SOME BASIC DEFINITIONS OF PROBABILITY

Experiment:




Experiment is an activity that is either observed or measured, such as tossing a coin, or
drawing a card.



Event (Outcome):



An event is a possible outcome of an experiment. For example, if the experiment is to
sample six lamps coming off a production line, an event could be to get one defective and
five good ones.



Elementary Events:



Elementary events are those types of events that cannot be broken into other events. For
example, suppose that the experiment is to roll a die. The elementary events for this
experiment are to roll a 1 or a 2, and so on, i.e., there are six elementary events (1, 2,
3, 4, 5, 6). Note that rolling an even number is an event, but it is not an elementary
event, because the even number can be broken down further into events 2, 4, and 6.



Sample Space:




A sample space is a complete set of all events of an experiment. The sample space for the
roll of a single die is 1, 2, 3, 4, 5, and 6. The sample space of the experiment of
tossing a coin three times is:



First toss.........T T T T H H H H

Second toss.....T T H H T T H H

Third toss........T H T H T H T H



Sample space can aid in finding probabilities. However, using the sample space to express
probabilities is hard when the sample space is large. Hence, we usually use other
approaches to determine probability.



Unions & Intersections:




An element qualifies for the union of X, Y if it is in either X or Y or in both X
and Y. For example, if X=(2, 8, 14, 18) and Y=(4, 6, 8, 10, 12), then the union of
(X,Y)=(2, 4, 6, 8, 10, 12, 14, 18). The key word indicating the union of two or more
events is or.

An element qualifies for the intersection of X,Y if it is in both X and Y.
For example, if X=(2, 8, 14, 18) and Y=(4, 6, 8, 10, 12), then the intersection of
(X,Y)=8. The key word indicating the intersection of two or more events is and. See
the following figures:




Mutually Exclusive Events:



Those events that cannot happen together are called mutually exclusive events. For
example, in the toss of a single coin, the events of heads and tails are mutually
exclusive. The probability of two mutually exclusive events occurring at the same time is
zero. See the following figure:


Independent Events:



Two or more events are called independent events when the occurrence or nonoccurrence of
one of the events does not affect the occurrence or nonoccurrence of the others. Thus,
when two events are independent, the probability of attaining the second event is the same
regardless of the outcome of the first event. For example, the probability of tossing a
head is always 0.5, regardless of what was tossed previously. Note that in these types of
experiments, the events are independent if sampling is done with replacement.



Collectively Exhaustive Events:



A list of collectively exhaustive events contains all possible elementary events for an
experiment. For example, for the die-tossing experiment, the set of events consists of 1,
2, 3, 4, 5, and 6. The set is collectively exhaustive because it includes all possible
outcomes. Thus, all sample spaces are collectively exhaustive.



Complementary Events:




The complement of an event such as A consists of all events not included in A. For
example, if in rolling a die, event A is getting an odd number, the complement of A is
getting an even number. Thus, the complement of event A contains whatever portion of the
sample space that event A does not contain. See the following figure:

APPROACHES OF ASSIGNING PROBABILITIES

There are three approaches of assigning probabilities, as follows:



1. Classical Approach:




Classical probability is predicated on the assumption that the outcomes of an experiment
are equally likely to happen. The classical probability utilizes rules and laws. It
involves an experiment. The following equation is used to assign classical probability:



P(X) = Number of favorable outcomes / Total number of possible
outcomes



Note that we can apply the classical probability when the events have the same chance of
occurring (called equally likely events), and the set of events are mutually exclusive and
collectively exhaustive.



2. Relative Frequency Approach:



Relative probability is based on cumulated historical data. The following equation is used
to assign this type of probability:




P(X) = Number of times an event occurred in the past/ Total number
of opportunities for the event to occur



Note that relative probability is not based on rules or laws but on what has happened in
the past. For example, your company wants to decide on the probability that its inspectors
are going to reject the next batch of raw materials from a supplier. Data collected from
your company record books show that the supplier had sent your company 80 batches in the
past, and inspectors had rejected 15 of them. By the method of relative probability, the
probability of the inspectors rejecting the next batch is 15/80, or 0.19. If the next
batch is rejected, the relative probability for the subsequent shipment would change to
16/81 = 0.20.



