Ice Fishing Season is Now Over

Posted on 8-April-2014 by mathscinotes

Every night I walk around a lake by my house, which is still mainly frozen. The weather is warmer now, but you can still see some signs of the ice fishing that went on this winter. Yesterday afternoon, I chuckled when I noticed a couple of lounge chairs on the frozen lake by my house (Figure 1). The idea of lounge chairs on a frozen lake strikes me as funny. Anyway, the ice fisherman had left their chairs by their fishing hole for the last few weeks. Unfortunately, the shore ice has started to melt and now they cannot get out onto the ice to recover them. I guess they will need to wait for the ice to completely melt and then try to recover their chairs from the lake bottom. At least it wasn't any of their ice fishing gear that ended up stranded, as I imagine they paid a huge amount of money for it, especially the drill. They always had something new each time I saw them, whether it was a new rod, a new reel, or even a new fishing suit to keep them warm. I must say, I am looking forward to seeing them out there again next season!

Unbeknownst to many, ice fishing is actually quite popular. It just offers you something different than what fishing on a normal lake does. To be honest, it's quite hard to explain if you've never taken part in it, or witnessed it before. Surprisingly, you have an extended list of equipment that you need to get before being successful at this downtime activity. It seems to me like those ice fishermen knew exactly what they needed to increase the likelihood of them catching the fish. Obviously, they would need to find a way to break the ice first, and I overheard them saying that they think it's time to look for a new ice auger, and how they are thinking about having a look at something like these Outdoorempire ice auger reviews to find the best one. All ice fishers want is the opportunity to catch some fish and they need the best quality equipment to make this happen. I can't wait to see if they did decide to buy a new auger; I'm practically on the edge of my seat just thinking about it. Roll on next season.

Figure 1: Lounge Chairs on a Frozen Lake.

Figure 1: Lounge Chairs on a Frozen Lake.

Posted in Humor | Tagged | 2 Comments

Inductive Car Sensor Math

Posted on 7-April-2014 by mathscinotes

Introduction

Figure 1: Inductive Car Sensor Under a Roadway.

Figure 1: Inductive Car Sensor Under a Roadway.


Spring has come to Minnesota and I want to build an outdoor electronics project -- this is one of my favorite activities. I have decided to build a car sensor for my cabin in northern Minnesota. This car sensor will detect when cars use my driveway. If a car comes up my driveway when I am not there, I would like a tweet sent to me. I will bury the car sensor under my gravel driveway so that it is unobtrusive.

Figure 1 shows an example of inductive car sensors buried under a roadway (source). I would like to build something similar for use at my cabin. I will review how I designed the sensor in this post.

Background

Induction Loop

Figure 2 shows how an inductive car sensor is built into a roadway (source).

Figure 2: Inductive Sensor Built into a Roadway.

Figure 2: Inductive Sensor Built into a Roadway.

I have seen quite a few of these cuts in asphalt around our local stoplights. They literally just cut the asphalt, bury the cables, and seal it up with an asphalt sealer.

Shortcomings

There are a number of shortcomings to this type of sensor. Here is a short list:

  • Bicycles and small vehicles are difficult to detect because they do not disturb the magnetic field sufficiently (source).

    I have read that some bicyclists attach magnets to bottoms of their shoes to trigger the sensors (source). Some people even mount magnets to their vehicles (source).

  • The sensor has to be large to detect vehicles with high ground clearance.

    The rule of thumb is that the maximum height of detection is 2/3 the length of the sensors shortest dimension (source). Since many of the vehicles in northern Minnesota are trucks with high ground clearance, I will need to have a large sensor.

  • The sensor's inductance is sensitive to temperature variations.

    Temperature change cause the wire length to vary, which changes the area of the loop and its inductance. Burying the sensor will help moderate some of the temperature change, but the rules are that you cannot bury the sensor more than 2 inches below the surface (), a depth which will not provide much temperature stabilization. This depth constraint is driven by the need to keep the sensor close enough to detect the underbodies of the cars that pass overhead, while providing physical protection for the wire and some level of temperature stabilization.

  • The sensor's inductance will vary if there is metal in the roadway

    Highways usually have reinforcement bars (aka rebar) to provide tensile strength to the road surface. Fortunately, my cabin driveway does not have any rebar.

