What I Learned this Month

Thank Alexander Graham Bell (1847-1922) for the introduction of the dB. Though best known for the invention of the telephone, he was even better know in his day for his research in studying the deaf and quantifying the sensation of hearing.

He noticed that our sensation of hearing is not linear with the power in a wave, but scales with the log of the power of the wave. Bell created a perceived loudness scale based on the log of the acoustic power in a sound wave.

For historical reasons, when we take the log of the ratio of powers, we refer to the units as Bels. Even though this is in reference to Alexander Bell, we drop one of the ”l”s and just call it the Bel scale. A Bel is ALWAYS the log of the ratio of two powers.

The threshold of hearing (TOH) is about 10^-12 W/m² of sound intensity. A normal conversation is about 10^-6 W/m². On the Bel scale, the conversation would be rated as log(10^-6/10^-12) = 6 Bels.

On the Bel scale, a value of 1 means a power is 10¹ or 10x higher than the reference base. A value of 3 Bels means the power is 10³ or 1000x higher than the reference base . Historically, we have come to use the Bel scale to measure all powers, such as light intensity and radio power, relative to some baseline value.

In general, the Bel scale is not very large. For example, the entire range of hearing goes from the TOH to about 10⁴ W/m², where the ear drum is perforated, or a total of 16 Bels. For such a large range of sensations, 16 is just not a very large number.

This is why it has become conventional to use not Bels but deciBels as the scale. A deci means 1/10th, so there are 10 deciBels in 1 Bel. We abbreviate this as dB. This means that we can write any power in terms of is dB value, relative to a reference level as: Power_in_dB = 10 x log(P/P₀). The factor of 10 is to convert the value in Bels into deciBels.

This sets the range of hearing to start at 0 dB at the TOH to 160 dB as damaging. More examples of sound levels can be found here.

A Saturn V Apollo rocket launch generated sound levels of 135 dB 1 mile away from the launch pad.

But, if we want to measure a quantity that is NOT a power, such as the amplitude of a wave, like a voltage or current, we can’t use the dB scale. It is only used for the log of the ratio of powers.

The work around is that if we want to measure the ratio of two voltages in dB, we actually measure the ratio of the powers in the waves. The power in a wave is the square of the amplitude. So, when we measure the ratio of two voltages in dB, we are really measuring the ratio of the powers in the voltages.

For example, an S-parameter is really the ratio of two amplitudes, not powers. When we calculate the magnitude of the S-parameter in dB, we use the factor of 20:

Sometimes it is confusing to figure out is the quantity we are looking at a power or an amplitude. For example, we often will see impedance measured in dBOhms. Is impedance an amplitude or a power? It turns out it is an amplitude. If you want to convert an impedance from dBOhms into Ohms, you need to use the factor of 20: Z = 10^(Z_dB/20) .

As long as you keep in mind that dB is ALWAYS the log of the ratio of powers, it’s pretty clear how to interpret the results.

For handy reference, keep in mind that a –3 dB drop in power means the power decreased by 50%. The amplitude decreased to only 70% of its initial value.

When you see a drop in amplitude of 50%, this is a drop in power of –6 db.

A signal that drops off by a factor of 10 in amplitude with each decade, linearly decreasing with frequency, for example, has a drop of a factor of 100 in power per decade, or –20 dB per decade.

Capacitance is the efficiency of storing a charge difference between two conductors at the expense of the voltage between them. A high capacitance means there is a lot of charge stored per volt between the conductors. This is described by the definition, C = Q/V. If the definition requires two conductors, what does it mean to talk about the capacitance of one conductor?

If you put some excess charge on a conductor, sitting in space, it will rise to a new voltage, compared to any other place measured as the reference. It will have a capacitance, the ratio of the charge added to its change in voltage.

It’s difficult to calculate the voltage generated on an isolated conductor except for simple geometries, like a sphere. The case of two concentric spheres is a classical problem in all freshmen EM classes. A great explanation is found here.

