Earth Altitude with Equivalent Pressure to Mars

Posted on 6-October-2012 by mathscinotes

Quote of the Day

In ordinary life we hardly realize that we receive a great deal more than we give, and that it is only with gratitude that life becomes rich.

— Dietrich Bonhoeffer, German theologian who died fighting against Hitler. This statement reminds me of the Benedictine saying from my youth that if you want to be happy, be grateful for something.

Introduction

I have been keeping close tabs on the Curiosity rover's progress on Mars. While I find many things about Mars interesting, I find the thin atmosphere of Mars especially interesting. If you are looking for some reading on the subject, the Wikipedia has a good article . The rover image in Figure 1 shows that Mars looks pretty bleak.

Figure 1: Rover Image Showing Marineris Volcano.

Figure 1: Mars Rover Image Showing Marineris Volcano.

During my reading, I have seen different values listed for the altitudes on Earth that have the same pressure as Mars' surface. Let's see if we can understand how these equivalent Earth altitudes are arrived at.

Background

On Earth, we usually talk about pressure at sea level. Mars does not have an ocean that we can use as an altitude reference. The Wikipedia gives two points of reference for the atmospheric pressure on Mars:

This is quite a range of values. The atmospheric pressure on the surface of Mars has dynamic range of 38.5 = 1155/30. To compute the dynamic range of the Earth's surface pressure, let's use the following two points:

  • Peak of Mount Everest (8,848 meters above sea level): 33,730 pascals
  • Dead Sea (423 meters below sea level): 106,200 pascals

This means that the dynamic range of air pressure at the Earth's surface is only 3.14 =106,200/33,730. We see that the dynamic range of air pressure on Earth is much less than we would encounter on Mars.

Let's find the altitudes on Earth with the same atmospheric pressures as Olympus Mons and Hellas Planetia.

Analysis

The quickest (and cheapest) way to find the altitudes we want is to go out to NASA's web site and download a table. Using this table, we can look up the altitudes that correspond to pressures of 0.3 millibars and 11.5 millibars. Those altitudes are:

  • 11.5 millibars ⇒ 30.125 km = 98,350 feet
  • 0.3 millibars ⇒ 57.150 km = 187,500 feet

Conclusion

The surface pressure on Mars is equivalent to the range of pressures on Earth at altitudes between ~30 km and ~60 km. That seems like pretty thin atmosphere. Since humans require pressure suits for altitudes above ~19 km (called the Armstrong limit), it looks like people will always be wearing pressure suits while walking about Mars. Too bad -- I actually kind of liked the scenario shown in the movie Robinson Crusoe on Mars (Figure 2).

Figure 2: Scene from Robinson Crusoe on Mars.

Figure 2: Scene from Robinson Crusoe on Mars.

I have to admit it -- Robinson Crusoe on Mars is one of my guilty pleasures.

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Posted in Astronomy | 36 Comments

Fuel Efficiency Math

Posted on 4-October-2012 by mathscinotes

A number of months ago, I wrote a blog post that analyzed the fuel efficiency claims of CSX, which was expressed in ton-miles per gallon. While doing some other efficiency work, I stumbled upon a web site that nicely summarized the efficiencies of a number of transportation modes. Here is a table that summarizes that data by transportation mode.

Table 1: Fuel Efficiency of Various Transportation Modes.
Transportation Mode Ton-Miles per Gallon of fuel
Semi-Trailer Trucks (half loaded) 90.5
Semi-Trailer Trucks (fully loaded) 186.6
Grain Trains (Iowa to West Coast) 437.0
Grain Trains (Iowa to New Orleans) 640.1
Barge (Iowa to New Orleans and return with 35% load) 544.5
Barge (Upper Mississippi Southbound) 953.0
Barge (Upper Mississippi Northbound with 37% load) 243.0
Small Ocean-Going Ship (>30K tons Deadweight) 574.84
Large Ocean-Going Ship (>100K tons Deadweight) 1043.4

Here is what I take from this data:

  • Full loads are much more efficient than partial loads.

    I have read that one of the ways that Walmart achieves such remarkable distribution efficiency is by making sure that every load is full. This makes sense.

  • Going downstream is easier than upstream.

    This also makes a lot of sense.

  • If you are going to ship freight on the ocean, use a large ship.

    The efficiency of large ships explains the large increase in the number of enormous container vessels over the years.

Posted in General Science | 4 Comments

How Much Radioactive Material is in a Smoke Alarm?