3. Subjective Approach:



The subjective probability is based on personal judgment, accumulation of knowledge, and
experience. For example, medical doctors sometimes assign subjective probabilities to the
length of life expectancy for people having cancer. Weather forecasting is another example
of subjective probability

fourier transform

The Fourier Transform is a generalization of the Fourier Series. Strictly speaking it only applies to continous and aperiodic functions, but the use of the impulse function allows the use of discrete signals.

The Fourier transform is defined as :

The inverse transform is defined as:

f we take the Fourier transform of a square pulse,


and apply the Fourier transform to it we get:

Since the pulse is zero everywhere except in the range
we can rewrite the equation as:

Note how the limits have changed from +/- infinity to +/- T/2, and that f(t) has disappeared because between the limits of +/- T/2 it has the value of one. Substituting the limits in then gives us:


Using Euler's expressions:

we can rewrite this as:

This is the sinc(x)

SAMPLING FUNCTION

function that is shown below. This function appears very frequently in Fourier transforms. It shows the main frequency content based around zerp frequency (d.c.) with progressively less and less energy in the higher frequencies



FOURIER TRANSFORM OF A AN EXPONENTIALLY DECAYING SIGNAL

basics of fourier transform

To solve many engineering problems, we need to know the response
of a Linear Time Invariation system to some
input signal. If the input signal can be broken up into simple signals
and we know how the system responds to these simple signals, then we can
predict how the system will behave to our input.

Therefore anything that can break a signal down into it's constituent
parts would be very useful. One such tool is the Fourier Series

With the exception of some mathematical curiosities, any periodic
signal of period T can be expanded into a trigonometric series of sine
and cosine functions, as long as it obeys the following conditions:

DIRICHLET CONDITIONS

1. 
   1)f(t) has finite number of maxima and minima within T



2. 
   2)f(t) has finite number of discontinuities within T, and



3. 3)It is necessary that
   

These three conditions mean that you are able to calculate the area under the graph.

f (1) is not true then the signal power goes to infinity and it is not absolutely integrable. If (2) is not true then once again it is not absolutely integrable, and (3) is simply reaffirming conditions (1) and (2).

If all of these are true then the signal can be represented as:


and the coefficients are:

In practice things don't seem as complicated as the theory. For a start you only need a finite number of harmonics to construct f(t) with an acceptable error.

The Fourier series can also be represented in a complex form which is more compact and is more convenient when dealing with complex signals.

If we make use of a trig identity

and we define
we get the complex form of the Fourier series to be

Alternators

Alternators are electromechanical device used in automobiles to convert mechanical energy to alternating current electrical energy. While some of the alternators use rotating magnetic field, some use linear .

Earlier alternators were developed by Michael Faraday and Hippolyte Pixii. Later Faraday developed the "rotating rectangle. Lord Kelvin and Sebastian Ferranti also developed early alternators. With time, alternators were designed for varying alternating-current frequencies.

Function of Alternators

The functioning of alternators is same as that of the DC generators. As the magnetic field around a conductor changes, a current is induced in the conductor. The rotating magnet or the rotor turns and it cuts the field across the conductors. Thus, an electrical current is produced which causes the rotor to turn. This rotating magnetic field causes an AC voltage in the stator windings.



In automobiles, alternators are used to charge the battery and to power a car's electric system when its engine is running. One of the biggest advantages of using alternators is that you do not have to use commutator. Therefore, the alternator becomes simpler, lighter, less expensive and rougher.

Alternators used in passenger vehicles and light trucks use Lundell or claw-pole field construction. Salient-pole alternators are used in larger vehicles. The modern day automotive alternators have a voltage regulator built into them. The efficiency of these alternators is limited by various factors like iron loss, fan-cooling loss, bearing loss, copper loss and the voltage drop in the diode bridges.

The automotive alternators used in buses are very large. The old automobiles with negligible lighting and electronic devices have 30-ampere alternator. The 70 amperes alternators are used in passenger cars and light trucks. The largest automotive alternators are mostly water-cooled or oil-cooled.

Alternators in Hybrid automobiles

Alternators in Hybrid automobiles are combined motor or generator that not only crank the internal combustion engine when starting, but also provides it with additional mechanical power for accelerating. They also charge a large storage battery when the vehicle is running at steady speed.

Fleming's Right Hand rule

Fleming's right hand rule (for generators) shows the direction ofinduced current flow when a conductor moves in a magnetic field.

The right hand is held with the thumb, first finger and second finger mutually at right angles, as shown in the diagram .

