## Temperature Sensing with a Bandgap Reference

Quote of the Day

Journalism is printing what someone else does not want printed; everything else is public relations.

## Introduction

Figure 1: Typical LMT70 Application Circuit. My application circuit will be VERY similar. (Source)

I have decided that my next home electronics project will be a precision thermometer that I can read over the Internet. I will be mounting the sensor at my cabin in Northern Minnesota, where winter temperatures can drop to -40 °C or lower. During the summer, temperatures can rise to nearly 40 °C. My plan is to connect the unit to a Raspberry Pie that I use to provide remote monitoring and control. I decided that I going to use a Texas Instruments' LMT70 precision temperature sensor, which uses a well-known circuit called a Brokaw bandgap reference to measure the temperature of its die.

This post documents how I familiarized myself with this part and how it works. My Mathcad and LTSpice source files are included here.

## Background

### Definitions

Breakout Board
Breakout boards are small PCBs on which you can mount an integrated circuit and that provides you readily accessible points for connecting the integrated circuit pads/balls to the outside world using pins and wires. This web page shows a good example of breakboard application.
Bandgap Reference
A bandgap voltage reference is a temperature-independent voltage reference circuit widely used in integrated circuits. It produces a fixed (constant) voltage regardless of power supply variations, temperature changes and circuit loading from a device. Normally, bandgap reference circuits cancel out two opposing variations caused by temperature. For temperature measurement, we will not be cancelling out the temperature variation – we will use the very predictable variation present in part of the circuit to measure the die temperature of the LMT70. In general, the die temperature is strongly correlated with the ambient temperature. The relationship between ambient temperature and die temperature is usually established empirically.
Proportional to Absolute Temperature
A circuit parameter (e.g. voltage) that is proportional to the LMT70's absolute die temperature, ie. temperature measured in Kelvin. Note that transistors do not work at absolute zero, but the linear response will extrapolate down to 0 K.

### Requirements

This is a home project, which means my requirements will be fairly simple:

• The sensor must be able to measure a range of temperature from -40 °C to 40 °C.

The LMT70 is capable of measuring from -55 °C to 150 °C, so it has plenty of dynamic range.

• I am looking for an accuracy of ±0.5 °C.

The LMT70 is rated for ±0.36 °C over the range of -55 °C to 150 °C. I need to ensure that the error introduced by my A/D conversion does not cause my overall error to exceed my requirement.

• I need to be able to mount the sensor onto a board that I can build.
• The sensor must be capable of being shutdown and only activated periodically.

This is my approach for minimizing the error due to self heating. I plan on having the sensor off most of the time and only turning it on for a few seconds every five minutes or so. This approach will minimize the error from self-heating.

### Brokaw Cell

The LMT70 uses a Brokaw bandgap reference circuit that can produce an output voltage that is proportional to the absolute temperature. The circuit is well-described on the Wikipedia, so I refer you there for more details. I do want to point out an excellent video presented by A. Paul Brokaw, the developer of the circuit. You rarely see a circuit design presented by its original developer, so this video is a treat.

### Voltage Proportional to Absolute Temperature

Equation 1 is describes how the output voltage from a Brokaw bandgap reference varies with absolute temperature. I derive Equation 1 in Figure 2. R1 and R2 are resistors shown in Figure 2.

 Eq. 1 $\displaystyle {{V}_{{PTAT}}}=\frac{{{{k}_{b}}\cdot T}}{{{{q}_{e}}}}\cdot 2\cdot \text{ln}\left( n \right)\cdot \frac{{R1}}{{R2}}$

where

Except for temperature T, all parameters on the right-hand side of Equation 1 are constants. Thus, Equation 1 describes a linear relationship between VPTAT and T.

## Analysis

### Output Voltage Derivation

Figure 2 shows my Brokaw reference circuit built using 2N2222 transistors and a generic high-gain opamp. You can see my Spice commands on the left for setting up the simulation – perform a 1 msec transient analysis at temperatures from 10 °C to 70 °C in steps of 10 °C. I also include my derivation of Equation 1 as it applied to the circuit in Figure 2.  In this Brokaw realization, the voltage sum of the VBE across three (n = 3) transistors (Q1, Q2, Q3) plus the voltage drop across R2 is forced to be equal to the VBE drop across Q2. The derivation is easily extended for any number transistors (n > 1).

