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Volt - unit of electrical potential or motive force - potential is required to send one ampere of current through one ohm of resistance Ohm - unit of resistance - one ohm is the resistance offered to the passage of one ampere when impelled by one volt

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ELECTRICAL ENGINEERING: The typical curriculum of an undergraduate electrical engineering student includes the subjects listed in Table 1.1. Although the distinction between some of these subjects is not always clear-cut, the table is sufficiently representative to serve our purposes. The aim of this book is to introduce the non-electrical engineering student to those aspects of electrical engineering that are likely to be most relevant to his or her professional career. Virtually all of the topics of Table 1.1 will be touched on in the book, with varying degrees of emphasis. The following example illustrates the pervasive presence of electrical, electronic, and electromechanical devices and systems in a very common application: the automobile. As you read Table 1.1 Electrical through the example, it will be instructive to refer to Table 1.1. Table 1.1 ELECTRICAL ENGINEERING

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Voltage and current:: Voltage attempts to make a current flow, and current will flow if the circuit is complete. Voltage is sometimes described as the 'push' or 'force' of the electricity, it isn't really a force but this may help you to imagine what is happening. It is possible to have voltage without current, but current cannot flow without voltage. Charge: electric charge, basic property of matter carried by some elementary particles. Electric charge, which can be positive or negative, occurs in discrete natural units and is neither created nor destroyed. Electric circuits: which are collections of circuit elements connected together, are the most fundamental structures of electrical engineering. A circuit is an interconnection of simple electrical devices that have at least one closed path in which current may flow. However, we may have to clarify to some of our readers what is meant by “current” and “electrical device,” a task that we shall undertake shortly. Circuits are important in electrical engineering because they process electrical signals, which carry energy and information; a signal can be any time varying electrical quantity. Engineering circuit analysis is a mathematical study of some useful interconnection of simple electrical devices.

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Alternating Current (AC): Most students of electricity begin their study with what is known as direct current (DC), which is electricity flowing in a constant direction, and/or possessing a voltage with constant polarity. DC is the kind of electricity made by a battery (with definite positive and negative terminals), or the kind of charge generated by rubbing certain types of materials against each other. As useful and as easy to understand as DC is, it is not the only “kind” of electricity in use. Certain sources of electricity (most notably, rotary electro-mechanical generators) naturally produce voltages alternating in polarity, reversing positive and negative over time. Either as a voltage switching polarity or as a current switching direction back and forth, this “kind” of electricity is known as Alternating Current (AC): Figure below

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Electric Potential and Voltage: When electrical forces act on a particle, it will possess potential energy. In order to describe the potential energy that a particle will have at a point x, the electric potential at point x is defined as where w(x) is the potential energy that a particle with charge q has when it is located at the position x. The zero point of potential energy can be chosen arbitrarily since only differences in energy have practical meaning. The point where electric potential is zero is known as the referencepoint or ground point, with respect to which potentials at other points are then described. The potential difference is known as the voltage expressed in volts (V) or joules per coulomb (J/C).If the potential at B is higher than that at A,

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AC Waveforms: When an alternator produces AC voltage, the voltage switches polarity over time, but does so in a very particular manner. When graphed over time, the “wave” traced by this voltage of alternating polarity from an alternator takes on a distinct shape, known as a sine wave: Figure below Graph of AC voltage over time (the sine wave) In the voltage plot from an electromechanical alternator, the change from one polarity to the other is a smooth one, the voltage level changing most rapidly at the zero (“crossover”) point and most slowly at its peak. If we were to graph the trigonometric function of “sine” over a horizontal range of 0 to 360 degrees, we would find the exact same pattern as in Table below. Trigonometric "sine" fuction...............

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Ohm's Law: relationship between current, voltage, and resistance is called Ohm's Law, discovered by Georg Simon Ohm and published in his 1827 paper, The Galvanic Circuit Investigated Mathematically. Ohm's principal discovery was that the amount of electric current through a metal conductor in a circuit is directly proportional to the voltage impressed across it, for any given temperature. Ohm expressed his discovery in the form of a simple equation, describing how voltage, current, and resistance interrelate: Applications of Ohm's Law: let's see how these equations might work to help us analyze simple circuits:

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RMS Value of an AC Waveform: The average value of an AC waveform is NOT the same value as that for a DC waveforms average value. This is because the AC waveform is constantly changing with time and the heating effect given by the formula ( P = I ².R ), will also be changing producing a positive power consumption. The equivalent average value for an alternating current system that provides the same power to the load as a DC equivalent circuit is called the "effective value". This effective power in an alternating current system is therefore equal to: ( I ².R.Average ). As power is proportional to current squared, the effective current, I will be equal to √ I ² Ave. Therefore, the effective current in an AC system is called the Root Mean Squared or R.M.S. value and RMS values are the DC equivalent values that provide the same power to the load. The effective or RMS value of an alternating current is measured in terms of the direct current value that produces the same heating effect in the same value resistance. The RMS value for any AC waveform can be found from the following modified average value formula.

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Kirchoff’s Laws: Arguably the most common and useful set of laws for solving electric circuits are the Kirchoff’s voltage and current laws. Several other useful relationships can be derived based on these laws. Kirchoff’s Voltage Law (KVL): "The sum of all the voltages (rises and drops) around a closed loop is equal to zero.” In other words, the algebraic sum of all voltage rises is equal to the algebraic sum of all the voltage drops around a closed loop. Example In each of the circuit diagrams in given figure, write the mesh equations using KVL.

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Generation of Three-phase Balanced Voltages: A polyphase system is basically an ac system composed of a certain number of single-phase ac systems having the same frequency and operating in sequence. Each phase of a polyphase system (i.e., the phase of each single-phase ac system) is displaced from the next by a certain angular interval. In any polyphase system, the value of the angular interval between each phase depends on the number of phases in the system. This manual covers the most common type of polyphase system, the three-phase system. Three-phase systems, also referred to as three-phase circuits, are polyphase systems that have three phases, as their name implies. They are no more complicated to solve than single-phase circuits. In the majority of cases, threephase circuits are symmetrical and have identical impedances in each of the circuit’s three branches (phases). Each branch can be treated exactly as a single-phase circuit, because a balanced three-phase circuit is simply a combination of three single-phase circuits. Therefore, voltage, current, and power relationships for three-phase circuits can be determined using the same basic equations and methods developed for single-phase circuits. Non-symmetrical, or unbalanced, three-phase circuits represent a special condition and their analysis is more complex. Unbalanced three-phase circuits are not covered in detail in this manual. A three-phase ac circuit is powered by three voltage sine waves having the same frequency and magnitude and which are displaced from each other by 120°. The phase shift between each voltage waveform of a three-phase ac power source is therefore 120° (360° ÷ 3 phases). Figure 1 shows an example of a simplified three-phase generator (alternator) producing three-phase ac power. A rotating magnetic field produced by a rotating magnet turns inside three identical coils of wire (windings) physically placed at a 120° angle from each other, thus producing three separate ac voltages (one per winding). Since the generator’s rotating magnet turns at a fixed speed, the frequency of the ac power that is produced is constant, and the three separate voltages attain the maximal voltage value one after the other at phase intervals of 120°.

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Power Factor: The term cos φ in Equation P_av = 1/2 (V_mI_mcosφ)is called the power factor and is an important parameter in determining the amount of actual power dissipated in the load. In practise, power factor is used to specify the characteristics of a load. For a purely resistive load φ = 0⁰, hence Unity Power Factor For a capacitive type load I leads V , hence Leading power factor

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Conductors and Insulators: Materials through which charge flows readily are called conductors. Examples include most metals, such as silver, gold, copper, and aluminum. Copper is used extensively for the conductive paths on electric circuit boards and for the fabrication of electrical wires. Insulators are materials that do not allow charge to move easily. Examples include glass, plastic, ceramics, and rubber. Electric current cannot be made to flow through an insulator, since a charge has great difficulty moving through it. One sees insulating (or dielectric) materials often wrapped around the center conducting core of a wire. one can say qualitatively that a conductor has a very low resistance to the flow of charge, whereas an insulator has a very high resistance to the flow of charge. Charge-conducting abilities of various materials vary in a wide range. Semiconductors fall in the middle between conductors and insulators, and have a moderate resistance to the flow of charge. Examples include silicon, germanium, and gallium arsenide.

