Interesting Historical Note About Cheese

Posted on 27-February-2013 by mathscinotes

Interesting Historical Note About Cheese

I like blog posts that go into the history of common things. This post does a nice job discussing cheese and its long history.

Posted in General Science | Tagged | Comments Off on Interesting Historical Note About Cheese

Battery Freezing Math

Posted on 26-February-2013 by mathscinotes

Quote of the Day

The best argument against democracy is a five-minute conversation with the average voter.

— Winston Churchill

Introduction

7.2 A-hr Sealed Lead Acid Battery.

Figure 1: 7.2 A-hr Sealed Lead Acid Battery
(Source). This battery is a workhorse products
for many markets, including telecom and home
security.

I live in a cold climate -- so cold that under certain circumstances we can freeze our lead-acid batteries (Figure 1). A customer who lives in my region called recently and was wondering if I thought any of his batteries would have frozen over the winter. A number of his Internet service subscribers have vacation homes that are unoccupied over the winter. All of these vacation home owners turn off their AC power for the winter. Since all of our Optical Network Terminals (ONT) are connected to Uninterruptible Power Sources (UPS), they will begin operating off of their battery when the AC power goes away. If the home owner does not disconnect the battery, the ONT will run discharge the battery. This is important because a discharged battery will freeze -- a charged battery will not freeze. A battery that has been frozen is very likely a dead battery and you will need a replacement battery.

Note that car and rv batteries rarely freeze because a car and rv battery is rarely fully discharged. These batteries will freeze if they are allowed to get cold enough when discharged. Here is a typical situation:

  • You have a car or rv with remote start and standard electronics. This suite of hardware puts a 100 mA parasitic load on the battery: 70 mA for the remote starter and 30 mA for the car computer.
  • The car or rv is parked in a cold garage for five days.
  • The battery discharges and the battery freezes.

This EXACT situation just happened to my son. He now must ensure he drives the car every few days to keep the battery charged.

Let's examine why my ONT customers need to worry about discharged batteries over the winter.

Background

Lead-acid batteries contain a solution of sulfuric acid (H2SO4) and water -- the solution is referred to as the battery's electrolyte. Adding a solute (in this case, H2SO4) to a solvent (in this case, H2O) will lower the freezing point of a solution. A fully charged battery has more H2SO4 than a discharged battery. The additional H2SO4 depresses the freezing point of the batteries electrolyte to around -70 °C. This is a temperature we do not see in Minnesota. However, a discharged battery's freezing point rises to ~-10 °C. Unfortunately, the temperature in Minnesota frequently drops below -10 °C.

Analysis

This blog post is going to focus on presenting empirical data. However, I do want to spend a bit of time discussing the various ways of expressing the concentration of battery acid.

Battery Acid Concentration

There are three common ways of expressing the concentration of battery acid.

  • Specific Gravity (symbolized by SG)

    Specific gravity compares the density of the battery electrolyte to that of water. Specific gravity is readily measured using a hydrometer, which almost every auto mechanic has used – even I have a hydrometer. I see inexpensive hydrometers for sale every time I go into an automotive parts store.

  • Mass Fraction (symbolized by w)

    The mass fraction expresses the acid concentration in terms of the mass of acid divided by the total mass of the acid -water mixture. This measure of concentration is a convenient measure because only a scale is required to mix up a properly measured solution. Unfortunately, there is no inexpensive instrument for measuring the mass fraction directly after it has been mixed. Once mixed, we use SG.

  • Molality (symbolized by m)

    Molality is the number of moles of solute per kg of solvent. The advantage of using molality as a measure of battery acid concentration is that you can create a properly mixed solution using only a scale. The problem is that there is no readily available instrument for measuring molality after it has been mixed. Again, we usually use SG.

The mass fraction and molality are related by the equation m=\frac{w}{{MM\cdot \left( {1-w} \right)}}, where MM is the molar mass of the solute (98 grams/mol for H2SO4). Specific gravity can be used to relate mass fraction to molarity (symbolized by M) by the equation M=\frac{w \cdot SG\left( w \right)}{MM}, where SG is assumed to be equal to the solution's density (close enough for most applications). I do not see molarity used by battery people, but chemists use it all the time. I cover these formulas in more depth in this post.

Cell Voltages: Open Circuit, Charging, and Discharging

Figure 2 (Source) shows the terminal voltages experienced by "12 V", 6-cell, lead-acid battery when at different levels of charge and discharge currents.

Figure 1: Voltages During Charging and Discharging for a 12 V Battery.

