The solution of the cubic equation

The Cubic Formula:-

(Solve Any 3rd Degree Polynomial Equation)

I'm putting this on the web because some students might find it interesting. It could easily be mentioned in many undergraduate math courses, though it doesn't seem to appear in most textbooks used for those courses. None of this material was discovered by me. -- ES
You should know that the solution of ax2+bx+c=0 is

There is an analogous formula for polynomials of degree three: The solution of ax3+bx2+cx+d=0 is

(A formula like this was first published by Cardano in 1545.) Or, more briefly,

x   =   {q + [q2 + (r-p2)3]1/2}1/3   +   {q - [q2 + (r-p2)3]1/2}1/3   +   p
where
p = -b/(3a),   

q = p3 + (bc-3ad)/(6a2),   

r = c/(3a)

But I do not recommend that you memorize these formulas.
Aside from the fact that it's too complicated, there are other reasons why we don't teach this formula to calculus students. One reason is that we're trying to avoid teaching them about complex numbers. Complex numbers (i.e., treating points on the plane as numbers) are a more advanced topic, best left for a more advanced course. But then the only numbers we're allowed to use in calculus are real numbers (i.e., the points on the line). That imposes some restrictions on us --- for instance, we can't take the square root of a negative number. Now, Cardan's formula has the drawback that it may bring such square roots into play in intermediate steps of computation, even when those numbers do not appear in the problem or its answer.
For instance, consider the cubic equation 

x3-15x-4=0.

 (This example was mentioned by Bombelli in his book in 1572.) That problem has real coefficients, and it has three real roots for its answers. (Hint: One of the roots is a small positive integer; now can you find all three roots?) But if we apply Cardano's formula to this example, we use a=1, b=0, c=-15, d=-4, and we find that we need to take the square root of -109 in the resulting computation. Ultimately, the square roots of negative numbers would cancel out later in the computation, but that computation can't be understood by a calculus student without additional discussion of complex numbers.
There is also an analogous formula for polynomials of degree 4, but it's much worse to write down; I won't even try here.
There is no analogous formula for polynomials of degree 5. I don't just mean that no one has found the formula yet; I mean that in 1826 Abel proved that there cannot be such a formula. The problem is that the functions   don't do enough of what you need for solving all 5th degree equations. (Imagine a calculator that is missing a few buttons; there are some kinds of calculations that you can't do on it.) You need at least one more function. One such function, for instance, is the inverse of the function f(x)=x5+x. (There are other functions that would also work, and some of them are more interesting to mathematicians for various reasons, but I like this one because it can be described in fairly elementary terms.) That function, together with the functions  and addition, subtraction, multiplication, and division is enough to give a formula for the solution of the general 5th degree polynomial equation in terms of the coefficients of the polynomial - i.e., the degree 5 analogue of the quadratic formula. But it's horribly complicated; I don't even want to think about writing it down.

Insulation discs calculation for HV and EHV

Insulation Coordination in Power System
Under Electrical Transmission

Insulation Coordination in Power System was introduced to arrange the electrical insulation levels of different components in the electrical power system including transmission network, in such a manner, that the failure of insulator, if occurs, confindes to the place where it would result in the least danmage of the system, easy to repair and replace, and results least disturbance to the power supply.

When any over voltage appears in the electrical power system, then there may be a chance of failure of its insulation system. Probability of failure of insulation, is high at the weakest insulation point nearest to the source of over voltage. In power system and transmission networks, insulation is provided to the all equipment and components. Insulators in some points are easily replaceable and repairable compared to other. Insulation in some points are not so easily replaceable and repairable and the replacement and repairing may be highly expensive and require long interruption of power. Moreover failure of insulator at these points may causes bigger part of electrical network to be out of service. So it is desirable that in situation of insulator failure, only the easily replaceable and repairable insulator fails. The overall aim of insulation coordination is to reduce to an economically and operationally acceptable level the cost and disturbance caused by insulation failure. In insulation coordination method, the insulation of the various parts of the system must be so graded that flash over if occurs it must be at intended points.