Figure 2: LTSpice Example Circuit and My Derivation of Equation 1.

### LTSpice Simulation

Figure 3 shows the results of my LTSpice simulation. I ran the simulation over a temperature range from 10 °C to 70 °C in increments of 10 °C.  In Figure 3, the red line corresponds to 10 °C and 70 °C corresponds to 70 °C. I could have ran the simulation over a wider range, but my interest here is expository – the parts have a guaranteed level of accuracy over a temperature range from -55 °C to 150 °C.

Figure 3: LTSpice Simulation of Brokaw Cell.

### Simulation Results Versus Theoretical Prediction

Figure 4 plots Equation 1 and my LTSpice simulation versus temperature. As you can see, the agreement is excellent.

Figure 4: LTSpice Graph of Brokaw Bandgap Reference VPTAT for Various Temperatures.

## Conclusion

I go through an analysis like this every time I use a part for the first time. I used a combination of Mathcad and LTSpice to develop simple models for predicting circuit behavior and optimizing my designs.

Posted in Electronics | 2 Comments

## Some PT Boat Statistics

Quote of the Day

The truth is, unless you let go, unless you forgive yourself, unless you forgive the situation, unless you realize that the situation is over, you cannot move forward.

— Steve Maraboli

## Introduction

Figure 1: PT-109 Crew. JFK is on the far right. (Source)

I was doing some reading about President John F. Kennedy (JFK) and was surprised to learn that he actually commanded three PT boats: PT-101, PT-109, and PT-59. His service on  PT-101 was very short. His next command, PT-109,  became famous because of its ramming and sinking by the Japanese destroyer Amagiri. Though injured, JFK was able to lead his surviving crew out of enemy-held territory. JFK also commanded PT-59, with one of its actions dramatized in the movie PT-109. During this action, PT-59 rescued US Marines stranded on a beach while under fire. JFK's service on PT-59 would normally have made it a significant piece of naval history, but an amazing sequence of bureaucratic screw-ups, including a typing error, caused it to be left to rot at its New York mooring.

Figure 2: PT-109 on board SS Joseph Stanton. (Source)

This bit of history got me curious about PT boats and their history. I decided to do some web scraping and pull together some statistics on PT boats into this post. As I read the history of PT boats, it became obvious that the US was scrambling during the early years of WW2 to put any type of craft they could into battle. Unlike steel destroyers and battleships, these boats were made of  plywood and mahogany. They depended on speed and "hit and run" tactics to survive.

The actual web scraping (source) was a bit complex: copied the page into Notepad++ and used a regex to clean things up, Power Query to the do the parsing and transformations, and tables were generated using Excel. My source files are here.

## Background

### What is a PT Boat?

The popular term PT boat came from their US Navy designation of Patrol, Torpedo. They were fast attack craft that were armed with torpedoes, depth charges, and machine guns. Their use was almost exclusively limited to WW2 – four boats were built during the Korean War. The PT boats were small, fast, and inexpensive to build because of their wooden construction. Their effectiveness was hampered by ineffective torpedoes (fixed later), limited armament, and lack of armor.

### What was their Role?

Figure 3: PT-32, one of the PT boats that evacuated MacArthur from the Philippines. (Source)

The PT boats were small and generally limited to coastal operations. In combat, they were most known for their commerce raiding during the Solomon Islands campaign , and were particularly effective at attacking Japanese barge traffic, which was critical to starving remote Japanese garrisons.

Early in WW2, four PT boats performed an important special operation by evacuating MacArthur and his staff and family from the Philippines (Figure 3).

## Analysis

### Who Built Them?

The US built 813 PT boats during WW2 and Korea (Figure 4). There were only 4 built during the Korean conflict (1951), so the 809 were built just prior to and during WW2.

Figure 4: PT Boat Manufacturers.