Figure 2: Voltages During Charging and Discharging for a 12 V Battery.

Because of the variation in battery terminal voltage with charge or discharge current, I will plot (Figure 3) the open-circuit terminal voltage. This will make the graph simpler.

Freezing Point and Open-Circuit Cell Voltage Versus Acid Concentration

Figure 3 (Source) shows both the electrolyte's freezing point and the open-circuit cell voltage as a function of mass fraction, specific gravity, and molality. We usually define a fully charged battery has having an electrolyte with a molality of 6.0 moles/kg. Similarly, a discharged battery is usually defined as a battery with an electrolyte having a molality of 2.0 moles/kg.

editedfinal Cell Voltage Versus Specific Gravity3
Figure 3(a): Battery Freezing Point Versus Specific Gravity. Figure 3(b): Cell Voltage Versus Specific Gravity.

Conclusion

I have dealt with this issue for a number of years. I thought it was worth documenting why the UPS batteries can freeze. The solution is simple -- disconnect the charged battery from the UPS. A lot of other things will freeze (e.g. the water/anti-freeze mixture often used to winterize vacation home plumbing) before that charged battery will freeze.

To cross-check my information, I also consulted additional sources, which I document here.

Appendix A: Primary Battery Source Material

In addition to the data presented in Figure 2, I also used the following table from Vinal's "Storage Batteries". (Google Books Reference)

Figure M: Battery Data from Vinal, 1951 (Source).Figure M: Battery Data from Vinal, 1951 (Source).

Figure 4: Battery Data from Vinal, 1951 (Source).

Appendix B: Additional Battery Source Material

I consulted numerous sources to corroborate the data presented here. Figure 5 shows data from Sandia, which has gathered together a surprising amount of lead-acid battery data.

Figure 1: Specific Gravity, Terminal Voltage, and State of Charge Data (Sandia Labs).

Figure 5: Specific Gravity, Terminal Voltage, and State of Charge Data (Sandia Labs). A 12 V battery has 6 cells. To derive the cell voltage, I divided the terminal voltage by 6.

To simplify comparison with Figure 3, I will reformat the Sandia data so that the state of charge and cell voltage are functions of specific gravity (Figure 6).

Figure M: Sandia Data Reformated As Function of Specific Gravity.

Figure 6: Sandia Data Reformatted As a Function of Specific Gravity.

The data in Figure 5 is similar to that shown in Figure 2. Since this is empirical data, I expect some differences between sources. If you are curious about how I generated Figure 5, see the Mathcad and PDF file here.

Figure 6 shows another set of data. This data is also consistent with the other data sets I found.

Figure M: Cell Voltage Versus Specific Gravity and Depth of Discharge (Source).

Figure 6: Cell Voltage Versus Specific Gravity and Depth of Discharge (Source).

Posted in Batteries, Electronics | Tagged , , | 17 Comments

Russian Meteor Characteristics

Posted on 20-February-2013 by mathscinotes

Quote of the Day

I write a lot of programs and I can’t claim to be typical but I can claim that I get a lot of them working for a large variety of things and I would find it harder if I had to spend all my time learning how to use somebody else’s routines. It’s much easier for me to learn a few basic concepts and then reuse code by text-editing the code that previously worked.

— Donald Knuth

Introduction

Figure 1: Chelyabinsk, Russia shown in red.

Figure 1: Location of Chelyabinsk, Russia, marked in red (Source).

A meteor exploded over Chelyabinsk, Russia (Figure 1), at 3:20 UTC on February 15, 2013. I have been reading accounts of the size, speed, and energy of the meteor. This post presents some simple calculations that verify the consistency of the expert's estimates on the meteor characteristics. Also, I thought it would also be interesting to look at the amount of overpressure required to cause the sort of damage that was seen after the meteor and to include some explanatory material as to how scientists determine the characteristics of a meteor.

Background

Meteor Trajectory

This meteor entered the Earth's atmosphere without warning. There are a number of reasons why it was not detected earlier. One reason is that it came from the direction of the Sun, which generates so much glare that it is difficult to see meteoroids coming in from that direction. Figure 2 illustrates its trajectory (Source). There is a program, called Sentinel, that will be attempting to place a telescope near the orbit of Venus that will allow us to see some meteoroids coming from the direction of the Sun by 2017.

Figure 1: Trajectory of the Russian Meteor.

Figure 2: Trajectory of the Chelyabinsk Meteor.

Coincidentally, asteroid 2012 DA14 passed by 16 hours before. Figure 3 shows another view of the orbits (Source).

