IEEE Consultants' Network of Long Island

Orthogonality - John Dunn, Consultant, Ambertec, P.E., P.C.

At the following URL, we find several definitions of the word "orthogonal", but the one I want to call attention to is the fifth definition, "statistically independent".  

Please see:  http://www.merriam-webster.com/dictionary/orthogonal

Back in my college days, this particular definition  would have helped me resolve a personal mental- block about mathematical analyses that involve more than three dimensions. How could there possibly be more than three dimensions? 

Some Number Stuff - John Dunn, Consultant, Ambertec, P.E., P.C.

Divisibility:

If any number ends in zero or five, then that number itself is evenly divisible by five with no remainder which is pretty much obvious.

What is somewhat less obvious however is that If the sum of the digits of any number is evenly divisible by three with no remainder, then the number itself is evenly divisible by three with no remainder.

For example, in the number 9638184, the sum of the digits is 39 which is evenly divisible by three with no remainder which means that the number 9638184 is itself is evenly divisible by three with no remainder.

Thus, 9638184 / 3 = 3212728 exactly with no remainder. 

Dynamic Microphones and SPL - John Dunn, Consultant, Ambertec, P.E., P.C.

Let us first consider some definitions regarding sound pressure level (SPL):

1 Microbar = 1 Dyne / cm². 

10 Microbars = 10 Dyne / cm² = 1 Pascal.

200 Microbars = 200 Dyne / cm² = 20 Pascals.

20 Micropascals = 0.0002 Dyne / cm² = Zero dB SPL

Zero dB SPL is often taken as the threshold of human hearing.

Thus: SPL of 28 Dynes / cm² --> 20 * log (28 / 0.0002) = 102.92 dB SPL  ~  103 dB SPL

which is the sound pressure level at which dynamic microphones are often specified for their output signal levels into their specified load impedances.

Padding Around with Attenuators - John Dunn, Consultant, Ambertec, P.E., P.C.

These two configurations of attenuator pad are undoubtedly familiar, but finding their resistance values for some particular dB of attenuation can be a little inconvenient. 

Rail Voltage Response to Current Pulses - John Dunn, Consultant, Ambertec, P.E., P.C.

Rail voltage responses to current pulses are analyzed using recursive differential equations and two different current pulse models on the worst case assumption that passive decoupling effects of bypass capacitors are all that we have working in our favor..

It is shown in this particular case that rail voltage excursions in response to current pulses of 3 nSec rise and fall times do not exceed 40 mV per Ampere of peak pulse current.

The recursive differential equations are as follows:

Bidirectional Logarithmic Compression - John Dunn, Consultant, Ambertec, P.E., P.C.

You can be confronted with some dependent variable that varies versus some independent variable where that variation just happens to be of the greatest interest when the independent variable is somewhere in the middle of a numerical range.

A graphic image of the relationship between the two variables may benefit if the x-axis of the presentation is given a bidirectional logarithmic compression along the lines (no pun intended) of the following examples.

Here, we look at x-axis plots of a variable from zero to 100 with the center value equaling 50. Vertical rises are shown upward at increments of 10:   

 
 

The larger you make the coefficient K2, the more expanded the x-axis becomes in the center and the more compressed at each end. Coefficient K1 then needs to be tailored for the appropriate width.

Architectural Columns - John Dunn, Consultant Ambertec, P.E., P.C.

I once read about this architect who had a project underway where a large open area was to be provided with vertical columns. He decided that the usual circular column cross section was not esthetically pleasing and that he wanted to use something else, but what that something else would be was elusive. Therefore, he sought the assistance of a consultant, in this case, a professor at a nearby university.

The professor came up with something right away.

The equation of a circle is x² + y² = Radius² for which the exponent of each variable is "2". The professor said to instead use the equation x^1.5 + y^1.5 = Radius^1.5 for which the cross section looks like this:   

 

Nines Complement Arithmetic - John Dunn, Consultant, Ambertec, P.E., P.C.

To subtract one number from another, you can do the usual arithmetic algorithm, but you can also use an addition process instead that can be handy for checking subtraction results when working by hand. This avoids doing the identical process with which you got your first result lest you inadvertently repeat making an error you might have made the first time.

Consider the following examples:

8718 - 4732 = 3986  This is completely standard arithmetic.

Now, replace the "4732" with its nines complement which in this case means "5267". Each digit of this new number is that which if added to the corresponding digit of the old number, totals to nine. Ergo, 4+5=9, 7+2=9, 3+6=9 and 2+7=9.

Capacitor Voltage Decay at Constant Power - John Dunn, Consultant, Ambertec, P.E., P.C.

The discharge curve of a capacitor's voltage versus time for a resistive load is an exponential decay curve of Tau = RC and for a constant current shunt, is a linear downward slope of dV/dt = i/C . However, the discharge curve for a constant power load is quite different from either of those.

An example of constant power decay, taken from real life, is shown here. The question at hand was for a 470 µF capacitor shunted by a constant power load of 714 watts, how long would it take the capacitor voltage to decay to 200 volts from various starting voltages.

In the sketch below, each value of time, "Tnnn", is the time value at 200 volts from having started at "nnn" volts. Thus, starting at 400 volts, the decay of capacitor voltage to 200 volts takes T400 = 39.5 mSec.

Using Recursive Differential Equations - John Dunn, Consultant, Ambertec, P.E., P.C.

This circuit is a simple RC network being charged from a DC source for which a descriptive, first order differential equation is written and is then put into a software loop to simulate the actual process under consideration: