### Wilkinson Power Divider

And now for a little applied physics!  The Wilkinson power divider shown to the left schematically (picture 1), is a cool little circuit that evenly divides a microwave signal at a specified design frequency and supplies it to two or more circuits downstream from itself.  In addition to evenly dividing the applied power from the source, the Wilkinson divider also protects each of the circuits it supplies from any reflected signals from the other supplied circuits.  The circuit design was first published in 1960 by Ernest Wilkinson[1].

A simplified diagram of the circuit is shown below.  It divides the input from the source down two conductors that are each cut to be exactly as long as one quarter of the wavelength of the microwave signal supplied by the source.  The power is automatically divided due to one of the properties that physicists love: symmetry.  Faced with no difference in the two paths it's presented with, the input microwave signal splits and half of its power travels down each path.  That's kind of cool, but if that's all that Wilkinson intended, he could have used any conductor length.

The quarter wavelength conductors provide an ingenious way of protecting each supplied circuit from the
other ones.  Under normal operation, there's no potential difference at the end of the divider where the resistor connects the two conductors, so no current flows through the resistor.  If one of the supplied circuits begins to force reflected power back into the divider, the resistor comes into play.  It immediately shunts part of the reflected power over to the other output.  That sounds bad, but wait for it.

The portion of the reflected signal, (step 1 in picture 3) that isn't shunted will travel back down the quarter wavelength conductor changing it's phase by 90 degrees, (a quarter wavelength), in the process, (step 2 in picture 3).  At the top of the divider it will travel the opposite direction down the other conductor and in so doing pick up another 90 degree phase shift.  At this point, it's picked up 180 degree phase shift and perfectly interferes with itself in the second conductor cancelling itself out! (step 3 in picture 3)

References:

1.  Original circuit design
Wilkinson E.J. (1960). An N-Way Hybrid Power Divider, IEEE Transactions on Microwave Theory and Techniques, 8 (1) 116-118. DOI:

http://ieeexplore.ieee.org/xpl/articleDetails.jsp?tp=&arnumber=1124668&contentType=Journals+%26+Magazines&sortType%3Dasc_p_Sequence%26filter%3DAND%28p_IS_Number%3A24846%29

jowdjbrown said…
A simplified diagram of the circuit is shown below. 300w ac dc switching adapter​

### Cool Math Tricks: Deriving the Divergence, (Del or Nabla) into New (Cylindrical) Coordinate Systems

The following is a pretty lengthy procedure, but converting the divergence, (nabla, del) operator between coordinate systems comes up pretty often. While there are tables for converting between common coordinate systems, there seem to be fewer explanations of the procedure for deriving the conversion, so here goes!

What do we actually want?

To convert the Cartesian nabla

to the nabla for another coordinate system, say… cylindrical coordinates.

What we’ll need:

1. The Cartesian Nabla:

2. A set of equations relating the Cartesian coordinates to cylindrical coordinates:

3. A set of equations relating the Cartesian basis vectors to the basis vectors of the new coordinate system:

How to do it:

Use the chain rule for differentiation to convert the derivatives with respect to the Cartesian variables to derivatives with respect to the cylindrical variables.

The chain rule can be used to convert a differential operator in terms of one variable into a series of differential operators in terms of othe…

### Lab Book 2014_07_10 More NaI Characterization

Summary: Much more plunking around with the NaI detector and sources today.  A Pb shield was built to eliminate cosmic ray muons as well as potassium 40 radiation from the concreted building.  The spectra are much cleaner, but still don't have the count rates or distinctive peaks that are expected.
New to the experiment?  Scroll to the bottom to see background and get caught up.
Lab Book Threshold for the QVT is currently set at -1.49 volts.  Remember to divide this by 100 to get the actual threshold voltage. A new spectrum recording the lines of all three sources, Cs 137, Co 60, and Sr 90, was started at approximately 10:55. Took data for about an hour.
Started the Cs 137 only spectrum at about 11:55 AM

Here’s the no-source background from yesterday
In comparison, here’s the 3 source spectrum from this morning.

The three source spectrum shows peak structure not exhibited by the background alone. I forgot to take scope pictures of the Cs137 run. I do however, have the printout, and…

### Unschooling Math Jams: Squaring Numbers in their own Base

Some of the most fun I have working on math with seven year-old No. 1 is discovering new things about math myself.  Last week, we discovered that square of any number in its own base is 100!  Pretty cool!  As usual we figured it out by talking rather than by writing things down, and as usual it was sheer happenstance that we figured it out at all.  Here’s how it went.

I've really been looking forward to working through multiplication ala binary numbers with seven year-old No. 1.  She kind of beat me to the punch though: in the last few weeks she's been learning her multiplication tables in base 10 on her own.  This became apparent when five year-old No. 2 decided he wanted to do some 'schoolwork' a few days back.

"I can sing that song... about the letters? all by myself now!"  2 meant the alphabet song.  His attitude towards academics is the ultimate in not retaining unnecessary facts, not even the name of the song :)

After 2 had worked his way through the so…