Figure 2: Orbits of Asteroid 2012 D14 and the Russion Meteor.

Figure 3: Orbits of Asteroid 2012 D14 and the Chelyabinsk Meteor.

Meteor Characteristics

There are numerous articles quoting different size and energy estimates. If you want to see how these characteristics are determined, see this document on how the meteor characteristics were determined for the Tagish Lake fireball in Canada. It is a bit technical, but does provide a complete description of the meteor analysis process.

For the Russian meteor characteristics, let's work with the press release from JPL. It states that the meteor has the following characteristics:

  • velocity: vMeteor = 44,000 miles/hour
  • mass: mMeteor = 7,000 to 10,000 tons
  • energy: EKE = 500 kilotons

Another press report stated the iron content of the meteor is 10 %, which is consistent with a ordinary chondrite meteor type H. Given this information, we can proceed to cross-check these figures.

Meteor Density

Assuming that the meteor is roughly spherical and we have an estimate of the mass, we can calculate the diameter if we know the density of the meteor. Let's assume the meteor is an ordinary chondrite type H. Figure 4 shows a table that lists density values for the different types of meteors (Source). An ordinary H chondrite type H has an average density of 3.4 gm/cm3.

Figure 1: Density Distribution of Meteorites.

Figure 4: Density Distribution of Meteorites.

Analysis

Meteor Diameter

Using the meteor mass and density numbers from above, we can compute an estimate for the meteor diameter using the approach shown in Figure 5.

Figure 3: Calculations for the Meteor Diameter.

Figure 5: Calculations for the Meteor Diameter.

My estimate of 16.8 meters for the meteor diameter is in the 15 meter to 17 meter diameter range stated in the JPL press release.

Meteor Energy

The Cold War left us with an infrastructure for measuring meteor characteristics as part of the Comprehensive Test Ban Treaty Organization. These folks run a network of sound sensing stations that can detect nuclear blasts. They can also track meteors, which gives us velocity information, and they can estimate the energy of the meteor by the loudness of the explosion. My calculations in Figure 6 show that the reported energy value of 500 kilotons and the meteor's velocity and mass estimates are consistent.

Figure 4: Calculations for the Kinetic Energy of the Meteor.

Figure 6: Calculations for the Kinetic Energy of the Meteor.

You do occasionally hear news about these world-wide sensor networks for monitoring nuclear test ban compliance in contexts other than meteors (e.g. the Vela Incident).

Overpressure

Exploding meteors can have a tremendous amount of energy. This energy can be comparable to that of a nuclear weapon. Like a nuclear weapon, an exploding meteor can generate blast effects due to overpressure. The Wikipedia defines overpressure as

Overpressure (or blast overpressure) is the pressure caused by a shock wave over and above normal atmospheric pressure. The shock wave may be caused by sonic boom or by explosion, and the resulting overpressure receives particular attention when measuring the effects of nuclear weapons or thermobaric bombs.

If you want to get some insight into the power of an air burst with energy comparable to a nuclear weapon, look at the following video of a UK nuclear test for a 1.8 megaton air burst (altitude = 8000 feet) at a range of 20 miles (Figure 7).

Figure 7: First British Nuclear Test.

Since the meteor blew out many windows and did some minor structural damage, we can estimate the amount of overpressure on the ground using data from Table 1 (Source), which gives an idea of the level of damage that can be generate by the overpressure associated with an air burst.

Table 1: Impacts of Peak Overpressure on Buildings and Humans. 
Peak Overpressure (psi) Effect on Structures Degree of Damage
0.15-0.22 Typical window glass breakage Moderate
0.5-1.1 Minor damage to some buildings Moderate
1.1-1.8 Panels of sheet metal buckled Moderate (broken)
1.8-2.9 Failure of concrete block walls Severe
Over 5.0 Collapse of wood framed buildings Severe
4-7 Serious damage to steel framed buildings Severe
6-9 Severe damage to reinforced concrete structures Moderate
10-12 Probable total destruction of most buildings Severe (collapse)

Conclusion

This was just a quick look at some science in the news. For more information on meteor air bursts, see this post.

Posted in Astronomy, Electronics | 1 Comment

Battery Outgassing Math

Posted on 12-February-2013 by mathscinotes

Quote of the Day

Madam, if you were my wife, I'd drink it!

- Winston Churchill's response to Lady Astor, who had said to him "If you were my husband, I'd poison your tea." Churchill and Lady Astor were famous for their feuding.

Introduction

Figure 1: Aftermath of a Hydrogen Gas Explosion in a Battery Vault.