• The Thumb represents the direction of Motion of the conductor.
• The First finger represents the direction of the Field.
• The Second finger represents the direction of the induced or generated Current (in the classical direction, from positive to negative).

NORMAL DISTRIBUTION

(GAUSSIAN)

A normal distribution has a bell-shaped density curve described by its mean 

and standard deviation
. The density curve is symmetrical,
centered about its mean, with its spread determined by its standard deviation.
The height of a normal density curve at a given point x is given by

---

The Standard
Normal curve, shown here, has mean 0 and standard deviation 1. If a dataset follows
a normal distribution, then about 68% of the observations will fall
within

of the mean , which in this case is with
the interval (-1,1). About 95% of the observations will fall within 2 standard deviations
of the mean, which is the interval (-2,2) for the standard normal, and about 99.7%
of the observations will fall within 3 standard deviations of the mean, which
corresponds to the interval (-3,3) in this case. Although it may appear as if a
normal distribution does not include any values beyond a certain interval, the density
is actually positive for all values, .
Data from any normal distribution may be transformed into data following the standard normal
distribution by subtracting the mean and dividing
by the standard deviation .

---

Generator

Principle of operation

The scientific principle on which generators operate was discovered almost simultaneously in about 1831 by the English chemist and physicist, Michael Faraday, and the American physicist, Joseph Henry. Imagine that a coil of wire is placed within a magnetic field, with the ends of the coil attached to some electrical device, such as a galvanometer. If the coil is rotated within the magnetic field, the galvanometer shows that a current has been induced within the coil. The magnitude of the induced current depends on three factors: the strength of the magnetic field, the length of the coil, and the speed with which the coil moves within the field.

In fact, it makes no difference as to whether the coil rotates within the magnetic field or the magnetic field is caused to rotate around the coil. The important factor is that the wire and the magnetic field are inmotion in relation to each other. In general, most DC generators have a stationary magnetic field and a rotating coil, while most AC generators have a stationary coil and a rotating magnetic field.

Dollar rate up but IT cos' currency woes see no end

The Weekend Story

 The rupee depreciation may have brought some cheer to IT companies but their currency woes are far from over. The dollar has appreciated against other currencies, including the euro and the pound that account for more than 20% of revenues for top-tier IT exporters. This has stoked fears that dollar revenues may be lower than the guidance projected by IT bellweather Infosys Technologies, and other firms in the sector. 

Dealers said a note by brokerage CLSA warning that Infosys may not be able to meet its dollar guidance for the second quarter and the full fiscal, sparked the sell-off in IT stocks on Friday , sending the BSE IT index tumbling 5%. 

“Infosys is most likely to miss its US dollar revenue guidance for both the September quarter and the full year FY2009. About 28% of invoices are denominated in GBP (pound), euro and A$ (Australian dollar ). The adverse movement of these currencies against the US$ is creating a ‘cross-currency’ headwind,” the note by the brokerage said. The steep fall shaved Rs 15,000 crore off the constituents of the BSE IT index. 

“Prima facie that (what is mentioned in the note) appears to be the reason for the fall. It is frightening that one note can spark off such a collapse ,” said one Mumbai-based IT analyst . According to the note, Infosys will need to revise its FY09 revenue growth guidance from 19-21 % to 17-19 % because of this. 

The company had given a 6% sequential growth guidance for the September 2008 quarter and the CLSA note said Infosys was most likely to miss this as well. 

“There has been double-digit dollar appreciation against the pound and Euro. This will obviously lead to lower dollar realisation and the dollar guidance could be missed. In rupee terms, this will be offset because of the double digit rupee depreciaton,” said Harit Shah, analyst with Angel Broking, adding that this was a currency issue and not related to any further drop in demand for IT. 

“The street has been expecting that the rupee fall from Rs 42 to Rs 45 (against the dollar) will be very good for IT firms. But what the street has not seen is that rupee has appreciated against the euro and pound, and so has the dollar appreciated against these currencies,” said Gaurav Dua, head of research at ShareKhan. 

DC MACHINE CONSTRUCTION

A typical DC generator or motor usually consists of: An armature core, an air gap, poles, and a yoke which form the magnetic circuit; an armature winding, a field winding, brushes and a commutator which form the electric circuit; and a frame, end bells, bearings, brush supports and a shaft which provide the mechanical support. See figure 8.