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Three-Phase, Four-Wire System: It is not necessary to have six wires from the three phase windings to the three loads, provided there is a common ‘ return ’ line. Each winding will have a ‘ start ’ (S) and a ‘ fi nish ’ (F) end. The common connection mentioned above is achieved by connecting the corresponding ends of the three phases together. For example, either the three ‘ F ’ ends or the three ‘ S ’ ends are commoned. This form of connection is shown in Fig.(given below) , and is known as a star or Y connection. With the resulting 4-wire system, the three loads also are connected in star confi guration. The three outer wires are called the lines, and the common wire in the centre is called the neutral. If the three loads were identical in every way (same impedance and phase angle), then the currents fl owing in the three lines would be identical. If the waveform and/or phasor diagrams for these currents were drawn, they would be identical in form to Figs. A and B . These three currents meet at the star point of the load. The resultant current returning down the neutral wire would therefore be zero. The load in this case is known as a balanced load, and the neutral is not strictly necessary. However it is diffi cult, in practice, to ensure that each of the three loads are exactly balanced. For this reason the neutral is left in place. Also, since it has to carry only the relatively small ‘ out-ofbalance ’ current, it is made half the cross-sectional area of the lines. Let us now consider one of the advantages of this system compared with both a single-phase system, and the three-phase 6-wire system. Suppose that three identical loads are to be supplied with 200 A each. The two lines from a single-phase alternator would have to carry the total 600 A required. If a 3-phase, 6-wire system was used, then each line would have to carry only 200 A. Thus, the conductor csa would only need to be 1/3 that for the single-phase system, but of course, being six lines would entail using the same total amount of conductor material. If a 4-wire, 3-phase system is used there will be a saving on conductor costs in the ratio of 3.5:6 (the 0.5 being due to the neutral). If the power has to be sent over long transmission lines, such as the National Grid System, then the 3-phase, 4-wire system yields an enormous saving in cable costs. This is one of the reasons why the power generating companies use three-phase, star-connected generators to supply the grid system.

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The Average and Effective Value of an AC Waveform: The average or mean value of a continuous DC voltage will always be equal to its maximum peak value as a DC voltage is constant. This average value will only change if the duty cycle of the DC voltage changes. In a pure sine wave if the average value is calculated over the full cycle, the average value would be equal to zero as the positive and negative halves will cancel each other out. So the average or mean value of an AC waveform is calculated or measured over a half cycle only and this is shown below. To find the average value of the waveform we need to calculate the area underneath the waveform using the mid-ordinate rule, trapezoidal rule or Simpson's rule found in mathematics. The approximate area under any irregular waveform can easily be found by simply using the mid-ordinate rule. The zero axis base line is divided up into any number of equal parts and in our simple example above this value was nine, ( V₁ to V₉ ). The more ordinate lines that are drawn the more accurate will be the final average or mean value. The average value will be the addition of all the instantaneous values added together and then divided by the total number. This is given as.

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star-delta transformation/delta-star transformation: Three branches in an electrical network can be connected in numbers of forms but most common among them is either star or delta form. In delta connection three branches are so connected that they form a closed loop that is they are mesh connected. As these three branches are connected nose to tail they forms an triangular closed loop, this configuration is referred as delta connection. On the other hand when either terminal of three branches are connected to a common point to form a Y like pattern is known as star connection. But these star and delta connections can be transformed from one form to other. For simplifying complex network, it is often required delta to star or star to delta transformation. In many circuit applications, we encounter components connected together in one of two ways to form a three-terminal network: the “Delta,” or Δ (also known as the “Pi,” or π) configuration, and the “Y” (also known as the “T”) configuration.

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Wye and delta configurations: The windings of a three-phase ac power source can be connected in either a wye configuration, or a delta configuration. The configuration names are derived from the appearance of the circuit drawings representing the configurations, i.e., the letter Y for the wye configuration and the Greek letter delta for the delta configuration. The connections for each configuration are shown in Figurea. Each type of configuration has definite electrical characteristics. As Figure a shows, in a wye-connected circuit, one end of each of the three windings (or phases) of the three-phase ac power source is connected to a common point called the neutral. No current flows in the neutral because the currents flowing in the three windings (i.e., the phase currents) cancel each other out when the system is balanced. Wye connected systems typically consist of three or four wires (these wires connect to points A, B, C, and N in FigureA(1)), depending on whether or not the neutral line is present. Figureb shows that, in a delta-connected circuit, the three windings of the three-phase ac power source are connected one to another, forming a triangle. The three line wires are connected to the three junction points of the circuit (points A, B, and C in Figureb). There is no point to which a neutral wire can be connected in a three-phase delta-connected circuit. Thus, deltaconnected systems are typically three-wire systems.

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Phase Difference: the Sinusoidal Waveform (Sine Wave) can be presented graphically in the time domain along an horizontal zero axis, and that sine waves have a positive maximum value at time π/2, a negative maximum value at time 3π/2, with zero values occurring along the baseline at 0, π and 2π. However, not all sinusoidal waveforms will pass exactly through the zero axis point at the same time, but may be "shifted" to the right or to the left of 0^o by some value when compared to another sine wave. For example, comparing a voltage waveform to that of a current waveform. This then produces an angular shift or Phase Difference between the two sinusoidal waveforms. Any sine wave that does not pass through zero at t = 0 has a phase shift. The phase difference or phase shift as it is also called of a sinusoidal waveform is the angle Φ (Greek letter Phi), in degrees or radians that the waveform has shifted from a certain reference point along the horizontal zero axis. In other words phase shift is the lateral difference between two or more waveforms along a common axis and sinusoidal waveforms of the same frequency can have a phase difference. The phase difference, Φ of an alternating waveform can vary from between 0 to its maximum time period, T of the waveform during one complete cycle and this can be anywhere along the horizontal axis between, Φ = 0 to 2π (radians) or Φ = 0 to 360^o depending upon the angular units used. Phase difference can also be expressed as a time shift of τ in seconds representing a fraction of the time period, T for example, 10mS or - 50uS but generally it is more common to express phase difference as an angular measurement.

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Kirchoff’s Current Law (KCL): "The algebraic sum of all the currents entering or leaving a node in an electric circuit is equal to zero.” In other words, the sum of currents entering is equal to the sum of currents leaving the node in an electric circuit. Example: For the circuit diagrams depicted in given figure, write the nodal equations. Figure(a) contains just one node excluding the reference, hence one equation is required.

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Phase rotation: Let's take the three-phase alternator design laid out earlier (Figure below) and watch what happens as the magnet rotates. Three-phase alternator The phase angle shift of 120^o is a function of the actual rotational angle shift of the three pairs of windings (Figure below). If the magnet is rotating clockwise, winding 3 will generate its peak instantaneous voltage exactly 120^o (of alternator shaft rotation) after winding 2, which will hits its peak 120^o after winding 1. The magnet passes by each pole pair at different positions in the rotational movement of the shaft. Where we decide to place the windings will dictate the amount of phase shift between the windings' AC voltage waveforms. If we make winding 1 our “reference” voltage source for phase angle (0^o), then winding 2 will have a phase angle of -120^o (120^o lagging, or 240^o leading) and winding 3 an angle of -240^o (or 120^o leading).

