New Zealand Complex Phone Line Impedance

Posted on 15-June-2016 by mathscinotes

Quote of the Day

I found one day in school a boy of medium size ill-treating a smaller boy. I expostulated, but he replied: 'The bigs hit me, so I hit the babies; that's fair.' In these words he epitomized the history of the human race.

— Bertrand Russell

Introduction

Figure 1: A Map of New Zealand.

Figure 1: A Map of New Zealand (Source).

I received an email today asking me about the phone line impedance  differences between New Zealand (Figure 1) and Australia.  This is an easy question to answer, and I wrote up a quick Mathcad worksheet to perform the calculation.

New Zealand is a country that I dream about visiting – I want to see Hobbiton. I loved watching The Lord of the Rings, and this put New Zealand on my bucket list.

My calculation worksheet and a PDF is available here.

Background

All the background material needed to understand this post is covered in my earlier post on Australian off-hook phone impedance.

Analysis

Reference Circuit

Figure 2 shows an excerpt from a web reference on the BT3 impedance model used by New Zealand.

Figure 2: New Zealand BT3 Reference Circuit.

Figure 2: New Zealand BT3 Reference Circuit (Source).

Analysis Setup

Figure 3 shows how I modeled the characteristic impedance for 26 AWG wire, which is commonly used in North America. I also show the impedance models for Australia and New Zealand.

Figure 3: Formulae for 26 AWG Characteristic Impedance, Plus Australia and New Zealand Reference Impedances.

Figure 3: Formulae for 26 AWG Characteristic Impedance, Plus Australia and New Zealand Reference Impedances.

Comparison with Australia and US Impedance Levels

Figure 4 shows my plot of the impedances for:

  • 26 AWG wire pair
  • North American 600 Ω and 900 Ω line impedance model
  • Australian line impedance model
  • New Zealand (BT3) line impedance model
FIgure 4: Plot of Telephony Impedances.

Figure 4: Plot of Telephony Impedances.

The plot shows that the New Zealand impedance characteristic is fairly similar to Australia's and that of a 26 AWG wire pair.

Conclusion

I have always wondered why so many countries use custom line impedances given that phone wiring is so similar around the world. To ensure that we pass the acceptance tests for each country, we must adapt our equipment to match the impedance requirements for each country. Today, this adaptation can be done using software to configure impedance settings on the Subscriber Line Interface Circuits (SLICs). Back in the old days, it required hardware modifications.

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Posted in Electronics, Telephones | 1 Comment

Tank Track Ground Pressure Examples

Posted on 10-June-2016 by mathscinotes

There are only two ways to live your life. One is as though nothing is a miracle. The other is as though everything is a miracle.

— Albert Einstein

Introduction

Figure 1: Good Photograph that Shows the Tank Track of an M48A5 Patton Tank (Source).

Figure 1: Good Photograph that Shows the Tank
Track of an M48A5 Patton Tank (Source).

I have been reading some military history on tank operations during the WW2 and the subject of the ground pressure exerted by the tank's tracks has figured prominently in the discussions on the Eastern Front. The T34/85 was mentioned as a particularly mobile tank because of its low ground pressure. Since I am working diligently on improving my web scraping skills, I decided to generate a short table of the ground pressures of some famous tanks.

Since most WW2 tanks were much lighter than modern tanks, I thought I was going to see a major difference in the ground pressures exerted by the tracks. It turns out that all the tanks I examined exert ground pressures in the range of 10 lbf/in2 to 16 lbf/in2. To make these numbers meaningful, you should compare them to something you see in everyday life. Consider the examples shown in Figure 2.

Figure 2: Everyday Examples of Ground Pressure.

Figure 2: Everyday Examples of Ground Pressure (Source).

So most tanks exert somewhat less ground pressure than people.

Background

Definitions

Track Ground Contact Length (lGC)
The area of the track in contact with the ground at any time.
Track Width (w)
Track width is simply the width of the track. You do need to be a bit careful though because most tanks have two types of track: (1) a narrow track used when being transported, and (2) a wider track used for combat. For this exercise here, I am only interested in the combat track width.
Ground Pressure (PGC)
The pressure exerted by the tank track on the ground. It is computed using the formula p_{GC}=\frac{W_{Tank}}{{2\cdot {{l}_{{GC}}}\cdot w}}.

Analysis

Table 1 shows the results of my web scraping and simple calculations. It looks like the Tiger 1 and M1A1 have the highest ground pressure of the tanks I examined. The T34/85 had very low ground pressure, which historians often cite as numerical support for their claims of its excellent mobility.