Analysis

Approach

There are an endless variety of formulas for computing the inductance of all sorts of shapes (example: Grover). For my analysis here, I will use the approach for single layer circular coils published by Lundin. This approach has been described as accurate by some of the online amateur radio references I respect.

Algorithm

Figure 3 shows my Mathcad implementation of Lundin's inductance formula.

Figure 3: Mathcad Implementation of Lundin's Inductance Formula.

Figure 3: Mathcad Implementation of Lundin's Inductance Formula.

Empirical Check

To try out this equation, I built an inductive vehicle sensor with the following characteristics:

  • Circular with 6-foot diameter
  • I wound the loop as a single-layer coil of four turns (four was recommended in this reference)
  • I used 16 gauge stranded wire.
  • The loop thickness was 0.25 inch.

Given these coil parameters, my calculations indicated that the inductance should be 120.4 ?H. My handheld inductance meter measured the inductance of my coil at 121 ?H, which is so close to my theoretical prediction that I was shocked. Figure 4 shows my calculations.

Figure 4: My Empirical Check of the Loop Inductance.

Figure 4: My Empirical Check of the Loop Inductance.

Car Test

I have several circuits that I want to try for detecting a car passing over the coil. These circuits all depend on measuring the coil's inductance change when a car passes over it. I decided to use my 2002 Subaru Legacy to test out my coil. Here is the data I read using my handheld inductance meter:

  • No car over the coil: L = 121 ?H
  • Car over the coil: L = 108 ?H

So I see about a 10% inductance reduction when the car passes over the coil. I need to make sure that my interface circuit will detect this difference while having a low false positive detection level.

Conclusion

Now that I have my sensor and I understand its characteristics, I can begin serious work on my sensor interface. I will report on that work in future blog posts. One interesting aspect of this type of sensor is that you can get an idea of the kind of vehicle that passed over the sensor by looking at how the inductance varies with time. Furthermore, I wonder what applications will be possible with my own car sensor. Should I pass the technology onto insurance companies or are they already happy with the systems a lot their cars have? It might be worth contacting some insurance companies and asking them about how my sensor could effect drives insurance premiums. If you're looking for the cheapest car insurance, by the way, Money Expert should be able to help. When it comes to car insurance though it can get a little bit confusing as there are loads of different types of insurance out there. And not just for cars either, you can get motorbike insurance or even van insurance. If you drive a van and need insurance then you might be interested in checking out this any driver van insurance here.

Appendix A: Vehicle Inductance Signatures

Figure 5 shows how the inductance characteristics vary for different types of vehicles (source).

Figure 6: Inductance Signatures of Different Vehicles Passing Over the Sensor.

Figure 6: Inductance Signatures of Different Vehicles Passing Over the Sensor.

Posted in Cabin, Electronics | 2 Comments

Kepler Planet Finding Probabilities

Posted on 29-March-2014 by mathscinotes

Quote of the Day

To finish a task, you first must start it.

— Gary Ronneberg

Introduction

Figure 1: Planetary Systems with Different Orientations.

Figure 1: Planetary Systems with Different Orientations.

One of my favorite online magazines is +Plus. I was reading an article in +Plus today about how the Kepler satellite finds exoplanets and the article mentioned a simple formula for the probability of Kepler being able to detect a planet. I thought it would be interesting to discuss this formula here because it involves a simple formula that provides insight into cutting-edge science.

The article was written by Marianne Freiberger, whose written work for +Plus is excellent. Actually, all their math writing is excellent. Check them out.

Background

Planetary Systems with Different Orientations

Figure 2 shows an image from NASA/JPL that shows two adjacent planetary systems in different orientations.

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

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

Kepler Planet-Hunting Concept

The basic idea behind the Kepler planet hunting concept is a satellite can detect the dip in optical power level seen from a star when a planet passes in front of it. A planet passing through our field of view is called a transit. Figure 3 illustrates this concept (source).

Figure 2: Planet Transit and Light Level.

Figure 3: Planet Transit and Light Level.

This approach does have shortcomings:

  • The light level dip is very small.

