Telephone Ring Trip Math

Posted on 27-May-2014 by mathscinotes

Quote of the Day

Talent is God given. Be humble. Fame is man-given. Be grateful. Conceit is self-given. Be careful.

— John Wooden

Introduction

Figure 1: My Phone At Work -- Always Going Off-Hook.

Figure 1: My Phone At Work -- Always Going Off-Hook.

As you can tell by my recent posts, I am doing quite a bit of traditional landline phone math lately. The problems are not very complex, but their resolution is important to delivering quality voice service. Today, I have been working on the circuitry that determines when a phone has gone off-hook.

Phone ringer systems determine when a phone has gone off-hook (see Figure 1) by measuring either the DC resistance or AC impedance magnitude on the phone line and comparing it to a threshold value called the ring trip resistance. The choice of measuring DC resistance or the AC impedance magnitude depends on the length of the line and I will discuss that in detail below. The setting of the ring trip threshold value is definitely a Goldilocks problem -- not too high, not too low -- it must be just right. The value you choose also depends on the country that will be deploying your phone system.

My staff asked me three questions about the setting of the ring trip resistance value:

  • What system characteristics determine the value of the ring trip resistance?
  • Does the value need to be different for other countries?
  • Why is the value different for Fiber-To-The-Home (FTTH) systems than for traditional landline phone system?

I thought these were good questions and worth discussing here.

Background

Country Focus

My discussion today will focus on phone systems in the US and Brazil. US phone standards are controlled by Telcordia, a company that has its roots in the Bell System. The telephone standards in Brazil are controlled by a federal agency, Anatel.

Brazil has similar telephony specifications to those of the US. There are two main differences:

  • The ring frequency in Brazil is 25 Hz and the ring frequency in the US is 20 Hz.

    25 Hz is very common around the world. I have no idea why the US uses 20 Hz. I am always amazed at how the telecommunications standards in each country are similar, but just different enough to require specialized hardware configurations.

  • Each country tests the ringer circuit's ring trip level differently.

    US specifications (GR-909) state that a ringer circuit must be able to ring a line load consisting of a resistance of 10 KΩ and in parallel with a capacitance of 6 μF. This load is equivalent to ~5.3 REN (see Appendix A). Brazil uses an even heavier load (820 Ω in series with 6.8 μF), which is equivalent to ~5.7 REN (see Appendix A).

Definitions

Ring Trip
The process of stopping the AC ring signal and connecting the voice path at the Central Office (CO) when the ringing telephone is answered (Source). This is equivalent to saying that the ringer circuit has detected the phone going off-hook.
Ring Trip Immunity Load
The minimum line impedance at which the ringer circuit is guaranteed to be able to apply ringing voltage without detecting a ring trip (i.e. off-hook) condition.
LSSGR
The acronym stands for Local Switching System General Requirements. This term refers to a number of Telcordia specifications for long reach, copper wire-based telephone systems. These systems use CO-based ringer circuits and can reach subscribers 20,000 feet or more from the CO.
GR-909
The telephone specification used in the United States for phone lengths of length 500 feet or less. This specification is commonly used with FTTH systems, which place their ringer circuits on the side of the home.
Ringer Equivalence Number (REN)
The electrical load placed on a phone line by the single electric bell of a Model 500 phone, which we call a REN. In the US, the number of these old phones allowed on a line is limited to 5, which is called a "5 REN load". In impedance terms, one REN is equivalent to 6930 Ω resistor in series with an 8 µF capacitor.

Telephone Ringing Process

For our discussion here, let's describe the telephone ringing process as follows:

  • The ringer circuit (in the central office or FTTH ONT) applies the ring voltage to the phone line.
  • The phone rings and the ringer circuit monitors the AC impedance (short-line per GR-909) or DC resistance (LSSGR) it measures on line.
  • If the AC impedance magnitude or DC resistance drops below the ring trip threshold, the ringer circuit stops ringing the phone and another circuit sends out the dial tone on the line.

While the total phone load in a home is limited to 5 REN, telephone standards require that ringer circuits must be able to ring a phone line without ring tripping when the phone line is loaded with slightly more than 5 REN. This margin is to ensure that the phone line can be rung even if the phone REN values vary a bit from their specified values. The load level at which the ringer circuit is guaranteed to ring a line without ring tripping is called the ring trip immunity load.

Note that modern phones usually use electronic ringers and put significantly less than one REN load on a phone line. This means you can put more than 5 modern phones on a line.