As you can see in Figure 4, the Elco and Higgins companies dominated the production of PT boats.

### Differences in PT Boat Construction

The bulk of the PT boats built were 78 or 80 feet long (Figure 5). Many of the smaller boats were eventually removed from combat roles and used as utility vessels. The four largest boats (89 ft, 94 ft, 98 ft, 105 ft) were aluminum boats built during the Korean conflict.

Figure 5: Length of PT Boats.

Excluding three prototypes, 531 PT boats were built for the US Navy and 279 were built for our allies (Figure 6).

Figure 6: PT Boat Allocations By Nation.

### What Happened to the US PT Boats?

Figure 7: PT boat burning in the Philippines. (Source)

Figure 7 shows that most of the US Navy PT boats survived the war and were were sold, scrapped, abandoned, or destroyed at the war's conclusion. Many of the destroyed boats met their end by being dragged onto beaches and burned (Source, Source). The non-combat losses were due to weather or grounding, which was a problem for boats running in uncharted shallow waters. Thirty-four boats were lost in combat, with one boat was lost due to ramming, JFK's PT-109.

Figure 7: Fate of the PT Boats.

## Conclusion

I am always amazed when I read about how both wooden boats and aircraft (ex. deHavilland Mosquito) played significant roles in WW2. Unfortunately, ships built of wood require careful maintenance and few PT boats survived into modern times. Here are a few links to the existing PT boats that I know of:

Figure 8 shows a PT boat running similar to PT 109 running at high speed.

Figure 8: PT-105, a PT boat very similar to PT-109. (Source)

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## Fact Checking Power Over Ethernet Marketing Math

Quote of the Day

In the case of good books, the point is not how many of them you can get through, but rather how many can get through to you.

Mortimer J. Adler. He wrote a book called How to Read a Book that helped me become an effective reader.

## Introduction

Figure 1: Example of very neat network cable bundles. In PoE applications, these cable bundles can experience significant self-heating, which will reduce their load capacity. (Source)

I was reading a blog by a cable manufacturer (Belden) this morning on the advantages of using Cat 6 cable over Cat 5e for network installations going forward (Figure 1 shows a great example of network cabling). Normally, I see the cable manufacturers recommending Cat 6 to customers because it will allow them to upgrade to 10 Gbps Ethernet, at least for runs less than 55 meters long.

The blog I read this morning took a bit different approach. It was encouraging customers to switch to Cat 6 because it consumes less power in Power over Ethernet (PoE) applications. It made the following testable claims:

• As much as 20% of the power through the cable can get “lost” in a 24-gauge Category 5e cable [relative to a Category 6 cable], leading to inefficiency.
• As we mentioned above, losing nearly one-fifth of the total power in a 24-gauge Category 5e cable may seem like a lot of power loss – and it is. But doing the math will show you that the total dollar amount comes out to be only around $7 per year. I will check these claims in this blog post. ## Background Table 1 summarizes some of the key characteristics of Cat5e and Cat6 cable. For this blog post, the important difference from a PoE standpoint is the wire gauge. Table 1: Key Characteristics of Cat5e and Cat6 Cable. Characteristic Cat5e Cat6 Max Bit Rate (bps) 1,000 10,000 Approximate Cost ($/foot) 0.3 0.5
Frequency Bandwidth (Mhz) 100 250
1000 BaseT Reach (m) 100 100
10000BaseT Reach (m)  –  55
Wire Gauge (AWG) 24 23

## Analysis

### Claim 1: 20% Power Loss on a PoE Line.

I was not able to confirm the 20% loss of the total power loss being attributable to the wire resistance. It is easy to confirm that as much as 15% of the total power loss is attributable to wire resistance. Figure 2 shows my calculations.  So I would say that there claim is close to true.

Figure 2: I calculate 15% for the maximum loss percentage.

### Claim 2: $7 Per Year Per PoE Line Cost. I was able to confirm their claim that each PoE line burns$7 per year in the wire resistance (Figure 3). That was a bit surprising.

Figure 3: Annual Electrical Cost for a Running a PoE Line in the US.