    Detecting a planet with the size and orbital of Earth about a large star requires a very sensitive instrument. I will compute the level of optical power reduction that the Earth transiting the Sun would provide to an alien observer.

  • The solar system we are looking at must be oriented toward our solar system so that its planets cross the face of the star.

    Many solar systems are not oriented relative to us in a way that their planets will ever transit the face of their star.

  • Because many planets orbit their stars slowly, many planets will require years of observation to find.

    Few planets have been imaged directly. Most are found by indirect methods that take years of observation to detect planets.

Analysis

Article's Equation

The +Plus article stated Equation 1 for the probability for the transit of a random orbit to be visible about a star.

Eq. 1 \displaystyle {{P}_{Transit}}=\frac{{{D}_{Star}}}{{{D}_{Orbit}}}

where

  • PTransit is the probability of planetary transit by random orbit.
  • DStar is the star's diameter.
  • DOrbit is diameter of the planet's orbit.

I had not seen this equation before and I thought it would be useful to derive it here.

Approach

My approach will be simple:

  • Develop a model for the different types of orbital orientations.
  • Determine the set of orbits that will transit the star when seen from Earth
  • Determine the probability of an orbit passing in front of a star when viewed from Earth

Orbit Orientation

Figure 3: Orbital Orientation Specification.

Figure 4: Orbital Orientation Specification.

Figure 4 illustrates how you can specify any orbit of radius R with two angles, θ and φ. These angles simply provide the orientation of a normal vector to the orbital reference plane.
I will first compute the probability of an orbit transiting a star by holding one of the angles, say θ, constant and letting φ vary across its range. It does not matter which θ I hold constant because we have spherical symmetry here. Since we have spherical symmetry and I arbitrarily chose the angle θ to hold constant, all the θs will have the same probability of transit. Since they all have the same probability of transit, the overall probability of transit is the same as for one orbit.

Transit Probability

Derivation

Figure 4 shows how we can compute the probability of a transit for a given set of orbits. I will break problem up into three parts:

    • Consider the set of all orbits of radius R that have a single angle parameter (θ) held constant.

Figure 5(A) shows this case. There are an infinite number of orbits with this orientation and we can see that probability that an orbit transits the star is related to the:

    • Diameter of the star
    • Diameter of the orbit
  • Compute the percentage of these orbits that will transit the star.

    Each orbit in the set can be described by its y-axis intercept. All orbits with a y-axis intercept less than the radius of the star (RStar) will transit the star. This means that the percentage of orbits that will transit the star is simply P_{Transit}= \frac{R_{Star}}{R_{Orbit}}=\frac{D_{Star}}{D_{Orbit}}, which I illustrate in Figure 5(B).

  • Observe each orbit set with a different θ will have the same probability of transit.

    Figure 5(C) illustrates a few different θ values. Since all θ values have the same probability of transit, this means the overall probability of transit is given by P_{Transit}= \frac{D_{Star}}{D_{Orbit}}.

Figure 4: Illustration of Different Orbital Orientations.

Figure 5: Illustration of Different Orbital Orientations.

Example

Figure 5: Probability of Detecting Earth.

Figure 6: Probability of Detecting Earth.

Figure 6 shows how we can compute probability that an alien observer would be able to use the transit method to detect Earth about the Sun. The percentage is ~0.5 %, which is pretty small. This means that to detect many planets, you will need to look at many stars. In fact, Kepler looks at thousands of stars. Here is a quote from the NASA's Kepler page.

Data from the US Naval Observatory digitization of the Palomar Observatory Sky Survey (USNO-A1.0) (Dave Monet, 1996), complete to MV[Visual Magnitude]=18, was used to determine that the actual number of stars with MV<14 of all spectral types and luminosity classes in the 105 deg2 FOV[Field of View] to be 223,000. About 61%, i.e., 136,000, are estimated to be main-sequence stars. Prior to launch high-resolution spectroscopy was performed to identify and eliminate the giant stars in the FOV. During the first year of the mission, the 25% most active of the dwarf stars were eliminated reducing the number to 100,000 useful target stars.

I do find it interesting that they filter out active dwarf stars. I guess that makes sense -- if the star's brightness is varying, how do you know if it was because of a planet transiting or just something that star does (see NASA page on this topic).