Ring Trip Detection Approaches

As mentioned earlier, there are two approaches to detecting that a phone has gone off-hook:

  • AC Impedance Magnitude Measurement

    This approach is simpler to implement but it turns out it only works for phone lines of short lengths (specified by GR-909 as 500 feet or less in length). It turns out that AC impedance magnitude detection is easy to implement in single-chip, Subscriber Line Interface Circuits (SLICs). If you are curious, you can check out this example of a SLIC.

  • DC Resistance Measurement

    This approach works for any length line, but is more difficult to implement in a small, cheap integrated circuit.

I only work with short-length phone lines and, therefore, I have only worked with the AC impedance magnitude approach for detecting ring trip.

Analysis

Copper Landline

Figure 2 shows my ring trip resistance calculations for a copper landline.

Figure 2: Long Line Ring Trip Thresholds, AC and DC.

Figure 2: Long Line Ring Trip Thresholds, AC and DC.

Table 1 summarizes my landline, ring trip resistance results.

Table 1: AC Impedance Magnitude and DC Resistance Ranges for Long Lines (LSSGR).

Central Office

AC

DC

On-hook

0 Ω to 1.24 kΩ

10 kΩ to infinity

Off-hook

0 Ω to 1.93 kΩ

0 Ω to 1.93 kΩ

As you can see in Table 1, the AC impedance magnitude ranges for a landline will not allow us to unambiguously detect a phone going off-hook. That is why long lines always detect off-hook using a DC resistance measurement.

Short Lines (e.g. FTTH)

Figure 3 shows my calculation for the on-hook and off-hook AC impedance magnitude and DC resistances for a short (<500 feet) phone line.

Figure 3: Calculations of Ring Trip Threshold for a Short Line.

Figure 3: Calculations of Ring Trip Threshold for a Short Line.

Table 2 summarizes my results.

Fiber/GR-909

AC

DC

On-hook

1.24 kΩ to infinity

10 kΩ to infinity

Off-hook

0 Ω to 500 Ω

0 Ω to 500 Ω

These calculations show that we have a clear difference between the AC impedance magnitude of an on-hook and off-hook phone. Since the AC impedance measurement works and is simpler, single-chip SLICs use this approach. Table 2 also shows that both AC and DC ring trip detection will work on a short line. However, the ring trip resistance value you use may be different for each approach.

Answers to Questions

I will now answer the questions I posed in the introduction:

  • What system characteristics determine the value of the ring trip resistance?

    The key factors are:

    • line length -- this tells us how much DC resistance will be added to the phone ringer load by the wire of the line.
    • AC impedance magnitude at which we must guarantee that we will not ring trip -- our ring trip resistance must be below this value (i.e. < 1.24 KΩ).
    • line leakage value -- our ring trip resistance must be below this value (i.e. < 10 KΩ).

    All this means that we have a fair range over which we can set the ring trip resistance and have it work reliably. You often see ring trip resistance values of ~1 KΩ for a GR-909-compliant system. The ring trip resistance values for a landline system are usually ~4 KΩ.

  • Does the value need to be different for other countries?

    Yes, the ring trip resistance value will vary depending on each country's requirements for ring trip immunity. The ring trip immunity load and the ring frequency determine the AC impedance magnitude.

  • Why is the value different for FTTH systems than for traditional landline phone system?

    FTTH systems are short, which means that there is little DC resistance (< 70 Ωs worst-case) added to the phone's on-hook impedance (~1.4 KΩ worst-case). Since the on-hook impedance (~1.4  KΩ worst-case) is so much different than the off-hook impedance (~500 Ω), it is easy to detect that a phone has gone off-hook. For a landline phone, the DC resistance can be so large (~1.5 KΩ) that it makes the impedance change caused by a phone going off-hook impossible to detect reliably. In this case, it is better to just measure the DC resistance.

Conclusion

I hope this discussion helps explain why phone circuits used in FTTH and landline systems behave differently with respect to ring trip. I sometimes have deployments scenarios where a service subscriber, usually a rancher or farmer, will want to run a phone line from an ONT to a remote barn that is 5000 feet from the ONT. Unfortunately, this will not work because the AC impedance measurement approach used in single-chip SLICs to detect ring trip is just not up to the task.

Appendix A: REN Equivalence Calculations

Figure 4 shows my estimates for the equivalent REN loads of the GR-909 and Anatel loads that must not cause a ring trip. I have included an impedance calculation that includes a frequency error in the ringing signal. This error is possible and must be included in a calculation of the worst-case loading.