## Conclusion

I read marketing claims all the time. Most of the time, there is some level of reality to them. In this case, one claim is close and the other is accurate. I was surprised at the cost yearly cost incurred because of the cable resistance.

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## Good Example of Learning Curve Labor Cost Reductions

Quote of the Day

There was this one time I remember I was at the Iowa State Fair, very much lost and confused. And a farmer came up to me and showed me around, helped me understand where I was inside the pig barn, at the time. And he said, 'I want to do this for you, because I hope if my son ever goes to New York someone will be kind enough to do the same for him.'

— Danny Freeman, journalist, talking about the kindness of the Iowa voters during the 2016 presidential campaign. You see this attitude throughout the Midwest, including Minnesota.

Figure 1: Plot of Labor Expended Per Ju88 Assembled.

I am always looking for examples of efficiencies that can be attributed to learning curve and production volumes. Figure 1 shows an example from an analysis of war production in Germany during WW2. This particular example focuses on the labor required to build a Ju 88 multi-role aircraft (Figure 2).

I normally model learning curve improvements using Wright's law, which says component cost linearly reduces with total quantity produced on a semilog chart. Figure 1 looks at labor hours per unit, which generally have a strong influence on product cost. Cost versus time is usually modeled using Moore's law , which says component cost linearly reduces with time on a semilog chart. Of course, quantity produced and time are usually related. There is some support for saying that Wright's law is probably more correct than Moore's law, but the differences are minor. I will dig up the production data for the Ju 88 and see if I can beat Figure 1 into the Wright model in a later post.

If Moore's model ideally represented reality, Figure 1 would show a straight line. However, reality is complicated – there are two anomalies (i.e. bumps) in the chart. The October 1939 anomaly reflects inefficiencies created when two new producers came online. The March-June 1941 anomaly represents problems caused when a new design of the Ju 88 was introduced.

Figure 2: Ju 88, a German WW2 multi-role combat aircraft.

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## Measuring a Chamfer Angle Using Gage Balls

Quote of the Day

The safest general characterization of the European philosophical tradition is that it consists of a series of footnotes to Plato.

— Alfred North Whitehead, Process and Reality, p. 39 [Free Press, 1979]

## Introduction

Figure 1: Chamfer Angle Measurement Example Using Two Gage Balls.

One metrology operation I have had to perform a number of times is measuring a chamfer angle precisely – Figure 1 shows today's example. Many items are chamfered – even in electronics. For example, edge connectors on printed circuit boards often need to be chamfered to ensure that they do not damage the connectors they are being inserted into.

Referring to Figure 1, you might think that chamfer measurements would be easy because you have a vertical edge and horizontal edge that you should be able to measure. Unfortunately, these edges are rarely straight. They often are rounded or irregular. This makes the chamfer angle measurement non-repeatable. Using gage balls eliminates any dependence on the determining precisely the location of an edge.

## Background

Equation 1 is the formula I use for determining a chamfer angle using two gage balls of different diameter.

 Eq. 1 $\displaystyle \theta \left( {{{R}_{1}},{{R}_{2}},{{M}_{1}},{{M}_{2}}} \right)=2\cdot \text{arctan}\left( {\frac{{{{R}_{1}}-{{R}_{2}}}}{{{{M}_{1}}-{{M}_{2}}-{{R}_{1}}+{{R}_{2}}}}} \right)$

where

• M1 is height measurement of ball 1 above the top surface.
• M2 is height measurement of ball 2 above the top surface.
• R1 is the radius of gage ball 1.
• R2 is the radius of gage ball 2.

I derive Equation 1 in the analysis section.

## Analysis

### Symbol Definitions

Figure 2 shows the variables that I defined for the angle measurement scenario of Figure 1.

### Derivation and Example Calculation

I am lazy this morning. I am sure there is a clever geometric derivation, but the quickest way to get a formula is to use Mathcad's symbolic processor to solve a simple system of equations. Figure 3 shows my derivation with Q = arctan(θ/2). I often initially do my trigonometric derivations sans trig functions because Mathcad will often generate overly complex answers, e.g. applying half-angle formulas to solve for θ.