Table 1: List of Famous Tanks and Their Track Characteristics
Tank Country Intro Year Weight
(lbf)		 Weight
(US ton)		 Track length
(in)		 Track Width
(in)		 Ground Pressure
(lbf/in2)		 Ground Pressure
(kg/cm2)
Mk IV Germany 1939 55,116 27.6 159.8 15.7 11.0 0.77
Panther Germany 1942 100,310 50.2 156.5 25.6 12.5 0.88
Tiger 1 Germany 1942 126,214 63.1 141.9 28.5 15.6 1.10
Tiger 2 Germany 1944 139,600 69.8 162.2 31.5 13.7 0.96
Leopard 2 Germany 1979 121,584 60.8 194.7 26.5 11.8 0.83
T34/85 Russia 1940 73,193 36.6 151.6 23.0 10.5 0.74
IS-2 Russia 1943 101,412 50.7 171.7 25.6 11.5 0.81
T54 Russia 1947 78,264 39.1 155.1 21.3 11.9 0.83
T72A Russia 1973 97,900 49.0 168.5 22.8 12.7 0.89
T80 Russia 1976 101,200 50.6 169.2 22.8 13.1 0.92
Churchill Mk VII UK 1944 86,240 43.1 149.6 22.0 13.1 0.92
Challenger 2 UK 1998 137,789 68.9 188.6 25.6 14.3 1.00
M4 USA 1942 66,800 33.4 147.0 16.6 13.7 0.96
M26 USA 1945 92,355 46.2 153.5 24.0 12.5 0.88
M48 USA 1953 99,000 49.5 157.0 28.0 11.3 0.79
M60 USA 1961 102,000 51.0 166.7 28.0 10.9 0.77
M1 USA 1980 120,000 60.0 180.0 25.0 13.3 0.94
M1A1 USA 1985 130,000 65.0 180.0 25.0 14.4 1.02
M1A2 USA 1990 139,000 69.5 180.0 25.0 15.4 1.09

Conclusion

It certainly does look like the T34/85 had excellent ground pressure numbers. When you are battling winter and mud along with an adversary, I can see where this metric could tip the balance.

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Posted in Military History | 12 Comments

Instrumentation Amplifier Gain Adjustment

Posted on 9-June-2016 by mathscinotes

Quote of the Day

Who is that woman? She can't be the woman who raised me.

— I heard a coworker say this about his mother. He was shocked that his mother acted differently in her role as a grandmother than she acted as a mother. Personally, I plan on being an extremely doting grandfather. I can't wait to get the talk about my behavior from my son.

Introduction

FIgure 1: Modifying the Gain of an LT1101 Instrumentation Amplifier.

Figure 1: Modifying the Gain of an LT1101
Instrumentation Amplifier. (Source)

I have an existing circuit for which I need to modify the front-end gain. The gain is provided by an LT1101, which is a common instrumentation amplifier. This part is normally used with one of its two fixed gain settings (10x, 100x). As commonly happens, I need to find a way to resolve an issue without making major changes to a circuit. The designers of the LT1101 provided you a way to modify the amplifiers gain by adding two resistors  to the circuit. Figure 1 shows the modified circuit, with the added resistors marked with red ovals and labeled Rx.

Background

The LT1101 allows you to configure its fixed gains as follows:

  • 10x Gain: leave pins 2 and 7 open.
  • 100x Gain: (1) short pin 2 to pin 1, (2) short pin 7 to pin 8.

For gains G such that 10 ≤ G ≤ 100, the LT1101 datasheet provides us Equation 1 for adjusting the gain to values above 10 by putting identical resistors between pins 2 and 1, and pins 7 and 8.

Eq. 1 G=10+\frac{{Rx}}{{R+\frac{{Rx}}{{90}}}}

where

  • G is the desired gain.
  • Rx is the external resistor being added.
  • R is the internal resistor that has a nominal value of 9.2 kΩ.

What I am doing in this post works fine for a small number of test circuit, but long term it would be better to use an instrumentation amplifier whose gain can be programmed using a single resistor (e.g. LT1789).

Normally, I would not post an analysis this simple, but this solution does illustrate some interesting techniques:

  • Use of an infix operator to represent the resistance of parallel resistors
  • Setting up and solving circuit equations
  • Using the partial fraction keyword to simplify an expression
  • Using a datasheet's block diagram to modify a circuit's performance

Analysis

Figure 2 shows my analysis of the LT1101's gain circuit. I applied Kirchhoff's voltage law to the internal opamp circuit. Note that VA is the output of the opamp labeled A in Figure 1.

I should mention that I wanted to convert the input voltages at the inverting (VN) and non-inverting pins (VP) into a common-mode (VC) and differential-mode (ΔV) representation. I did this through the substitutions {{V}_{P}}={{V}_{C}}+\frac{{\Delta V}}{2} and {{V}_{N}}={{V}_{C}}-\frac{{\Delta V}}{2}. With this substitution, the VC term was cancelled out and only the ΔV term was left.

Figure 2: Derivation of the Equation 1 and Determining Rx for G=20.

Figure 2: Derivation of the Equation 1 and Determining Rx for G=20.

I simulated this circuit modification using LTSpice and it worked perfectly. I now need to go into the lab and try it out.

Conclusion

This post just provides a simple demonstration of basic resistor calculations using Mathcad. While there is nothing complicated here, this example does show how easy it is to document your calculations using this type of tool. I can return to this circuit analysis years later, and I will easily be able to determine what I was doing.

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Posted in Electronics | Comments Off on Instrumentation Amplifier Gain Adjustment

Educating Children Is Expensive, But Ignorance Costs More

Posted on 8-June-2016 by mathscinotes

Quote of the Day

Time not spent investing in yourself carries an opportunity cost, rendering you at a competitive disadvantage compared to the others who are maintaining the priority of self-advancement.

