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“Wye Knot”

May12
2013 Leave a Comment Written by M. Eric Carr

I came across an interesting puzzle a while ago (stated in one of the excellent Feynman Lectures on Physics): What is the resistance between two opposite corners of a cube made from 1-ohm resistors?

At first, the problem looks like it might just be an exercise in resistance simplification using the well-known laws of parallel and serial resistances: Serial resistances add linearly, and parallel resistances follow 1/R = 1/R1 + 1/R2 etc. Unfortunately, this one isn’t quite so simple — three resistors meet at each vertex of the cube, and no two are directly parallel to or in series with each other.

A cube of 1-ohm resistors. What is the equivalent resistance between opposite corners?

One solution to this problem is to use the Wye-Delta conversion formula. Each of the corners of the cube is the center node of three resistors in a wye configuration.  These three resistors can be replaced, and the central node removed, by following the formula:

Wye-Delta conversion formula

(formula from Wikipedia)

Rp in the formula is the sum of products — that is, R1R2 + R2R3 + R1R3.

Delta and Wye configurations. (Click for larger.)

(image from Wikipedia)

Using this formula, corners of the cube where three 1-ohm resistors meet can be removed; the three outer vertices of this wye are then connected by the equivalent network of one 3-ohm resistor between each pair of vertices (see image).

 

The cube, with one (shaded) corner transformed. Note the 3-ohm resistors.

One way you can think about this is that, initially, the resistance between any two nodes of the wye is 2 ohms (the inbound resistor plus the outbound resistor.) In order for the delta network to be equivalent, this resistance must be the same; the value of the delta resistor connecting those two nodes, in parallel with the sum of the other two delta resistors (for the long path around the triangle), must equal the original value of the resistance between the two nodes of the wye. For a wye of 1-ohm resistors, this is 3 ohms, since a 3-ohm resistor in parallel with a 6-ohm equivalent will equal two ohms. (1/3 + 1/6 = 1/2)

You can continue reducing the network in this way — but it turns out that symmetry and a little common sense provide an easier way to solve the puzzle.

The cube, with equipotential points shown, assuming resistance measured between blue and red. (Click for larger.)

Suppose that in the cube above, a voltage source is connected between the blue and red nodes. The resulting current can be used to determine the overall resistance. To calculate this without resulting to Wye-Delta conversions, note that all of the “purple” nodes are at the same voltage (due to symmetry), and all of the “green” nodes are at the same voltage (also due to symmetry). Because of this, the three purple nodes and the three green nodes can therefore be connected by wires in order to simplify the analysis. Because no voltage difference exists between the nodes, no current will flow across these wires, so the nature of the circuit hasn’t been fundamentally changed.

Here is a schematic of the original cube configuration…

A schematic for a circuit electrically equivalent to the Resistor Cube. (Click for larger.)

…and here is a schematic of the cube with the equipotential connections in place.

The same schematic, with connections added between the equipotential points. (Click for larger.)

The problem now becomes trivially easy. The blue node is connected to the green node via three 1-ohm resistors; the green node is connected to the purple node via six 1-ohm resistors; and the purple node is connected to the red node via another three 1-ohm resistors.

The equivalent circuit -- a simple series-parallel network. (Click for larger.)

The total circuit resistance is therefore 1/3 + 1/6 + 1/3 ohms, or 5/6 ohm.

 

 

 

 

Posted in Analog, EET201, Electronics, Fundamentals, Math

How To Use A Digital Multimeter (DMM)

Feb01
2013 Leave a Comment Written by M. Eric Carr

Digital Multimeters (DMMs) are quite versatile devices. Typically, they can be used to measure voltage, current, and resistance. Many DMMs are also capable of other measurements, such as continuity (an extension of the resistance function, actually) and frequency. Some are also capable of basic math operations on the measurements performed, including null offset and maximum/minimum readings. Here is a brief guide to using a DMM — specifically, an HP/Agilent 34401A. This type of meter is a “bench” meter — meaning that it is intended for use on an electronics workbench. It is somewhat portable, and includes a handle, but is significantly bulkier (and a bit more fragile) than handheld meters, so it is usually found on a workbench or in a lab. On the plus side, it is far more accurate than most handheld meters.