Armature Core or Stack

The armature stack is made up thin magnetic steel laminations stamped from sheet steel with a blanking die. Slots are punched in the lamination with a slot die. Sometimes these two operations are done as one. The laminations are welded, riveted, bolted or bonded together.

Armature Winding

The armature winding is the winding, which fits in the armature slots and is eventually connected to the commutator. It either generates or receives the voltage depending on whether the unit is a generator or motor. The armature winding usually consists of copper wire, either round or rectangular and is insulated from the armature stack.

Field Poles

The pole cores can be made from solid steel castings or from laminations. At the air gap, the pole usually fans out into what is known as a pole head or pole shoe. This is done to reduce the reluctance of the air gap. Normally the field coils are formed and placed on the pole cores and then the whole assembly is mounted to the yoke.

Field Coils

The field coils are those windings, which are located on the poles and set up the magnetic fields in the machine. They also usually consist of copper wire are insulated from the poles. The field coils may be either shunt windings (in parallel with the armature winding) or series windings (in series with the armature winding) or a combination of both.

Yoke

The yoke is a circular steel ring, which supports the field, poles mechanically and provides the necessary magnetic path between the pole. The yoke can be solid or laminated. In many DC machines, the yoke also serves as the frame.

Commutator

The commutator is the mechanical rectifier, which changes the AC voltage of the rotating conductors to DC voltage. It consists of a number of segments normally equal to the number of slots. The segments or commutator bars are made of silver bearing copper and are separated from each other by mica insulation.

Brushes and Brush Holders

Brushes conduct the current from the commutator to the external circuit. There are many types of brushes. A brush holder is usually a metal box that is rectangular in shape. The brush holder has a spring that holds the brush in contact with the commutator. Each brush usually has a flexible copper shunt or pigtail, which extends to the lead wires. Often, the entire brush assembly is insulated from the frame and is made movable as a unit about the commutator to allow for adjustment.

Interpoles

Interpoles are similar to the main field poles and located on the yoke between the main field poles. They have windings in series with the armature winding. Interpoles have the function of reducing the armature reaction effect in the commutating zone. They eliminate the need to shift the brush assembly.

Frame, End Bells, Shaft, and Bearings

The frame and end bells are usually steel, aluminum or magnesium castings used to enclose and support the basic machine parts. The armature is mounted on a steel shaft, which is supported between two bearings. The bearings are either sleeve, ball or roller type. They are normally lubricated by grease or oil.

Back End, Front End

The load end of the motor is the Back End. The opposite load end, most often the commutator end, is the Front End of the motor.

UNIT STEP&IMPULSE FUNCTION

UNIT STEP FUNCTION


The Heaviside Unit Step Function defines functions encountering ideal On/Off

The Heaviside unit step function turns on a function at t0


The switch (change) at t0 is in fact an impulse, i.e., the Dirac delta function.



DIRAC DELTA OR UNIT IMPULSE FUNCTION



The Dirac delta function works like a sampling gate at t0,


The effect of the sampling gate accumulated through the domain t is the unit step function.

LISTEN TO SQUARE SIGNAL(CLICK LISTEN)

LISTEN

LISTEN

Analog Signal

A signal is any kind of physical quantity that conveys information. Audible speech is certainly a kind of signal, as it conveys the thoughts (information) of one person to another through the physical medium of sound. Hand gestures are signals, too, conveying information by means of light. This text is another kind of signal, interpreted by your English-trained mind as information about electric circuits. In this chapter, the word signal will be used primarily in reference to an electrical quantity of voltage or current that is used torepresent or signify some other physical quantity.

An analog signal is a kind of signal that is continuously variable, as opposed to having a limited number of steps along its range (called digital). A well-known example of analog vs. digital is that of clocks: analog being the type with pointers that slowly rotate around a circular scale, and digital being the type with decimal number displays or a "second-hand" that jerks rather than smoothly rotates. The analog clock has no physical limit to how finely it can display the time, as its "hands" move in a smooth, pauseless fashion. The digital clock, on the other hand, cannot convey any unit of time smaller than what its display will allow for. The type of clock with a "second-hand" that jerks in 1-second intervals is a digital device with a minimum resolution of one second.

Both analog and digital signals find application in modern electronics, and the distinctions between these two basic forms of information is something to be covered in much greater detail later in this book. For now, I will limit the scope of this discussion to analog signals, since the systems using them tend to be of simpler design.

