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Control Decimal Vs Binary Numeration
Counting from zero to twenty using four different kinds of numeration systems: hash marks, Roman numerals, decimal, and binary:
System: Hash Marks Roman Decimal Binary
------- ---------- ----- ------- ------
Zero n/a n/a 0 0
One | I 1 1
Two || II 2 10
Three ||| III 3 11
Four |||| IV 4 100
Five /|||/ V 5 101
Six /|||/ | VI 6 110
Seven /|||/ || VII 7 111
Eight /|||/ ||| VIII 8 1000
Nine /|||/ |||| IX 9 1001
Ten /|||/ /|||/ X 10 1010
Eleven /|||/ /|||/ | XI 11 1011
Twelve /|||/ /|||/ || XII 12 1100
Thirteen /|||/ /|||/ ||| XIII 13 1101
Fourteen /|||/ /|||/ |||| XIV 14 1110
Fifteen /|||/ /|||/ /|||/ XV 15 1111
Sixteen /|||/ /|||/ /|||/ | XVI 16 10000
Seventeen /|||/ /|||/ /|||/ || XVII 17 10001
Eighteen /|||/ /|||/ /|||/ ||| XVIII 18 10010
Nineteen /|||/ /|||/ /|||/ |||| XIX 19 10011
Twenty /|||/ /|||/ /|||/ /|||/ XX 20 10100
Neither hash marks nor the Roman system are very practical for symbolizing large numbers. Obviously, place-weighted systems such as decimal and binary are more efficient for the task. Notice, though, how much shorter decimal notation is over binary notation, for the same number of quantities. What takes five bits in binary notation only takes two digits in decimal notation. This raises an interesting question regarding different numeration systems: how large of a number can be represented with a limited number of cipher positions, or places? With the crude hash-mark system, the number of places IS the largest number that can be represented, since one hash mark "place" is required for every integer step. For place-weighted systems of numeration, however, the answer is found by taking base of the numeration system (10 for decimal, 2 for binary) and raising it to the power of the number of places. For example, 5 digits in a decimal numeration system can represent 100,000 different integer number values, from 0 to 99,999 (10 to the 5th power = 100,000). 8 bits in a binary numeration system can represent 256 different integer number values, from 0 to 11111111 (binary), or 0 to 255 (decimal), because 2 to the 8th power equals 256. With each additional place position to the number field, the capacity for representing numbers increases by a factor of the base (10 for decimal, 2 for binary).
An interesting footnote for this topic is the one of the first electronic digital computers, the Eniac. The designers of the Eniac chose to represent numbers in decimal form, digitally, using a series of circuits called "ring counters" instead of just going with the binary numeration system, in an effort to minimize the number of circuits required to represent and calculate very large numbers. This approach turned out to be counter-productive, and virtually all digital computers since then have been purely binary in design. To convert a number in binary numeration to its equivalent in decimal form, all you have to do is calculate the sum of all the products of bits with their respective place-weight constants. To illustrate:
Convert 110011012 to decimal form:
bits = 1 1 0 0 1 1 0 1
. - - - - - - - -
weight = 1 6 3 1 8 4 2 1
(in decimal 2 4 2 6
notation) 8
The bit on the far right side is called the Least Significant Bit (LSB), because it stands in the place of the lowest weight (the one's place). The bit on the far left side is called the Most Significant Bit (MSB), because it stands in the place of the highest weight (the one hundred twenty-eight's place). Remember, a bit value of "1" means that the respective place weight gets added to the total value, and a bit value of "0" means that the respective place weight does not get added to the total value. With the above example, we have:
12810 + 6410 + 810 + 410 + 110 = 20510
If we encounter a binary number with a dot (.), called a "binary point" instead of a decimal point, we follow the same procedure, realizing that each place weight to the right of the point is one-half the value of the one to the left of it (just as each place weight to the right of a decimal point is one-tenth the weight of the one to the left of it). For example:
Convert 101.0112 to decimal form:
.
bits = 1 0 1 . 0 1 1
. - - - - - - -
weight = 4 2 1 1 1 1
(in decimal / / /
notation) 2 4 8
410 + 110 + 0.2510 + 0.12510 = 5.37510
Saturday, March 3, 2012
Control The mathematical model of the magnetic levitation system
Model of the magnetic levitation system
The magnetic levitation system considered in this paper consists of a ferromagnetic ball suspended in a voltage-controlled magnetic field. Only the vertical motion is considered.
The objective is to keep the ball at a prescribed reference level. The schematic diagram of the system is shown in Figure 2.1. The dynamic model of the system can be written as
Schematic diagram of the magnetic levitation system.
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The mathematical model of the electromagnetic levitation system
One can build the mathematical model of the levitation system by writing
appropriate differential equations in accordance to the typical mechanical- and electrical principles. The way the components are appreciated in the approaching mode can lead to simpler or more complex alternatives.
The formula for the energetic balance within the system is:
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Basic mathematical model of magnetic levitation
In terms of the design, chosen part of the magnetic levitation model was simulated. We used the software for multidisciplinary system simulation – Dynast [9]. In the Fig. 6, a basic simulation model is shown. One of the significant parts of the model is the feedback circuit. It Decreasing / increasing of the duty cycle of the PWM
Approaching / taking away of the levitating object Decreasing / increasing of the output voltage of the Hall Effect sensor The place stabilization of the levitating object
The variation of the attraction force of the electromagnet means we simulated the output signal of the amplifier with the adjustable gain. According to the Fig. 6, there is a derivation block which is realized by passive components – resistors and capacitors. These components influence stability of the model. On this account, we have optimized the derivation block according to the results of the simulation. After the realization of the electromagnetic levitation device, we have compared real levels of the chosen parameters of the constructed device with the simulation model. The results of the simulation approximatelly correspondence with the real device.
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Model of the magnetic levitation system
The system dynamics describing the behaviour of the moving
ball is derived from the Newton’s laws:
where z denotes the position of the ball (as indicated figure 2),
m its mass, g the acceleration of gravity, i the coil current and
Fmag(i,z) the electromagnetic force applied to the ball. d
denotes a bounded perturbation.
Drawing up the energy balance of the whole system and under
the assumption that the magnetic core is non saturated (which
occurs because of the air gap), the electromagnetic force can
be expressed as following:
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