Posted on 28-September-2012 by mathscinotes

Quote of the Day

MacArthur could never see another sun, or even a moon for that matter, in the heavens, as long as he was the sun.

- Dwight Eisenhower on Douglas MacArthur

Introduction

I was watching this Youtube video below on Americium and they mentioned that Americium is used in small quantities in smoke detectors. I thought it would be a nice mathematics exercise to compute the mass of Americium-241 in a smoke detector. I also thought it would be another good excuse to try out Mathcad Prime 2.0.

Warning on the video - Youtube often puts advertisements at the beginning. I have no control over what they put there.

Background

There is a very complete discussion of smoke detector operation at this web site, but I will give a brief description of how they work here. For a more complete discussion of Americium and its history, try this document.

The radioactive source produces alpha particles that ionize the air in a small chamber, which makes the air conductive. This conductivity can be sensed by electronics. When the air is clear of smoke, the ionized air conducts a given amount of electrical current. During a fire, smoke particles enter the chamber and reduce the conductivity of the air. The electronics sense this change in conductivity and activate the alarm. Hopefully, you will never be in a situation where your home is on fire but, if you are, having an alarm in place and a proper fire risk assessment strategy could be crucial in protecting your home and saving lives - look here to get more info on how to do a fire risk assessment such as this.

This post will examine the amount of the radioactive material used in the home smoke detectors.

Analysis

Figure 1 shows a photograph of the smoke sensor from a home smoke detector I found on the Wikipedia.

Figure 1: Wikipedia Photograph of the Radiation Source within a Smoke Detector.

Figure 1: Wikipedia Photograph of the Radiation Source within a Smoke Detector.

We see that the smoke sensor is actually labeled with the phrase "1.0 µCi 37k Bq". This means that the radioactive source generates 1 µCurie of radiation, which equals 37,000 Bequerels (Bq). A Bq means a single decay event, which for Americium-241 means the generation of an alpha particle.

Figure 2 shows my calculations for determining the mass of Americium-241 needed to make 37,000 Bq.

Figure 2: Americium 241 Mass Calculations.

Figure 2: Americium 241 Mass Calculations.

I calculate that there must be 0.29 µgrams of Americium-241 in the smoke detector. This agrees with the value given from other sources, like here.

Conclusion

I am amazed that people can actually measure out 0.29 micrograms of anything for a consumer product. As far as Mathcad Prime 2.0 goes, I am becoming more comfortable with it. Mathcad 15 has been a good friend for the last couple of years, but I am excited by where I see Mathcad Prime going. I am looking forward to seeing what is in Mathcad Prime 3.0, which is scheduled for release next year. All this radiation isn't all that good for you and this is why you should make sure you invest in the best smoke detector on the market to ensure your safety is number one.

Posted in Electronics, General Science | 5 Comments

Variable Voltage Power Supply Control Input Design Example

Posted on 27-September-2012 by mathscinotes

Introduction

An engineer in my group is learning Mathcad and he asked me if could I show him how to use Mathcad to design the control input for a variable voltage power supply. After looking at the problem, I decided this would be a nice test case for my first use of Mathcad Prime 2.0.

Background

Let's begin the exercise by looking at how we will be controlling the output voltage of the power supply. Take a close look at Figure 1.

Figure 1: Block Diagram of the Control Input of the Power Supply.

Figure 1: Block Diagram of the Control Input of the Power Supply.

On the left-hand side of Figure 1, I show the control input for standard fixed voltage power supply. For this case, the power supply sets the output voltage (labeled OUT) to a value that will maintain a voltage on the feedback pin (labeled FB) of 1.23 V. On the right-hand side of Figure 1, I show the control input for a variable voltage supply. In this case, I sum the output voltage from Digital-to-Analog Converter (DAC) onto the FB pin along with a scaled version of the output voltage. As I increase the DAC voltage, the output voltage must drop to maintain a 1.23 V level on the FB pin. Similarly, a decrease in the DAC voltage means the output voltage must increase to compensate.

The design requires that I select three resistor values: R1, R2, and R3. There is a bit of algebra associated with determining the values of these resistors. It turns out that I saw a designer using trial and error to determine these values. Since we have a tool like Mathcad available, I thought this problem would make a nice demonstration of the power available in a computer algebra system.