Figure 4: My Estimate for the Equivalent REN Load of the GR-909 and Anatel "No Trip" Loads.

Figure 4: My Estimate for the Equivalent REN Load of the GR-909 and Anatel "No Trip" Loads.

Appendix B: GR-909 Ring Trip Immunity Specifications

Figure 5 shows the GR-909 ring trip immunity requirements.

Figure 5: GR-909 Ring Trip Immunity Specification Snippet.

Figure 5: GR-909 Ring Trip Immunity Specification Snippet.

Appendix C: Anatel Ring Trip Immunity Specification Snippet

I grabbed the following quote from an Anatel specification on their ring trip immunity -- it is in Portuguese.

O aviso sonoro para o terminal de voz deve ser acionada quando este for submetido a um sinal de chamada conforme especificado no Art. 11, para linhas de 0 a 3 km, com uma impedancia em paralelo ao terminal de 6,8uF + 820 ohms conforme figura 7B desta consulta.

Using Google Translate, I believe the following statement is a reasonable translation.

The ringer in the voice terminal must be engaged when it is subjected to a signal as specified in Art. 11, for lines 0 to 3 km, with an impedance in parallel to terminal 6.8uF + 820 ohms as figure 7B of this consultation.

Note the 3 km line length limit.

Posted in Electronics, Telephones | 5 Comments

Measuring Telephone Ring Power

Posted on 17-May-2014 by mathscinotes

Quote of the Day

I think if I've learned anything about friendship, it's to hang in, stay connected, fight for them, and let them fight for you. Don't walk away, don't be distracted, don't be too busy or tired, don't take them for granted. Friends are part of the glue that holds life and faith together. Powerful stuff.

— Jon Katz

Introduction

Figure 1: Old School Telephone..

Figure 1: Old Telephone (Wikipedia).

I have been looking at some power data for telephone circuits today and this data provided a useful empirical check on the theoretical calculations that I have done elsewhere. When I showed the data to some other engineers, they had some good questions that I thought would be worth covering here.

In this post, I am just interested in the on-hook ringing power that is required by phone company specifications to ring a phone line loaded with a 5 Ringer Equivalence Number (REN) impedance -- one REN is the impedance of an old phone similar to that shown in Figure 1. So the standards assume a phone line loaded with five phones. In the old days of party lines, this might be five different homes. Today, it might be a single home with five phones on a single line.

As I will show below (Figure 4), there can also a momentary surge above 3 W for less than 100 milliseconds (ms) when the ringing signal is active and the call recipient has just picked up the phone, an event known as "ring trip". I will address the value of that surge in another post.

Background

What happens when you ring a phone?

Here are the basics of ringing an old landline phone:

  • A old-style phone contains an electromechanical bell that rings when an AC voltage is put upon it.
  • The AC voltage is applied by a circuit called a Subscriber Line Interface Circuit (SLIC) that can be in the service provider's central office or in a box somewhere near your home.
  • The phone rings with a cadence of 2 seconds of ringing followed by 4 seconds of silence.

These electromechanical bells present in old-style phones present a substantial load to a phone circuit. Newer phones present a minimal load. Of course, we need to make sure that our phone circuits can ring both old and new phones.

Ring Voltage

The ring voltage can be described as follows:

  • The AC voltage has a frequency of 20 Hz (North America).

    Various ring frequencies are used around the world -- 25 Hz is the most common and 20 Hz is used in North America.

  • The ring wave shape in an ONT is usually trapezoidal.

    Trapezoids can be used when ringing phone lines that run short distances and are not bundled with other phone lines. Trapezoidal signals have lower crest factor and provide more RMS voltage for a given peak level. Unfortunately, trapezoids have more harmonic content than sinusoids and will cause interference issues when operating phone lines over long distances in a bundle of other phone lines.

  • The voltage has an amplitude of ~85 V (typical for the ring voltage generated near the home from an ONT on a short-distance line.)

    Short phone lines are often rung with a peak voltage around 85 V. Long lines are rung at voltages above 100 V to compensate for line losses.

  • The RMS value of the AC voltage is ~65 V.

    This is another commonly seen voltage value. The GR-909 requirement is 40 VRMS, but more is better to ensure old phones ring loud enough.

Figure 2 shows an example of a trapezoidal waveform.

Figure 2: Trapezoidal Waveform.

