust the battery's rated lifetime.

Eq. 1 {{T}_{Life\_Profile}}=\frac{{{T}_{Life\_At\_25{}^\circ C}}}{K}

where

  • K is the nominal lifetime conversion factor [unitless]
  • TLife_At_25°C is the battery manufacturer's rated lifetime for the battery at a reference temperature (usually 25 °C) [years]
  • TLife_Profile is the estimated battery lifetime for the customer's temperature profile [years]

For the case of the temperature holding a finite number of discrete values, Equation 2 can be used to compute the conversion factor.

Eq. 2 K=\sum\limits_{i=1}^{N}{\frac{ t{{\%}_{i}}}{A({{T}_{i}})}}

where

  • K is the nominal lifetime conversion factor [unitless]
  • Ti is the temperature during the ith time interval [°C]
  • t%i is the percentage of the battery lifetime at a temperature of Ti [hours]
  • A(Ti) is aging function that corresponds to the battery in question [unitless]

For the case of a continuously varying temperature environment, Equation 3 must be used.

Eq. 3 K=\frac{1}{{{\tau }_{0}}}\cdot \int\limits_{0}^{{{\tau }_{0}}}{\frac{dt}{A(t)}}

where

  • ti is time [hours]
  • τ 0 is the interval of integration over the profile time [hours]
  • T is the duration for the application of the temperature profile [hours]
  • K is the duration of the time interval with a temperature of Ti [hours]

You can view dt/{{\tau }_{0}} as representing the percentage of time represented by a differential length of profile time. As in the discrete case, you can view this as the computing of a weighted average.

Analysis

Discrete Temperature Distribution

Figure 1 shows my evaluation evaluation of Equation 2 for the discrete temperature profile used in the paper by McCluer. I obtained his result, so I have verified my Mathcad implementation.

Figure 1: Analysis of a Discrete Temperature Distribution.

Figure 1: Analysis of a Discrete Temperature Distribution.

Continuous Temperature Distribution

Figure 2 shows my evaluation of Equation 3 for a both the discrete temperature profile used in McCleur's paper and for a continuous temperature variation based on a sine wave (I could have used anything). I am assuming that every day has the same temperature variation. Thus, the percentage of time spent at each temperature is the same for each day and for all the days in total. If I wish to use a period of time longer than a day, I simply extend my interval of integration over the period of time desired.

Figure 2: Aging Analysis for a Continuous Temperature Variation.

Figure 2: Aging Analysis for a Continuous Temperature Variation.

Conclusion

I recently went through this exercise while assisting a customer in a desert climate. They were trying to estimate the effect of temperature on their battery replacement costs. My analysis gave them a rationale approach for determining their operational costs.

Posted in Batteries, Electronics | 1 Comment

Asteroid Belt Mass Distribution Analysis

Posted on 3-March-2013 by mathscinotes

Introduction

The meteor explosion over Russia really interests me and I have been reading as much about it as I can. While reading about meteors and asteroids, I encountered the following statement.

The total mass of the asteroid belt is estimated to be 2.8×1021 to 3.2×1021 kilograms, which is just 4% of the mass of the Moon.

I like to fact check things I read on the web or in the news. Checking this statement is a nice application of simple data analysis in Excel. Let's do a quick fact check here.

Background

The Wikipedia has a great article on the asteroid belt and I cannot improve on their article for basic information. What I need is data on all the known objects in the asteroid belt. It turns out that the folks at the Jet Propulsion Laboratory (JPL) have a great database of all the known asteroid belt objects. I will use that database to generate a list of all the known asteroid belt objects with a diameter greater than or equal to 1 km. I chose the 1 km lower bound to keep the amount of data small.

Analysis

My analysis is approximate -- think of it as a Fermi problem-type analysis. My analysis approach is simple:

  • Download a paper that contains asteroid densities and compute a weighted average density to use for my analysis -- see the Appendix of this post for my average density analysis analysis
  • Download all the main belt objects with diameters (D) greater than or equal to 1 km from JPL website
  • Determine the volume of each asteroid using the formula M=\frac{4}{3}\cdot \pi \cdot {{\left( \frac{D}{2} \right)}^{2}}\cdot \rho , where ρ is my estimate of the average density of an asteroid (2.46 gm/cm3)
  • Put all the data into a table and sort them by mass
  • Compute the percentage of mass that each asteroid represents in the asteroid belt

Table 1 shows the first 10 entries of my table. There are actually 2039 asteroids in my complete table. Understand that I only have a very approximate analysis here. If you want to see a list of the most massive 18 asteroids using the most accurate numbers known, see the Wikipedia. I have included the whole Excel file in the Appendix for this post.

Table 1: Top Ten Asteroids in Terms of Mass.
Name Diameter(km) Estimated Mass(1019 kg) Main Belt Mass(%)
Ceres 952.40 111.32 41.1%
Pallas 545.00 20.86 7.7%
Vesta 530.00 19.18 7.1%
Hygiea 407.12 8.70 3.2%
Davida (1903 LU) 326.06 4.47 1.6%
Interamnia (1910 KU) 316.62 4.09 1.5%
Europa 302.50 3.57 1.3%
Euphrosyne 255.90 2.16 0.8%
Eunomia 255.33 2.14 0.8%
Psyche 253.16 2.09 0.8%

When I added up all the masses in my list, I got a number slightly less than 4% of the mass of the Moon -- fact confirmed.

Conclusion

I was a bit surprised that the asteroid belt contains so little material. Think about -- the total mass in the asteroid belt is less than 4% of the Moon's mass. That really is not very much.

Appendix

I put my Excel file out here for the curious.

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