The Agilent 34401 Digital Multimeter (DMM). (Click for larger.)

 

Measurement of voltage

The most basic function of a DMM is as a basic voltmeter. Most DMMs are autoranging, meaning that they automatically select the correct voltage range (millivolts, volts, etc) based on the measurement being made at the time. To measure DC voltage with the 34401A:

  • Turn on the meter using the power switch on the left end of the front panel.
  • Press the “DC V” button (if you will be measuring DC voltages.)
  • Connect the negative probe (typically black in color) to the rightmost black “LO” port. (The two ports in the left column are used for four-wire resistance measurements, which are not covered in this guide.)
  • Connect the positive probe (typically red) to the upper right “HI” port.
  • Connect the probes to the circuit under test. Voltage is measured in parallel with a component, so you would not break the circuit to measure voltage.
  • Read the voltage data on the meter.

Connect the voltmeter in parallel with the resistor to measure volts. (Click for larger.)

 

Measurement of current

Another useful function of a DMM is as an ammeter — a meter designed to measure current flow. Since current flow is measured within a conductor, the meter must be actually inserted in the circuit in order to measure this flow. (There are “clamp” ammeters which do not work this way, but they are not covered here.) Because of this requirement, measuring current in a circuit is done differently than measuring voltage — the connections are made differently. Instead of simply clipping on to a circuit, the DMM replaces one of the wires in the circuit, and the current is made to flow through the meter. It is important to make sure that the meter replaces a wire of the circuit in this way, or else it could cause a short circuit. (That is, you need to break a connection, then insert the DMM in the gap.)
To measure DC current with the 34401A:

  • Turn on the meter
  • Press the blue SHIFT button once, then press the DC V button. This will put the meter into “DC I” (DC current) mode.
  • Connect the negative lead to the right “LO” port
  • Connect the positive lead to the right “I” port (at the lower right).
  • Disconnect power to the circuit to be tested
  • Choose a wire in your circuit where you will measure current. Disconnect it.
  • Connect the black lead to one end of where the wire was (ideally, its more negative end.)
  • Connect the red lead to the other end of where the wire was (ideally, its more positive end.)
  • Reconnect power to the circuit to be tested.
  • Read the current value on the meter’s display. (Note mA or A, as well.)

Connect the DMM in series with the resistor to measure current. (Click for larger.)

 

Measurement of resistance

DMMs can also function as ohmmeters, allowing the measurement of resistance. Unlike measurements of voltage and current, this must be done with power disconnected from the circuit. (In fact, since parallel resistances can affect the measurement, resistance is usually measured with the component disconnected from the circuit, at least at one end.) When measuring resistance, the meter passes a small amount of current through the resistor and measures the resulting voltage. To measure resistance with the 34401A:

  • Turn on the meter
  • Press the “Ω 2W” button.
  • Connect the negative lead to the right “LO” port
  • Connect the positive probe (typically red) to the upper right “HI” port.
  • Disconnect at least one end of the resistor to be tested from the circuit.
  • Connect one probe to one lead of the resistor to be tested.
  • Connect the other probe to the resistor’s other lead. (Direction doesn’t matter.)
  • Read the resistance value on the meter’s display. (Note mΩ, Ω, kΩ, or MΩ.)

To measure resistance, disconnect the power supply and connect the DMM across the resistor to be measured. (Click for larger.)

 

The 34401A has many other measurement capabilities (frequency, continuity, four-wire resistance, diode check, etc.) It can even connect to a computer via RS232 or GPIB for automated measurements.

 

Posted in Analog, Drexel, EET201, Electronics, Fundamentals, Tools

BMP file format

Jan02
2013 Leave a Comment Written by M. Eric Carr

Since uncompressed bitmaps are a one-to-one representation of pixels in an image, they are one of the simplest formats to generate, if you are writing your own code. Here is a description (plus FreeBASIC code, which works well enough as pseudocode) of how to output a .bmp file, given an array of pixels.

In order to make the process as straightforward as possible, I will make several (hopefully reasonable) assumptions:

  • The image exists in memory, in a format (an array, for instance) which can be easily read in random-access order;
  • A color depth of 24 bits is used (eight bits each for Red, Green, and Blue);
  • BITMAPCOREHEADER (the simplest header) is used; and
  • The height and width of the image are reasonable and known.
    (I will call these variables XSIZE and YSIZE.)