Figure 2: Trapezoidal Waveform.

Phone Load

In the United States, the load of a phone is specified in units of Ringer Equivalence Number (REN). According to telephone electronics lore, the one REN represents the load of an old Model 500 phone (Figure 3).

Figure 1: Model 500 Phone (Wikipedia).

Figure 1: Model 500 Phone (Wikipedia).

Modern phones typically have REN values of ~0.2. Because you can have multiple phones on a line, a phone circuit is designed to drive as many as five Model 500 phones at the same time. This is referred to as a "5 REN" load.

In many ways, the 5 REN load requirement is archaic. However, there are millions of old phones still in use around the world and the phone service subscribers expect their old phones to work.

Analysis

There were three questions raised by my staff when we reviewed the phone power data:

  • What was the measured average power during ringing and why was it lower than my theoretical prediction of 3 W?
  • Why is the period of the power draw twice that of the ringing signal?
  • What does complex power mean anyway?

I will address these questions below.

Empirical Data

Figure 4 shows the raw oscilloscope image with some comments added. I am measuring the current and voltage from the power adapter. I know my telephone power converter will convert the adapter power with 85% efficiency into the ring voltage and current.

We will use this data to learn a few things about the power needed to ring a phone.

Figure 4: Measured Ring Power for a Telephone.

Figure 4: Measured Ring Power for a Telephone.

Average Ring Power Usage

Figure 5 shows my calculation for the average power dissipated during ringing. My measured value of 2.6 W is fairly close to the 3 W I have computed theoretically.

Figure 5: Updated Ring Power Calculation.

Figure 5: Updated Ring Power Calculation.

The reason the measured value is a bit low is because my theoretical calculations assumed a ring voltage with an RMS value of 65 V, which is the nominal SLIC output voltage under moderate load conditions. Under a heavy load, that number actually drops to 60 V because of internal losses associated with this load level's high current.

Figure 6 shows my updated theoretical calculation of 2.55 W for a lower RMS voltage (only the real part is important for this discussion). This is reasonably close to the 2.6 W I measure in the lab.

Figure 6: Ring Power Calculation with Lower RMS Ring Voltage.

Figure 6: Ring Power Calculation with Lower RMS Ring Voltage.

Power Frequency

While we ring the phone at 20 Hz, the power signal has a 40 Hz frequency. Why?

Let's begin by examining the simpler case of a resistor. The power generated from an AC voltage into a resistor will reach its peak during both the positive and negative peaks of the voltage/current waveforms. So power peaks twice during every period of the AC signal.

I can perform a quick demonstration of this fact by graphing the power into a simple resistive load -- the same result holds for a complex load. Figure 7 shows my demonstration of why the phone power frequency is twice the frequency of the input voltage and current signals.

Figure 7: Power into a Resistor is Twice That of the Voltage/Current Waveforms.

Figure 7: Power into a Resistor is Twice That of the Voltage/Current Waveforms.

The average power for the voltage and current waveforms in Figure 7 is 1 W ({{P}_{Average}}={}^{{{V}_{0}}\cdot {{I}_{0}}}\!\!\diagup\!\!{}_{2}\;={{V}_{RMS}}\cdot {{I}_{RMS}}=1\text{ W}). In Figure 7, the power function is always positive. That will change in the following paragraphs as we add phase shift between the voltage and current.

What does a complex value of power mean?

A complex power value simply means that the power dissipated by the load is not always positive. A negative power being dissipated in a load means the load is providing power to the driving circuit for some portion of the AC cycle. This power comes from the stored energy within the reactive components within the ringer circuit.

Figure 8 shows how phase shifting the current affects the results shown in Figure 7 and the average output power. Notice how the power function goes negative. That represents power being returned to the ring voltage generator. Not all generator circuits like to be driven and you need to be careful about the amount of imaginary power you have in your system.

Figure 8: Example of Complex Power.

Figure 8: Example of Complex Power.

Note how the average power is now reduced to 0.707 W simply because of the introduction of a phase shift.

As shown in the Figure 8, the complex power calculation provides me the same answer as a conventional average power calculation, but it also provides me insight into the imaginary power side of the problem.

Conclusion

When I first looked at the data, I didn't think there was much special there. However, a number of people had questions about the three aspects I discussed in this post. I documented those aspects here because I thought others might find the information useful.

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Yet Another Thermistor Discussion

Posted on 15-May-2014 by mathscinotes

Quote of the Day

Vision is the art of seeing things invisible.