Analysis

Requirements

The requirements are pretty basic:

  • The maximum power supply output voltage is 60 V.
  • The minimum power supply output voltage is 20 V.
  • The maximum DAC voltage is 2.5 V.
  • The minimum DAC voltage is 0 V.
  • The feedback pin must be maintained at 1.23 V.

Calculation

Figure 2 shows my Mathcad Prime 2.0 worksheet.

Figure 2: Screenshot of My Power Supply Control Input Design Worksheet.

Figure 2: Screenshot of My Power Supply Control Input Design Worksheet.


This analysis shows that my three resistor values are:

  • R1 = 317.8 KΩ
  • R2 = 10.0 KΩ
  • R3 = 19.9 KΩ

Conclusion

This project worked out well. It was a good exercise for Mathcad Prime 2.0. I liked the fact that it let me use units in the numerical solver. The interface was a fairly straightforward extension of the Mathcad 15.0 interface. I will continue to try Mathcad Prime on further exercises.

Posted in Electronics | 2 Comments

Habitable Planet Math

Posted on 25-September-2012 by mathscinotes

Introduction

I was reading the news the other day when I stumbled upon an article about exoplanets -- planets that are not in our solar system. The article referenced this web page that provided a Figure of Merit (FOM) for the similarity of an exoplanet to Earth. I am going to examine this FOM to determine what I can learn about it.

The subject of exoplanets has become pretty interesting in recent years. When I was a boy, all the scientific articles at the time described the search for exoplanets as hopeless. Today, scientists regularly find exoplanets by detecting the wobble planets induced in a star or the dimming of a star when a planet passes in front of it. Isn't it strange how something can go from impossible to commonplace in one generation?

I like this web page because it actually includes data that we can perform mathematical experiments on. This allows me to check my knowledge -- if I can compute their results using their data then I probably understand what they are doing.

Background

Once a scientist has detected an exoplanet, one of the first questions is -- could it support life? But what does it mean for a planet to support life? We know the Earth supports life, so if the exoplanet is physically similar to Earth then it might also support life. This is where the FOM comes into play.

FOMs are used all the time in engineering. They allow us to rank implementations relative to things like cost, performance, portability and any other criteria we deem important. In the case of an exoplanet, this web page defines the following criteria as important for habitability:

  • radius

    Exoplanets have a wide range of sizes. Even in our own solar system, the Earth is small relative to the Jupiter and Saturn. There is some evidence that planets need to be near Earth-size to form a liquid core and magnetic field.

  • density

    The Earth is considered a rocky planet. A planet similar to the Earth should have a similar density.

  • surface temperature

    Earth-like biology lives in a very narrow temperature range. Because temperature is so important, this criterion should be weighted higher than the others.

  • escape velocity

    Escape velocity is related to the acceleration due to gravity.

You could easily come up with other parameters to use for our planet habitability FOM. Obvious things are:

  • presence of water
  • presence of oxygen
  • presence of an atmosphere

The problem is that these characteristics are not easily measurable. The FOM discussed here can be computed using readily available parameters. The FOM can then be used to identify the exoplanets that should have priority for further evaluation.

Analysis

FOM Definition

Equation 1 shows the Earth Similarity Index (ESI) used by the Planet Habitability Lab of the University of Puerto Rico.

Eq. 1 ESI\triangleq \prod\limits_{i=1}^{n}{{{\left( 1-\left| \frac{{{x}_{i}}-{{x}_{i0}}}{{{x}_{i}}+{{x}_{i0}}} \right| \right)}^{\frac{{{w}_{i}}}{n}}}}

where

  • xi is the value of the ith planetary parameter (e.g. surface temperature)
  • xi0 is the value of the ith planetary parameter for Earth (our reference)
  • wi is the weighting exponent assigned to the ith planetary parameter (arbitrary value indicating relative value)
  • n is the number of planetary parameters

Equation 1 is used to define three ESI FOMs:

  • Interior ESI

    This ESI is based on the exoplanet's radius (weight exponent = 0.57) and density (weight exponent = 1.07).

  • Surface ESI

    This ESI is based on the exoplanet's surface temperature (weight exponent = 5.58) and escape velocity (weight exponent = 0.70).

  • Global ESI

    A composite of both the interior and exterior ESI's.

I will demonstrate how to compute the values of these ESI's using Mathcad.

Equation 1 is interesting because the weighting factors show the relative priorities that the astronomers assign to the planetary parameters. For example, surface temperature is given a very high weight, while the radius of the planet is given much less weight. This makes sense because the biochemical reactions that we know of can exist in only a narrow temperature range.