Amplitude of the Dip

The optical power we see from a star will dip slightly when a planet passes between the star and the Earth. As an example, we can estimate the optical power dip for the Earth in front of our Sun by realizing that when the Earth passes between the sun and the observer, the total visible optical power will dip by the ratio of the circular area of the Earth to the circular area of the Sun. Figure 7 illustrates how this dip can be computed.

Figure 3: Illustration of Dip Calculation.

Figure 7: Illustration of Dip Calculation.

The calculations in Figure 8 show show that dip for the Earth transiting the Sun is about 1 part in 10000 (as stated in Marianne's article).

Figure 4: Computation of Sun Dimming From Earth's Transit.

Figure 8: Computation of Sun Dimming From Earth's Transit.

Conclusion

I have been reading about the Kepler planet-finder for years, but I have never worked through any numbers on what it is doing. This was a good exercise that provided me some insight as to the difficulty of the problem they are dealing with.

Posted in Astronomy | 1 Comment

Discussion of Math Associated with MH370 Search

Posted on 26-March-2014 by mathscinotes

Here are a couple of web sources on how the course of MH370 was determined.

Some of these transponder-based, distance analyses reminds me a bit of the Oboe navigation system used during WW2.

The Doppler shift work is interesting and I quote here from the Wikipedia. One good aspect of this approach is that it is easily testable using actual aircraft tracking data.

After establishing the "corridors" where the aircraft may have been located at the time of the final handshake, Inmarsat proceeded to further analyze the signals received by its ground stations during the handshakes with Flight 370. This analysis focused on the frequency of the signal expected from the aircraft and the actual frequency of the signal received, known as the burst frequency offset. The burst frequency offset results from the change in frequency of an electromagnetic wave to an observer due to the motion of the source, known as the doppler effect. A baseline of system characteristics for the aircraft, satellite, and ground station could be calculated from several messages sent by Flight 370 while on the ground in Kuala Lumpur and during the early stage of the flight when its location was known.The burst frequency offset would change not only based on the aircraft's airspeed, but also its position along the identified arcs and direction. To test its theory, Inmarsat calculated positions for six Boeing 777 aircraft flying in various directions on the same day and compared those calculations to actual positions, resulting in a good match. Using the results of that analysis, there was a good correlation of the expected and actual burst frequency offsets with the aircraft flying south over the Indian Ocean (along the southern corridor at the time of last transmission) and poor correlation if the aircraft flew north. After factoring in the satellite's small movements in relation to the earth, which were not taken into account in the earlier calculations, the northern corridor was ruled out completely.

Posted in Navigation | Comments Off on Discussion of Math Associated with MH370 Search

6000 Meter Depth Rating on Ocean Search Gear

Posted on 23-March-2014 by mathscinotes

Quote of the Day

The sea is notoriously unforgiving, but it reserves its harshest penalties for those who venture beneath its surface.

— U.S. Navy veteran A.J. Hill

I have been watching the news coverage of the search for the missing Boeing 777 (aka flight MH370) over the southern Indian Ocean. I have heard quite a bit of news coverage referring to oceanographic search gear that will operate to 6000 meters depth (example 1, example 2). Here is one quote (source):

One system, called a "Towed Pinger Locator", is towed behind ships and is used to listen for downed Navy and commercial aircraft at depths of up to 20,000 feet (6000 meters), according to the U.S. Navy's website.

Ocean search gear depth standards are based on the percentage of the ocean bottom that the gear can explore. In the case of a 6000 meter rating, that gear can cover 95% of the Earth's ocean-covered surface. The key to understanding where this coverage percentage comes from is to look at the hypsograph of the Earth (Figure 1).

Figure 1: Hypsograph of the Earth (Source: Wikipedia).

Figure 1: Hypsograph of the Earth (Source: Wikipedia).

A hypsograph plots the proportion of land/sea area that exists at each elevation/depth. You cannot read the 95% directly off of Figure 1. You need to compute a conditional probability. I illustrate this calculation in Equation 1.