— David Brooks quoting Victoria Buehler, who was describing the attitude of  Millennials.

My sister claims that this animated GIF perfectly summarizes what it feels like to have two daughters at university. In my case, the cost of two sons going to university was no less expensive.

Figure 1: Raising Daughters.

Figure 1: My Sister's View of Putting Two Daughters Through University. The scene is from the movie Spanky, which starred Spanky MacFarland.

Paying school bills is certainly not the only expensive item for parents. I often tell people that the most expensive movie I ever took my sons to was The Mighty Ducks – it cost me about $4k per year for a number of years. This was roughly the cost back in the 1990s of having two kids in traveling hockey. After my sons saw that movie, they were determined to become hockey players.

I looked back on it all fondly now. My wife and I will be grandparents this fall, and we look forward to helping our son and his wife with our grandchild. I have fond memories of my grandparents, and I smile when I think of  Deena Nataf saying that "Grandparents and grandchildren have a special relationship because they share a common enemy" 😉

If I play my cards right, it won't be that many years before I have the kid designing Raspberry Pie-based hardware …

Posted in Humor, Personal | Comments Off on Educating Children Is Expensive, But Ignorance Costs More

A Lesson Learned from Lab Testing a Colpitts Oscillator

Posted on 7-June-2016 by mathscinotes

Quote of the Day

The successful warrior is the average man with laser-like focus.

— Bruce Lee, martial artist, actor and filmmaker. I firmly believe that "average" people are capable of great things by focusing on what they are doing.

Introduction

Figure 1: Colpitts Oscillator Prototype.Figure 1: Colpitts Oscillator Prototype.

Figure 1: Colpitts Oscillator Prototype.

It is the weekend and time to build my personal circuits. Some of my recent personal work has been focused on detecting cars in my cabin's driveway by using  a loop of wire that experience an inductance change when a car passes over it. This weekend I built the Colpitts oscillator I discussed in this post, which is part of a car detector improvement I am considering. The circuit I am using now is not as sensitive as I would like, and I am hoping a few changes will allow me to detect smaller vehicles, like ATVs.

This new circuit is so simple that I expected it would work with no problems – the gods of electronics do not look kindly on this form of hubris. Of course, the oscillator would not oscillate. During my troubleshooting, I re-learned a couple of lessons that I thought would be worth sharing here. I have noticed that a number of other engineers have had startup problems with Colpitts oscillators and their issues were similar to mine.

The lessons I re-learned are simple and timeless:

  • Inductors are not the same as their mathematical idealization.

    In this case, I used a 100 µH inductor with 1.7 Ω of series resistance to simulate my loop of wire. My mathematical analysis and early simulations ignored the inductor's series resistance. It turns out that the gain required to make the circuit oscillate is strongly dependent on this series resistance. The lesson here is that I need to include the series resistance in future simulation efforts involving inductors.

  • All component values are subject to tolerance variation.

    I built my prototype (Figure 1) using some old 10% tolerance components I had laying around. It just so happened that their actual values all were in a direction that caused me to require more loop gain than I had predicted in my analysis using nominal component values. The lesson here is to make sure that I understand the sensitivity of my circuits to these component variations.

Ultimately, the problem was resolved by increasing the value of one resistor – the challenge was understanding why my simulation was off by so much. In the end, my simulation results matched the performance of my actual circuit. I now feel I can move forward since I have a firm grasp on the key performance-determining parameters.

For those who wish to experiment with my simulation, my LTSpice source is here.

Background

Previous Work

Please review my work in this post. It shows the initial circuit that I built assuming all nominal values. While the simulation worked fine, the actual circuit failed to oscillate.

Test Circuit

Figure 2 shows the actual circuit I built that worked with measured values. The key part modification required to ensure oscillation was changing R2, from 3 kΩ to 4.7 kΩ. The 3 kΩ value simply did not provide sufficient gain to make the real circuit oscillate. You will also see that I increased my capacitor values. This was not needed to ensure oscillation, but rather to reduce the oscillation frequency a bit.

This circuit and my simulation matched within reasonable expectations. While not shown in the circuit, the inductor has 1.7 Ω of series resistance in its model.

Figure M: Colpitts Circuit I Built in the Lab.

Figure 2: Colpitts Circuit I Built in the Lab.

Simulation Results

Figure 3 shows my simulation results. The simulation shows an oscillation frequency of 91.4 kHz.

Figure M: Simulation Results for the Circuit of Figure M.

Figure 3: Simulation Results for the Circuit of Figure 3.

Lab Results

Figure 4 shows the sinusoid present at Node 4 (Figure 2). The oscillation frequency is 94.3 kHz, which is relatively close (3% difference) to my simulation result of 91.4 kHz. The prototype waveform has a peak-to-peak amplitude of ~2 V, while the simulation had 2.3 V. This also is relatively close. These amplitude values are strongly determined by the limiting characteristics of the opamp, which are not well-modeled in the simulations.

Figure 3: Opamp Output Voltage.

Figure 4: Opamp Output Voltage.

