Thermistors

Thermistor is a portmanteau of the words thermal and resistor. A thermistor is a type of resistor used to measure temperature changes, relying on the change in its resistance with changing temperature. The temperature coefficient of a thermistor is defined as change in the resistance for 1 K change in temperature.

Semi conductors are classified as posistor and a thermistor based on their temperature coefficient of resistance. A posistor has a positive coefficient of temperature where as a thermistor has a negative coefficient of temperature. That is as the temperature increases the resistance decreases for a thermistor. Thermistors are semiconducting ceramics composed of mixtures of several metal oxides such as those of Cobalt, Magnesium, Manganese, Nickel, Tin, Titanium, Uranium, Zinc and Aluminum. Some NTC thermistors are crystallized from semiconducting material such as Silicon and Germanium.

Electrical circuitry is colder at start-up than after running for a length of time. NTC thermistors are used to take advantage of this to protect the circuitry from the surge in electrical flow that accompanies start-up. Because the resistance of NTC thermistors varies gradually with temperature, they are also used as temperature measuring devices.

The variation of resistance with temperature can be expressed linearly for a small change of temperature. However for higher temperatures, the resistance / temperature curve must be described in more detail. So, Steinhart Hart equation is widely used,

1/T = a + b ln(R) + c (ln(R))^3 where T is the temperature and R is the resistance. Further a, b, c are Steinhart-Hart parameters which change from device to device.

The above equation can be approximated for a temperature T as 1/T = a + b ln(R). When T = T0 then R = R0, we get

1/T - 1/T0 = B (ln(Rt) - ln(R0))

This equation on simplification gives Rt = R0 exp B (1/T - 1/T0) where,
R0 - Resistance at T0 K in ohms
Rt - Resistance at T K in ohms
B - a constant depending on the material of the device

The value of B is given by B = E/K where,

E - energy gap in eV
K - Boltzmans constant (8.625 x 10^(-5) eV/K)

If T tends to infinity then R tends to A which is the thermistors resistance as temperature approaches infinity. Therefore, the temperature resistance relations can be written as Rt = A exp B/T.

The temperature coefficient of resistance is usually expressed as

alpha = dR/dT (1/R)
= (A exp B/T) (-B/T^2) (1/A exp (B/T))
= -B/T^2

where T is the absolute temperature in Kelvin.

Towers Of Hanoi

The Towers of Hanoi puzzle was published in 1883 by French mathematician Edouard Lucas, under the pen-name N. Lucas de Siam. The Towers of Hanoi puzzle basically consists of three towers. A specific number of discs are placed on one of the towers, such that the discs are placed in the ascending order of their size from top to bottom. The objective of the game is to move the discs to another tower such that they still follow the same order. Further only one disc can be moved at a time and a bigger disc cannot stand on a smaller disc.

Consider the following scenario. Here we have three towers. The first one will have three discs arranged in ascending order from top to bottom. The other two towers will be empty. The ending scenario could be - The towers 1 and 2 will be empty and the third one will have the three discs arranged in ascending order from top to bottom. The challenge of Tower of Hanoi is to achieve this with the constraint that only one disc can be moved at a time and a bigger disc cannot stand on a smaller disc. The choice of starting tower and destination tower is possible and also the number of discs can be chosen. The more the number of discs, the challenge becomes more difficult.

Legends

There is a legend related with the game which states that in Benares, during the reign of the Emperor Fo Hi, there was a temple with a dome which marked the center of the world. In this temple there contained a large room with three time-worn posts in it surrounded by 64 golden discs. The priests of Brahma, acting out the command of an ancient prophecy, have been moving these discs, in accordance with the rules of the puzzle. According to the legend, when the last move of the puzzle is completed the world will end. The puzzle is therefore also known as Tower of Brahma puzzle.

If the legend were true, and if the priests were able to move discs at a rate of 1 per second, using the smallest number of moves, it would take them 2^64 - 1 seconds or roughly 585 billion years.

There are many variations on this legend. For instance, in some stories the temple is a monastery and the priests are monks. The temple or monastery may be said to be in different parts of the world - including Hanoi, Vietnam and may be associated with any religion. In some versions, other elements are introduced, such as the fact that the tower was created at the beginning of the world, or that the priests or monks may make only one move per day.

Mathematical Significance

The problem of the Towers of Hanoi is isomorphic to finding a Hamiltonian path on a n-hypercube. The problem is solved by a remarkably simple recursive procedure. Further the terms of the sequence resembles the binary carry sequence. Amazingly, the number of discs moved after the kth step is the same as the element which needs to be added or deleted in the kth addend of the Ryser formula.

The number of steps required for n discs is 2^(n) - 1 which is the Mersenne numbers. Further this relates the Pascal's triangle to the Hanoi graph.

Significance in Computers

The problem of Towers of Hanoi is one the common methods of finding the speed of the computer. A code is written to solve the problem and the time taken for the computer to execute this code is used to determine the speed of the computer. You can try out the Towers of Hanoi here.

Vocabulary 03

• Aggrandize - To extend or exaggerate
  The media always tries to aggrandize any news it gets.
• Alienate - To isolate oneself from others or another person from oneself
  After finding that Ramu was a thief, he was alienated by all his friends.
• Amalgamate - To merge or combine into a single thing
  The two companies were amalgamated as they were not functioning properly.
• Ambiguous - Unclear or capable of having more than one meaning
  It was ambiguous what the lawyer was trying to prove in the court.
• Ambivalent - Characterized by uncertainty; unable to decide between opposites
  The jury was ambivalent when it was asked to choose the winners.
• Ameliorate - To enhance or improve something
  After the company had been taken over, it was ameliorated by the new CEO.
• Amenable - Responsive to suggestion; willing
  My boss was amenable to the suggestion I made.
• Amiable - Friendly and pleasant
  The students of my class were amiable to me when I first met them.
• Amortize - To reduce gradually over a period of time
  The prices of Televisions have amortized from the past 10 to 15 years.
• Anachronistic - Out of order; chronologically misplaced
  The death of the king was anachronistic in the textbook.