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Concept of Phasor: A phase vector ("phasor") is a representation of a sine wavewhose amplitude (A), phase (θ), and frequency (ω) are time-invariant. It is a subset of a more general concept called analytic representation. Phasors reduce the dependencies on these parameters to three independent factors, thereby simplifying certain kinds of calculations. In particular the frequency factor, which also includes the time-dependence of the sine wave, is often common to all the components of a linear combination of sine waves. Using phasors, it can be factored out, leaving just the static amplitude and phase information to be combined algebraically (rather than trigonometrically). Similarly, linear differential equationscan be reduced to algebraicones. The term phasor therefore often refers to just those two factors. In older texts, a phasor is also referred to as a sinor. Addition of two out-of-phase sinusoidal signals is rather complicated in the time domain. An example could be the sum of voltages across a series connection of a resistor and an inductor. Phasors simplify this analysis by considering only the amplitude and phase components of the sine wave. Moreover, they can be solved using complex algebra or treated vectorially using a vector diagram. Consider a vector quantity A in the complex plane as shown in Figure (a). Then

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passive element: The element which receives energy (or absorbs energy) and then either converts it into heat (R) or stored it in an electric (C) or magnetic (L ) field is called passive element. Active element: The elements that supply energy to the circuit is called active element. Examples of active elements include voltage and current sources, generators, and electronic devices that require power supplies. A transistor is an active circuit element, meaning that it can amplify power of a signal. On the other hand, transformer is not an active element because it does not amplify the power level and power remains same both in primary and secondary sides. Transformer is an example of passive element

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Distinction between line and phase voltages, and line and phase currents: The voltage produced by a single winding of a three-phase circuit is called the line-to-neutral voltage, or simply the phase voltage, E_phase In a wye-connected three-phase ac power source, the phase voltage is measured between the neutral line and any one of points A, B, and C. This results in the following three distinct phase voltages: E_A-N, E_B-N, and E_C-N. The voltage between any two windings of a three-phase circuit is called the lineto line voltage, or simply the line voltage E_line. In a wye-connected three-phase ac power source, the line voltage is √3 (approximately 1.73) times greater than the phase voltage (i.e., E_line = √3 E_phase). In a delta-connected three-phase ac power source, the voltage between any two windings is the same as the voltage across the third winding of the source (i.e.,E_line = E_phase), as shows Figure 3b. In both cases, this results in the following three distinct line voltages: E_A-B, E_B-C, and E_C-A. The three line wires (wires connected to points A, B, and C) and the neutral wire of a three-phase power system are usually available for connection to the load, which can be connected in either a wye configuration or a delta configuration. The two types of circuit connections are illustrated in Figure 1. Circuit analysis demonstrates that the voltage (line voltage) between any two line wires, or lines, in a wye-connected load is √3 times greater than the voltage (phase voltage) across each load resistor. Furthermore, the line current l_ine flowing in each line of the power source is equal to the phase current I_phase flowing in each load resistor. On the other hand, in a delta-connected load, the voltage (phase voltage) across each load resistor is equal to the line voltage of the source. Also, the line current is √3 times greater than the current (phase current) in each load resistor. The phase current in a delta connected load is therefore √3 times smaller than the line current.

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Notice that I'm assuming a perfectly reactive inductor and capacitor, with impedance phase angles of exactly 90 and -90^o, respectively. Although real components won't be perfect in this regard, they should be fairly close. For simplicity, I'll assume perfectly reactive inductors and capacitors from now on in my example calculations except where noted otherwise. Since the above example circuit is a series circuit, we know that the total circuit impedance is equal to the sum of the individuals, so:

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we tend to associate the word "positive" with "surplus" and "negative" with "deficiency," the standard label for electron charge does seem backward. Because of this, many engineers decided to retain the old concept of electricity with "positive" referring to a surplus of charge, and label charge flow (current) accordingly. This became known as conventional flow notation: Others chose to designate charge flow according to the actual motion of electrons in a circuit. This form of symbology became known as electron flow notation:

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Three-phase Y and Delta configurations: Initially we explored the idea of three-phase power systems by connecting three voltage sources together in what is commonly known as the “Y” (or “star”) configuration. This configuration of voltage sources is characterized by a common connection point joining one side of each source. (Figure below) Three-phase “Y” connection has three voltage sources connected to a common point.

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Generation of Sinusoidal (AC) Voltage Waveform: if a single wire conductor is moved or rotated within a stationary magnetic field, an "EMF", (Electro-Motive Force) will be induced within the conductor due to this movement. From this tutorial we learnt that a relationship exists between Electricity and Magnetism giving us, as Michael Faraday discovered the effect of "Electromagnetic Induction" and it is this basic principal that is used to generate a Sinusoidal Waveform. In the Electromagnetic Induction, tutorial we said that when a single wire conductor moves through a permanent magnetic field thereby cutting its lines of flux, an EMF is induced in it. However, if the conductor moves in parallel with the magnetic field in the case of points A and B, no lines of flux are cut and no EMF is induced into the conductor, but if the conductor moves at right angles to the magnetic field as in the case of points C and D, the maximum amount of magnetic flux is cut producing the maximum amount of induced EMF.

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Polarity of voltage drops: We can trace the direction that electrons will flow in the same circuit by starting at the negative (-) terminal and following through to the positive ( ) terminal of the battery, the only source of voltage in the circuit. From this we can see that the electrons are moving counter-clockwise, from point 6 to 5 to 4 to 3 to 2 to 1 and back to 6 again. As the current encounters the 5 Ω resistance, voltage is dropped across the resistor's ends. The polarity of this voltage drop is negative (-) at point 4 with respect to positive ( ) at point 3. We can mark the polarity of the resistor's voltage drop with these negative and positive symbols, in accordance with the direction of current (whichever end of the resistor the current is entering is negative with respect to the end of the resistor it is exiting:

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Power in balanced three-phase circuits: The formulas for calculating active, reactive, and apparent power in balanced three-phase circuits are the same as those used for single-phase circuits. Based on the formula for power in a single-phase circuit, the active power dissipated in each phase of either a wye- or delta-connected load is equal to: where P_phase is the active power dissipated in each phase of a three-phase circuit, expressed in watts (W) I_phase is the phase voltage across each phase of a three-phase circuit, expressed in volts (V) E_phase is the phase current flowing in each phase of a three-phase circuit, expressed in amperes (A) φ is the angle between the phase voltage and current in each phase of a three-phase circuit, expressed in degrees (°)

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AC inductor circuits: Inductors do not behave the same as resistors. Whereas resistors simply oppose the flow of electrons through them (by dropping a voltage directly proportional to the current), inductors oppose changes in current through them, by dropping a voltage directly proportional to the rate of change of current. In accordance with Lenz's Law, this induced voltage is always of such a polarity as to try to maintain current at its present value. That is, if current is increasing in magnitude, the induced voltage will “push against” the electron flow; if current is decreasing, the polarity will reverse and “push with” the electron flow to oppose the decrease. This opposition to current change is called reactance, rather than resistance. Expressed mathematically, the relationship between the voltage dropped across the inductor and rate of current change through the inductor is as such:

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Thevenin's Theorem: Thevenin's Theorem states that it is possible to simplify any linear circuit, no matter how complex, to an equivalent circuit with just a single voltage source and series resistance connected to a load. The qualification of "linear" is identical to that found in the Superposition Theorem, where all the underlying equations must be linear (no exponents or roots). If we're dealing with passive components (such as resistors, and later, inductors and capacitors), this is true. However, there are some components (especially certain gas-discharge and semiconductor components) which are nonlinear: that is, their opposition to current changes with voltage and/or current. As such, we would call circuits containing these types of components, nonlinear circuits. Thevenin's Theorem is especially useful in analyzing power systems and other circuits where one particular resistor in the circuit (called the "load" resistor) is subject to change, and re-calculation of the circuit is necessary with each trial value of load resistance, to determine voltage across it and current through it. Let's take another look at our example circuit:

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Power Transmission and Distribution: The structure of a power system can be divided into generation (G), transmission, (T), and distribution (D) facilities, as shown in Figure (A).An ac three-phase generating system provides the electric energy; this energy is transported over a transmission network, designed to carry power at high or extrahigh voltages over long distances from generators to bulk power substations and major load points; the subtransmission network is a medium to high-voltage network whose purpose is to transport power over shorter distances from bulk power substations to distribution substations. The transmission and subtransmission systems are meshed networks with multiplepath structure so that more than one path exists from one point to another to increase the reliability of the transmission system. The transmission system, in general, consists of overhead transmission lines (on transmission towers), transformers to step up or step down voltage levels, substations, and various protective devices such as circuit breakers, relays, and communication and control mechanisms. Figure A shows a typical electric-power distribution system and its components. Below the subtransmission level, starting with the distribution substation, the distribution system usually consists of distribution transformers, primary distribution lines or main feeders, lateral feeders, distribution transformers, secondary distribution circuits, and customers’ connections with metering. Depending on the size of their power demand, customers may be connected to the transmission system, the subtransmission system, or the primary or secondary distribution circuit. fig(A)Typical power-system structure. In central business districts of large urban areas, the primary distribution circuits consist of underground cables which are used to interconnect the distribution transformers in an electric network. With this exception, the primary system is most often radial. However, for additional reliability and backup capability, a loop-radial configuration is frequently used. The main feeder is looped through the load area and brought back to the substation, and the two ends of the loop are connected to the substation by two separate circuit breakers. For normal operation, selected sectionalizing switches are opened so as to form a radial configuration. Under fault conditions, the faulted section is isolated and the rest of the loop is used to supply the unaffected customers.