As an example, I use Equation 1 to determine the angle in Example 1. In my function evaluation, I use the fact that the gage ball diameter, D, is 1/2 the radius, R.

Figure 3: Derivation of Equation 1 and Application to Figure 1 Example.

## Conclusion

This is a relatively simple formula that is useful for precision angle measurement of a chamfer angle using two gage balls or roller gages of different diameter.

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## Calorie Per Acre Improvements in Staple Crops Over Time

Quote of the Day

We have invaded space with our rocket and for the first time. We have used space as a bridge between two points on the earth; we have proved rocket propulsion practicable for space travel. This third day of October, 1942, is the first of a new era of transportation, that of space travel.

General Walter Dornberger, project leader for the V2 rocket program during WW2. He made this statement after the first successful test launch of a A4 (aka V2) rocket.

## Introduction

Figure 1: Calories Per Acre For Some Staple Crops. This is my plot of USDA data. Note that the apple data only goes back a few years.

My family has strong agricultural roots – mainly in dairy and potato farming – and our holiday conversations frequently turn to discussions of crop yields (bushels per acre or lbs per acre). As I listened to the discussion between my brothers on this year's crop yields, I realized that the yield numbers they were quoting were much higher today than we saw as children. This made me curious, and I decide to go out to the US Department of Agriculture's National Agricultural Statistics Service crop database and download CSV files on the yield of some key staple crops  for processing by Power Query (i.e. recently renamed Get and Transform). I will be using this file to train  my staff on defining Power Query functions. No macros were used in this analysis.

I am most interested in determining which staple crop produces the most food value per acre, with food value defined as calories per acre. When I was a boy, I was told that sugar cane produced the most calories per acre. Recently, I have had various farmers tell me that apples, corn, or potatoes produce the most calories per acre.

Figure 1 provides us the answer. I would make the following observations about Figure 1.

• Apples are not even close to winning the calories per acre contest.

Rice, corn, sugarcane, or potatoes all outpace apples. Note that the USDA did not have apple yield data that went back in time very far.

• You can argue either corn or potatoes win the calorie per acre race.

The corn crops show more yield variability than potato crops. I would guess that potatoes have more consistent yield because they are usually irrigated. I would also guess would be that corn would win over potatoes on a calories per acre per cost of production metric because irrigation is expensive.

• Sugarcane did produce the most calories per acre in my youth (1960s and 1970s).

I find it interesting that while sugarcane has experienced yield improvement since the 1940s, it did not improve at the same rate as corn and potatoes.

• The rise in the yields of rice, corn, and potatoes since 1940 is remarkable.

I have to believe that this yield increase is because of the application of technology to agriculture that occurred after WW2.

• Notice how there was almost no yield growth prior to 1940.

I will do a bit more research to try and determine what happened after 1940 that was not happening for many decades prior.

The rest of this post covers how I generated Figure 1. For those interested in following my work, here is my source. You should be able to unzip my workbook and data files to any location and have it work. The data files are unmodified downloads from the USDA web page. The graph work is routine – Power Query is the interesting part.

## Background

### Definitions

yield
Crop yield refers to either volume of a crop per unit acre or the mass of a crop per unit area of land cultivated.
staple crop
A staple food, or simply a staple, is a food that is eaten routinely and in such quantities that it constitutes a dominant portion of a standard diet for a given people, supplying a large fraction of energy needs and generally forming a significant proportion of the intake of other nutrients as well.
food calorie
Food calories are measure in units of kilocalories (kcal).

### Baseline

I decided to look at the calories per acre for the following crops:

• potatoes
• wheat
• rice
• apples
• corn
• sugarcane
• soybeans

All my data is based on US national averages – there is quite a bit of variability between the states. Note that some US crops yields are measured by volume, and I needed to convert these volumetric units to mass units for the energy calculation by using their densities. I obtain the densities from the table shown in Appendix B.

### Horrific Units

This analysis involved some of the screwiest units I have ever used:

I am afraid these units are still commonly used in US agriculture. I cover a calorie calculation example in Appendix A. I also include a video in Appendix C that shows how complex creating sugar from sugarcane is – I was impressed with the amount of work required.