For proper understanding the insulation coordination we have to understand first, some basic terminologies of the electrical power system. Let us have a discussion.

Nominal System Voltage:-

Nominal System Voltage is the phase to phase voltage of the system for which the system is normally designed. Such as 11KV, 33KV, 132KV, 220KV, 400KV systems.

Maximum System Voltage:-

Maximum System Voltage is the maximum allowable power frequency voltage which can occurs may be for long time during no load or low load condition of the power system. It is also measured in phase to phase manner.

Generally maximum system voltage is 110% of corresponding nominal system voltage up to voltage level of 220 KV, and for 400 KV and above it is 105%.

Factor of Earthing:-

This is the ratio of the highest rms phase to earth power frequency voltage on a sound phase during an earth fault to the rms phase to phase power frequency voltage which would be obtained at the selected location without the fault.

This ratio characterizes, in general terms, the earthing conditions of a system as viewed from the selected fault location.

Effectively Earthed System:-

A system is said to be effectively earthed if the factor of earthing does not exceed 80% and non-effectively earthed if it does.
Factor of earthing is 100% for an isolated neutral system, while it is 57.7% (1/√3 = 0.577) for solidly earthed system.

Insulation Level:-

Every electrical equipment has to undergo different abnormal transient over voltage situation in different times during its total service life period. The equipment may have to withstand lightning impulses, switching impulses and/or short duration power frequency over voltages. Depending upon the maximum level of impulse voltages and short duration power frequency over voltages that one power system component can withstand, the insulation level of high voltage power system is determined.
During determining the insulation level of the system rated less than 300 KV, the lightning impulse withstand voltage and short duration power frequency withstand voltage are considered.

Lightning Impulse Voltage:-

The system disturbances occur due to natural lightning, can be represented by three different basic wave shapes. If a lightning impulse voltage travels some distance along the transmission line before it reaches to a insulator its wave shaped approaches to full wave, and this wave is referred as 1.2/50 wave. If during travelling, the lightning disturbance wave causes flash over across an insulator the shape of the wave becomes chopped wave. If a lightning stroke hits directly on the insulator then the lightning impulse voltage may rise steep until it is relieved by flash over, causing sudden, very steep collapse in voltage. These three waves are quite different in duration and in shapes.

Switching Impulse:-

During switching operation there may be uni-polar voltage appears in the system. The wave form of which may be periodically damped or oscillating one. Switching impulse wave form has steep front and long damped oscillating tale.
Short Duration Power Frequency Withstand Voltage

Short duration power frequency withstand voltage is the prescribed rms value of sinusoidal power frequency voltage that the electrical equipment shall withstand for a specific period of time normally 60 seconds.

Protection Level Voltage of Protective Device:-

Over voltage protective device like surge arrestors or lightning arrestors are designed to withstand a certain level of transient over voltage beyond which the devices drain the surge energy to the ground and therefore maintain the level of transient over voltage up to a specific level. Thus transient over voltage can not exceed that level. The protection level of over voltage protective device is the highest peak voltage value which should not be exceeded at the terminals of over voltage protective device when switching impulses and lightening impulses are applied.

Now let us discuss the insulation coordination methods one by one-

Using Shield Wire or Earth Wire:-

Lightning surge in over head transmission line may be caused due to direct hits of lightening strokes. It can be protected by providing a shield wire or earth wire at a suitable height from the top conductor of transmission line. If the conducting shield wire is properly connected to transmission tower body and the tower is properly earthed then direct lightning strokes can be avoided from all the conductors come under the protective angle of earth wire. Over head earth wire or ground wire or shield wire is also used to over the electrical substation to protect different electrical equipment from lightning strokes.
Conventional Method of Insulation Coordination

As we discussed above that a component of electrical power system may suffer from different level of transient voltage stresses, switching impulse voltage and lightning impulse voltage. The maximum amplitude of transient over voltages reach the components, can be limited by using protecting device like lightning arrestor in the system. If we maintain the insulation level of all the power system component above the protection level of protective device, then ideally there will be no chance of breakdown of insulation of any component. Since the transient over voltage reaches at the insulation after crossing the surge protective devices will have amplitude equals to protection level voltage and protection level voltage < impulse insulation level of the components.
Generally, the impulse insulation level is established at 15 to 25% above the protective level voltage of protective devices.