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Measurement of Power in Three phase circuit: Real power in balanced (or unbalanced ) 3 phase circuits can be measured by using two wattmeters connected in any two lines of a three-phase three wire system. In Fig.4(a) the current coils are connected in lines A and C, with the voltage coil reference connections in line B. The current °owing through the current coil of the wattmeter connected in line A is I_A and that in line C is I_C. Similarly, the voltage applied to the voltage coils is V _ab and V _cb (and not V _bc) respectively. Their readings will be: From Fig.3 (a or b) angle between Vab and I_A is (30 θ) and between V_cb and Ic is (30-θ). Therefore the wattmeter readings will be:

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Power in AC Circuits: From DC circuit analysis, power is given by the following relationship: P = V I Watts Therefore the instantaneous power is given by p(t) = v(t) · i(t). Let i(t) = I_m sin(ωt β) be the current flowing through an impedance, Z, then the voltage across the impedance will be v(t) = V_m sin(ωt α). Therefore p(t) = VmIm sin(ωt α) sin(ωt β)

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Branch current method: The first and most straightforward network analysis technique is called the Branch Current Method. In this method, we assume directions of currents in a network, then write equations describing their relationships to each other through Kirchhoff's and Ohm's Laws. Once we have one equation for every unknown current, we can solve the simultaneous equations and determine all currents, and therefore all voltage drops in the network. Let's use this circuit to illustrate the method:

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Introduction to Power System: Thomas A. Edison’s work in 1878 on the electric light led to the concept of a centrally located power station with distributed electric power for lighting in a surrounding area. The opening of the historic Pearl Street Station in New York City on September 4, 1882, with dc generators (dynamos) driven by steam engines, marked the beginning of the electric utility industry. Edison’s dc systems expanded with the development of three-wire 220-V dc systems. But as transmission distances and loads continued to grow, voltage problems were encountered. With the advent of William Stanley’s development of a commercially practical transformer in 1885, alternating current became more attractive than direct current because of the ability to transmit power at high voltage with corresponding lower current and lower line-voltage drops. The first single-phase ac line (21 km at 4 kV) in the United States operated in 1889 between Oregon City and Portland. Nikola Tesla’s work in 1888 on electric machines made evident the advantages of polyphase over single-phase systems. The first three-phase line (12 km at 2.3 kV) in the United States became operational in California during 1893. The three-phase induction motor conceived by Tesla became the workhorse of the industry. Today the two standard frequencies for the generation, transmission, and distribution of electric power in the world are 60 Hz (in the United States, Canada, Japan, and Brazil) and 50 Hz (in Europe, the former Soviet Republics, South America except Brazil, India, and also Japan). Relatively speaking, the 60-Hz power-system apparatus is generally smaller in size and lighter in weight than the corresponding 50-Hz equipment with the same ratings. On the other hand, transmission lines and transformers have lower reactances at 50 Hz than at 60 Hz. Along with increases in load growth, there have been continuing increases in the size of generating units and in steam temperatures and pressure, leading to savings in fuel costs and overall operating costs. Ac transmission voltages in the United States have also been rising steadily: 115, 138, 161, 230, 345, 500, and now 765 kV. Ultrahigh voltages (UHV) above 1000 kV are now being studied. Some of the reasons for increased transmission voltages are:

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Dynamometer type Wattmeter: Principle: When a current carrying moving coil is placed in a magnetic field produced by the current carrying fixed coil, a mechanical force is exerted on the coil sides of the moving coil and deflection takes place. when the field produced by the current carrying moving coil(F_r) tries to come in line with the field produced by the current carrying fixed coil (F_m), a deflecting torque is exerted on the moving system.

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Power Factor Correction: As mentioned above, to minimise I²R losses in AC networks, it may be necessary to manipulate the power factor. This can be achieved by connecting a purely reactive impedance, Z' in parallel with the load impedance as depicted in Figure below: (figure)Power factor correction by introducing a pure reactance in parallel with the load impedance.

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DC Sources: In general, there are two main types of DC sources An independent source produces its own voltage and current through some chemical reaction and does not depend on any other voltage or current variable in the circuit. The output of a dependent source, on the other hand, is subject to a certain parameter (voltage or current) change in a circuit element. Herein, the discussion shall be confined to independent sources only. DC Voltage Source: This can be further subcategorised into ideal and non-ideal sources.

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Magnetic Circuit: In this lesson, we shall acquaint the reader, primarily with the basic concepts of magnetic circuit and methods of solving it. Biot-Savart law for calculating magnetic field due to a known current distribution although fundamental and general in nature, requires an integration to be evaluated which sometimes become an uphill task. Fortunately, due to the specific nature of the problem, Ampere’s circuital law (much easier to apply) is adopted for calculating field in the core of a magnetic circuit. You will also understand the importance of B-H curve of a magnetic material and its use. The concept and analysis of linear and non linear magnetic circuit will be explained. The lesson will conclude with some worked out examples. Key Words: mmf, flux, flux density, mean length, permeability, reluctance. Before really starting, let us look at some magnetic circuits shown in the following figures.

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Working Principles of PMMC: It has been mentioned that the interaction between the induced field and the field produced by the permanent magnet causes a deflecting torque, which results in rotation of the coil. The deflecting torque produced is described below in mathematical form: Deflecting Torque: If the coil is carrying a current of , the force on a coil side = BIL N (newton, N). .....EQ(1) .......EQ(2)

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Quality Factor and Bandwidth of a Resonant Circuit: The Q, quality factor, of a resonant circuit is a measure of the “goodness” or quality of a resonant circuit. A higher value for this figure of merit corresponds to a more narrow bandwidth, which is desirable in many applications. More formally, Q is the ration of power stored to power dissipated in the circuit reactance and resistance, respectively: Q = P_stored/P_dissipated = I²X/I²R Q = X/R

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Source Transformation: In an electric circuit, it is often convenient to have a voltage source rather than a current source (e.g. in mesh analysis) or vice versa. This is made possible using source transformations. It should be noted that only practical voltage and current sources can be transformed. In other words, a Th´evenin’s equivalent circuit is transformed into a Norton’s one or vice versa. The parameters used in the source transformation are given as follows. Any load resistance, RL will have the same voltage across, and current through it when connected across the terminals of either source.

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Auto-Transformer: So far we have considered a 2-winding transformer as a means for changing the level of a given voltage to a desired voltage level. It may be recalled that a 2-winding transformer has two separate magnetically coupled coils with no electrical connection between them. In this lesson we shall show that change of level of voltage can also be done quite effectively by using a single coil only. The idea is rather simple to understand. Suppose you have a single coil of 200 turns (= N_BC) wound over a iron core as shown in figure 27.1. If we apply an a.c voltage of 400 V, 50 Hz to the coil (between points B and C), voltage per turn will be 400/200 = 2 V. If we take out a wire from one end of the coil say C and take out another wire tapped from any arbitrary point E, we would expect some voltage available between points E and C. The magnitude of the voltage will obviously be 2 × N_EC where N_EC is the number of turns between points E and C. If tapping has been taken in such a way that N_EC = 100, voltage between E and C would be 200 V. Thus we have been able to change 400 V input voltage to a 200 V output voltage by using a single coil only. Such transformers having a single coil with suitable tapings are called autotransformers. It is possible to connect a conventional 2-winding transformer as an autotransformer or one can develop an autotransformer as a single unit.\ fig(A) transformer with a single coil 2-winding transformer as Autotransformer: Suppose we have a single phase 200V/100V, 50Hz, 10kVA two winding transformer with polarity markings. Then the coils can be connected in various ways to have voltage ratios other than 2 also, as shown in figure (B)

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Introduction of measuring instruments: Measuring instruments are classified according to both the quantity measured by the instrument and the principle of operation. Three general principles of operation are available: (i) electromagnetic, which utilizes the magnetic effects of electric currents; (ii) electrostatic, which utilizes the forces between electrically-charged conductors; (iii) electro-thermic, which utilizes the heating effect. Electric measuring instruments and meters are used to indicate directly the value of current, voltage, power or energy. In this lesson, we will consider an electromechanical meter (input is as an electrical signal results mechanical force or torque as an output) that can be connected with additional suitable components in order to act as an ammeters and a voltmeter. The most common analogue instrument or meter is the permanent magnet moving coil instrument and it is used for measuring a dc current or voltage of a electric circuit. On the other hand, the indications of alternating current ammeters and voltmeters must represent the RMS values of the current, or voltage, respectively, applied to the instrument.