Figure 5 shows the output voltage of my opamp. As the simulation shows, this is a highly distorted signal. The way to stop this distortion is by controlling the waveform amplitude using a separate, more gentle nonlinearity. If I decide to pursue a less distorted output waveform, that would be a topic for a separate post.

Figure 5: Opamp Output Voltage.

Figure 5: Opamp Output Voltage.

Conclusion

I am quite happy with this circuit. It is part of a larger circuit for which I am designing a Printed Circuit Board (PCB). I expect to have this PCB ready for release in about 4 weeks. I will report on my test results when I get that PCB back, stuffed with parts, and tested.

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Posted in Cabin, Electronics | Comments Off on A Lesson Learned from Lab Testing a Colpitts Oscillator

Effect of Centrifugal Force on Weight

Posted on 6-June-2016 by mathscinotes

Quote of the Day

If you are not annoying someone you are not doing anything new.

— Michael Stainer

Introduction

Figure 1: North American X-15 Hypersonic Rocket-Powered Aircraft.

Figure 1: North American X-15 Hypersonic
Rocket-Powered Aircraft.  (Source)

I love to read Quora, and I often see interesting factoids there that I inspire me to put pencil to paper and verify them. This week, I read a response to the question "What does the pilot of a supersonic fighter feel when flying at Mach 3 at 40,000 feet?" I found one of the answers particularly interesting because of how the respondent generalized the question to make it more interesting. I love when people take a basic question and turn it into a more interesting question.

The response that intrigued me began by pointing out that an airplane flying at 40,000 feet with a velocity of Mach 3 would probably be melting – the same answer given by other respondents. The author then changed the question by increasing the airplane's speed to Mach 8, its altitude to 70,000 feet, and its location to the equator. I quote part of his response here.

Actually, as the velocity increases, a pilot or passenger would feel slightly less weight, because the Earth's curvature becomes very important. This gets a little unusual because things start to matter that you would not expect are relevant. For example, if you are flying straight east over the equator at Mach 8, you would feel only 87% of your weight, but if you were flying west at the same point, you would feel 93% of your weight. The difference is due to the Earth's rotation.

Since only a few manned research aircraft have even come close to this speed (e.g. Figure 1), I do not expect to be able to perform an actual test. However, a little math will confirm his statement. Let's dig in …

Background

Definitions

Mach Number (symbol: M)
A dimensionless quantity representing a fluid's velocity relative to the speed of sound, i.e. M = \frac{v_{Fluid}}{c_{Sound}}, where vFluid in this case is the air speed of the aircraft, and cSound is the speed of sound at that altitude. (Source)
Hypersonic
In aerodynamics, a hypersonic speed is one that is highly supersonic. Since the 1970s, the term has generally been assumed to refer to speeds of Mach 5 and above. (Source)

Approach

Here is my approach to duplicating the respondent's results:

  • determine the velocity represented by Mach 8 at 70,000 feet.

    Mach number varies with altitude, and we need to determine the velocity in m/s for an airplane at 70,0000 feet with a speed of Mach 8.

  • compute the velocity of an object on the Earth's surface at the equator.

    Even an object sitting stationary on the equator is actually being carried along by the rotation of the Earth. This movement causes the weight of an object to be measurably less at the equator than at the pole by the factor 288/289, which I will demonstrate in my analysis.

  • compute the reduction in apparent gravity on an object due to centrifugal force on an airplane traveling easterly at Mach 8 and 70,000 feet.

    When moving easterly, the airplane's velocity adds to the intrinsic velocity of an object being carried along with the Earth's rotation.

  • compute the reduction in apparent gravity on an object due to centrifugal force on an airplane traveling westerly  at Mach 8 and 70,000 feet.

    When moving westerly, the airplane's velocity subtracts from the intrinsic velocity of an object being carried along with the Earth's rotation.

  • the weight reduction at the equator while flying at altitude is relative to the person's sea level weight at the pole.

    At the poles, a person has no weight reduction due to centrifugal force.

Analysis

Figure 2 shows my mathematical work that duplicates the results cited above (87% and 94%), with the final numbers marked with light-yellow highlights.

Figure 2: My Analysis of the Weight Reduction at the Equator.

Figure 2: My Analysis of the Weight Reduction at the Equator.

Mach number versus altitude Google Answers Discussion

Conclusion

I was surprised that high-speed, high-altitude flight could experience enough centrifugal force to have  a measurable effect on the weight of a person. At first glance, I found the weight reductions stated to be almost unbelievable – however, a bit of physics shows that the numbers are correct.

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Posted in Physics | 7 Comments

Computing Lux Level Given A Lamp's Power Spectrum

Posted on 2-June-2016 by mathscinotes

Quote of the Day

Experience is not what happens to a man; it is what a man does with what happens to him.

— Aldous Huxley

Introduction

Figure 1: Xenon Lamp Irradiance Example (Source).

Figure 1: Xenon Lamp Irradiance Example (Source).

I have two engineering interns working in my group this summer. I love having interns around, but they do require more management oversight. I was an intern at Medtronic when I was at university and found the intern experience wonderful. I owe a great debt of gratitude to Ron Meister and Jim Sinclair for their patient support of a very naive young man. Today, I follow their lead and try to support my profession by helping young folks get started.

