CO-RE of Electrical Engineering

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The book 'CO-RE OF ELECTRICAL ENGINEERING' has been written with this objective. It is hoped that it will serve as a handy document for quickly finding the important constants and the relations needed to solve an Electrical Engineering problem in hand, or to develop othter inter-relationships. The subjects covered are circuits and networks, field theory, electrical engineering materials, instruments and measurements, control systems, electrical machines, and power systems. The coverage of constants and the formulas are, of course, not exhaustive, but those which are basic in nature, or are quite often needed have been included. No detailed derivations of the relations have been given. Only very brief explanations or the key points of some of the relations have been presented. It has been assumed that the reader knows and understands the subject. He only needs to refresh.

1m= 1650763.73 wavelengths in vacuum of radiation (2p 10 - 5d5 ) ofKr 86 . 1 kg- International prototype kilogram. 1 s = 9192631770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state ofthe caesium-133 atom. 1 A- The current which when flowing in each of the two infinitely long parallel conductors of negligible cross-section 1 metre apart in vacuum will produce on each a force of 2 x10- 7 N/m. 1 K - The kelvin is 11273.16 of the thermodynamic temperature of the triple point of water.

Sound pressure level in db= 20 lg (pip), wherep is the sound pressure in 1J bar, andp0 equals 2 x 10-4 IJ bar. (1 IJ bar= 1 dyne I cm2 ). The sound pressure Pu is the threshold of hearing. A pressure of about 1000 dynes I cm2 causes serious discomfort or damage to the ear.

In chemistry, the generalization that there is a recurring patt~rn in the properties of the elements when they are arranged in order of increasing atomic number---,i.e., the. total number of protons in the atomic nucleus, is depicted in the form of periodic table. The periods (horizontal rows) of the periodic table illustrate these relationships. 

The plane angle e ( in radians) subtended at point 0 by a small length M is ; where r r is the length of the straight line joining 0 and P, P being a point at the middle of M, in is a unit vector normal to 11l at point?, and i,. is a unit vector along OP. Min and i,. are to be in the same plane.

The plane angle e ( in radians) subtended at point 0 by a small length M is ; where r r is the length of the straight line joining 0 and P, P being a point at the middle of M, in is a unit vector normal to 11l at point?, and i,. is a unit vector along OP. Min and i,. are to be in the same plane.

The passive elements are-Resistor, Inductor, and Capacitor. The resistance of an element is a measure of the opposition offered to the flow of electric current through it. In all the relations listed below, the symbols have their usual meanings in context with the topic being considered.

For sinusoidal time variations of voltages and currents, we analyze the circuits, for steadystate conditions, through the use ofphasors ofthe corresponding quantities. Using phasors we 0 are able to suppress the time variation, and hence the convenience in analyzing. Yet we obtain the results as if we analyzed the circuits in time domain.

Assuming zero initial conditions, for a series R-L circuit, the expression for the resulting current on sudden application of a de voltage Vis i(t) = ~ [ 1- e -~). Tis the time constant ofthe circuit and equals LIR.

The Laplace variable s is a complex quantity; as such all the functions, expressions, and equations involving s are also complex. However, for ease, no bold face typing is used for them. The laplace transform ofthe excitation, which may either be a voltage signal, or a current signal applied to a pair of terminals; will be denoted by E(s), and that of the response which again can be a voltage across some element, or a current through an element, by R(s).

A polynomial in s with real coefficients, which has none of its roots in the right half of - the complex s-plane, and the roots on thejw-axis are all simple; is called a Hurwitz polynomial. Thus, condition (i), and the first part of condition (ii) are satisfied ifD(s) is a Hurwitz polynomial. ·Therefore, for a function F(s) to be a positive real function, its denominator polynomial D(s) must be a Hurwitz polynomial, and the second part of condition (ii) along with condition (iii) should also be satisfied.

The vector quantities-like force F, electric field E, conduction current density J, unit vector i etc. These quantities are space directed. The phasors to represent the quantities varying sinusoidally with time-like the current phasor I, the voltage phasor V etc.

The electric field intensity Eat a point Pis the force per unit charge and its direction is the direction ofthe force. The test charge in determiningE should be as small as possible so that it does not alter the distribution of charges, whose very electric field we are finding. If the source charges are held firmly in position, then the test charge may be of any value.

A static charge, as well as a moving charge, experience force in electric field. In magnetic field, a static charge experiences no force , but a moving charge experiences. Thus, in the absence of electric field, if a moving charge experiences a force, then a magnetic field is present.

Faraday's law also constitutes one of the Maxwell's equations. Faraday's law implies that a time-varying magnetic field gives rise to electric field. It may be recalled that the Ampere's law brought out the fact that a time-varying electric field gives rise to magnetic field.