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The Cosine Waveform: So we now know that if a waveform is "shifted" to the right or left of 0^o when compared to another sine wave the expression for this waveform becomes A_m sin(ωt ± Φ). But if the waveform crosses the horizontal zero axis with a positive going slope 90^o or π/2 radians before the reference waveform, the waveform is called a Cosine Waveform and the expression becomes. The Cosine Wave, simply called "cos", is as important as the sine wave in electrical engineering. The cosine wave has the same shape as its sine wave counterpart that is it is a sinusoidal function, but is shifted by 90^o or one full quarter of a period ahead of it.

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Mesh current method: The Mesh Current Method is quite similar to the Branch Current method in that it uses simultaneous equations, Kirchhoff's Voltage Law, and Ohm's Law to determine unknown currents in a network. It differs from the Branch Current method in that it does not use Kirchhoff's Current Law, and it is usually able to solve a circuit with less unknown variables and less simultaneous equations, which is especially nice if you're forced to solve without a calculator. Let's see how this method works on the same example problem:

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Practical transformer: A practical transformer will differ from an ideal transformer in many ways. For example the core material will have finite permeability, there will be eddy current and hysteresis losses taking place in the core, there will be leakage fluxes, and finite winding resistances. We shall gradually bring the realities one by one and modify the ideal transformer to represent those factors. Consider a transformer which requires a finite magnetizing current for establishing flux in the core. In that case, the transformer will draw this current Im even under no load condition. The level of flux in the core is decided by the voltage, frequency and number of turns of the primary and does not depend upon the nature of the core material used which is apparent from the following equation:

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A multi-range ammeters: An ammeter is required to measure the current in a circuit and it therefore connected in series with the components carrying the current. If the ammeter resistance is not very much smaller than the load resistance, the load current can be substantially altered by the inclusion of the ammeter in the circuit. To operate a moving coil instrument around a current level 50ma is impractical owing to the bulk and weight of the coil that would be required. So, it is necessary to extend the meter-range shunts (in case of ammeters) and multipliers (in case of volt meters) are used in the following manner. For higher range ammeters a low resistance made up of manganin (low temperature coefficient of resistance) is connected in parallel to the moving coil (see Fig.A) and instrument may be calibrated to read directly to the total current. ................fig (A) They are called shunts. The movement of PMMC instrument is not inherently insensitive to temperature, but it may be temperature-compensated by the appropriate use of series and shunt resistors of copper and manganin. Both the magnetic field strength and spring-tension decrease with an increase in temperature. On other side, the coil resistance

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Series resistor-inductor circuits: Now we will mix the two components together in series form and investigate the effects. Take this circuit as an example to work with: (Figure below)

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Introduction to network theorems: Anyone who's studied geometry should be familiar with the concept of a theorem: a relatively simple rule used to solve a problem, derived from a more intensive analysis using fundamental rules of mathematics. At least hypothetically, any problem in math can be solved just by using the simple rules of arithmetic (in fact, this is how modern digital computers carry out the most complex mathematical calculations: by repeating many cycles of additions and subtractions!), but human beings aren't as consistent or as fast as a digital computer. We need "shortcut" methods in order to avoid procedural errors. In electric network analysis, the fundamental rules are Ohm's Law and Kirchhoff's Laws. While these humble laws may be applied to analyze just about any circuit configuration (even if we have to resort to complex algebra to handle multiple unknowns), there are some "shortcut" methods of analysis to make the math easier for the average human.

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ideal transformer: Transformers are one of the most important components of any power system. It basically changes the level of voltages from one value to the other at constant frequency. Being a static machine the efficiency of a transformer could be as high as 99%. Big generating stations are located at hundreds or more km away from the load center (where the power will be actually consumed). Long transmission lines carry the power to the load centre from the generating stations. Generator is a rotating machines and the level of voltage at which it generates power is limited to several kilo volts only a typical value is 11 kV. To transmit large amount of power (several thousands of mega watts) at this voltage level means large amount of current has to flow through the transmission lines. The cross sectional area of the conductor of the lines accordingly should be large. Hence cost involved in transmitting a given amount of power rises many folds. Not only that, the transmission lines has their own resistances. This huge amount of current will cause tremendous amount of power loss or I2r loss in the lines. This loss will simply heat the lines and becomes a wasteful energy. In other words, efficiency of transmission becomes poor and cost involved is high. The above problems may addressed if we could transmit power at a very high voltage say, at 200 kV or 400 kV or even higher at 800 kV. But as pointed out earlier, a generator is incapable of generating voltage at these level due to its own practical limitation. The solution to this problem is to use an appropriate step-up transformer at the generating station to bring the transmission voltage level at the desired value as depicted in figure A where for simplicity single phase system is shown to understand the basic idea. Obviously when power reaches the load centre, one has to step down the voltage to suitable and safe values by using transformers. Thus transformers are an integral part in any modern power system. Transformers are located in places called substations. In cities or towns you must have noticed transformers are installed on poles – these are called pole mounted distribution transformers. These type of transformers change voltage level typically from 3-phase, 6 kV to 3-phase 440 V line to line.

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Multi-range voltmeter: A dc voltmeter is constructed by a connecting a resistor in series with a PMMC instrument. Unlike an ammeter, a voltmeter should have a very high resistancese R_se and it is normally connected in parallel with the circuit where the voltage is to be measured (see Fig.A). To minimize voltmeter loading, the voltmeter operating current should be very small i.e., the resistance connected in series with the coil should be high. ..........fig(A) The moving coil instruments can be suitably modified to act either as an ammeter or as a voltmeter. For multi-range voltmeters the arrangement is as follows: In Fig.42.5 any one of several multiplier resistors is selected by means of a rotary switch. Unlike the case of the ammeter, the rotary switch used with the voltmeter should be a break-before make type, that is, moving contact should disconnect from one terminal before connecting to the next terminal.

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Parallel R, L, and C: We can take the same components from the series circuit and rearrange them into a parallel configuration for an easy example circuit: The fact that these components are connected in parallel instead of series now has absolutely no effect on their individual impedances. So long as the power supply is the same frequency as before, the inductive and capacitive reactances will not have changed at all:

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Maximum Power Transfer Theorem: The Maximum Power Transfer Theorem is not so much a means of analysis as it is an aid to system design. Simply stated, the maximum amount of power will be dissipated by a load resistance when that load resistance is equal to the Thevenin/Norton resistance of the network supplying the power. If the load resistance is lower or higher than the Thevenin/Norton resistance of the source network, its dissipated power will be less than maximum. This is essentially what is aimed for in stereo system design, where speaker "impedance" is matched to amplifier "impedance" for maximum sound power output. Impedance, the overall opposition to AC and DC current, is very similar to resistance, and must be equal between source and load for the greatest amount of power to be transferred to the load. A load impedance that is too high will result in low power output. A load impedance that is too low will not only result in low power output, but possibly overheating of the amplifier due to the power dissipated in its internal (Thevenin or Norton) impedance.