— Jonathan Swift, Thoughts on Various Subjects, 1711.

Introduction

I received a question yesterday on a variation on my simple thermistor linearizer discussion in this blog post. In that post, I wanted to accomplish two things:

  • linearize the output voltage from a thermistor/resistor divider circuit about a specific temperature.
  • The circuit needed an output voltage that increased with increasing temperature.

The question I received yesterday was closely related in that they wanted to:

  • linearize the output voltage from a thermistor/resistor divider circuit about a specific temperature.
  • The circuit needs a decreasing output voltage with increasing temperature.

I thought enough people may want to see this slight variation that it would be worth putting into a separate post. It turns out the optimal linearization value for RS is exactly the same as for my original post. This result makes sense if you think about it a bit because a nearly linear output across one resistor in the circuit implies a linear voltage across the other. You can derive this result rigorously using the same approach I used in the Appendix of my original post.

For those interested in a slightly more sophisticated circuit, see the short note I published in EDN magazine.

Background

All the background material from the original post is true here. Please see the original post for background details.

Analysis

My discussion here will be brief because I am just expanding on my previous post.

Circuit

Figure 1 shows the circuit in question. The thermistor and the resistor positions have been swapped.

Figure 1: Thermistor Circuit with Decreasing Voltage Versus Temperature Characteristic.

Figure 1: Thermistor Circuit with Decreasing Voltage Versus Temperature Characteristic.

Example

Figure 2 is a rework of my example from my previous post.

Figure 2: Worked Example.

Figure 2: Worked Example.

Conclusion

Just a quick post to answer a specific question that I thought others may be interested in.

Appendix A: Derivation of Optimal Value of RS

Figure 3 shows an abbreviated derivation -- I did not include much detail.

Figure 3: Derivation of Optimal Resistor Value.

Figure 3: Derivation of Optimal Resistor Value.

Posted in Electronics | Comments Off on Yet Another Thermistor Discussion

Bisecting an Outside Wall Corner Angle

Posted on 5-May-2014 by mathscinotes

Quote of the Day

I have always imagined that paradise will be a kind of library.

— Jorge Luis Borges

Introduction

Figure 1: Putting Up Crown Molding.

Figure 1: Putting Up Crown Molding.

I have just returned from putting up crown molding at brother's house. It is always fun working with my brothers. In many ways, we are little different today than we were 40 years ago. During this task, I encountered a wall corner that was not square. Let's talk about how you can measure and bisect this angle. You need to bisect the angle when you want to cut the molding to fit around the corner (Figure 1).

Background

Why Do You Need to Bisect the Angle?

You need to cut the molding at an angle that bisects the wall angle because the two molding cross-sections must be identical to match up when you join them together.

Why Isn't the Wall Angle 90°?

Some walls are designed to meet at angles other than 90°. Most walls are designed to meet at 90° but are not exactly at 90° because of how drywall is installed. Figure 2 shows a detailed drawing of a drywall corner constructed using corner bead.

Figure 2: Drywall Corner Detail.

Figure 2: Drywall Corner Detail.

The walls do not meet at 90° because

  • The wall framing is not perfectly square (i.e. 90°)
  • The process of skimming the wall intersection with drywall compound results in a bulge at the joint.

Analysis

Approach

I have used my approach for so long that I have forgotten where I learned it. Here is a link that does a nice job describing the method I use. The approach is straightforward:

  • Place two flat pieces of wood of the same width along both walls as shown in Figure 3 (use thin wood so it does not teeter as much).
  • Draw lines along both sides of the upper piece of wood onto the bottom piece of wood.
  • Take the wood pieces down from the ceiling
  • Draw a diagonal connecting the two lines from above as shown in Figure 4. The diagonal is the angle bisector.
  • Cut the molding on the angle bisector.
Figure 3: Angle Measurement Approach.

Figure 3: Angle Measurement Approach.

Figure 4: Line Drawing Illustrating Measurement Procedure.

Figure 4: Line Drawing Illustrating Measurement Procedure.

Geometric Proof of the Method

Figure 5 shows how we can prove that we have bisected the angle using geometrical methods. There is nothing special in the proof, just remember your basic properties of isosceles triangles.

Figure 5: Geometric Proof.

Figure 5: Geometric Proof.

Conclusion

I wish all my walls met at 90°, but they don't. This method has allowed me to put up crown molding with just a little bit of trouble -- there always is some finessing that has to be done.