A uniform plane wave is one for which E (and H too) is same everywhere in a plane perpendicular to the direction of travel of the wave. If the direction oftravel ofthe wave is x-direction, then the uniform planes are parallel to the y-z plane. In such planes, there is no variation of E, or H withy, and z.

The parallel conducting planes are situated parallel to they-z plane for wave propagation between parallel conducting planes, and the distance between these planes is a. For rectangular waveguides, there are parallel plates which are here taken as parallel to thex-z plane, and the distance between these plates is being denoted by b.

A transmission line is a distributed parameter circuit. The parameters used, i.e, R, L, z, G, C, yare all line parameters per unit length. At the receiving end we take x = 0, and at the sending end, x = l; thus x increases in the direction from R-end to S-end. The relations given are for steady state ac conditions, and apply to single-phase lines, and also to the three-phase lines on per phase basis.

To understand the physical properties of a material and the effect ofvarious external variable parameters on its properties, one need to examine the material on atomic scale. Simplified atomic models suffice for the purpose. Bohr's model is considered acceptable.

True value of a quantity is the average of a very large number of measurements made ofthat quantity, committing no gross errors and using instruments known to have no systematic errors.

We are required to measure several electrical parameters like resistance, inductance, quality factor, capacitance, loss angle, frequency etc. The ranges of their values are also very wide. One may estimate the values of impedances by simply applying the Ohm's law. However, the loading effects of the instruments, the instrument errors may cause the measurement to be inaccurate. The comparison method and the bridge methods on the other hand are free from instrument errors and are considered superior. These give more accurate results provided the components which are used in the measurement are ofhigh quality and their values are accurately known.

In all walks of life, we always wish to keep one or more quantities or items of our interest at the desired levels. Because of various disturbances caused by society, environment, associated mechanisms etc., the quantities of our interest, may not be at the levels desired. We have to build something into the system so that the quantities in question stay at the desired levels, or within a narrow range of the desired levels. This 'building something into the system' is controlling the quantities of interest (henceforth called controlled quantities or variables) in a manner to keep them at the intended levels.

The transfer function description of a control system is useful for frequency domain analysis of the system. The time domain re8ponse oftlw system as found using transfer function does not include the effect of initial conditions as the transfer function is defined with zero initial conditions.

A control system may include control elements of several types, viz.-mechanical, electrical, electromechanical, hydraulic, pneumatic, and various other types. To find the overall transfer function, or to develop the state equations of the control system, one should first obtain the governing equations relating the variables of interest of individual elements.

Stability of a system means that its response does not increase indefinitely for any finite input to it. For any bounded input, its response remains bounded. Mathematically, it can be stated as I c(t) I ~ M ~ oo fort?. t 0 , if I r(t) I ~ N ~ oo fort?. t 0 .

Because of the greater computational effort involved in the time domain analysis, it is usually difficult to obtain time response of the feedback control systems. Frequency response is often obtained by means of graphical methods and the time domain behavior is found from the relationships between the frequency response and the time response. As such it is pertinent to study frequency response of the control systems.

A control system with only its minimum required components seldom gives satisfactory performance. Compensating networks are often needed in the forward/ feedback path to improve the performance of the system.

A magnetic field is that in which a moving charge, a current-carrying element, and a magnetic needle, all experience a force. The source of magnetic field is basically electric current. The magnetic effect of a given current element will have different values in different media/materials. The effect is represented graphically by lines, called the field- or the flux lines; the tangent to which at any point gives the direction along which the magnetic needle will orient when placed at that point, and the relative strength of the effect is depicted by the relative closeness of the lines. A magnetic flux line closes on itself

A transformer works on the principle of mutual induction between two circuits linked by common magnetic flux.

An electromechanical energy conversion process involves a coupling field between the electrical system and the mechanical system; and the tendency of the energy in the · coupling field to release itself and do work is the basis of the conversion process. The interchange of energy between an electrical system and a mechanical system is possible if the energy in the coupling field is influenced by the configuration of the mechanical elements. The electromechanical energy conversion occurs if the change in flux in the coupling field is associated with mechanical motion.

In a double layer winding, the number of coils is an integer multiple of the number of slots. As the junction ofthe finish of one coil and the start ofthe next coil is connected to a commutator segment, the number of commutator segments equals the number of coils. As such, a de machine armature winding is a closed winding.

Induction machines are more often used as motors than as generators and the induction motors are the most widely used type of motors. These work on the principle of mutual induction like transformers, but here the induction is by rotating flux rather than by pulsating flux. As already stated, the primary, or the armature is on the stator, and the secondary on the rotor. The armature has a balanced three-phase winding. When the armature is connected to a balanced three-phase supply, a rotating mmf of constant amplitude rotating at synchronous speed is produced in the air gap. ·The rotor conductors are cut by this rotating mmf, hence, an emf at slip frequency is induced in the rotor circuit. The rotor circuit being closed, rotor currents flow . The field produced by the armature currents and that produced by the rotor currents interact to develop the torque.