Eq. 1 \displaystyle P\left( {\text{Prob}\le \text{6000 m}}/{\text{ProbOcean}}\; \right)=\frac{P\left( \text{Prob}\le \text{6000 m}\cap \text{ProbOcean} \right)}{P\left( \text{ProbOcean} \right)}
\displaystyle =\frac{97.0\%-29.2\%}{100\%-29.2\%}=\frac{67.8\%}{70.8\%}=95.8\%

where

This result is close enough to 95% for my purposes -- certainly when you consider the kind of data I am using.

I have great sympathy for those building gear that works down to 6000 meters -- this is a difficult task. The deep ocean is very unforgiving.

Posted in Underwater | 1 Comment

Frost Depth Deeper Than Normal This Winter

Posted on 16-March-2014 by mathscinotes

Quote of the Day

Maybe not today..., maybe not tomorrow..., but someday... soon; ... and for the rest of your life ...

- Rick Blaine, Casablanca.

I was in my home town of Osseo this weekend to do some computer work for my mother. As I drove through town, I noticed many signs of water system maintenance (see Figure 1). This is a sign that the winter this year was a hard one - the soil froze so deep that the water pipes froze and burst. This can bring about different types of leaks to your home and surrounding areas such as slab leaks, detecting slab leaks are important and there are signs to look out for in your area, for instance, cracks, raised water bills, hot spots on the floor, etc. If this happens, contact a plumber local to you. You can search for a plumber by clicking plumber near me here.

Frozen water pipes happen every year. Households who don't take precautions often get their pipes frozen and the ice expanding causes them to burst and leak. You would then have to get your leak fixed and remember not to do the same the next year. It is also important to remember that this can happen to households in countries from all over the world, so you should always have the relevant precautions and knowledge in place in case it does. For example, if you live in Sydney, then you may want to have a look at the causes of leaky pipes in Sydney homes to put yourselves in the best possible position in having well-working pipes for the winter season. So what's the big deal now? Well, this year it's the actual main water pipes that are freezing; not the small household ones.

Figure 1: Osseo Road Work.

Figure 1: Osseo Road Work.

In my region, state regulations require that we bury water pipes 48 inches deep. Since the frost depth in the southern counties of Minnesota is stated as 42 inches, I have never seen problems like this before. However, this year the frost depth went down to 5 feet, hence the problems with pipes freezing in Osseo. The northern counties of Minnesota do have a 5 foot frost depth specification.

The Osseo water pipes that froze were all in the streets. We had a massive amount of snow this year, which insulated the pipes that run under the lawns and into homes. However, the streets were kept clear of snow and there was no insulating layer of snow on the streets to prevent the frost from going very deep.

Figure 2: Wikipedia Picture of an ONT.

Figure 2: Wikipedia Picture of an ONT.

I frequently have to deal with issues related to frost depth and fiber optic deployments. In clay soils, I see as much as 9 inches of frost heave. I have seen this frost movement tear the fiber out of the Optical Network Terminal (ONT).

Frost heave can damage much more substantial structures than an ONT. I had a conversation with a building inspector who said that he had seen Sonotube foundation piers split in half by frost heave. Figure 3 shows my sketch of the scenario he described. The frozen soil grabs the upper section of the pier and pulls it up, while the footing stays in place. This tears the pier in two.

Figure 3: Illustration of How A Pier is Broken By Frost Heave.

Figure 3: Illustration of How A Pier is Broken By Frost Heave.

I have included an old US government frost depth (or line) graph in Figure 4 so that you can see the frost depth in your area.

Figure 4: Old Weather Bureau Contour Map of Frost Depth.

Figure 4: Old Weather Bureau Contour Map of Frost Depth.

Posted in Construction | 1 Comment

Using a D-Size Battery String for a UPS Energy Source

Posted on 15-March-2014 by mathscinotes

Quote of the Day

If there is one thing more than another for which I admire you, it is your original discovery of the Ten Commandments.

— Thomas B. Reed to the youthful Theodore Roosevelt about his self-righteousness

Introduction

Figure 1: Picture of a Standard D-Cell (Wikipedia).

Figure 1: Picture of a Standard D-Cell (Wikipedia).