## Analysis

The analysis is straightforward and you can see how it is done by looking at my source. Here is my approach:

• download crop yield CSV files from the USDA National Agricultural Statistics Service.
• put together table of conversions from volumetric units to mass units.
• put together table of conversion for calories by mass unit of each crop.
• use Power Query joins to merge data and conversions.
• add column using formula to convert yields to calories.
• plot the data.

## Conclusion

I wonder if how far into the future these yield increases can continue. It would be interesting to know how much of this increase is attributable to improved techniques (e.g. fertilizer, irrigation) and how much is attributable to improved genetics. As the world's population increases, these yield increases will become more and more critical – our crop lands are limited and under much pressure.

If you want to see come other writings that confirm some of my calculations, see the blog and the newspaper article it references.

## Appendix A: Unit Conversions for Sugarcane Calories/Acre.

Figure 2 shows an example of the sugarcane calories per acre calculation.

Figure 2: Sugarcane Calories Per Acre Calculation.

## Appendix B: Crop Mass Densities.

I used the crop densities list in Figure 3 for a number of the calculations (Source).

Figure 3: Density of Staple Crops.

## Appendix C: Video of Sugar Cane Processing.

Figure 4 shows how sugar cane is processed into granular sugar. The process is quite complex.

 Figure 2: Good Video Briefing on Sugarcane Processing.

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## Marine Management Philosophy Versus Engineering Management Philosophy

Quote of the Day

Left unsung, the noblest deed will die.

General Mattis, quoting the Greek poet Pindar. He was explaining how embedded journalists could help the US Marines tell their story.

Figure 1: General Mattis. (Source).

I have been reading an article about our new Secretary of Defense James Mattis, a US Marine general who is well known for his thoughtful insights on the world situation. He is also viewed as a superb manager, which means I want to learn as much from his experience as possible. In the article, Mattis has listed some of his management credos. I thought I would look at coming up with corresponding credos for managing a civilian engineering team (Table 1).

Table 1: US Marine Versus Engineering Credos
General Mattis' View of a Marine My View of an Engineer
Attitude is a weapon. Attitude is often your most power tool.
Everyone fills sandbags in this outfit. No one is above any job.
If a Marine or a unit is screwing up, hug them a little more. If someone is screwing up, for gosh sake help – unless it’s a competitor, where we just get out of their way.
There are only two types of people on the battlefield: hunters and the hunted. There are only two types of engineers: those that make things happen, and those who have things happen to them.
He encouraged simplicity in planning, and speed, surprise, and security in execution. He encouraged simplicity in planning, and speed, surprise, and security in execution.
The two qualities I look for most in my Marines are initiative and aggressiveness. The two qualities I look for most in my engineers are initiative and a can-do attitude.
Remember, Orville Wright flew an airplane without a pilot’s license. Remember, Orville Wright flew an airplane without a pilot’s license.
No better friend, no worse enemy. No better partner, no more fierce competitor.
Treat every day as if it were your last day of peace. Treat everyday as an opportunity to improve our competitive position.
This is not some JV, bush-league outfit. We’re the Marines. This is not some JV, bush-league outfit. We are industry-leaders and we need to act like it.
I have been accused of making my subordinates my equals, and I happily stand guilty. I have been accused of making my subordinates my equals, and I happily stand guilty.
I don’t want us to put someone in front of the the media that is going to have their second childhood. I only want tough Marines in front of the camera. Customer relations are for Sales and Marketing folks – customers might overdose on reality if they dealt directly with development engineers.
Engage your brain before you engage your trigger. Engage your brain before you engage your mouth – if I were King of the World, I would make this happen.
The number-one authority you have as a leader is your moral authority and your number one power is expectation. The number-one authority you have as a leader is your moral authority and your number one power is expectation.

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## Tribble Math

Quote of the Day

Politics is ethics done in public.