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Efficiency of transformer: In a practical transformer we have seen mainly two types of major losses namely core and copper losses occur. These losses are wasted as heat and temperature of the transformer rises. Therefore output power of the transformer will be always less than the input power drawn by the primary from the source and efficiency is defined as We have seen that from no load to the full load condition the core loss, Pcore remains practically constant since the level of flux remains practically same. On the other hand we know that the winding currents depend upon the degree of loading and copper loss directly depends upon the square of the current and not a constant from no load to full load condition. We shall write a general expression for efficiency for the transformer operating at x per unit loading and delivering power to a known power factor load. Let,

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Basic principle operation of Moving-iron Instruments: We have mentioned earlier that the instruments are classified according to the principles of operation. Furthermore, each class may be subdivided according to the nature of the movable system and method by which the operating torque is produced. Specifically, the electromagnetic instruments are sub-classes as (i) moving-iron instruments (ii) electro-dynamic or dynamometer instruments, (iii) induction instruments. In this section, we will discuss briefly the basic principle of moving-iron instruments that are generally used to measure alternating voltages and currents. In moving –iron instruments the movable system consists of one or more pieces of specially-shaped soft iron, which are so pivoted as to be acted upon by the magnetic field produced by the current in coil. There are two general types of moving-iron instruments namely (i) Repulsion (or double iron) type (ii) Attraction (or single-iron) type. The brief description of different components of a moving-iron instrument is given below.

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Series-parallel R, L, and C: Take this series-parallel circuit for example: The first order of business, as usual, is to determine values of impedance (Z) for all components based on the frequency of the AC power source. To do this, we need to first determine values of reactance (X) for all inductors and capacitors, then convert reactance (X) and resistance (R) figures into proper impedance (Z) form:

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Bilateral element: Conduction of current in both directions in an element (example: Resistance; Inductance; Capacitance) with same magnitude is termed as bilateral element. Unilateral Element: Conduction of current in one direction is termed as unilateral (example: Diode, Transistor) element.

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B-H Characteristics : A magnetic material is identified and characterized by its B – H characteristic. In free space or in air the relationship between the two is linear and the constant of proportionality is the permeability μ0. If B is plotted against H, it will be straight a line. However, for most of the materials the relationship is not linear and is as shown in figure (A). A brief outline for experimental determination of B-H characteristic of a given material is given now. First of all a sample magnetic circuit (with the given material) is fabricated with known dimensions and number of turns. Make a circuit arrangement such as shown in Figure 21.8, to increase the current from 0 to some safe maximum value. Apart from ammeter reading one should record the amount of flux produced in the core by using a flux meter-let us not bother how this meter works. Now corresponding to this current, calculate H=NI/l and B=φ/a and tabulate them. Thus we have several pair of H & B values for different values of currents. Now by choosing H to be the x axis B to be the y axis and plotting the above values one gets a typical B-H curve as shown in Figure (A)below.

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General Theory Permanent Magnet Moving Coil (PMMC) Instruments: The general theory of moving-coil instruments may be dealt with considering a rectangular coil of N turns, free to rotate about a vertical axis. fig.A fig..B Fig. (given above) shows the basic construction of a PMMC instrument. A moving coil instrument consists basically of a permanent magnet to provide a magnetic field and a small lightweight coil is wound on a rectangular soft iron core that is free to rotate around

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Representation of Sinusoidal Signal by a Phasor: A sinusoidal quantity, i.e. current, , is taken up as an example. In Fig. a, the length, OP, along the x-axis, represents the maximum value of the current I_m, on a certain scale. It is being rotated in the anti-clockwise direction at an angular speed, ω , and takes up a position, OA after a time t (or angle, θ = ω t , with the x-axis). The vertical projection of OA is plotted in the right hand side of the above figure with respect to the angle θ . It will generate a sine wave (Fig. b), as OA is at an angle, θ with the x-axis, as stated earlier. The vertical projection of OA along y-axis is OC = AB =Iθ=i_m sinθ, which is the instantaneous value of the current at any time t or angle θ . The angle θ is in rad., i.e. θ = ω t . The angular speed, ω is in rad/s, i.e.ω = 2π f , where f is the frequency in H_z or cycles/sec. Thus, i= I_m sinθ =I_m sinωt=I_msin 2πft So, OP represents the phasor with respect to the above current, i.

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loop and nodal methods of analysis: We have seen that using Kirchhoff’s laws and Ohm’s law we can analyze any circuit to determine the operating conditions (the currents and voltages). The challenge of formal circuit analysis is to derive the smallest set of simultaneous equations that completely define the operating characteristics of a circuit. In this lecture we will develop two very powerful methods for analyzing any circuit: The node method and the mesh method. These methods are based on the systematic application of Kirchhoff’s laws. We will explain the steps required to obtain the solution by considering the circuit example shown on Figure 1. Node Method: A voltage is always defined as the potential difference between two points. When we talk about the voltage at a certain point of a circuit we imply that the measurement is performed between that point and some other point in the circuit. In most cases that other point is referred to as ground. The node method or the node voltage method, is a very powerful approach for circuit analysis and it is based on the application of KCL, KVL and Ohm’s law. The procedure for analyzing a circuit with the node method is based on the following steps.

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Eddy current: Look at the Figure (A) where a rectangular core of magnetic material is shown along with the exciting coil wrapped around it. Without any loss of generality, one may consider this to be a part of a magnetic circuit. If the coil is excited from a sinusoidal source, exciting current flowing will be sinusoidal too. Now put your attention to any of the cross section of the core and imagine any arbitrary rectangular closed path abcd. An emf will be induced in the path abcd following Faraday’s law. Here of course we don’t require a switch S to close the path because the path is closed by itself by the conducting magnetic material (say iron). Therefore a circulating current i_eddy will result. The direction of ieddy is shown at the instant when B(t) is increasing with time. It is important to note here that to calculate induced voltage in the path, the value of flux to be taken is the flux enclosed by the path i.e., φmax=Bmax*area of the loop abcd. The magnitude of the eddy current will be limited by the path resistance, R_path neglecting reactance effect. Eddy current will therefore cause power loss in R_path and heating of the core. To calculate the total eddy current loss in the material we have to add all the power losses of different eddy paths covering the whole cross section. Use of thin plates or laminations for core: We must see that the power loss due to eddy current is minimized so that heating of the core is reduced and efficiency of the machine or the apparatus is increased. It is obvious if the cross sectional area of the eddy path is reduced then eddy voltage induced too will be reduced (E_eddy ∞ area), hence eddy loss will be less. This can be achieved by using several thin electrically insulated plates (called laminations) stacked together to form the core instead a solid block of iron. The idea is depicted in the Figure (B) where the plates have been shown for clarity, rather separated from each other. While assembling the core the laminations are kept closely pact. Conclusion is that solid block of iron should not be used to construct the core when exciting current will be ac. However, if exciting current is dc, the core need not be laminated. fig.(A)

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Shunts and Multipliers for MI instruments: For moving-iron ammeters: For the circuit shown in Fig.), let R_m and L_mare respectively the resistance and inductance of the coil and R_sh and L_sh the corresponding values for shunt. fig.(A) The ratio of currents in two parallel branches is

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Review of Resistance, Reactance, and Impedance: Resistance is essentially friction against the motion of electrons. It is present in all conductors to some extent (except superconductors!), most notably in resistors. When alternating current goes through a resistance, a voltage drop is produced that is in-phase with the current. Resistance is mathematically symbolized by the letter “R” and is measured in the unit of ohms (Ω). Reactance is essentially inertia against the motion of electrons. It is present anywhere electric or magnetic fields are developed in proportion to applied voltage or current, respectively; but most notably in capacitors and inductors. When alternating current goes through a pure reactance, a voltage drop is produced that is 90^o out of phase with the current. Reactance is mathematically symbolized by the letter “X” and is measured in the unit of ohms (Ω). Impedance is a comprehensive expression of any and all forms of opposition to electron flow, including both resistance and reactance. It is present in all circuits, and in all components. When alternating current goes through an impedance, a voltage drop is produced that is somewhere between 0^o and 90^o out of phase with the current. Impedance is mathematically symbolized by the letter “Z” and is measured in the unit of ohms (Ω), in complex form.