I am looking at supporting an Uninterruptible Power Source (UPS) that uses twelve D-size alkaline batteries for its energy source. Telecom service providers like D-size alkaline batteries because their subscribers can purchase them at many local grocery stores. They contain no harmful material and can be tossed in the trash. Service providers currently use a Sealed Lead-Acid (SLA) batteries. SLA batteries are not as widely available and and they contain lead, which must be recycled. However, SLA batteries are convenient because you only need one battery and it is fairly compact.

I thought it would be useful to show you the kind of mathematics that goes into evaluating a battery for use in a UPS application. This is my first application for a D-size batteries. My only contact with them goes back to the late 80s when my children had a CD player that used six D-size batteries. I just remember the batteries as being really big.

Background

Size of a D-Battery

Figure 2: Mechanical Drawing of a D-Size Battery.

Figure 2: Mechanical Drawing of a D-Size Battery.

Figure 2 shows a dimensioned drawing for a D-size battery. D-size batteries are the largest of the cylindrical batteries that are commonly available in grocery stores in the United States. In addition to being large in volume, they are also fairly heavy at 144 grams (5.1 ounces).

The primary reason that some service providers want to use D-size batteries is to ensure that their customers have easy access to replacement batteries. In the US, D-sized batteries are available at many stores (e.g. grocery, gas stations, department, hardware, automotive, etc). However, SLA batteries are not nearly as available − people usually buy them online or at some hardware stores. However, hardware stores are not nearly as common as grocery stores or gas stations.

Figure 3 shows how the D-size battery compares in size with other commonly seen battery sizes.

Figure 3: Comparison of Common Battery Sizes (Wikipedia).

Figure 3: Comparison of Common Battery Sizes (Wikipedia).

The use of D-size batteries for backup power in Fiber-to-the-Home (FTTH) applications has been done before. Marconi used them for an early system. For reasons that I have never fully understood, that system was not commercially successful. However, those folks (like George BuAbbud) got a lot of things right in their design. It was a case of being a little too far ahead of where the market was.

Critical Battery Voltages

There are three battery voltage specifications that will be important for our discussion.

Nominal Cell Voltage (VNom)
This is the labeled battery voltage. For an alkaline battery, it is 1.5 V.
Open Circuit Voltage (VOC)
The open circuit voltage of a battery is the cell voltage of a fully charged battery with no load. For an alkaline battery, this voltage varies from 1.50 V to 1.65 V. The variation is caused by minor differences in concentrations of the materials used in the specific battery.
Cutoff Voltage (VCO)
The cutoff voltage is the cell voltage at which the battery is considered fully discharged.

For the discussion to follow, I will assume that VOC = VNom.

Energy Capacity

One of my tasks is to estimate the run-time of an Optical Network Terminal (ONT) when it is powered by twelve D-size batteries connected in series. The ONT is in backup mode and it will apply a constant load of 9 W to the battery string. My analysis will be based on Figure 4 from this battery specification.

Figure 4: Run-Time of a D-Size Battery As a Function of Load Current and Cut-Off Voltage.

Figure 4: Run-Time of a D-Size Battery As a Function of Load Current and Cut-Off Voltage.

Capacity Retention

Most of the work I do involve Telcordia specifications that mandate a backup time of 10 hours from a new battery and 8 hours from an aged battery. Thus, I plan on replacing the batteries when they have lost 20% of their capacity. The capacity loss from an alkaline battery is often stated to be 3 %/year (Figure 4 - Source).

Figure 5: Alkaline Capacity Degradation with Time.

Figure 5: Alkaline Capacity Degradation with Time.

Analysis

Battery Capacity as a Function of Cut-Off Voltage

Approach

My analysis will be very similar to that performed in this blog post on laser slope efficiency. I am most interested in determining the run time of this twelve-battery pack as a function of the cut-off voltage. The cut-off voltage is important to me because my power supply can only run down to ~10 V, which means my cut-off voltage will be \displaystyle {{V}_{CO}}={}^{10\text{ V}}\!\!\diagup\!\!{}_{12}\;=0.833\text{ V}.

My analysis approach is simple.

  • Digitize the curves in Figure 4 using Dagra.
  • Create a two-dimensional interpolation of the digitized data from Figure 4 using Mathcad.
  • Create a graph of run-time versus cut-off voltage.