Posted in Construction, Geometry | Comments Off on Bisecting an Outside Wall Corner Angle

ITU 100 GHz Frequency Grid Math

Posted on 24-April-2014 by mathscinotes

Quote of the Day

A hero is someone who has given his or her life to something bigger than oneself.

— Joseph Campbell

Introduction

Figure 1: Typical Optical Fiber Cable.

Figure 1: Typical Optical Fiber Cable.

A physicist in my group and I were having a discussion about how the wavelengths (i.e. colors) for lasers are specified by an international standard and I thought this discussion would provide a nice example of a differential approximation. The widespread deployment of fiber optic cable (see Figure 1, Wikipedia) is a game changer for networking and may be our most important new infrastructure -- remember that high-speed wireless depends on cell towers interconnected with fiber optic cables.

Analysis

Fiber optic cable is an incredible media for transmitting light. We are greatly increasing the amount of information that we can transfer over fiber by adding additional wavelengths of light onto the fiber. Our discussion this morning centered on the Dense Wavelength Division Multiplexing (DWDM) wavelengths specified in ITU-T G.694.1. This standard specifies laser wavelengths in terms of a frequency grid. Adjacent wavelengths in the grid are separated in frequency by 100 GHz. Each wavelength is referred to as a channel.

I normally think of light in terms of wavelength and not frequency. The wavelength and frequency of light are related by Equation 1.

Eq. 1 \displaystyle \lambda =\frac{c}{\nu }

where

  • λ is wavelength of the light channel.
  • ν is is the frequency of the light channel.
  • c is speed of light in a vacuum.

I have seen engineers use Equation 2 to approximate the wavelength difference between two wavelengths separated by a defined frequency difference.

Eq. 2 \displaystyle d\lambda =d\left( \frac{c}{\nu } \right)=-\frac{c}{{{\nu }^{2}}}\cdot d\nu =-\frac{{{\lambda }^{2}}}{c}\cdot d\nu

where

  • dν is is the frequency difference between adjacent wavelengths.

As you can see from Equation 2, holding the frequency difference between adjacent λ's constant means that dλ will vary for each wavelength. I have seen engineers incorrectly assume that the dλ is constant for all wavelengths -- not true -- only the frequency difference is fixed.

Using Mathcad, we can easily compute the wavelengths associated with each frequency (Figure 2).

Figure 2: Computing the ITU Frequency Grid Using Mathcad.

Figure 2: Computing the ITU Frequency Grid Using Mathcad.

I have included that actual grid specification in Appendix A. It is identical to what I generated in Mathcad.

Conclusion

Just a quick post to illustrate a quick use of differentials.

Appendix A: ITU 100 GHz Frequency Grid.

Table 1: ITU 100 GHz Frequency Grid.

Channel Frequency (GHz) Wavelength (nm)
1 190,100 1577.03
2 190,200 1576.2
3 190,300 1575.37
4 190,400 1574.54
5 190,500 1573.71
6 190,600 1572.89
7 190,700 1572.06
8 190,800 1571.24
9 190,900 1570.42
10 191,000 1569.59
11 191,100 1568.77
12 191,200 1567.95
13 191,300 1567.13
14 191,400 1566.31
15 191,500 1565.5
16 191,600 1564.68
17 191,700 1563.86
18 191,800 1563.05
19 191,900 1562.23
20 192,000 1561.42
21 192,100 1560.61
22 192,200 1559.79
23 192,300 1558.98
24 192,400 1558.17
25 192,500 1557.36
26 192,600 1556.55
27 192,700 1555.75
28 192,800 1554.94
29 192,900 1554.13
30 193,000 1553.33
31 193,100 1552.52
32 193,200 1551.72
33 193,300 1550.92
34 193,400 1550.12
35 193,500 1549.32
36 193,600 1548.51
37 193,700 1547.72
38 193,800 1546.92
39 193,900 1546.12
40 194,000 1545.32
41 194,100 1544.53
42 194,200 1543.73
43 194,300 1542.94
44 194,400 1542.14
45 194,500 1541.35
46 194,600 1540.56
47 194,700 1539.77
48 194,800 1538.98
49 194,900 1538.19
50 195,000 1537.4
51 195,100 1536.61
52 195,200 1535.82
53 195,300 1535.04
54 195,400 1534.25
55 195,500 1533.47
56 195,600 1532.68
57 195,700 1531.9
58 195,800 1531.12
59 195,900 1530.33
60 196,000 1529.55
61 196,100 1528.77
62 196,200 1527.99
63 196,300 1527.22
64 196,400 1526.44
65 196,500 1525.66
66 196,600 1524.89
67 196,700 1524.11
68 196,800 1523.34
69 196,900 1522.56
70 197,000 1521.79
71 197,100 1521.02
72 197,200 1520.25
73 197,300 1519.48
Posted in Fiber Optics | Comments Off on ITU 100 GHz Frequency Grid Math

555 Timer Math

Posted on 10-April-2014 by mathscinotes

Quote of the Day

With hard work, difficult material can be grasped. Step by step, incrementally, the novice can become the master.