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Norton's Theorem: Norton's Theorem states that it is possible to simplify any linear circuit, no matter how complex, to an equivalent circuit with just a single current source and parallel resistance connected to a load. Just as with Thevenin's Theorem, the qualification of "linear" is identical to that found in the Superposition Theorem: all underlying equations must be linear (no exponents or roots). Contrasting our original example circuit against the Norton equivalent: it looks something like this:

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Inductor: One can make an inductor L, by having several turns N, wound over a core as shown in figure(A). In an ideal inductor, as we all know, no power loss takes place. Therefore, we must use a very good magnetic material having negligible B-H loop area. Also we must see that the operating point lies in the linear zone of the B-H characteristic in order to get a constant value of the inductance. This means μ_r may be assumed to be constant. To make eddy current loss vanishingly small, let us assume the lamination thickness is extremely small and the core material has a very high resistivity ρ. Under these assumptions let us derive an expression for the inductance L, in order to have a feeling on the factors it will depend upon. Let us recall that inductance of a coil is defined as the flux linkage with the coil when 1 A flows through it. fig(A) an inductor

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Construction of Moving-iron Instruments: The deflecting torque in any moving-iron instrument is due to forces on a small piece of magnetically ‘soft’ iron that is magnetized by a coil carrying the operating current. In repulsion (Fig.42.7) type moving–iron instrument consists of two cylindrical soft iron vanes mounted within a fixed current-carrying coil. One iron vane is held fixed to the coil frame and other is free to rotate, carrying with it the pointer shaft. Two irons lie in the magnetic field produced by the coil that consists of only few turns if the instrument is an ammeter or of many turns if the instrument is a voltmeter. Current in the coil induces both vanes to become magnetized and repulsion between the similarly magnetized vanes produces a proportional rotation. The deflecting torque is proportional to the square of the current in the coil, making the instrument reading is a true ‘RMS’ quantity Rotation is opposed by a hairspring that produces the restoring torque. Only the fixed coil carries load current, and it is constructed so as to withstand high transient current. Moving iron instruments having scales that are nonlinear and somewhat crowded in the lower range of calibration. Another type of instrument that is usually classed with the attractive types of instrument is shown in Fig.42.8. ----fig(A) repultion type---------fig(B)attraction type This instrument consists of a few soft iron discs (B) that are fixed to the spindle (D), pivoted in jeweled bearings. The spindle (D) also carries a pointer (P), a balance weight (W₁), a controlling weight (W₂) and a damping piston (E), which moves in a curved fixed cylinder (F). The special shape of the moving-iron discs is for obtaining a scale of suitable form.

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Phasor representation of Voltage and Current: The voltage and current waveforms are given as: It can be seen from the waveforms (Fig. b) of the two sinusoidal quantities – voltage and current, that the voltage, V lags the current I, which means that the positive maximum value of the voltage is reached earlier by an angle, φ , as compared to the positive maximum value of the current. In phasor notation as described earlier, the voltage and current are represented by OP and OQ (Fig. a) respectively, the length of which are proportional to voltage, V and current, I in different scales as applicable to each one. The voltage phasor, OP (V) lags the current phasor, OQ (I) by the angleφ , as two phasors rotate in the anticlockwise direction as stated earlier, whereas the angleφ is also measured in the anticlockwise direction. In other words, the current phasor (I) leads the voltage phasor (V).

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equivalent circuit: Approximate equivalent circuit is widely used to predict the performance of a transformer such as estimating regulation and efficiency. Instead of using the equivalent circuit showing both the sides, it is always advantageous to use the equivalent circuit referred to a particular side and analyse it. Actual quantities of current and voltage of the other side then can be calculated by multiplying or dividing as the case may be with appropriate factors involving turns ratio a. Transfer of parameters (impedances) and quantities (voltages and currents) from one side to the other should be done carefully. Suppose a transformer has turns ratio a, a = N1/N2 = V1/V2 where, N1 and N2 are respectively the primary and secondary turns and V1 and V2 are respectively the primary and secondary rated voltages. The rules for transferring parameters and quantities are summarized below. In spite of all these, gross mistakes in calculating the transferred values can be identified by remembering the following facts. 1. A current referred to LV side, will be higher compared to its value referred to HV side. 2. A voltage referred to LV side, will be lower compared to its value referred to HV side. 3. An impedance referred to LV side, will be lower compared to its value referred to HV side.

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Inductor quirks: In an ideal case, an inductor acts as a purely reactive device. That is, its opposition to AC current is strictly based on inductive reaction to changes in current, and not electron friction as is the case with resistive components. However, inductors are not quite so pure in their reactive behavior. To begin with, they're made of wire, and we know that all wire possesses some measurable amount of resistance (unless its superconducting wire). This built-in resistance acts as though it were connected in series with the perfect inductance of the coil, like this: (Figure below) Equivalent circuit of a real inductor:

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Analysis of Series magnetic circuit: Consider first a simple magnetic circuit, shown in Figure (A) with a single core material having uniform cross sectional area A and mean length of flux path l. Reluctance offered to the flow of flux is ℜ. The corresponding electrical representation is rather simple. Due to the fact that NI=φℜ=HI, the equivalent electrical circuit is also drawn beside the magnetic circuit. Polarity of mmf is decided on the basis of the direction of the flux which is clockwise inside the core in this case. Although in the actual magnetic circuit there is no physical connection of the winding and the core, in the electrical circuit representation mmf and reluctance are shown to be connected. One should not feel disturbed by this as because the relationship between mmf and flux prompted us to draw an electrical equivalent to facilitate easier calculation and neat visualization of the actual problem. fig (A) Let us now consider another magnetic circuit which is similar to the earlier one but has a small air gap of length l_g as shown in Figure (B) and note that it is a series circuit involving two mediums, namely (i) iron and (ii) air. It is a series circuit because same flux (φ) has to flow through the mediums. Hence total reluctance will be the sum of reluctances of iron and air()airiℜ=ℜ ℜ.

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Simple parallel (tank circuit) resonance: A condition of resonance will be experienced in a tank circuit when the reactances of the capacitor and inductor are equal to each other. Because inductive reactance increases with increasing frequency and capacitive reactance decreases with increasing frequency, there will only be one frequency where these two reactances will be equal. In the above circuit, we have a 10 μF capacitor and a 100 mH inductor. Since we know the equations for determining the reactance of each at a given frequency, and we're looking for that point where the two reactances are equal to each other, we can set the two reactance formulae equal to each other and solve for frequency algebraically:

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Introduction of D.C Machines: D.C machines were first developed and used extensively in spite of its complexities in the construction. The generated voltage in a coil when rotated relative to a magnetic field, is inherently alternating in nature. To convert this A.C voltage into a D.C voltage we therefore need a unit after the coil terminals. This unit comprises of a number commutator segments attached to the shaft of the rotor and a pair of suitably placed stationary carbon brushes touching the commutator segments. Commutator segments together with the fixed brushes do the necessary rectification from A.C to D.C and hence sometimes called mechanical rectifier. Constructional Features: Figure (A) shows a sectional view of a 4-pole D.C machine. The length of the machine is perpendicular to the paper. Stator has got 4 numbers of projected poles with coils wound over it. These coils may be connected in series in order that consecutive poles produce opposite polarities (i.e., N-S-N-S) when excited from a source. Double layer lap or wave windings are generally used for armature. Essentially all the armature coils are connected in series forming a closed armature circuit. However as the coils are distributed, the resultant voltage acting in the closed path is zero thereby ensuring no circulating current in the armature. The junctions of two consecutive coils are terminated on to the commutator segments. Stationary carbon brushes are placed physically under the center of the stator poles touching the rotating commutator segments. fig(A) sectional diagram of a D.C machines

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Ampere’s circuital law: Application of Ampere’s circuital law in magnetic circuit: Ampere’s circuital law is quite handy in determining field strength within a core of a magnetic material. Due to application of mmf, the tiny dipole magnets of the core are aligned one after the other in a somewhat disciplined manner. The contour of the lines of force resembles the shape the material. The situation is somewhat similar to flow of water through an arbitrary shaped pipe. Flow path is constrained to be the shape of the bent pipe. For an example, look at the sectional view (figure A & B) of a toroidal magnetic circuit with N number of turns wound uniformly as shown below. When the coil carries a current i, magnetic lines of forces will be created and they will be confined within the core as the permeability of the core is many (order of thousands) times more than air.