Figure 1: Aftermath of a Hydrogen Gas Explosion in a Battery Vault.

I recently have received a number of questions about the outgassing of hydrogen gas that can occur from lead-acid batteries when they are being overcharged. I thought it would be useful to review what is happening when a battery is outgassing. When being charged, batteries can release enough hydrogen gas to create an explosive hazard. Consider this report and Figure 1 as an example as to what can happen. With lead-acid batteries, hydrogen gas can be generated at any time, but charging is when the greatest challenges are faced. You may also check this out to learn more about hydrogen gas detection.

Background

Basic Chemistry

When you think about it, a lead-acid battery being charged looks a lot like a water electrolysis setup. You can split water into its hydrogen and oxygen components by applying an electrical potential greater than 1.48 V to water. Figure 2 shows a typical electrolysis setup (source).

Figure 2: Example of an Electrolysis Setup.

Figure 2: Example of an Electrolysis Setup.

Figure 2 also describes a battery being charged -- there are two terminals that are separated by water (plus some H2SO4) and the terminals have a voltage applied to them. Figure 3 shows a cross-section diagram of a lead-acid battery. Figures 2 and 3 are very similar.

Figure 3: Battery Cross Section Diagram.

Figure 3: Battery Cross Section Diagram.

Oxygen is generated at the positive terminal and hydrogen is generated at the negative terminal. Since we normally charge lead-acid batteries at a potential higher than 2.2 V, we always get some electrolysis along with the charging. That is why some batteries need to have their water replenished frequently. Those that do not need to have their water replenished incorporate some mechanism for gas recombination (see AGM and Gel Battery).

Equation 1 shows the basic electrolysis reactions.

Eq. 1 \displaystyle 2{{H}_{2}}O\to {{O}_{2}}\uparrow +4H+4{{e}^{-}} reaction at the positive electrode
\displaystyle 2{{H}_{2}}O+2e\bar{\ }\to ~{{H}_{2}}\uparrow \text{ }+\text{ }2\left( OH \right)\bar{\ } reaction at the negative electrode

How to Abuse a Lead Acid Battery

There actually are standards for how to abuse a battery (i.e. force it to runaway and outgas). The one I am most familiar with is the Induced Destructive Overcharge Test in IEC standard 952-1:1988. Here is a useful reference that discusses how to perform the test.

Example of Battery Abuse

It does not take much searching on the web to find examples of a lead acid battery that has undergone thermal runaway. Figure 4 shows an example of battery damage as the result of thermal runaway. I have seen some cases where the battery case got so hot that it melted.

Figure 4: Example of an Battery That Has Exploded (Wikipedia).

Figure 4: Example of an Battery That Has Exploded (Wikipedia).

Remember that overcharging, outgassing, and thermal runaway are all related. Thermal runaway is not a good thing to have to happen.

Analysis

Battery Example

The discussion that follows is about the Panasonic LC-127R2P 12V/7.2Ah, which is a very commonly used sealed lead-acid battery. I show this battery in Figure 5. This battery is composed of 6 cells connected in series. Each cell nominally generates 2.0 V, but the exact voltage varies with the batteries state of charge and can be anywhere from 1.75 V to 2.25 V when discharging. As this battery is filled with Sulfuric acid, this is a dangerous chemical for anyone to deal with, especially if it is released into the atmosphere. With this being said, it may be worth checking out a site like Storemasta, in the hopes of finding out more about chemical like Sulfuric acid.

Figure 5: Sealed Lead Acid Battery Example.

Figure 5: Sealed Lead Acid Battery Example. This is an AGM (Absorbed or Advanced Glass Mat) battery. The mat is sandwiched between the plates and is saturated with sulfuric acid. The mat retains the acid in the event of a breakage, making the battery spill-proof. You can also get AGM Solar Batteries.

It really is a workhorse product -- I have never had any issue with it. Like all lead-acid batteries, you just need to treat it nicely.

Rate of Outgassing

I thought it would be a good exercise to show how much hydrogen and oxygen a battery can generate. Note that most references focus on the generation of hydrogen because that is the gas that is flammable. IEEE 484 is the standard governing the installation practices for lead-acid batteries and it states that

5.4 Ventilation

... Maximum hydrogen evolution rate is 0.127 mL/s per charging ampere per cell at 25 °C and standard pressure (760 mmHg). The worst-case condition exists when forcing maximum current into a fully charged battery. ...