## Introduction

Figure 1: Captain Kirk with Tribbles. (Source)

Over the 2016 New Year's holiday, a number of television stations were showing the remastered original Star Trek episodes. While you generally do not hear much math in a Star Trek episode, the "Trouble with Tribbles" actually had two statements to which a bit math can be applied. You can search around the Internet and find solutions for one of the statements, but I wanted to work through the math myself. It is interesting that a script writer must have actually worked through the details during time before calculators.

My Mathcad worksheet and a PDF are included here.

## Background

### Math Statement #1

Here is a quote from the episode (Source) that sets up the math problem.

KIRK: Here. Let me try it.
(But he can't open it either, so he tried one of the overhead doors instead. That does open, and a whole load of tribbles fall out, burying the gallant Captain up to his shoulders. Spock examines one.)
SPOCK: They seem to be gorged.
BARIS: Gorged? On my grain? Kirk, I am going to hold you responsible. There must be thousands of them.
KIRK: Hundreds of thousands.
SPOCK: One million seven hundred seventy one thousand five hundred sixty one. That's assuming one tribble, multiplying with an average litter of ten, producing a new generation every twelve hours over a period of three days.
KIRK: That's assuming they got here three days ago.
SPOCK: And allowing for the amount of grain consumed and the volume of the storage compartment.

My plan here is to duplicate the Spock's math here.

### Math Statement #2

The other math statement, which left much information out, was how long it would take Cyrano Jones to cleanup all the dead tribbles – 17.9 years.

KIRK: There is one thing you can do.
JONES: Yes?
KIRK: Pick up every tribble on the space station. If you do that, I'll speak to Mister Lurry about returning your spaceship.
JONES: It would take years.
SPOCK: Seventeen point nine, to be exact.
JONES: Seventeen point nine years.
KIRK: Consider it Job security.
JONES: Captain, you're a hard man. All right! All right!

I am going to treat this statement as a Fermi problem and estimate the average tribble cleanup rate that Cyrano Jones will need to maintain to clean up all the tribbles from Math Problem #1 within 17.9 years.

### Solution of a First-Order Difference Equation

Math problem #1 can be expressed as a first-order difference equation, which has the solution shown in Figure 2.

Figure 2: Solution of a First-Order Differential Equation. (Source)

## Analysis

### Math Statement #1 Solution

Figure 3 shows my solution for the number of tribbles after 3 days.

Figure 3: Math Problem #1 Solution.

### Math Statement #2 Solution

I will estimate the tribble cleanup rate that Cyrano Jones will need to maintain of 17.9 years in order to clean up all the tribbles from problem #1. Based on the assumptions laid out in Figure 4, Cyrano Jones will need to average picking up 35 tribble per hour over the 17.9 years in order to clean all the tribbles up.

Figure 4: Math Problem #2 Solution.

## Conclusion

This was good problem that left me pleased knowing the episode author, David Gerrold, must know a bit of math.

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Posted in Science Fiction | 4 Comments

## Old-School Selling of the Exploration of Mars

Quote of the Day

Late to bed, early to rise, work like hell, and advertise.

— Saying on Wernher Von Braun's desk. He was not going to repeat the mistakes of other rocket pioneers who shunned publicity.

Figure 1: Wernher von Braun in 1960. (Source)

My youngest son has been fascinated with Elon Musk's plans for colonizing Mars. He is not that different from his old man because in my youth Wernher von Braun (Figure 1) had me captivated with his plans for human-crewed missions to Mars. As I described von Braun's plans for exploring Mars to my son, I realized the both Musk and von Braun applied similar state-of-the-art marketing approaches.

You might think that Musk's use of animation and visionary storytelling were unprecedented for selling a technical program – not so. I would argue that Musk is following the von Braun playbook almost exactly.

Von Braun also applied a media full-court press for exploring both the Moon and Mars:

• Wrote a science fiction novel, The Mars Project, with an amazing appendix containing the math and physics behind the mission plan. True, many engineers argued with von Braun about the details but he saw those arguments as unimportant – the details looked convincing. I re-read this book a couple of years ago. While the science fiction story is marginal, the vision just exudes credibility.
• Wrote a popular science book with Willy Ley called The Conquest of the Moon that described how the Moon would be be explored. The Moon was always portrayed as a stepping stone on the way to the real goal, Mars.  As a boy, I loved this book and read it several times.
• Teamed with famous space artist Chesley Bonestell on some beautiful space-related art that appeared in various magazines (e.g Collier's) back in the 1950s. Here is one of my favorites.