We start with two concentric spheres. In this geometry, one sphere is inside the other. The capacitance of this configuration can be easily calculated and is

Suppose we make the outer sphere bigger and bigger, effectively moving it farther and farther away. The value of b gets larger and larger and 1/b goes to zero. If the outer sphere is more than 10 x the radius of the inner sphere, the value of 1/b is less than 10% the value of of 1/a and has only a small impact. In this extreme case, when the outer sphere is very far away- like to the floor, or the walls of the room, the capacitance of the inner sphere, to any metal far away, is related to the size of the smaller sphere, a.

The capacitance of a small, isolated sphere is just 4 x pi x epsilon zero x a. Using values of 0.225 pF/inch for epsilon zero, the capacitance of a sphere, with radius a is: C in pF = 1.4 x D, with D the diameter of the sphere in inches.

This is a startling result. It says that a piece of metal floating in space has a capacitance to any far away surface and it is roughly related to its diameter. If it is other than a sphere, it is a little hard to calculate, so to use the luxury of a simple estimate, we have to assume it is a spherical shape.

For example, if we have a cable sticking out from a computer that is 3 inches long, maybe its equivalent to a sphere with a diameter of about 1 inch. By nature of its size, it will have a capacitance to any other surface, far away, of about 2 pF.

This is a very good estimate of the capacitance between the shield of a cable to the floor. It’s the fringe electric fields of external cables to the floor that is the return path for common currents on the cable. It’s the impedance of this path that usually determines how much common current flows on the cable shield.

If there is 2 pF of fringe field capacitance, at 100 MHz, the impedance is roughly 1 kOhm. If the ground bounce noise on a plane, that drives the common currents, is just 100 mV, the common currents will be on the order of 0.1 v/1k Ohm = 100 uA. It only takes 3 uA of common current to fail an FCC class B certification test so we see how easy it is for ground bounce to cause EMC problems.

In the last few months, I’ve had occasion to participate in a number of question and answer events.

On Jan 26, 2012, I hosted the first online Chat with Printed Circuit University. In the hour and a half live session, I answered about 20 questions, all of which were recorded and posted.

What most participants did not know was that at almost the moment the chat room opened, I had a water pipe burst in my basement and I was rushing between my computer to quickly read a question and furiously type an answer, to finding the main water shut off valve in the house, mopping up the floor and finding the number for an emergency plumber. This was one occasion I was glad the chat was only by written word and not a video feed!

At the most recent DesignCon 2012, I moderated a panel discussion, “Ask the experts,… anything goes.” We had seven industry experts field questions from an audience of about 75 sitting around us on the show floor in the ChipHead Theater. In our brief 45 minutes together, we covered questions about return currents to the future of copper vs optical interconnects.

In the months of January and February, I taught a total of 14 different classes around the world, in Switzerland, Germany, the States, Malaysia, Singapore and China, and visited with over 800 students. After each class, I was always inundated with students hungry for answers to their specific questions.

I usually receive a dozen emails each week from former students or folks who read my book and still have a question. While I try to answer each note, I am finding that many of the questions are similar.

I’ve decide to introduce a new feature in my blog which will be Frequency Asked Questions (FAQs). I’ve created a new static page on my blog which will be the running list of questions folks send to me and I will select specific questions every so often to answer in my blog. When I post an answer, I will link it to the question on the static page.

I invite you to submit your questions to me at DoctorIsIn@beTheSignal.com and I will add it to my list and try to make a point of answering it for you. I am going to keep the question source anonymous so you should feel free to ask anything you want. The better I can understand your question, the better I will be able to answer it.

I’m giving some thought to creating some sort of office hours which might be a Google+ video chat room or some other live video chat event. If you have some suggestions, drop me a note: DoctorIsIn@beTheSignal.com

Hope to see you in one of my upcoming classes.