The hydrogen evolution rate is important to know because the Lower Explosive Limit (LEL) concentration for H2 gas is 4%. Knowing the rate of hydrogen gas generation and the volume of the battery enclosure allows us to determine the amount of ventilation required for an explosion-proof installation. We can show where this result comes from applying a bit of basic chemistry. Figure 6 shows my derivation. In this derivation, I show how to compute the H2, O2, and total gas (H2 and O2) generation rates per cell.

Figure 6: Derivation of IEEE 950 Value for H2 Gas Generation Per Amp of Current.

Figure 6: Derivation of IEEE 484 Value for H2 Gas Generation Per Amp of Current. A Total Gas Volume Value is Also Generated For Comparison with Panasonic Data in Figure 7.

Using the derivation of Figure 6, Equation 2 states the complete equation for the total gas generation rate (O2 and H2) from a battery composed of multiple cells.

Eq. 2 {{R}_{Gas}}={{R}_{O2}}+{{R}_{H2}}=11.4\frac{\text{mL}}{\text{minute}\cdot \text{cell}\cdot \text{Ampere}}\cdot {{N}_{Cells}}\cdot {{I}_{Charge}}

where

  • RGas is the total gas generation rate.
  • NCell is the number of cells in the battery.
  • ICharge is the charging current.

While Equation 2 is stated for the total gas generation rate, the same basic equation form holds for O2 and H2 individually, just change the constant term from 11.4 mL/(min ·A ·cell) to:

  • 7.6 mL/(min ·A ·cell) for H2 generation only
  • 3.8 mL/(min ·A ·cell) for O2 generation only

Empirical Data

Figure 7 shows the outgassing graph for the Panasonic LC-127R2P 12V/7.2Ah Sealed Lead Acid Battery. This graph shows the total amount of gas generated, which means both O2 and H2. You can see from the handwriting on the graph that this is not an official graph -- I got this from an electrochemist with Panasonic who had measured the outgassing characteristic. The little black arrows on the graph indicate which axis the individual curve corresponds to. The x-axis is stated in units of "CA". CA describes the charging current as a fraction of the A-hr capacity of the battery (the A-hr capacity is treated as a current value). Using the 7.2 A-hr battery for an example, 0.1 CA = 0.1 ·7.2 A = 0.72 A charging current.

Figure 7: Hydrogen Gas Emissions from a 7.2 A-hr Sealed Lead Acid Battery.

Figure 7: Hydrogen Gas Emissions from a 7.2 A-hr Sealed Lead Acid Battery.

Figure 7 deserves one other comment -- I have no idea why the battery vendor's electrochemist used a logarithmic x-axis. Gas generation is linear with charging current. The use of a logarithmic axis makes it look like something nonlinear is going on. In the following discussion, I will take the data and re-plot it on a linear graph (Figure 8).

Theoretical Versus Empirical

Figure 8 shows a comparison of the measured gas generation rate versus the predicted gas generation rate. Notice that the theoretical rate is higher than the measured rate. This is because 100% of the charging current into the battery does not go into electrolysis -- some must go into charging. To recover the water lost because of electrolysis, sealed lead acid batteries contain one of two types of gas recombining technology which will ensure that low levels of generated gas will be recombined into water. However, the theoretical rate is useful because it established an upper bound for the amount of H2 that can be generated.

Figure 8: Comparison of Theoretical Versus Empirical Gas Generation Rates.


Figure 8: Comparison of Theoretical Versus Empirical Gas Generation Rates.

Conclusion

Lead-acid battery outgassing is one of the least understood characteristics of this chemistry. Hopefully this note helps explain how outgassing works and how to estimate the amount of hydrogen generated.

References

Good Paper on Battery Outgassing
Ventilation Example

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Posted in Batteries, Electronics | 20 Comments

AC Backup Power Using A Simulated Sine Wave

Posted on 3-February-2013 by mathscinotes

Quote of the Day

Whenever you want to marry someone, go have lunch with his ex-wife.

- Shelly Winter

Introduction

Figure 1: Typical AC Signal At Wall Outlet.

Figure 1: Typical AC Signal At Wall Outlet (Source). The flat-top distortion is common – especially when many loads are switching power supplies.

A customer today requested an AC-output Uninterruptible Power Source (UPS) for our indoor Optical Network Termination (ONT) products. Product Management asked us to consider this unit from CyberPower. This unit, like most other AC-output UPS hardware, does not generate an actual sine wave output like the local power utility (Figure 1 – sine waves are what comes out of an AC outlet). Instead, it generates an output that is often referred to as a simulated sine wave (also known as a pseudo-sine wave, quasi-sine wave, modified sine wave, or Pulse-Width Modulated [PWM]). I thought it would be useful to look at the mathematics behind this type of output.