What do astronomers know about the exoplanets they discover?

I am not an astronomer, so I am inferring this information from my reading. Astronomers appear to get reasonably accurate values for the following planetary characteristics:

  • Revolution period around the star

    This measurement is fairly accurate when planets are detected by the wobble method.

  • Mass

    Bigger planets make bigger wobbles. Astronomers will have good information on the mass of the star that the exoplanet orbits. This should allow them to obtain good mass estimates.

  • Planet Radius

    This measurement can be made using the transit method. The bigger the planet, the more the star's light is obstructed.

  • Orbit Eccentricity

    The wobble characteristics should be able to give you this information.

  • Albedo

    See this web page for a discussion of how astronomers can get an indirect measurement of albedo. The albedo measurement is important for getting a good estimate of the exoplanet's temperature.

Applying a bit of celestial mechanics can then give you the following information:

  • Orbit Radius

    Knowing the period of revolution, Kepler's law will give you the major axis distance of the planet's orbit.

  • Density

    Straightforward calculation given planetary mass and radius.

  • Surface Gravity

    See the gravitational acceleration article on the Wikipedia.

  • Escape Velocity

    See the escape velocity article on the Wikipedia.

  • Surface Temperature

    See the effective temperature article on the Wikipedia. I cannot do a better job explaining the concept than they did. Note that determining the surface temperature requires knowing information about the albedo of the planet. Albedo data was not given on this web site, so I will not attempt to compute the surface temperature.

Figure 1 shows the equations that I did use to compute orbit radius, density, escape velocity, and surface gravity.

Figure 1: Equations Used to Compute Useful Exoplanet Parameters.

Figure 1: Equations Used to Compute Useful Exoplanet Parameters.

Appendix A contains a PDF file that shows the formulas of Figure 1 being used to compute the information required for the ESI calculation. As expected, I obtained the same results as the web page.

Calculation Method

I thought I would try to recompute their ESI results for the bodies in our solar system, which the web page authors also did. This means that I can compare my results to theirs. To keep my calculations simple, I only kept the data for bodies that rotate about the Sun, which allowed me to only have to deal with one value for Kepler's constant. The method of calculation would be identical for bodies that do not rotate around the Sun, but Kepler's constant would be different because the rotation is not about the Sun.

I downloaded their data files into Mathcad and computed the ESI using the Mathcad program shown in Figure 2.

Figure 2: Mathcad Program Used to Compute ESI Values.

Figure 2: Mathcad Program Used to Compute ESI Values.

Internal ESI Calculation

Figure 3 shows my calculation for the internal ESI value.

Figure 3: Internal ESI Results.

Figure 3: Internal ESI Results.

Surface ESI Calculation

Figure 4 shows my calculation for the surface ESI value.

Figure 4: Surface ESI Results.

Figure 4: Surface ESI Results.

Global ESI Calculation

Figure 5 shows my calculation for the global ESI value.

Figure 6: Global ESI Results.

Figure 6: Global ESI Results.

Conclusion

I was able to reconstruct the original web page results for the ESI of the various bodies in our solar system using their data plus simple celestial mechanics. This sort of calculation might be interesting for a high-school or early college science student.

Appendix A: PDF Version of My Calculations

People often ask me for copies of my original work. Since Mathcad is not as common as it should be, I will first include a PDF version of my work here.

PDF Version of My Mathcad Work

Here is a link to the raw Mathcad 15 file. It is an XML file -- just save it to your disk and run Mathcad on it.

Posted in Astronomy | 1 Comment

The Ultimate Lego-Based Analog Computer

Posted on 25-September-2012 by mathscinotes

I find this video incredible. I have heard of people reconstructing the Antikythera mechanism and I have even seen a reproduction in Bozeman, Montana. However, I have never heard of one built out of Legos before.

Posted in General Science | 2 Comments

John Wayne Math

Posted on 25-September-2012 by mathscinotes

My oldest son and I have been watching some old cowboy movies -- John Wayne has figured prominently in these movies. While watching "Stagecoach", the question came up as to how many movies John Wayne has made. I grabbed the movie lists from a number of sites on the web (here, here, here, and here). The lists were all different. I simply threw the lists into Excel, cleaned up the names so they all agreed, and removed duplicates. Here is the list. I filtered out all of his television work, but left in some movie narration. The Duke performed in 168 movies over a 50 year period. Unfortunately, not all of the movies survive. This is quite a list.