Statistical Methods of Insulation Coordination:-

At higher transmission voltages, the length of the insulator strings and the clearance in air do not increase linearly with voltage but approximately to V1.6. The required number of insulator disc in suspension string for different over voltages is shown below. It is seen that increase in the number of disc is only slight for 220 KV system, with the increase in the over voltage factor from 2 to 3.5 but that there is a ra pid increase in the 750 kV system. Thus, while it may be economically feasible to protect the lower voltage lines up to an over voltage factor of 3.5(say), it is definitely not economically feasible to have an over voltage factor of more than about 2 to 2.5 on the higher voltage lines. In the higher voltage systems, it is the switching over voltages that is predominant. However, these may be controlled by proper design of switching devices.

Proximity effect in conductors

In a conductor carrying alternating current, if currents are flowing through one or more other nearby conductors, such as within a closely wound coil of wire, the distribution of current within the first conductor will be constrained to smaller regions. The resulting current crowding is termed as the proximity effect. This crowding gives an increase in the effective resistance of the circuit, which increases with frequency.

A changing magnetic field will influence the distribution of an electric current flowing within an electrical conductor, by electromagnetic induction. When an alternating current (AC) flows through an isolated conductor, it creates an associated alternating magnetic field around it. The alternating magnetic field induces eddy currents in adjacent conductors, altering the overall distribution of current flowing through them. The result is that the current is concentrated in the areas of the conductor furthest away from nearby conductors carrying current in the same direction.

The proximity effect can significantly increase the AC resistance of adjacent conductors when compared to its resistance to a DC current. The effect increases with frequency. At higher frequencies, the AC resistance of a conductor can easily exceed ten times its DC resistance.

Example

For example, if two wires carrying the same alternating current lie parallel to one another, as would be found in a coil used in an inductor or transformer, the magnetic field of one wire will induce longitudinal eddy currents in the adjacent wire, that flow in long loops along the wire, in the same direction as the main current on the side of the wire facing away from the other wire, and back in the opposite direction on the side of the wire facing the other wire. Thus the eddy current will reinforce the main current on the side facing away from the first wire, and oppose the main current on the side facing the first wire. The net effect is to redistribute the current in the cross section of the wire into a thin strip on the side facing away from the other wire. Since the current is concentrated into a smaller area of the wire, the resistance is increased.

Similarly, in two adjacent conductors carrying alternating currents flowing in opposite directions, such as are found in power cables and pairs of bus bars, the current in each conductor is concentrated into a strip on the side facing the other conductor.

Effects

The additional resistance increases power losses which, in power circuits, can generate undesirable heating. Proximity and skin effect significantly complicate the design of efficient transformers and inductors operating at high frequencies, used for example in switched-mode power supplies.

In radio frequency tuned circuits used in radio equipment, proximity and skin effect losses in the inductor reduce the Q factor, broadening the bandwidth. To minimize this, special construction is used in radio frequency inductors. The winding is usually limited to a single layer, and often the turns are spaced apart to separate the conductors. In multilayer coils, the successive layers are wound in a crisscross pattern to avoid having wires lying parallel to one another; these are sometimes referred to as "basket-weave" or "honeycomb" coils. Since the current flows on the surface of the conductor, high frequency coils are sometimes silver-plated, or made of litz wire.

Skin Effect in Conductors

When a conductor is carrying dc current , it is uniformly distributed over the cross section of the conductor . However an alternting current flowing through the conductor does not distribute uniformly, rather it has the tendency to concetrate near the surface of the conductor. This is known as Skin effect.
  Thus , skin effect is the tendency of an alternting current to concentrate near the surface of the conductor.
   