Figure 2: Bonestell Visualization of Mars Exploration Effort. (Source)

• Gave presentations on space travel to any group that was willing to listen. I still remember the science teachers in my high school talking about when von Braun helicoptered in to give them a seminar on the space program. He must have given a motivational talk that rivaled anything that Vince Lombardi ever gave a football team.
• Teamed with Disney on making some marvelous animated films on how the Mars mission would be be implemented. Fortunately, all of this animation appears to be available on Youtube.
 Figure 3: Disney-Produced Program on Exploring Space.

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## Tapered Side Angle Measurement

Quote of the Day

The secret to living well and longer is: eat half, walk double, laugh triple, and love without measure.

— Tibetan Proverb

## Introduction

Figure 1: Side Angle Measurement of Slot. (Source)

I thought I was done with my metrology review when I encountered an excellent set of discussions at the Hobby-Machinist web site. They advertise themselves as "The Friendly Machinist Forum," and all signs indicate that is true. In addition to excellent tutorials, there are some first-rate metrology discussions on that site, and I want to document a few of the examples that are shown there.

Figure 1 shows a slot with angled sides with the angle being measuring using gage balls or roller gages.

## Analysis

### Derivation

Figure 2: Reference Drawing Showing Critical Variables for Measuring One Angle.

The math behind this example is almost identical to that presented in this post, where I present Equation 1 as the angle solution for the measurement scenario of Figure 2.

The differences are created by my distance measurements, L1 and L2, are doubled in Figure 1 because I am measuring two angles at once. This means that I can easily adapted Equation 1 to the configuration of Figure 1 with a simple substitution.

 Eq. 1 $\displaystyle \theta \left( {{{L}_{1}},{{L}_{2}},{{D}_{1}},{{D}_{2}}} \right)=2\cdot \text{arctan}\left( {\frac{1}{2}\cdot \frac{{{{D}_{1}}-{{D}_{2}}}}{{{{L}_{1}}-\frac{{{{D}_{1}}}}{2}-\left( {{{L}_{2}}-\frac{{{{D}_{2}}}}{2}} \right)}}} \right)$

where

• L1 is the distance from reference to outside edge of roller gage.
• L2 is the distance from reference to outside edge of roller gage.
• D1 diameter of the first roller gage.
• D2 diameter of the second roller gage.
• θ is the angle of the drill hole relative to the surface that is drilled.

I can redefine my symbols as shown in Figure 3 and generate a new formula, Equation 2, that applies to Figure 1.

Figure 3: Variable Definitions for Figure 1.

To obtain Equation 1 from Equation 2, you apply the substitutions ${{{L}'}_{1}}\leftarrow \frac{{{{L}_{1}}}}{2}$ and ${{{L}'}_{2}}\leftarrow \frac{{{{L}_{2}}}}{2}$.

 Eq. 2 $\displaystyle \theta \left( {{{{{L}'}}_{1}},{{{{L}'}}_{2}},{{D}_{1}},{{D}_{2}}} \right)=2\cdot \text{arctan}\left( {\frac{{{{D}_{1}}-{{D}_{2}}}}{{{{{{L}'}}_{1}}-{{{{L}'}}_{2}}-\left( {{{D}_{1}}-{{D}_{2}}} \right)}}} \right)$

where

• L'1 is the distance between the small diameter gages.
• L'2 is the distance between the large diameter gages.
• D1 diameter of the first roller gage.
• D2 diameter of the second roller gage.
• θ is the side angle relative to the slot base.

### Figure 1 Example

Figure 4 shows how to work the example of Figure 1. My formula results agree with those obtained from the scale drawing.

Figure 4: Example of Figure 1 Worked Using Equation 2.

## Conclusion

My plan is to continue to build up my metrology examples so that I have as complete a set as I can put together.

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Posted in Metrology | 2 Comments