— Joshua Waitzkin. World Tai Chi champion and subject of the book 'Searching for Bobby Fischer'. He is working with Khan Academy to promote learning through hard work.

Introduction

Figure 1: Classic 555 in an 8-Pin DIP.

Figure 1: Classic 555 in an 8-Pin DIP.

I have never used a 555 timer for a home project and now I have several applications for this handy device up at my cabin in Northern Minnesota. I thought I would cover some of the basics in this post. Sure this material is covered in other places, but I need to work all the details myself to really understand a part. I like to document my learning here so that others can share in it -- and help me find any errors in my work.

I first encountered the 555 when I was at university back in the 1970s. Eng Hoyme, the father of a very good friend, had designed a very impressive electronic organ using a bank of 555 timers. Figure 1 shows the NE555 in a DIP option − this is how I first saw it. It was always interesting for me to go into his basement and see all the stuff he was building. His oldest son had an old Altair 8800 system down their as well. This was my first contact with microcontrollers. It was exposure to their passion for electronics that helped light a fire in me for electronics − a passion that is even bigger today.

Background

Scope

There are an enormous range of applications for this part. My post here will cover the use of the 555 timer in a basic astable multivibrator configuration. My cabin application requires a low-frequency oscillator (27 kHz) that does not need to be very stable. The 555 is ideally suited for this type of application.

My objective for this blog post is to:

  • Derive an expression for the frequency of a 555 timer when used as an astable oscillator.
  • Derive an expression for the duty cycle of the 555 timer output when running as an astable oscillator.
  • Create a Mathcad routine for finding component values that simultaneously meet my requirements for duty cycle and frequency.

Some Definitions

Let's define a few of the terms that I will be using.

Duty Cycle (DC)
Duty cycle is the percentage of one period in which a signal is active. In my case, active means that VOut = ~VCC.
Time Constant (τ)
A time constant is the parameter characterizing the response to a step input of a first-order, linear time-invariant (LTI) system. Physically, the constant represents the time it takes the system's step response to reach 1-1/e \approx 63.2\% of its final (asymptotic) value. For an RC system like this, \tau = R \cdot C.
Threshold Voltage (VThresh)
As I use the term here, threshold voltage is the voltage at which a comparator will switch.

Block Diagram

Figure 1 shows a block diagram of the 555 that I pulled from the Wikipedia and modified slightly.

Figure 2: Block Diagram of the 555 Timer

Figure 2: Block Diagram of the 555 Timer

We can glean numerous bits of information from Figure 2. I will highlight a few:

  • The comparators are biased using 3, 5KΩ resistors.

    I have heard it said that these three 5 KΩ resistors are the source of the name "555".

  • The CONT pin gives you direct control of the upper comparator's threshold.

    You cannot independently control the lower comparator's threshold, but you can use a resistor divider on the THRESH pin to give you similar control.

  • The inputs of the two comparators are configured differently.

    The upper comparator is configured to reset the internal flip-flop when the THRESH pin exceeds the high-level threshold. The lower comparator is configured to set the internal flip-flop when the TRIG pin is less than the high-level threshold.

It is amazing how flexible this simple architecture has proven. I have seen hundreds of different applications for this simple part.

Basic Astable Operation

Figure 3 shows a 555 timer hooked-up in the standard astable configuration (Source).

Figure 3: Standard 555 Astable Configuration.

Figure 3: Standard 555 Astable Configuration.

The frequency of the OUT signal is set by the charge and discharge time of the RC circuit setup by R1, R2, and C. Charging of C is done by R1 and R2. Discharging of C is done by the DIS (DISCH in Figure 2) pin using R2 alone. Discharge time is only affected by R2 because the DIS pin will go to 0 V during the discharge cycle. This isolates R1 from the rest of the circuit.