Due to skin effect the effective area of cross section of the conductor through which current flows is reduced consequently, the resistance of the conductor is slightly increased when carrying an alternating current.

Cause of skin effect
The cause of skin effect can be easily explained. A solid conductor may be thought to be consisting of a large number of strands, each carrying a small part of current. The inductance of each strand will vary according to its position. Thus, the strands near the centre are surrounded by a greater magnetic flux and hence have larger inductance than that near the surface. The high reactance of the inner stranded causes the alternating current to flow near the surface of conductor. This flowing of current near the conductor surface is the skin effect.

The factors on the skin effect depends:-
i, Nature of material.
ii, Diameter of wire- directly proportional.
iii, frequency- directly proportional.
iv, Shape of wire- less for stranded conductor than the solid conductor.

Note: Skin effect is negligible for frequency less than 50HZ and conductor diameter less than 1 cm

Introduction to 'C'

INTRODUCTION

      C is a general-purpose programming language. It has been closely associated with the UNIX operating system where it was developed, since both the system and most of the programs that run on it are written in C. The language, however, is not tied to any one operating system or machine; and although it has been called a ``system programming language'' because it is useful for writing compilers and operating systems, it has been used equally well to write major programs in many different domains. Many of the important ideas of C stem from the language BCPL, developed by Martin Richards. The influence of BCPL on C proceeded indirectly through the language B, which was written by Ken Thompson in 1970 for the first UNIX system on the DEC PDP-7. BCPL and B are ``typeless'' languages. By contrast, C provides a variety of data types. The fundamental types are characters, and integers and floating point numbers of several sizes. In addition, there is a hierarchy of derived data types created with pointers, arrays, structures and unions. Expressions are formed from operators and operands; any expression, including an assignment or a function call, can be a statement. Pointers provide for machine-independent address arithmetic. C provides the fundamental control-flow constructions required for well-structured programs: statement grouping, decision making (if-else), selecting one of a set of possible values (switch), looping with the termination test at the top (while, for) or at the bottom (do), and early loop exit (break). Functions may return values of basic types, structures, unions, or pointers. Any function may be called recursively. Local variables are typically ``automatic'', or created anew with each invocation. Function definitions may not be nested but variables may be declared in a block-structured fashion. The functions of a C program may exist in separate source files that are compiled separately. Variables may be internal to a function, external but known only within a single source file, or visible to the entire program. A preprocessing step performs macro substitution on program text, inclusion of other source files, and conditional compilation. C is a relatively ``low-level'' language. This characterization is not pejorative; it simply means that C deals with the same sort of objects that most computers do, namely characters, numbers, and addresses. These may be combined and moved about with the arithmetic and logical operators implemented by real machines.

         C provides no operations to deal directly with composite objects such as character strings, sets, lists or arrays. There are no operations that manipulate an entire array or string, although structures may be copied as a unit. The language does not define any storage allocation facility other than static definition and the stack discipline provided by the local variables of functions; there is no heap or garbage collection. Finally, C itself provides no input/output facilities; there are no READ or WRITE statements, and no built-in file access methods. All of these higher-level mechanisms must be provided by explicitly called functions. Most C implementations have included a reasonably standard collection of such functions. Similarly, C offers only straightforward, single-thread control flow: tests, loops, grouping, and subprograms, but not multiprogramming, parallel operations, synchronization, or coroutines. Although the absence of some of these features may seem like a grave deficiency, (``You mean I have to call a function to compare two character strings?''), keeping the language down to modest size has real benefits. Since C is relatively small, it can be described in small space, and learned quickly. A programmer can reasonably expect to know and understand and indeed regularly use the entire language. For many years, the definition of C was the reference manual in the first edition of The C Programming Language.