This backup power unit converts the DC voltage from the battery into a simulated sine wave that will imitate the behavior of a real sine wave for most loads. A pure sine wave inverter can be used in a lot of different ways so are worth looking into. A device that performs this operation is called an inverter. You might wonder why the backup power folks chose to put out a simulated sine wave rather than a true sine wave. Like most things in the engineering, the issue is one of cost. Another option one could look into is the Switch Mode Power Supply (SMPS). This is more cost-effective compared to a pure sine-wave inverter. You could find clear instructions that will assist you in troubleshooting switch mode power supplies at a faster rate on many websites.

Generating a pure sine wave is relatively expensive when compared to generating an approximation. For an example of a circuit for generating a bipolar (i.e. both positive and negative voltage levels) PWM output, see the Wikipedia. In my experience, the simulated sine wave works almost as well as a real sine wave for most applications. I have had a couple of instances that involve radio receivers where the simulated sine wave did not work well. In each case, the audio output was contaminated with noise that would not have existed with a real sine wave. I discuss the inverter that caused these issues in Appendix A. Other than those two cases, I have been able to successfully power everything from lights to computers using a simulated sine wave.

Background

Calling the output of this UPS a "simulated sine wave" is a bit of a stretch. The power waveform from an outlet is a sine wave (purple waveform in Figure 2). Observe that the "simulated sine wave" (reddish-brown waveform in Figure 2) does not look like a sine wave at all -- it is a form of pulse-width modulation. As is shown in Figure 2, the power supply output has only three output levels: +VA, 0 V, and -VA, where VA is amplitude of pulse-width modulation. Notice also that three time values are shown in Figure 2:

  • TH: the time during each period that the voltage output stays at VA
  • TL: the time during each period that the voltage output stays at -VA
  • T: the period of the signal

For the simulated sine wave to have zero DC level like that of the utilities' AC waveform, TH= TL. We will define the duty cycle to be \displaystyle D\triangleq {}^{\left( {{T}_{H}}+{{T}_{L}} \right)}\!\!\diagup\!\!{}_{T}\;.

Figure 1: Sine Wave Versus Simulated sine Wave.

Figure 2: Sine Wave Versus Simulated Sine Wave.

These backup units work by creating a pulse-width modulated output that has:

  • an amplitude comparable to the amplitude of a standard AC sine wave VA
  • an RMS value (VRMS) that is comparable to a standard AC sine wave
  • the same period as the standard AC sine wave (60 Hz in North America)

We will use a bit of mathematics to derive the relationship between VA, D, and VRMS.

Analysis

Requirements

Most AC-input powered equipment will work if they are presented with a simulated sine wave voltage waveform that has the same:

  • VA as the utility power sine wave
  • VRMS as the utility power sine wave
  • T as the utility power sine wave

We will derive expressions that will allow us to determine the required VA and VRMS values in terms of the nominal power line voltage and the duty cycle.

Derivation

Voltage Amplitude

Determining the required voltage amplitude is the easy. In the US, the nominal AC voltage at the power meter is set by ANSI C84.1-2011 at 120 V (RMS). The amplitude of a sinusoid is related to its RMS value by Equation 1.

Eq. 1 \displaystyle {{V}_{A}}=\sqrt{2}\cdot {{V}_{RMS}}=\sqrt{2}\cdot 120\text{ V=170 V}

So we should see our inverter putting out a voltage with an amplitude around 170 V.

RMS Voltage

Equation 2 shows how we can derive the relationship between VA, D, and VRMS.

Eq. 2 \displaystyle {{V}_{RMS}}=\sqrt{\frac{1}{T}\cdot \int\limits_{0}^{T}{V{{(t)}^{2}}\cdot dt}} Definition of RMS voltage
\displaystyle {{V}_{RMS}}=\sqrt{\frac{1}{\frac{T}{2}}\cdot \int\limits_{0}^{T/2}{V{{(t)}^{2}}\cdot dt}} Positive and negative pulses are symmetrical, so I will just integrate over half a period
\displaystyle {{V}_{RMS}}=\sqrt{\frac{1}{\frac{T}{2}}\cdot \int\limits_{T/4-D\cdot T/4}^{T/4+D\cdot T/4}{V_{A}^{2}\cdot dt}} The function is a constant during the on-portion of the duty cycle, zero otherwise
\displaystyle {{V}_{RMS}}=\sqrt{\frac{2}{T}\cdot V_{A}^{2}\cdot D\cdot \frac{T}{2}}=\sqrt{D}\cdot {{V}_{A}} Evaluate the integral
\displaystyle D={{\left( \frac{{{V}_{RMS}}}{{{V}_{A}}} \right)}^{2}}={{\left( \frac{1}{\sqrt{2}} \right)}^{2}}=\frac{1}{2} Compute D for a North American power system

This means that an inverter that generates a simulated sine wave with 170 V amplitude and a duty cycle of 50% will provide the same peak voltage amplitude and RMS voltage as is delivered from a North American wall outlet. For many applications, that is good enough.