Values given here are in hexadecimal. All multi-byte values in BMP files are little-endian.

Bitmap files consist of three parts:

  • The bitmap header;
  • The DIB header; and
  • The pixel data array.

The bitmap header is straightforward enough:

  • Two bytes to denote a bitmap file: ASCII “BM”, or hex 0×42 0x4D
  • Four bytes, representing the size of the BMP file in bytes.
  • Two bytes reserved: these can safely be 0×00 0×00.
  • Two more bytes reserved: these can also be 0×00 0×00.
  • Four bytes for the offset address of the pixel data.
    (If using BITMAPCOREHEADER, this is 0x1A 0×00.)

The BITMAPCOREHEADER is next, and also relatively simple:

  • Four bytes for the size of the header (14 bytes, so 0x0E 0×00 0×00 0×00).
  • Two bytes for the image width in pixels;
  • Two bytes for the image height in pixels;
  • Two bytes for the number of pixel planes (must be 0×01 0×00); and
  • Two bytes for the bits per pixel (0×18 0×00 in our 24-bit example.)

The last structure is the bitmap data itself. This is three bytes per pixel, with each row padded to the next multiple of four bytes, as needed. Each pixel is in BBGGRR order.

We now have all of the pieces of information we need to create the bitmap file…

  • Output ASCII “BM” (0×42 0x4D)to the file
  • Calculate the size of the file:
  • – 14 bytes for the bitmap header, plus
  • – 12 bytes for the DIB header, plus
  • – The number of rows (image height) times the row size in bytes.
    (The row size is the image width, times three, rounded up to
    the next multiple of four, if needed.)
  • Write this figure to the file, in little-endian hex, using four bytes.
  • Write 0×00 0×00 0×00 0×00 to the file, for the two reserved fields.
  • Write 0x1A 0×00 (representing the 26-byte offset for the start of data.)
  • Write 0x0C 0×00 0×00 0×00 (representing the DIB header size.)
  • Write XSIZE in two-byte little-endian hex.
  • Write YSIZE in two-byte little-endian hex.
  • Write 0×01 0×00 for the number of pixel planes;
  • Write 0×18 0×00 to represent 24 bits per pixel.
  • Loop over the number of rows (YSIZE) in the image:
  • – For each pixel in that row, write its image value in little-endian hex (BBGGRR)
  • – At the end of the row, pad it with zero to three extra bytes to make the number of bytes in the row a multiple of four.
  • Close the file. You’re done!

Here is an example in FreeBASIC, which produces a simple 3×3 bitmap.



Posted in BASIC, Coding

A Confusion of COM Ports

Nov20
2012 Leave a Comment Written by M. Eric Carr

I like serial-port communications, but this is ridiculous.

Whenever a new USB serial device is connected to a Windows computer (and/or apparently whenever an existing USB serial device is moved to a new USB port), Windows assigns it a new COM port number. Back in the good old days, there was generally COM1 and COM2 — typically for a mouse and modem — and the addresses were set in hardware with jumpers or DIP switches.

Fast-forward to today’s environment, including Arduinos, Bluetooth serial devices, USB-to-serial cables, and others. I’m up to COM29 with no end in sight. Most of these are the ghosts of various devices plugged in to various USB ports. The drivers are all still installed, and they’ll be redetected when plugged back in.

Deleting unused COM ports in Windows isn’t as straightforward as perhaps it should be, since Device Manager doesn’t list non-present COM ports by default. Fortunately, an article on Microsoft’s Knowledge Base shows how to change this. Here’s how:

  • Open a command prompt
  • Type      set devmgr_show_nonpresent_devices=1    <enter>
  • Type      start devmgmt.msc  <enter> to start Device Manager
  • Click     View –> Show Hidden Devices.
  • The COM ports are listed under “Ports (COM & LPT).” Have at ‘em!

Incidentally, this solved the mysterious   “avrdude: stk500_getsync(): not in sync: resp=0×86″ error message I was getting when uploading an Arduino sketch. Apparently even the Arduino IDE had had enough.

 

Posted in Arduino, System Administration
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