I have assigned my interns a couple of simple design tasks – one of the tasks involves performing measurements on that amount ambient light that leaks into our enclosures. This task is going to provide data that will drive the design of a simple ambient light sensor for use within our enclosures to tell if the door has been opened. Open door sensors are a notorious source of false alarms. We used to use mechanical/magnetic sensors, but they proved to be unreliable. We are now looking at using ambient light sensors.

One of the interns asked how to convert our ambient light power measurements into lux (visual units). I thought I would rework a xenon flash lamp example I found on the web using Mathcad. This will give him a tool that he can use in his work here this summer.

Background

Requirements

lumen (symbol: lm)
The lumen unit of luminous flux, a measure of the total quantity of visible light emitted by a source. Luminous flux differs from power (radiant flux) in that radiant flux includes all electromagnetic waves emitted, while luminous flux is weighted according to a model of the human eye's sensitivity to various wavelengths. Lumens are related to lux in that one lux is one lumen per square meter.
lux (symbol: lx)
The lux is one lumen per square meter (lm/m2). There is no single conversion factor between lx and W/m2; there is a different conversion factor for every wavelength, and it is not possible to make a conversion unless one knows the spectral composition of the light.
luminous efficiency
Luminous efficacy is a measure of how well a light source produces visible light. It is the ratio of luminous flux to power. The concept of luminous efficiency is complicated by the fact that the eye has two types of receptors: rods and cones. Rods provide our low light vision and their luminous efficiency is described by the scotopic luminous efficiency curve. Cones are used at normal light levels and their luminous efficiency is described by the photopic luminous efficiency curve. My work here will focus on the normal light level (photopic) region of vision.
Figure 2 shows the photopic luminous efficiency curve as defined by the International Commission on Illumination (CIE).
Figure 2: Photopic Luminious Efficiency Curve (Source).

Figure 2: Photopic Luminous Efficiency Curve (Source).

Figure 2 show us that the eye is most sensitive to green light (555 nm). You can achieve a given visual light level with the least power using green light – all other colors will require more power.

Key Formula

The basic math required here is shown in Equation 1, which is just the sum of all the lamp's visual power weighted by the eye's sensitivity.

Eq. 1 {{E}_{v}}=683.002\cdot \int\limits_{{{{\lambda }_{{\min }}}}}^{{{{\lambda }_{{\max }}}}}{{y(\lambda )\cdot J(\lambda )\cdot d\lambda }}

where

  • Ev is the illuminance (i.e. visual brightness) measured in lux
  • y(λ) is the photopic luminosity function.
  • J(λ) is the spectral irradiance of the light – I think of this as the power spectral density of the light.
  • λmax is longest visual wavelength.
  • λin is the shortest visual wavelength.

For 555 nm light, the amount of illuminance is a simply unit conversion using 683.002 lux per W/m2.

Analysis

My Mathcad source and its PDF are included here.

Photopic Luminosity Capture

Figure 3 shows my capture of the photopic luminosity function using Dagra. The curve-fit luminance function that I will be using is highlighted in green.

Figure 3: Capture of the Photopic Luminosity Function.

Figure 3: Capture of the Photopic Luminosity Function.

Capture of Lamp Irradiance

Figure 4 shows my capture of the xenon lamp's irradiance (i.e. power spectrum). The curve-fit irradiance function that I will be using is highlighted in green.

Figure 4: Capture of Xenon Flash Lamp Irradiance.

Figure 4: Capture of Xenon Flash Lamp Irradiance.

Evaluation of Equation 1

Given the data captures in Figures 3 and 4, the evaluation of Equation 1 is simple. My result of 274 lux is quite close to the listed result of 270 lux. The error is from how I captured the data from the images.

Figure 5: Evaluation of Equation 1 for the Xenon Flash Lamp.

Figure 5: Evaluation of Equation 1 for the Xenon Flash Lamp.

Conclusion

This is just a quick illustration of how a common engineering task is made simple through the use of a computer algebra system. Some days I am stunned at how I work today compared to 30+ years ago. The tools today are so much better than what I had when I started out – handheld calculators and graph paper.

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Posted in optics | Comments Off on Computing Lux Level Given A Lamp's Power Spectrum

Backup Power For My Cabin

Posted on 1-June-2016 by mathscinotes

Quote of the Day

Today a man owns a jackass worth 50 dollars and he is entitled to vote; but before the next election the jackass dies. The man in the mean time has become more experienced, his knowledge of the principles of government, and his acquaintance with mankind, are more extensive, and he is therefore better qualified to make a proper selection of rulers—but the jackass is dead and the man cannot vote. Now gentlemen, pray inform me, in whom is the right of suffrage? In the man or in the jackass?

— Benjamin Franklin on the practice of allowing only property owners to vote. Ben Franklin provided support for the voting rights of every citizen and not just of property owners.

Introduction

FIgure 1: Tesla Powerwall Battery Pack (Source).

Figure 1: Tesla Powerwall Battery Pack (Source).

My wife and I are in the process of designing our northern Minnesota retirement cabin – the current structure is too primitive for any extended stay. Because power is unreliable in the northern woods, I am researching whole-home battery backup options. One possible option is the Tesla Powerwall, which provides 6.5 kW-hr of energy per battery pack. You can increase capacity by adding battery packs as you need.