"The Searchers" is still my favorite cowboy movie.

Table 1: John Wayne Movies
168 The Shootist 1976
167 Brannigan 1975
166 Rooster Cogburn 1975
165 McQ 1974
164 Cahill U.S. Marshal 1973
163 The Train Robbers 1973
162 The Cowboys 1972
161 Big Jake 1971
160 Chisum 1970
159 Rio Lobo 1970
158 The Undefeated 1969
157 True Grit 1969
156 Hellfighters 1968
155 The Green Berets 1968
154 The War Wagon 1967
153 Cast a Giant Shadow 1966
152 El Dorado 1966
151 In Harm's Way 1965
150 The Greatest Story Ever Told 1965
149 The Sons of Katie Elder 1965
148 Circus World 1964
147 Donovan's Reef 1963
146 McLintock! 1963
145 Hatari! 1962
144 How the West Was Won 1962
143 The Longest Day 1962
142 The Man Who Shot Liberty Valance 1962
141 The Comancheros 1961
140 North to Alaska 1960
139 The Alamo 1960
138 Rio Bravo 1959
137 The Horse Soldiers 1959
136 I Married a Woman 1958
135 The Barbarian and the Geisha 1958
134 Jet Pilot 1957
133 Legend of the Lost 1957
132 The Wings of Eagles 1957
131 The Conqueror 1956
130 The Searchers 1956
129 Blood Alley 1955
128 The Sea Chase 1955
127 The High and the Mighty 1954
126 Hondo 1953
125 Island in the Sky 1953
124 Trouble Along the Way 1953
123 Big Jim McLain 1952
122 Miracle in Motion 1952
121 The Quiet Man 1952
120 Flying Leathernecks 1951
119 Operation Pacific 1951
118 Rio Grande 1950
117 Sands of Iwo Jima 1949
116 She Wore a Yellow Ribbon 1949
115 The Fighting Kentuckian 1949
114 Fort Apache 1948
113 Red River 1948
112 Three Godfathers 1948
111 Wake of the Red Witch 1948
110 Angel and the Badman 1947
109 Tycoon 1947
108 Without Reservations 1946
107 Back to Bataan 1945
106 Dakota 1945
105 Flame of Barbary Coast 1945
104 They Were Expendable 1945
103 Tall in the Saddle 1944
102 The Fighting Seabees 1944
101 A Lady Takes a Chance 1943
100 In Old Oklahoma 1943
99 Flying Tigers 1942
98 In Old California 1942
97 Lady for a Night 1942
96 Pittsburgh 1942
95 Reap the Wild Wind 1942
94 Reunion in France 1942
93 The Spoilers 1942
92 A Man Betrayed 1941
91 Lady from Louisiana 1941
90 The Shepherd of the Hills 1941
89 Dark Command 1940
88 Seven Sinners 1940
87 The Long Voyage Home 1940
86 Three Faces West 1940
85 Allegheny Uprising 1939
84 New Frontier 1939
83 Stagecoach 1939
82 The Night Riders 1939
81 Three Texas Steers 1939
80 Wyoming Outlaw 1939
79 Overland Stage Raiders 1938
78 Pals of the Saddle 1938
77 Red River Range 1938
76 Santa Fe Stampede 1938
75 Adventure's End 1937
74 Born to the West 1937
73 California Straight Ahead! 1937
72 I Cover the War 1937
71 Idol of the Crowds 1937
70 Conflict 1936
69 King of the Pecos 1936
68 Sea Spoilers 1936
67 The Lawless Nineties 1936
66 The Lonely Trail 1936
65 The Oregon Trail 1936
64 Winds of the Wasteland 1936
63 Lawless Range 1935
62 Paradise Canyon 1935
61 Rainbow Valley 1935
60 Texas Terror 1935
59 The Dawn Rider 1935
58 The Desert Trail 1935
57 The New Frontier 1935
56 Westward Ho 1935
55 Blue Steel 1934
54 'Neath the Arizona Skies 1934
53 Randy Rides Alone 1934
52 The Lawless Frontier 1934
51 The Lucky Texan 1934
50 The Man from Utah 1934
49 The Star Packer 1934
48 The Trail Beyond 1934
47 West of the Divide 1934
46 Baby Face 1933
45 Central Airport 1933
44 College Coach 1933
43 His Private Secretary 1933
42 Riders of Destiny 1933
41 Sagebrush Trail 1933
40 Somewhere in Sonora 1933
39 The Life of Jimmy Dolan 1933
38 The Man from Monterey 1933
37 The Telegraph Trail 1933
36 The Three Musketeers 1933
35 Haunted Gold 1932
34 Lady and Gent 1932
33 Ride Him, Cowboy 1932
32 Running Hollywood 1932
31 Texas Cyclone 1932
30 That's My Boy 1932
29 The Big Stampede 1932
28 The Hurricane Express 1932
27 The Shadow of the Eagle 1932
26 Two-Fisted Law 1932
25 Arizona 1931
24 Girls Demand Excitement 1931
23 Maker of Men 1931
22 The Deceiver 1931
21 The Range Feud 1931
20 Three Girls Lost 1931
19 Born Reckless 1930
18 Cheer Up and Smile 1930
17 Men Without Women 1930
16 Rough Romance 1930
15 The Big Trail 1930
14 Salute 1929
13 Speakeasy 1929
12 The Black Watch 1929
11 The Forward Pass 1929
10 Words and Music 1929
9 Four Sons 1928
8 Hangman's House 1928
7 Mother Machree 1928
6 Noah's Ark 1928
5 Annie Laurie 1927
4 The Drop Kick 1927
3 The Great K & A Train Robbery 1927
2 Bardelys the Magnificent 1926
1 Brown of Harvard 1926
Posted in Personal | Comments Off on John Wayne Math