The circuit of Figure 3 will always have a duty cycle greater than 50%. You can understand this by observing that OUT = "1" (high voltage) when R1 and R2 are charging C from the low threshold level to the high threshold level. OUT = "0" (low voltage) when C is being discharged from the high threshold to the low threshold through R2. The charge time constant is always longer than the discharge time constant, so the "1" time is always longer than the "0" time. This is equivalent to saying the duty cycle is always greater than 50%. I include a more formal derivation here.

There is an alternative circuit that will give you a 50% duty cycle output. Observe that this circuit uses the OUT pin to discharge C rather than the DIS pin. Thus, the same resistor value is used for both charging and discharging. The disadvantage of this circuit is that the output current drive must be shared between the timing circuit and the external load. This is not a good thing for some applications because the external load may change the circuit's oscillation frequency significantly.

Basic RC Circuit Modeling

The voltage across a capacitor in an RC circuit can be modeled using Equation 1.

Eq. 1 \displaystyle {{v}_{C}}\left( t \right)={{V}_{Final}}+\left( {{V}_{Initial}}-{{V}_{Final}} \right)\cdot {{e}^{-\frac{t}{R \cdot C}}}

where

  • vC(t) is the voltage across the capacitor versus time.
  • VInitial is the voltage across the capacitor at t = 0.
  • VFinal is the voltage across the capacitor at t = ∞.
  • R is the resistance of the resistor that is charging the capacitor.
  • C is the capacitance of the capacitor being charged.

For a derivation, see this book.

Analysis

Oscillation Frequency

You can determine the oscillation frequency by applying Equation 1 to determine the rise and fall times of the voltage on capacitor C in Figure 3. Figure 4 shows my calculations.

Figure 4: Derivation of Frequency Formula.

Figure 4: Derivation of Frequency Formula.

Duty Cycle

My particular application requires that the 555 output have a specific duty cycle. We can compute the duty cycle from the circuit of Figure 3 using the formula shown in Figure 5.

Figure 5: Derivation of Duty Cycle Equation.

Figure 5: Derivation of Duty Cycle Equation.

Resistor Values in Terms of Frequency, Duty Cycle, and Capacitor Value.

I can use the formulas developed in Figures 4 and 5 to compute values for R1 and R2 in terms of the oscillator frequency, duty cycle, and the capacitance value. The derivation is shown in Figure 6.

For those requiring more details, I provide another example and a spreadsheet implementation of these formulas in the comments section of this post.

Figure 6: Derivation of Resistor Value Expressions.

Figure 6: Derivation of Resistor Value Expressions.

Check Against Simulation

To verify my equations, I simulated my example circuit using LTSpice. Figure 7 shows the circuit I simulated.

Figure 7: Schematic of Astable Circuit in LTSpice.

Figure 7: Schematic of Astable Circuit in LTSpice.

Figure 8 shows the output waveform from the circuit in Figure 7.

Figure 8: Output Waveform from 555 in Astable Configuration.

Figure 8: Output Waveform from 555 in Astable Configuration.

Figure 9 shows the error analysis of my measured values from my theoretical predictions.

Figure 9: Error Analysis.

Figure 9: Error Analysis.

I consider these errors pretty typical considering the modeling assumptions.

Conclusion

I would say that the majority of 555 circuits use some form of the astable configuration. It was a useful exercise for me to develop equations that will allow me to compute the required resistor values for a given oscillation frequency, capacitance value, and duty cycle.

Appendix A

Figure 10 shows a 555 astable circuit with 50% duty cycle (source -- a first-rate web page on the 555).

Figure 10: 50% Duty Cycle 555 Circuit.

Figure 10: 50% Duty Cycle 555 Circuit.

As I mentioned above, I cannot see using this circuit often because I prefer to keep my load circuit separate from my timing circuit. This makes the circuit's frequency more predictable.

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Posted in Electronics | 8 Comments

A Lone Engineer in a Marketing Meeting

Posted on 9-April-2014 by mathscinotes

Quote of the Day

Listening to NPR is like listening to your mother telling you to clean your room.

— Julia Sweeney

My staff is passing this video link around that illustrates what it is like being the lone engineer in marketing meeting. There is quite a bit of truth here.

I cannot tell you how often I have been told that I do not understand the "big" picture. In one case, I was informed that my "reality-based thinking" was limiting me. The comment to the engineer about his limitations because he is specialist in narrow area really hit home.

Posted in Humor, Management | 2 Comments

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