Backup power systems are important for remote residences. For example, a friend of mine recently put a whole-home backup system in his cabin, and he almost immediately put it to use after a major storm knocked out power for many hours. The storm unleashed a tremendous amount of rain during the power outage, and my friend needed to run his sump pump continuously during the storm to keep his basement clear of water. His backup system allowed him to enjoy his home and to preserve its value.

I want a backup system that can meet the same scenario he faced. As with all my requirement determination efforts, I begin by defining my use cases – in this situation, there is just one use case.

  • Power outage lasts for 8 hours.
  • Rain is causing me to run a sump power continuously.
  • I need to keep a small number of lights on – all are LED-based.
  • I need to keep a refrigerator running – I want to ensure that I have access to food.
  • I need to keep a gas-heated furnace with electric blower motor running during the outage.
  • I need to keep an well-water pump running – fortunately the well pump only needs to operate intermittently.
  • I do plan on having a powered generator, but I would like to avoid running that for short power outages (i.e. less than 8 hour).

In this post, I will estimate my energy needs and determine if one Powerwall would meet my needs or if I would need multiple units. A single Powerwall and inverter is estimated to cost $7340– ouch! I will need to compare this cost to that of an equivalent lead-acid backup system. That will be the subject of another post.

Background

Limitations of My Analysis

I am ignoring a number of important concerns:

  • Electric motor surge currents

    The usual rule of thumb is that a motor's starting current draw is 4x its nominal current draw. This surge can be accommodated a number of ways that do not require adding more batteries. However, surge current is a real problem that must be dealt with.

  • Temperature-dependent battery capacity

    Battery capacity decreases with temperature. I need to decide how I am going to deal with keeping the batteries from getting cold. I could bury them in a Controlled Environment Vault (CEV). However, they are very expensive.

  • Battery aging

    All batteries age because of various corrosion processes that occur within the cells. This aging is accelerated by high temperature.

This post documents the quick analysis I performed to scope my energy needs relative to what the Powerwall delivers.

Refrigerator Average Power Estimate

The US Energy Information Administration (EIA) requires the listing of the average annual energy usage of appliances, including refrigerators. I grabbed the estimated annual energy usage for a Samsung refrigerator and converted the annual energy usage into a power as shown in Figure 2.

Figure 2: Computing the Average Power of a Refrigerator.

Figure 2: Computing the Average Power of a Samsung Refrigerator.

Analysis

Figure 3 shows how I computed my emergency power requirements and determined how long one and two Powerwalls would be able to provide that level of power. These calculations show that I need two Powerwalls to meet my 8 hour requirement. Since Powerwalls are very expensive, I will need to repeat this calculation for a lead-acid alternative – I want to use the most cost effective solution.

Figure 3: My Calculations for the Backup Time Available Using One or Two Powerwalls.

Figure 3: My Calculations for the Backup Time Available Using One and Two Powerwalls.

Conclusion

This was just a quick calculation to estimate the kind of backup time I could expect from one and two Powerwalls. I need to compare this cost with the equivalent cost from a lead-acid battery pack. I assume that the lead acid-based system will be much larger and significantly cheaper. A little math will reveal that answer shortly.

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Posted in Batteries, Electronics | Comments Off on Backup Power For My Cabin

Quick Look at a Colpitts Oscillator

Posted on 26-May-2016 by mathscinotes

Quote of the Day

The most beautiful people we have known are those who have known defeat, known suffering, known struggle, known loss, and have found their way out of the depths. These persons have an appreciation, a sensitivity, and an understanding of life that fills them with compassion, gentleness, and a deep loving concern. Beautiful people do not just happen.

— Elisabeth Kubler-Ross

Introduction

Figure 1: Interesting Colpitts Oscillator Circuit.

Figure 1: Interesting Colpitts Oscillator Circuit (Source).

We have Memorial Day off from work and that can only mean one thing – time to work on some circuits for home projects. I have another inductive sensor project for which I want to generate a frequency that depends on the inductance value. The Colpitts oscillator is a good circuit for this type of application.

This is a quick note to document how I analyzed a circuit that I found on a tutorial and verified a statement made on the tutorial about the minimum gain required to startup the oscillation. I also derived an expression for the oscillation frequency.

My Mathcad, Mathcad PDF, and LTSpice source are here.

Background

There are excellent Colpitts oscillator references available:

Analysis

Node Labeling

Figure 2 shows how I labeled the circuit nodes for analysis. It also shows where I opened the feedback loop (i.e. at the output).

Figure 2: Open Feedback Loop Version of the Circuit.

Figure 2: Open Feedback Loop Version of the Circuit.

Circuit Analysis

Figure 3 shows my circuit analysis.

Figure 3: Circuit Analysis.

Figure 3: Circuit Analysis.