Another Young Couple Starting Their Life Together ...

Posted on 13-September-2012 by mathscinotes

I am going to be sentimental for a while. Thirty-three years ago I married the love of my life. We have two sons. One son is getting married this week to a wonderful girl he met while going to university in Montana. I am now sure that I will never get him back. I guess that is the way things are supposed to work. I think back to all the time spent helping him with homework, teaching him to ride a bicycle, coaching his soccer practice, and taking him to hockey practice. All the things that dads do. It has all been about bringing him to this point. He is now on his own -- an accountant at a hospital who is marrying a nurse. My son is doing very well as an accountant, but he dreams of owning his own firm at some point in the future. This could easily become a reality if he keeps working hard.

I was never sure about having children. I was worried that I did not know much about being a dad. My dad died when I was fourteen -- I was the oldest of five children. My mother was a secretary who did not drive. We lived in a small agricultural town and the townsfolk wanted to help. All we had to do was work. They also offered me and one of my brothers a job -- that is how small towns are. I worked at that job until I completed my engineering degree. While my mother did her best, my brothers and sister have always felt that something was missing without our father being there.

We were not alone in these feelings. Once, while I was delivering newspapers, the wealthiest man in town stopped me and told me that we must call him if we ever needed anything. With tears in his eyes, he told me that his dad had died in a hunting accident when he was a boy and that he knew what we were going through. That image has stuck with me my whole life. He knew. I couldn't begin to imagine what the grieving process would have been like for him as a young child.

My worries about being a dad evaporated when the boys came -- I soon realized that being a dad just required love. I could not have had two better sons. They are now both fine young men. My father would be very proud of how they turned out. I appreciate that every day I get to do things that he never was able to do -- things like attending his son's wedding.

I can only hope my son and his bride will be as happy as my wife and I. Things were not always easy. Early in my career, I was a contractor for the US Navy, and I was away from home for long periods of time. My wife had to handle two rambunctious boys on her own. While away from home, I would often think of those two boys with their smiling, toothless grins. They were not small long enough. All parents need to remember that children are only children for a very short period of time. They are gone before you know it.

My son and his bride will have their own adventures. That is the way of the world. Somehow it all leaves me both happy and sad. I guess that is what being a dad is about. For me, I will now have a daughter to love. This will be another new experience for me and I will try to cherish every moment.

Posted in Personal | 1 Comment

A Good Analog Computation Example

Posted on 31-August-2012 by mathscinotes

Introduction

I am always looking for real-world examples of analog computation and this blog post will discuss one of the best examples of analog computation that I found. I found this little gem in EDN magazines' Design Ideas section, which is a great place to look for clever analog solutions for real problems.

The circuit that I am going to review here is shown in Figure 1. During my analysis, I will break the circuit down into sub-circuits and then analyze the sub-circuits.

Figure 1: EDN Circuit For Measuring Available Wind Power.

Figure 1: EDN Circuit For Measuring Available Wind Power.