With many physical quantities, especially electrical, analog variability is easy to come by. If such a physical quantity is used as a signal medium, it will be able to represent variations of information with almost unlimited resolution.

Review:

•  Signal is any kind of detectable quantity used to communicate information.
• An analog signal is a signal that can be continuously, or infinitely, varied to represent any small amount of change.

OSCILLOSCOPE CONTROLS

1--On/Off. Do not use the wall plug as an on/off switch. Good switches help to control electrical transients which can be harmful to sensitive circuit components.
2- Intensity. Adjust the brightness of the trace until you can just see all the details of the waveform. If the trace is too bright you will not get the best data, your eyes will get very tired, and you could damage the scope.
3- Focus. Rotate this button until the trace is sharp.
4- Beam finder. If you do not find a trace, push this button. The screen will display what quadrant the trace is in. You can then use the horizontal (#10) and vertical controls (#15) to move the trace to the middle of the screen.
5- Triggering source and mode. You will use the scope to observe signals that repeat frequently. The scope must start the sweep at the same point on the waveform every time in order to produce a stable image on the screen. This function is called "triggering". For many common applications you should the source switch on "internal" and the mode switch to "auto". This lets the scope decide when to trigger.
6- Trigger Slope. Usually the signal voltage will equal the triggering voltage twice, once going up and once coming down. A trigger slope control enables you to select which voltage the scope will trigger on.
7- Trigger Level. This sets an internal voltage which is compared to the voltage of the input signal. When the input signal voltage equals the trigger voltage, the scope triggers. If you get an image that seems to be a superposition of many waves, turn the level knob back and forth slowly until you get a stable image.
8- Sweep calibration. This enables you to change the horizontal scale. Unless this knob is turned all the way clockwise, the scope is not calibrated and your data will be worthless. Turn this knob clockwise until it clicks and check it frequently as you take data.
9- Sweep. This determines the horizontal scale for the oscillograph. The scale is read in the upper white window. Its units are seconds/division. See Timebase illustration
10- Horizontal position. This enables you to move the signal back and forth along the X-axis. This determines, in effect, the value the signal will have at the origin.
11- Channel select. Most oscilloscopes are dual trace. This means that they can display two signals at once, which is why there are two signal ports and two sensitivity controls.
12- Signal ports. There is one signal port for each channel. It is a BNC connector for this oscilloscope.
13- Sensitivity calibration. This knob is used to change the vertical scale. If it is not turned all the way clockwise, the scope will be uncalibrated and your data will be worthless. Check this knob frequently as you take data.
14- Sensitivity. This determines the vertical scale. It is read in the left hand white window. The units are volts/division. See Voltage Sensitivity illustration
15- Vertical position. This knob controls the vertical position of the trace. You will find it very convenient when you are setting or reading voltages.
16- AC/DC select. When this is set to "AC" the DC part of the signal is filtered out by a capacitor placed in series between the signal input and the scope. When the selector is set to "ground", the beam will move to zero volts. When the selector is set to "DC", the entire signal will be displayed on the scope.

Imp. Note on Various Functions

The Gaussian density function is the most important of all the densities and it enters into the areas of science and engineering.This importance stems from its accurate description of many practical and significant real-world quantities, especially when such quantities are the result of many small independent random effects acting to create the quantity of interest.

The Binomial density can be applied to the Bernoulli trial experiment.It applies to many games of chance,detection problems in radar and sonar, and many experiments having only two possible outcomes on any given trial.

The Rayleigh density function describes the envelope of one type of noise when passed through a band pass filter.It is also important in analysis of errors in various measurement systems.

The Exponential density is useful in describing raindrop sizes when a large number of rainstorm measurements are made.It is also known to approximately describe the fluctuations in signal strength recieved by radar from certain types of aircraft.

The Uniform density finds a number of practical uses.A particularly important application is in the quantization of signal samples prior to encoding in digital communication systems.Quantization amounts to "rounding off" the actual sample to the nearest of a large number of discrete "quantum levels". The errors introduced in the round-off process are uniformly distributed.

transformer

THE CE AMPLIFIER COUPLING CAPACITANCES(CLICK ON THE PIC FOR CLARITY)

(CE AMPLIFIER CIRCUIT)






( COUPLING)

(NUMERICAL EXAMPLE )