Real UPS Output Example

Lab Note: My oscilloscope did not want to display the whole waveform on the screen, so I ended up running the output through a resistor divider that reduced its level by a factor of 33.8. This division factor allowed me to display everything with a minimum of fuss. So all voltages you see going forward will need to be multiplied by 33.8 to get their actual value.

Let's take a detailed look at the output voltage from a CyberPower CP550HG, which is a commonly used backup unit for PCs. Figure 3 shows an oscilloscope screenshot of the unloaded output from the CP550HG. I was lazy today and did not do any analysis of my own on the signal -- I just let the oscilloscope compute things like RMS voltage and period.

Figure 2: Oscilloscope Screenshot of Unloaded CP550HG Output.

Figure 3: Oscilloscope Screenshot of Unloaded CP550HG Output.

The key characteristics from this plot are:

  • Duty cycle of 49% (50% target)
  • Frequency of 60.770 Hz (60 Hz target)
  • Peak amplitude of 181 V = 33.8·10.7V/2 (170V target)
  • RMS output voltage of 125.2 V = 33.8·3.704 V (120 V target)

The numbers measured are all within the range I would have expected for this unit. Since these are numbers were measured with zero load, I would expect them to be a bit high and they are.

See Appendix A for the data I measured from an inverter I use to run my computer when I am in my car. This is a very cheap unit and its numbers are a bit different because their design goals were different.

Conclusion

I was able to derive the amplitude and duty cycle required for a simulated sine wave inverter to behave similarly to a North American 120 V power waveform and I verified this derivation with an actual test case from the lab. The values I measured were all as I would have expected.

Appendix A

Here is the Black and Decker inverter I use in my car (Figure 4). Overall, it has worked well for me over the past few years. However, it did corrupt the audio output from a couple of radios that I was trying out.

Figure 3: Black and Decker 400 W Inverter.

Figure 4: Black and Decker 400 W Inverter.

Here is a plot of its output (Figure 5).

Figure 4: Output From Black and Decker Inverter.

Figure 5: Output From Black and Decker Inverter.

It has a lower peak voltage than the CyberPower. They compensate for this by making the duty cycle longer, which increases the RMS voltage. This is the unit that has caused me trouble powering some radio gear.

I would summarize its characteristics as follows.

  • Duty cycle of 80%
  • Frequency of 59.6 Hz
  • Peak amplitude of 136.9 V
  • RMS voltage of 117.6 V

Since this unit does not have the same peak voltage as a utility sine wave, I would expect it may have some issues driving some types of loads and I have seen issues (mentioned above). They probably made the decision to reduce the output voltage because generating higher voltages costs more than generating lower voltages -- and cost drives everything in consumer products.

Posted in Electronics | 5 Comments

Great discussion of Saturn's Polar Hexagon

Posted on 2-February-2013 by mathscinotes

Quote of the Day

Limited funds are a blessing, not a curse. Nothing encourages creative thinking in quite the same way.

— L. Yau

Figure 1: Saturn's Hexagon as seen from Cassini Satellite.

Figure 1: Saturn's Hexagon as seen from Cassini Satellite. (Source)

This particular post discusses a very interesting laboratory experiment that illustrates how the hexagonal region at Saturn's pole may form.

Great discussion of Saturn's Polar Hexagon

Posted in Astronomy | Tagged , | Comments Off on Great discussion of Saturn's Polar Hexagon

Age Distribution of US Health Care Expenditures

Posted on 25-January-2013 by mathscinotes

Quote of the Day

The beginnings and endings of all human undertakings are untidy.

- John Galsworthy

Figure 1: Health Care Expenditure Percentage By Age Group.

Figure 1: Health Care Expenditure Percentage By Age Group.