This circuit generates a voltage that is proportional to the wind power currently available. It does this using two sensors:

  • anemometer/wind turbine

    I usually think of four rotating cups whose motion generates a signal with a frequency proportional to wind speed, which is how this circuit represents wind speed.

  • base-emitter junction of a transistor

    The base emitter junction's voltage variation with temperature provides an analog for the temperature variation of the air's density.

This review needs to cover a lot of technical territory so let's dig in ...

Background

For background on windmills and how they work, see this web site. The key equation for computing the maximum normalized power from a windmill is given by Equation 1. The normalized power is defined as the available watts per unit area of wind turbine.

Eq. 1 \displaystyle P=\frac{1}{2}\cdot A\cdot {{\rho }_{Air}}\cdot v_{_{Air}}^{3}\Rightarrow {P}'=\frac{P}{A}=\frac{1}{2}\cdot {{\rho }_{Air}}\cdot v_{_{Air}}^{3}

where

  • ρAir is the density of air, which is a function of temperature and pressure.
  • vAir is the air velocity.
  • A is the area of the wind turbine.
  • P′ is the watts per unit area of the wind turbine.

Our objective in this post is to analyze the circuit shown in Figure 1 and demonstrate how that circuit implements Equation 1.

Before we do any electronics design, we need to beat Equation 1 into a form that can be implemented using electrical components. Figure 2 goes through this derivation.

Figure 2: Rework of Equation 1 into am Electronics-Friendly Form.

Figure 2: Rework of Equation 1 into am Electronics-Friendly Form.

Analysis

Requirements

The circuit designer (Woodward) appears to have worked to the following requirements:

  • The circuit is generate 1 V of output for every 1 kW/m2of available wind power per unit area.

    The circuit can produce a wide range of values. A value needs to be chosen in order to determine concrete part values.

  • The circuit is to use a single power supply voltage.

    This circuit provides a nice illustration of designing an analog circuit for single supply operation. A one-supply design is normally preferred over a multi-supply design because it is cheaper. The designer used parts based on the 4000 series of CMOS devices. This is a very old family, nonetheless, many designers have a fondness for this family of digital parts for analog applications. See Appendix B for details on using these parts in analog applications.

  • The anemometer measuring the wind speed generates a signal with a frequency variation of 10 Hz per 1 m/s of wind velocity.

    The circuit can be adapted to various types of anemometers. We need to pick a specific conversion factor in order to pick specific components. Appendix C gives examples of anemometers that would work for this circuit.

  • The circuit will compensate for air density variations with temperature.

    It turns out that this compensation is relatively simple. Appendix A contains a derivation of the calibration equation presented in the designer's original article.

For this analysis, I will break the circuit up into three sub-circuits:

  • Forward-Biased Diode

    The forward diode voltage drop will be shown to have a temperature variation very similar to that of air.

  • Frequency-to-Voltage Conversion

    This circuit will be used to multiply the forward diode voltage drop times the frequency of the signal from an anemometer.

  • Level Shift and Amplify Stage

    This circuit removes a DC bias and properly scales the output signal level.

Forward-Biased Diode Voltage and the Density of Air

Figure 3 shows how the density variation for air on a percentage basis is similar to the percentage forward voltage variation across a diode or base-emitter junction.

Figure 3: Variation of Air Density with Temperature Compared to a Diode's Variation.

Figure 3: Variation of Air Density with Temperature Compared to a Diode's Variation.

Note that the molecular weight of air is 28.97 gm/mol, which is computed at this web site.

Voltage-to-Frequency Converter Section Operation

Figure 4 summarizes how the frequency-to-voltage converter works.

Figure 4: Voltage to Frequency Converter Subsection Operation.

Figure 4: Voltage to Frequency Converter Subsection Operation.

As shown in Figure 4, the frequency-to-voltage converter circuit generates an output with ripple on it. This ripple will be filtered out by the low-pass filter incorporated into the Level Shift and Amplify sub-circuit.

Figure 5 shows how I will represent the frequency-to-voltage converter as a circuit element.

Figure 5: Symbolic Representation of the Frequency-to-Voltage Converter.

Figure 5: Symbolic Representation of the Frequency-to-Voltage Converter.

The Ref pin shown in Figure 5 deserves some comment. It connects to the positive input pin of the operational amplifier. In a system with bipolar supplies, the Ref pin would be connected to ground. Because this is a single-power supply application, the Ref pin will be connected midway between ground and the supply voltage value. The single-supply setup will product a VOUT with a DC bias. This bias is removed by the Level Shift and Amplify stage.