    In 1983, the American National Standards Institute (ANSI) established a committee to provide a modern, comprehensive definition of C. The resulting definition, the ANSI standard, or ``ANSI C'', was completed in late 1988. Most of the features of the standard are already supported by modern compilers. The standard is based on the original reference manual. The language is relatively little changed; one of the goals of the standard was to make sure that most existing programs would remain valid, or, failing that, that compilers could produce warnings of new behavior. For most programmers, the most important change is the new syntax for declaring and defining functions. A function declaration can now include a description of the arguments of the function; the definition syntax changes to match. This extra information makes it much easier for compilers to detect errors caused by mismatched arguments; in our experience, it is a very useful addition to the language. There are other small-scale language changes. Structure assignment and enumerations, which had been widely available, are now officially part of the language. Floating-point computations may now be done in single precision. The properties of arithmetic, especially for unsigned types, are clarified. The preprocessor is more elaborate. Most of these changes will have only minor effects on most programmers. A second significant contribution of the standard is the definition of a library to accompany C. It specifies functions for accessing the operating system (for instance, to read and write files), formatted input and output, memory allocation, string manipulation, and the like. A collection of standard headers provides uniform access to declarations of functions in data types. Programs that use this library to interact with a host system are assured of compatible behavior. Most of the library is closely modeled on the ``standard I/O library'' of the UNIX system. This library was described in the first edition, and has been widely used on other systems as well. Again, most programmers will not see much change. Because the data types and control structures provided by C are supported directly by most computers, the run-time library required to implement self-contained programs is tiny. The standard library functions are only called explicitly, so they can be avoided if they are not needed. Most can be written in C, and except for the operating system details they conceal, are themselves portable.

     Although C matches the capabilities of many computers, it is independent of any particular machine architecture. With a little care it is easy to write portable programs, that is, programs that can be run without change on a variety of hardware. The standard makes portability issues explicit, and prescribes a set of constants that characterize the machine on which the program is run. C is not a strongly-typed language, but as it has evolved, its type-checking has been strengthened. The original definition of C frowned on, but permitted, the interchange of pointers and integers; this has long since been eliminated, and the standard now requires the proper declarations and explicit conversions that had already been enforced by good compilers. The new function declarations are another step in this direction.

        Compilers will warn of most type errors, and there is no automatic conversion of incompatible data types. Nevertheless, C retains the basic philosophy that programmers know what they are doing; it only requires that they state their intentions explicitly. C, like any other language, has its blemishes. Some of the operators have the wrong precedence; some parts of the syntax could be better. Nonetheless, C has proven to ben an extremely effective and expressive language for a wide variety of programming applications.

Black start in Hydropower Plant and Steam Plant

A black start is the process of restoring an electric power station or a part of an electric grid to operation without relying on the external transmission network.

Normally, the electric power used within the plant is provided from the station's own generators. If all of the plant's main generators are shut down, station service power is provided by drawing power from the grid through the plant's transmission line. However, during a wide-area outage, off-site power supply from the grid will not be available. In the absence of grid power, a so-called black start needs to be performed to bootstrap the power grid into operation.

To provide a black start, some power stations have small diesel generators, normally called the black start diesel generator (BSDG), which can be used to start larger generators (of several megawatts capacity), which in turn can be used to start the main power station generators. Generating plants using steam turbines require station service power of up to 10% of their capacity for boiler feedwater pumps, boiler forced-draft combustion air blowers, and for fuel preparation. It is uneconomical to provide such a large standby capacity at each station, so black-start power must be provided over designated tie lines from another station. Often hydroelectric power plants are designated as the black-start sources to restore network interconnections. A hydroelectric station needs very little initial power to start (just enough to open the intake gates and the excitation of the generator field coils), and can put a large block of power on line very quickly to allow start-up of fossil-fueled or nuclear stations. Certain types of combustion turbine can be configured for black start, providing another option in places without suitable hydroelectric plants.

Download free Engineering Books and Notes

Welcome to our new site!!!

Here you can download free Engineering ebooks, PDFs and notes.

You must be the member of this blog in order to download the ebooks, PDFs and notes for free.