I have been wondering why US health care costs are so high. I just read this report that claims that the US spends a far greater percentage of its health care dollars on the elderly than other countries. Increasing medical costs are common for the elderly because as we get older, we can develop various age-related illnesses. It's totally natural. When we get to a certain age, these illnesses can take over our lives. We might need to look for help from a Lynchburg home care agency or even try and find a more permanent place of residence that can provide assistance. Care homes are a popular choice for many of the elderly as they receive around-the-clock care and are able to enjoy the social elements that come with living with others in a similar situation to them. Many of us also choose to take out life insurance policies to cover the cost of any medical bills that may need to be paid after we pass. Life insurance policies and payments can vary, and you can click here to discover quotes if you need life insurance for yourself. But I always wondered if healthcare for the elderly is essential, why is it so expensive? I decided to plot the data from the report myself to make viewing the data a bit easier -- see Figure 1. The expenditures are normalized to the costs for people in the 50 to 64 year old age range (arbitrarily given a value of 1 unit). Here is what the chart tells me:

  • For citizens less than 65 years old, the distribution of US health care spending is not markedly different from other countries.
  • The US, Canada, and UK seem to be doing something distinctly different for their elderly than the other nations in the study, with a comparison between the Kew Gardens Aged Care and other care clinics able to be drawn.
  • The US expense distribution is markedly different than all other countries for people 65 year old and older.

For those who think that the problem is the percentage of old people in the US, remember that Japan has the oldest population in the world, yet their health care expense distribution is much flatter than that of the US. In actual fact, the US has favorable demographics compared to other nations in the study, whether that be for people who would need to access pharmacy services (such as these: https://southwestcare.org/services/support-services/pharmacy-services/) or other resources! Whilst there is no doubt we have some great services in this country when it comes to medical needs, it is interesting to see it through comparisons to other nations.

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Posted in Financial | 4 Comments

Great Article on Recreating Shackleton's Whiskey

Posted on 19-January-2013 by mathscinotes

Quote of the Day

Play is the only way the highest intelligence of humankind can unfold.

— Joseph Chilton Pearce

Figure 1: Shackleton's Whiskey Bottle. (Source)

Figure 1: One of Shackleton's Whiskey Bottle. (Source)

While I am not a drinker, I do admire the impressive technology that Scottish distillers used to duplicate the lost recipe for Shackleton's whiskey.  They were even able to identify where the peat used in its formulation came from.

Great Article on Recreating Shackleton's Whiskey

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Fond Memories of Fast-Moving Milk Bottles and Toy Boats

Posted on 19-January-2013 by mathscinotes

Quote of the Day

Write freely and as rapidly as possible and throw the whole thing on paper. Never correct or rewrite until the whole thing is down. Rewrite in process is usually found to be an excuse for not going on.

— John Steinbeck

Figure 1: Machine Filling Milk Bottles. (Source)

Figure 1: Machine Filling Milk Bottles. (Source)

Last night, I was watching a show on the Science Channel called How It's Made. This episode reminded me of an old movie I was shown in elementary school that started my interest in science and math. The message here is that you never know what will stimulate the interest of a child.

Last night on "How It's Made" they were showing machines filling bottles with a beverage. This show stimulated my memories from first-grade when I was shown a 16mm movie about how milk came to our door every morning. I still can remember almost every detail of this movie. The story started with a cow eating grass and ended with a bottle of milk on a kitchen table. What really caught my eye was the filling of the milk bottles by machines. It seemed like magic the way the bottles were put into a fast-moving row, filled while moving in a circle, and then capped. I became driven to learn how those machines worked and to build my own machines. I began my quest by asking Santa for an erector set. Things just grew from there.

Back in those days, every house in the town of Osseo had a small box that a milkman would fill with glass bottles of milk. The picture below is very similar to the milk box that my family had.

As the oldest of five children, one of my jobs was to move the milk from the box to the refrigerator. Seeing the magic that brought milk to our door made the mundane seem really interesting.

Youtube has many videos showing bottles being filled. I cannot find the original movie that I saw, but the video embedded below is similar in content to the bottle-filling sequence from the movie I saw originally.

While I am on the subject of children's videos, I noticed that Paddle to the Sea is also on Youtube. This is a very cute video that also made an impression on me. I actually tried to carve my own little wooden boat after seeing this movie back in second-grade.

Posted in General Science, Osseo, Personal | 1 Comment

The Joys of Electrostatic Discharge

Posted on 7-January-2013 by mathscinotes

Quote of the Day

The only people with whom you should try to get even are those who have helped you.

— John E. Southard

I had to laugh at this Youtube video this morning. This poor guy shows you exactly how NOT to handle hardware during ElectroStatic Discharge (ESD) tests.

Posted in Electronics | 3 Comments