Derivation

In Figure 4, I derive expressions for (1) the oscillation frequency, and (2) the critical amplifier gain needed to ensure the circuit oscillates. The analysis process was straightforward:

  • set VIN = V3
  • set s = j·ω because I am only interested in the steady-state sinusoidal solution.
  • set the imaginary part of the loop equation to zero and solve for the frequency at which the phase shift is zero – a required condition for oscillation. This will give me my oscillation frequency.
  • set the real part of the loop equation equal to one and solve for R2 – this effectively sets the gain condition for oscillation. Since the gain is given by R2/R1, this will allow me to solve for the required gain in terms of R1. This gain will be a minimum,  and we need to exceed that gain slightly to ensure the startup of the oscillator.
Figure 4: Derivation of Critical Formulae.

Figure 4: Derivation of Critical Formulae.

I computed a minimum gain of ~2.7 and an oscillation frequency of 11. 7 kHz. My simulation (below) confirmed these results.

When people discuss Colpitts oscillators, they usually state that the oscillation frequency is given by the equation {{f}_{{Standard}}}\left( {{{C}_{1}},{{C}_{2}},L} \right)=\frac{1}{{2\cdot \pi }}\sqrt{{\frac{{{{C}_{1}}+{{C}_{2}}}}{{{{C}_{1}}\cdot {{C}_{2}}\cdot L}}}}. This is not the equation I derive in Figure 4 \displaystyle \left( {{{f}_{{Mine}}}\left( {{{C}_{1}},{{C}_{2}},L} \right)=\frac{1}{{2\cdot \pi }}\sqrt{{\frac{{L+{{R}_{1}}\cdot {{R}_{3}}\cdot \left( {{{C}_{1}}+{{C}_{2}}} \right)}}{{{{R}_{1}}\cdot {{R}_{3}}\cdot {{C}_{1}}\cdot {{C}_{2}}\cdot L}}}}} \right). This is because most Colpitts discussions are about current source-based, transistor designs versus the voltage source-based, opamp design considered here. This difference in topology makes the circuit equations slightly different. I show in Figure 5 that when {{R}_{1}}\cdot {{R}_{3}}\cdot \left( {{{C}_{1}}+{{C}_{2}}} \right)>>L (usually the case), the standard Colpitts equation and my equation are approximately equal.

Figure M: Comparison

Figure 5: Transistor and Opamp Colpitts Frequency Formulas are Approximately the Same.

Simulation

The nodes in my simulation are labeled differently than in my mathematical work above. LTSpice wants to name the nodes in a manner that it finds convenient.

LTSpice Circuit

Figure 6 shows my implementation of the circuit using LTSpice. To force the oscillator to start, I set node 2 = 0.3 V at startup with the Initial Condition command (IC) – in real circuits, noise and the power supply ramp on startup drives the oscillator into action.

Figure 5: My LTSpice Version.

Figure 6: My LTSpice Version.

Results

Figure 7 shows my simulation results. The oscillation frequency is slightly lower (11.5 kHz) than the mathematical analysis showed because the analysis ignored the opamp's frequency response – the phase shift of the amplifier contributes to the where the circuit oscillates.

While not shown here, I tried various gains, and I confirmed that I needed an amplifier gain of ~3 to get the oscillator to startup.

Figure 6: Simulation Results.

Figure 7: Simulation Results. Observe that the sinusoid at node 4 is pretty clean compared to the amplifier output.

I should note that all oscillators have some level of distortion caused by their amplitude-limiting circuitry, which is intrinsically a nonlinear function. For example, Bill Hewlett used a lamp with a nonlinear resistance characteristic to stabilize the old HP200's amplitude level – as elegant a solution as I can imagine.

In this case, amplitude stability is maintained because of nonlinearities in the amplifier, which is NOT the way to achieve a quality sinusoid. However, I am not building a lab grade project here – purely a home project.

Conclusion

I will be building this circuit this weekend. I am hopeful that the analysis and simulation matches reality. My first step is ALWAYS to get a valid mathematical analysis and working simulation before I build anything.

Posted in Electronics | 5 Comments

Dynamical Parallax Examples

Posted on 25-May-2016 by mathscinotes

Quote of the Day

Be the change that you wish to see in the world.

— Mahatma Gandhi

Introduction

Figure 1: Book Cover of Parallax.

Figure 1: Book Cover of Parallax (Source).

Years ago, I read the book Parallax (Figure 1) and really enjoyed the tale of how 19th century astronomers measured the distance to the nearest stars. This measurement was critical to providing scientists some idea as to the scale of the universe.

The book Parallax describes how simple trigonometry, along with the introduction of  large telescopes coupled to precision measurement gear, could be used to determine the distance to a star by measuring the angular shift of that star as the Earth revolved around the Sun – a method called trigonometric or stellar parallax. During my recent perusing of the Wikipedia, I discovered that there was an alternative form of parallax measurement, called dynamical parallax, that allows one to estimate the distance to stars that are beyond the limits of trigonometric parallax.

In this post, I will verify my understanding of dynamical parallax by implementing the algorithm in Mathcad and applying it to nearby star systems for which we have accurate trigonometric data. I can then compare my dynamical estimates with the more accurate trigonometric measurements.

My Mathcad source and its PDF are here.