Level Shift and Amplify

Figure 6 shows the final stage of the circuit, which takes the output of the frequency-to-voltage converters and provides some amplification and removes the 2.5 V bias.

Figure 6: Output Circuit for Level Shift and Amplify Stage.

Figure 6: Output Circuit for Level Shift and Amplify Stage.

The component values can be selected as shown in Figure 7.

Figure 7: Component Selection for Output Circuit Stage.

Figure 7: Component Selection for Output Circuit Stage.

Entire Circuit

Figure 8 shows the whole circuit from my point of view.

Figure 8: Whole Circuit from a Block Diagram Viewpoint.

Figure 8: Whole Circuit from a Block Diagram Viewpoint.

We can determine the components required as shown in Figure 9.

Figure 9: Check of Final Component Values.

Figure 9: Check of Final Component Values.

Conclusion

I went through this circuit in excruciating detail because I thought it does a nice job of illustrating the kind of interplay between physics and electronics that often occurs in analog sensor applications. Also, I have a circuit application that I am working on that will use a circuit related to this one and I wanted to review this work before I pressed on with my circuit.

Appendix A: Derivation of Calibration Equation

The original article contains an equation that is useful for calibration. I derive his expression in Figure 10.

Figure 10: Derivation of Circuit Calibration Equation.

Figure 10: Derivation of Circuit Calibration Equation.

Appendix B: Designing Linear Circuits with 4000 Series CMOS Parts

There are quite a few designers who still use 4000 series parts (in this case, 74HC4000 series). See this document for details on applying these digital parts in an analog application.

Appendix C: Example of an Anemometer with 10 Hz per m/sec Output

I thought it was worthwhile showing some anemometers with 10 Hz per m/sec output. Both examples are powered.

Posted in Electronics | Comments Off on A Good Analog Computation Example

Book Review:"Iron Men and Tin Fish"

Posted on 29-August-2012 by mathscinotes

I just finished reading the book "Iron Men and Tin Fish" by Anthony Newpower (ISBN 978-1-59114-623-0). It is a short book that does a really nice job of covering the use of torpedoes during World War 2. The author looks at torpedoes from the Japanese, American, German, and British perspectives. I particularly like how he sectioned off each story with the use of chapters, as it allows the reader to decipher a difference between each set of torpedoes. Book Layout is very important for books like this, and especially when it concerns history. It provides more of an experience for the readers. As a result, you can truly immerse yourself into what the author has found. His observations can be distilled to these few points:

  • The Japanese had the best torpedoes during the war.

    They developed a superior set of vehicles and the tactics to apply them. It is amazing how the Allies underestimated the importance of the torpedo to Japanese naval strategy.

  • The British, American, and German navies all tried to sense the presence of a ship using magnetic field-sensing exploders. All failed.

    The magnetic field of the Earth proved to be too variable to count on.

  • The British figured out that their magnetic field-sensing exploder system did not work before the war and ended up producing effective and reliable torpedoes.

    They had an excellent test program and acted on the results. Their primary torpedo of WWII was the MK VIII, which was used to sink the Argentine light cruiser Belgrano during the Falkland's War.

  • The Germans saw the exact same problem (and some additional ones) at the start of the war and fixed it after a number of months of internal bureaucratic squabbling.

    Eventually, the Germans captured a British torpedo and ended up copying the British contact exploder.

  • The Americans had almost exactly the same problems as the Germans, but they tried to blame the crews rather than fix the problems.

    This was an excellent view into a management structure that was overly bureaucratic and unwilling to admit error. Eventually, an admiral did take responsibility, but only after years of time and many men dying. During the effort at fixing their torpedoes, the Americans ended up copying a German electric torpedo that was discovered on a beach. This helped alleviate the problem.

There have been a number of movies made on this topic. One pretty good one is "Operation Pacific." There have also been a number of magazine articles on the topic. For example, this magazine article discusses how Einstein worked on the problem and quickly came to the correct conclusion. No one listened to him -- very unfortunate.

As an engineer who is now in management, I always find these case studies of dysfunctional organizations interesting. I also find it interesting how most of the management lessons that I have learned are "negative" -- things that I will make sure that I do not do in my group.

Posted in History of Science and Technology, Underwater | Comments Off on Book Review:"Iron Men and Tin Fish"