The construction of a synchronous machine has already been described briefly. Here we consider various performance features ofthe machine. We shall first consider a three-phase cylindrical rotor machine, operating as generator under steady state condition. Saturation of iron, the losses in it, and the harmonics are neglected. It implies that the proportionality law can be applied to B- H relation, and that the flux density space wave is a sine wave.

Energy is neither available, nor is utilized in electric form. Yet the naturally available energy is converted to Electric form (ac). It is so because it is a very neat form of energy and can be sent to any nook or corner, near or far. It is the most desired form of energy.

The bulk electric power is usually generated at locations quite far away from the load centers. There is a network of lines and transformers between the generation point and the power consumption point. Between the two, there may be primary (hv) and secondary (lv) transmission network; and primary and secondary distribution network. Following are the typical voltage levels at various stages.

A tariff is simply a way of charging the consumers for supplying electricity to them. The 'way' has to be so devised that the annual fixed costs on account of the capital investment in the plant, transmission system, and the distribution system; and the annual operating cost to generate energy are recovered from the consumers in an equitable manner.

Consider that a transmission line conductor having weight per unit length as w is freely supported at two support points, P 1 and ? 2 ; ? 1 being on left of ? 2 . ? 1 andP2 may be at the same level, or at different levels. The horizontal distance between ? 1 and ? 2 is L . Because of the weight of the line, the line is in tension, and acquires the shape of a catenary. The coordinate axes are chosen to lie in the plane of the line with lowest point on the line as the origin 0 . The horizontal tangent to the line at point 0 and pointing to right is thex-axis and the vertical line through 0 is they-axis. Let the coordinates of a pointPon the line be (x,y), and the line length (curve) from 0 toP be s. The coordinates of the support points P 1 and ? 2 are (xI' y 1) and (x2, y2 ) respectively. The tension at 0 isH, and that at Pis T

The numerical value of any quantity expressed as a fraction of base (reference) value of that quantity, is its per unit (pu) value.

When in an area, the population density is high and the ovt>rhead wires become a potential hazard, or the overhead lines are not acceptable to the area as it mars its good looks, we resort to the use of underground cables for distribution, and in some cases for transmission too; even though cables are several times more expensive than the overhead lines.

Transmission lines are the highways of electric power. Large blocks of power glide along the line from one point to the other. As such one need to understand the behaviour of the lines under various operating conditions. Here, first we wish to examine the steady-state performance of a line under sinusoidal excitation and balanced conditions

A symmetrical fault is a balanced three-phase fault. It may be a direct fault, i.e. , a dead shortcircuit between the three phases at the fault point; or it may involve a balanced fault impedance, i.e., the fault introduces an impedancezfin each phase between the fault point and the shortcircuit point. A symmetrical fault can be analyzed on per phase basis; for transient, as well as for the steady state conditions. When analyzed this way, the transient state solution, however, shall not include the offset component.

A balanced three-phase system can always be analyzed as a single-phase circuit. An unbalanced system ofknown impedances can be analyzed directly by applying the usual methods of circuit analysis treating the three phase circuits as three circuits, and solving the equations simultaneously. But it would be more convenient if one could use the balanced system analysis approach.

Connecting the star p ints of the transformers, generators, and other star-connected equipment in the power system to the general mass of the earth is called grounding the neutrals. This connection to the ear h may be a direct connection, or a connection through some impedance. In either case we say that the neutral is grounded. If the neutral is not connected thus, we say it is an isolated o ungrounded neutral.

Stability of a power system is that attribute of the system due to which, on the occurrence of a disturbance, it develops counter forces within it so as to overcome the destabilizing effect of the disturbance, and it is able to continue operating normally, or in equilibrium.

The structure obtained after we replace each element of a network by a line segment, is the graph of the network. In the graph, the nature of the network elements is lost, but the connectivity is preserved. It has the same number of elements and nodes as the original network has. By assigning directions to the various elements, we get the oriented or the directed graph. 

For a three-phase elementpq, the number of quantities increase by a factor of three.

Load flow analysis in power systems is carried out to find -out the system bus voltages. The system network parameters and the injections at the buses are specified. In our normal nodal method of circuit analysis, the circuit parameters and the current sources at various nodes are specified, nodal equations are written and we can straight way solve these linear equations for node voltages.

As the loads on the power plants vary, so does their power generations. How should the plant load be distributed amongst the running units? There is an infinite number of combinations of loads on the units to meet a given load.

In power systems, frequency is very closely monitored and kept as close to the nominal value as possible, though a variation of3% is permitted. Any deviation from the nominal value affects the life of turbine blades of thermal units. The variations in frequency also cause variations in speeds of induction motors and synchronous clocks.