Digitized Version of Figure 4

Figure 6 shows my digitization process.

Figure 6: Digitized Version of Figure 4.

Figure 6: Digitized Version of Figure 4.

Figure 7 shows how my digitized data looks when plotted in the format of Figure 4. They look identical.

Figure 7: Plot of My Digitized Data.

Figure 7: Plot of My Digitized Data.

Generate 2-Dimensional Interpolation

Figure 8 shows the Mathcad program that I used to perform the 2-dimensional interpolation.

Figure 8: 2-Dimensional Interpolation.

Figure 8: 2-Dimensional Interpolation.

Plot of Run-Time Versus Cut-Off Voltage

Figure 9 shows my graph of run-time versus cut-off voltage. My chart of battery performance assumes a constant load current. Because the battery voltage is always reducing and my ONT's power supply draws a constant power level (i.e. switched mode power converter), the current into my ONT is always varying. I will approximate this current with an average value.

Figure 9: Run-Time Versus Cut-Off Voltage.

Figure 9: Run-Time Versus Cut-Off Voltage.

This chart shows that I will have no problem meeting my 10 hour run-time requirement with a cut-off voltage set at 0.833 V per battery. I can even raise my power supply's cut-off voltage a bit if that will reduce my product cost.

Replacement Time

The replacement time is determined by the 3 % loss of capacity per year. Since I must replace the battery when the loss is 20 %, this means that I will get 6.66 years of service from this battery string before I must replace it. I typically expect to replace batteries every five years. So the D-size alkaline batteries have plenty of retained capacity.

Conclusion

The D-size alkaline battery can be an effective backup energy source for an ONT. My analysis here was strictly technical, but ultimately it will be economics that determines whether the market moves this way. I still have this analysis to perform.

Posted in Batteries, Electronics | Comments Off on Using a D-Size Battery String for a UPS Energy Source

A Simple Frequency-to-Voltage Converter

Posted on 13-March-2014 by mathscinotes

Quote of the Day

We are what we repeatedly do. Excellence, then, is not an act, but a habit.

- Aristotle

Introduction

Figure 1: Explanation for Why I Need a Voltage-to-Frequency Converter.

Figure 1: Explanation for Why I Need a Voltage-to-Frequency Converter.

I often build small electronics projects for use around my home. I recently put together a small sensor interface that generates an output sine wave with a frequency proportional to the sensor's inductance. At my receiver, I want to convert that frequency to a voltage that is proportional to the sensor's inductance. I want to send out a text message when the value of the sensor's inductance passes through a threshold level – I am detecting the presence of a car (Figure 1).

Since this is a home project, I want to keep it simple and cheap. This blog post documents how I designed this circuit.

Background

The circuit I used is very slight variation on a charge pump circuit I saw years ago in the great little book called "Electronic Designer's Handbook" by Thomas Hemingway (Figure 13.5). This little gem can be viewed online here. The only change I made to this circuit was to shift the voltage levels so that my input signal can be positive.

I will derive an expression for the input frequency-to-output voltage transfer function that is a bit more accurate than what Hemingway derives. However, this is his circuit and his book is still worth reading for those interested in transistor-level design. As you can see, this book was first published in 1966 and I still use it nearly fifty years later.

Analysis

I will focus on my simulation results in this blog post section. Appendix A contains measured data from a working circuit in the field.

Schematic

Figure 2 shows an LTSpice schematic of the circuit I put together.

Figure 1: Simple Frequency-to-Voltage Converter.

Figure 2: Simple Frequency-to-Voltage Converter.

Theory of Operation

Qualitative Description

The circuit operation is fairly straightforward.

  • When the input signal goes low, C1 charges through D1. It is critical that C1 charge quickly relative to the period. This circuit depends on the transfer a fixed quantity of charge at the same rate as the input signal. This means that the charge transfer rate (i.e. current) is directly proportional to the input frequency.
  • When the input signal goes high, C1 discharges through Q1 into C2. C2 will have enough capacitance to ensure that it changes little when C1 discharges into it. This will provide the filtering we need to for a smooth output voltage.
  • The RC filter composed of R1 and C2. R is critical in determining the gain (i.e. slope) of the output voltage versus frequency curve. At equilibrium, the charge lost through R1 for each input cycle exactly equals the charge on C1.