Background

Overview

Distant stars have an annual parallax shift that is so small that it cannot be accurately measured. However, the separation between binary stars, their individual brightness (i.e. magnitude), and orbital period frequently are measurable. We can use Kepler's 3rd law and a luminosity-based estimate of the total mass of the binary stars to compute the orbit's semi-major axis.  Given the size of the orbit, we can compute the distance to the star knowing the angle subtended by the orbit and our estimate of the orbit size.

This distance estimate can be be iteratively refined by comparing the measured angle width of the orbit to what our orbital radius and distance measurements would indicate.

The Wikipedia has a good description of the overall process, which I quote below.

With this technique, the masses of the two stars in a binary system are estimated, usually as the mass of the Sun. Then, using Kepler's laws of celestial mechanics, the distance between the stars is calculated. Once this distance is found, their distance from the observer can be found via the arc subtended in the sky, giving a preliminary distance measurement. From this measurement and the apparent magnitudes of both stars, the luminosities can be found, and from the mass–luminosity relationship, the masses of each star. These masses are used to re-calculate the separation distance, and the process is repeated. The process is iterated many times, and accuracies as high as 5% can be achieved.

Definitions

Stellar Parallax
Stellar parallax is parallax on an interstellar scale: the apparent shift of position of any nearby star (or other object) against the background of distant objects. Created by the different orbital positions of Earth, the extremely small observed shift is largest at time intervals of about six months, when Earth arrives at exactly opposite sides of the Sun in its orbit, giving a baseline distance of about two astronomical units (AU) between observations. The parallax itself is considered to be half of this maximum, about equivalent to the observational shift that would occur due to the different positions of Earth and the Sun, a baseline of one astronomical unit (Source).
Because of the exceeding small size of the angles being measured, accurate stellar parallax measurements are limited to distances of less than 30 parsecs using Earth-based telescopes and 300 parsecs using space-based telescopes. (Source)
Dynamical Parallax
In astronomy, the distance to a visual binary star may be estimated from the masses of its two components, the size of their orbit, and the period of their orbit about one another. A dynamical parallax is an (annual) parallax which is computed from such an estimated distance. (Source)
Dynamical parallax can be applied to stars that are closer than 500 parsecs to Earth. (Source)
Mass-Luminosity Relation
In astrophysics, the mass–luminosity relation is an equation giving the relationship between a star's mass and its luminosity (Source). I discussed a commonly used mass-luminosity relation in this post.

Analysis

Caveat

The following derivation will ignore the inclination of the binary star system. Figure 2 shows how orbits can be in any orientation relative to our line of sight. This nasty reality can be included in the model, but I first want to work the simple case of a binary system that perpendicular or "flat on" to our line of sight.

Figure 2: Image of Actual Solar Systems with Different Orientations.

Figure 2: Image of Actual Solar Systems with Different Orientations (JPL/NASA).

Algorithm

Figure 3 shows a flowchart for the dynamical parallax algorithm as described in the Wikipedia.

Figure M: Dynamical Parallax Algorithm.

Figure 3: Dynamical Parallax Algorithm.

Implementation

Algorithm Setup

Figure 4 shows the units and constants that I will use for evaluating my dynamical parallax implementation on some reference star systems.

Figure M: Algorithm Setup.

Figure 4: Algorithm Setup.

Dynamical Parallax Book Parallax Mass Luminosity Relation Alternate Binary Star Calculation Types of Binary Stars Finding the Orbit Inclination

Orbital Radius

Figure 5 shows how to use Kepler's 3rd law to determine the radius of an orbit assuming a total mass estimate and revolution period.

Figure M: Formula for Calculating the Radius of the Orbit.

Figure 5: Formula for Calculating the Radius of the Orbit.

Mass-Luminosity Relation

Luminosity-Mass Relationship

Figure 6 shows the Wikipedia's formula for the luminosity of star of a given mass. In fact, I actually need the inverse of the Wikipedia's function (i.e. mass given luminosity) – a function I call "f" in Figure 6. I computed the inverse using Mathcad's root function.

I also include a function that computes the luminosity of a star given the star's apparent magnitude.

Figure M: Mass-Luminosity Relation.

Figure 6: Mass-Luminosity Relation.

Dynamic Parallax Function

Figure 7 is a Mathcad function that implements my understanding of the dynamical parallax algorithm. I pass the data into the function as a vector – this allows me to apply the algorithm as a post-fix operator, which I often find convenient.

Figure M: Dynamical Parallax Algorithm.

Figure 7: Dynamical Parallax Algorithm.

Mass-Luminosity Relation

Test Cases

Figure 8 shows four stars that I chose binary star systems composed of stars that were about the size of the Sun. The mass-luminosity relationship is relatively accurate for stars of this size.

Figure M: Dynamical Parallax Test Cases.

Figure 8: Dynamical Parallax Test Cases.

Alpha Centauri Procyon 70 Ophiuchi Eta Cassiopeiae

Conclusion

I was able to use the dynamical parallax algorithm to determine the range to a number of nearby stars that have been analyzed using the more accurate trigonometric parallax approach. My results are reasonably close considering the accuracy of the luminosity-mass curve for stars.

The analysis shown here is just one way of determining the distance to a binary system using orbital information. You can also measure the orbital velocities of the component stars using the Doppler shift and applying a modified version of Kepler's 3rd law. That will be a subject for a later post.

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