I included my simulation results below that clearly shows each of these operations occurring.

If you seek a more detailed narrative description of how this circuit work, see the discussion comments below. A reader put together an excellent theory of operation on the circuit.

Mathematical Details

Figure 3 shows my derivation of a formula for the frequency-to-voltage conversion process.

Figure 2: Derivation of Frequency-to-Voltage Conversion.

Figure 3: Derivation of Frequency-to-Voltage Conversion.

Simulation Results

Single Simulation Result

Figure 4 shows my simulation results. To see these results clearly, you need to click on the image. The circuit operates exactly as described above.

Figure 3: Simulation Results.

Figure 4: Simulation Results.

Linearity Measurements

Figure 5 shows the linearity that I measured on the simulator. I predicted the slope of my voltage conversion to be 1.08 in Figure 2. My simulation shows 1.095, which means an error of ~1.4%. Not too bad.

Figure 4: Plot of the Linearity of the Frequency-to-Voltage Conversion.

Figure 5: Plot of the Linearity of the Frequency-to-Voltage Conversion.

Conclusion

This circuit is a critical part of a small home sensor system that I have put together. Here is a quick description of my sensor system.

  • I built a metal sensor that is composed of a coil of wire (i.e. inductor) whose inductance decreases when a metal object comes near -- the inductance decreases because of how currents are induced in the metal object. This complex electrical process involved the use of specialist electronic tools such as wire strippers (wirestriper).
  • I built an oscillator which puts out a digital signal with a frequency that is proportional to the inductance of my wire coil.
  • The circuit described here converts the digital signal's frequency to a voltage level.
  • I use a Schmitt trigger to generate an alarm when the voltage from my frequency-to-voltage converter exceeds a value that I have set.

Appendix A: Some Empirical Data from a Circuit I Built

Figure 6 shows 3 data points from a version of this circuit that I am using for a home project. This particular application had C1 = 0.001 µF and R1 = 4 k?. You can see that the response is quite linear, the measured and predicted slopes were within experimental error, and it is working well in my application.

Figure 5: Three Data Points from a Quick Lab Test I Did.

Figure 6: Three Data Points from a Quick Lab Test I Did.

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Posted in Electronics | 41 Comments

Population Occurrence Frequency Math

Posted on 8-March-2014 by mathscinotes

Quote of the Day

Do what you can, with what you have, where you are.

— Theodore Roosevelt

I was reading an article on autism and the article mentioned that autism is much more prevalent in boys than girls. This article states that

The Centers for Disease Control and Prevention has stated that about 1 out of every 88 children are diagnosed with ASD [Autism Spectrum Disorder]. The disorder also affects 1 in every 54 boys, while in girls the rate is 1 in every 252.

I thought I would take a quick look at how the value "1 out of every 88 children" is computed.

In the childhood years, the number of boys slightly exceeds the number of girls. The Wikipedia has table of the gender ratios in each country. This article I am reading is about the United States, and our gender ratio for children under 15 is listed as 104 males to 100 females.

I can use Mathcad to compute the overall population ratio given the girl and boy autism ratios, and the ratio of boys to girls. Figure 1 shows my calculation. These calculations are rough because actual populations ratios are not integral. All of the ratios listed in the article are approximate.

Figure 1: Calculation of Overall Populaton Ratio for Autism.

Figure 1: Calculation of Overall Populaton Ratio for Autism.

So I see where the ratio 1 in 88 children comes from.

I know parents who are dealing autism, Asperberger's syndrome, and ADHD. I have great sympathy for their situation. Some correlation has been found with the prevalence of Asperberger's syndrome with engineers and mathematicians. I have no doubt about this − I have plenty of anecdotal evidence.

Posted in General Science, Health | Comments Off on Population Occurrence Frequency Math

Paper Mechanism Project I Have to Build

Posted on 2-March-2014 by mathscinotes

I think this interrupted motion paper machine is really interesting. This is on my list for a project. If you want to see other projects like this, got Robert Ives' web site.

Posted in Paper Machines | Comments Off on Paper Mechanism Project I Have to Build