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Simple series resonance: A similar effect happens in series inductive/capacitive circuits. When a state of resonance is reached (capacitive and inductive reactances equal), the two impedances cancel each other out and the total impedance drops to zero! At 159.155 Hz the following values are valid:

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D.C machine Armature Winding: Armature winding of a D.C machine is always closed and of double layer type. Closed winding essentially means that all the coils are connected in series forming a closed circuit. The junctions of the consecutive coils are terminated on copper bars called commutator segments. Each commutator segment is insulated from the adjacent segments by mica insulation. For reasonable understanding of armature winding, let us first get acquainted with the following terminologies • A coil has two coil sides occupying two distinct specified slots. Generally two maximize induced voltage in a coil, the spacing between them should be close to 180° electrical. This essentially means if at a given time one coil side is under the center of the north pole, the other coil side should be under the center of the south pole. • Coil span is nothing but the spacing between the two coil sides of a coil. The spacing is expressed in terms of number of slots between the sides. If S be the total number of slots and P be the total number of poles then coil span is S/P. For 20 slots, 4 poles winding, coil span is 5. Let the slots be numbered serially as 1, 2,…, 20. If one coil side is placed in slot number 3, the other coil side of the coil must occupy slot number 8 (= 3 5). • A Double layer winding means that each slot will house two coil sides (obviously belonging to two different coils). Physically one coil side is placed in the lower portion of the slot while the other is placed above it. It is because of this reason such an arrangement of the winding is called a double layer winding. In the n th slot, coil side in the upper deck is numbered as n and the coil side in the lower deck is numbered as n'. In the 5th slot upper coil side is numbered as 5 and the lower coil side is numbered 5'. In the winding diagram, upper coil side is shown with firm line while the lower coil side is shown with dashed line. Remembering that a coil has two coil sides, for a double layer winding total number of coils must be equal to the total number of slots. • Numbering a coil: A coil is so shaped, that when it is placed in appropriate slots, one coil side will be in the upper deck and the other side will be in the lower deck. Suppose S = 20 and P = 4, then coil span is 5. Let the upper coil side of this coil be placed in slot number 6, the other coil side must be in the lower deck of slot number 11. The coil should now be identified as (5 - 11'). In other words coil sides of a coil are numbered depending on the slot numbers in which these are placed. A typical single turn and multi turn coils are shown in figure (A)

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Introduction of Eddy Current & Hysteresis Losses: we assumed the exciting current to be constant d.c. We also came to know how to calculate flux (φ) or flux density (B) in the core for a constant exciting current. When the exciting current is a function of time, it is expected that flux (φ) or flux density (B) will be functions of time too, since φ produced depends on i. In addition if the current is also alternating in nature then both the magnitude of the flux and its direction will change in time. The magnetic material is now therefore subjected to a time varying field instead of steady constant field with d.c excitation. Let: Voltage induced in a stationary coil placed in a time varying field: If normal to the area of a coil, a time varying field φ(t) exists as in figure (A), then an emf is induced in the coil. This emf will appear across the free ends 1 & 2 of the coil. Whenever we talk about some voltage or emf, two things are important, namely the magnitude of the voltage and its polarity. Faraday’s law tells us about the both. Mathematically it is written as e(t) = -N(dφ/dt)

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Susceptance and Admittance: In the study of DC circuits, the student of electricity comes across a term meaning the opposite of resistance: conductance. It is a useful term when exploring the mathematical formula for parallel resistances: R_parallel = 1 / (1/R₁ 1/R₂ . . . 1/R_n). Unlike resistance, which diminishes as more parallel components are included in the circuit, conductance simply adds. Mathematically, conductance is the reciprocal of resistance, and each 1/R term in the "parallel resistance formula" is actually a conductance. Whereas the term "resistance" denotes the amount of opposition to flowing electrons in a circuit, "conductance" represents the ease of which electrons may flow. Resistance is the measure of how much a circuit resists current, while conductance is the measure of how much a circuit conducts current. Conductance used to be measured in the unit of mhos, or "ohms" spelled backward. Now, the proper unit of measurement is Siemens. When symbolized in a mathematical formula, the proper letter to use for conductance is "G". Reactive components such as inductors and capacitors oppose the flow of electrons with respect to time, rather than with a constant, unchanging friction as resistors do. We call this time-based opposition, reactance, and like resistance we also measure it in the unit of ohms.

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EMF Equation:Consider a D.C generator whose field coil is excited to produce a flux density distribution along the air gap and the armature is driven by a prime mover at constant speed as shown in figure (A). fig(A)Pole pitch & area on armature surface per pole. Let us assume a p polar d.c generator is driven (by a prime mover) at n rps. The excitation of the stator field is such that it produces a φ Wb flux per pole. Also let z be the total number of armature conductors and a be the number of parallel paths in the armature circuit. In general, as discussed in the earlier section the magnitude of the voltage from one conductor to another is likely to very since flux density distribution is trapezoidal in nature. Therefore, total average voltage across the brushes is calculated on the basis of average flux density Bav. If D and L are the rotor diameter and the length of the machine in meters then area under each pole is (πD/p)L. Hence average flux density in the gap is given by

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Reluctance & permeance: Let us now try to derive a relationship between flux produced φ and mmf Ni applied for linear case.

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Generator types & its Characteristics: D.C generators may be classified as (i) separately excited generator, (ii) shunt generator, and (iii) series generator and (iv) compound generator. In a separately excited generator field winding is energised from a separate voltage source in order to produce flux in the machine. So long the machine operates in unsaturated condition the flux produced will be proportional to the field current. In order to implement shunt connection, the field winding is connected in parallel with the armature. It will be shown that subject to fulfillment of certain conditions, the machine may have sufficient field current developed on its own by virtue of its shunt connection. In series d.c machine, there is one field winding wound over the main poles with fewer turns and large cross sectional area. Series winding is meant to be connected in series with the armature and naturally to be designed for rated armature current. Obviously there will be practically no voltage or very small voltage due to residual field under no load condition (Ia = 0). However, field gets strengthened as load will develop rated voltage across the armature with reverse polarity, is connected and terminal voltage increases. Variation in load resistance causes the terminal voltage to vary. Terminal voltage will start falling, when saturation sets in and armature reaction effect becomes pronounced at large load current. Hence, series generators are not used for delivering power at constant voltage. Series generator found application in boosting up voltage in d.c transmission system. A compound generator has two separate field coils wound over the field poles. The coil having large number of turns and thinner cross sectional area is called the shunt field coil and the other coil having few number of turns and large cross sectional area is called the series field coil. Series coil is generally connected in series with the armature while the shunt field coil is connected in parallel with the armature. If series coil is left alone without any connection, then it becomes a shunt machine with the other coil connected in parallel. Placement of field coils for shunt, series and compound generators are shown in figure 38.1. Will develop rated voltage across the armature with reverse polarity.

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Analysis of series-parallel magnetic circuit: We now take up the following magnetic circuit (Figure A) which appears to be not so straight forward as the previous cases. As a first step to solve this circuit, we would like to draw its equivalent electrical representation. Vertical links of the core are called limbs and the horizontal links are called yoke of the magnetic circuit. In the figure PU, QT and RS are the limbs whereas PQ, QR, UT and TS are the yokes. It is customary to fix up the corner points P,Q,R etc from the given physical dimensions, joining of which will give you the mean length of the flux paths. fig(A) If the coil carries a current I in the direction shown, flux φ, produced in the first limb will be in the upward direction. Same φ is constrained to move along the yoke PQ. At point Q, two parallel paths are available to φ for its onwards journey namely (i) the central limb QT and (ii) the yoke QR. In other words, φ will be divided into two components φ1 and φ2 as shown with the obvious condition φ = φ1 φ2. The relative values of these components will be decided by respective reluctances of the paths. φ1 and φ2 once again recombine at point T and completes the path. Now in the path TUPQ flux φ is same, it is made of same material and has same cross sectional area A, then its reluctance ℜTU PQ ∞ TU PQlA. In the central limb, flux is same (φ1), however it encounters two materials, one is iron (QM and WT) and the other is a small air gap (MW). The reluctance of the air gap 0ggl=μAℜ. The two reluctances ℜQM and ℜWT of the magnetic material may however be combined into a single reluctance as ℜ1 = ℜQM ℜWT. The portion of the magnetic circuit which carries flux φ2 can be represented by a single reluctance ℜQRST ∞ QRSTlA. Instead of carrying on with long suffixes let us call ℜQRST to be ℜ2. To write down the basic equations let us redraw the electrical equivalence of the above magnetic circuit below (Figure B):

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Torque equation: Whenever armature carries current in presence of flux, conductor experiences force which gives rise to the electromagnetic torque. In this section we shall derive an expression for the electromagnetic torque Te developed in a d.c machine. Obviously Te will be developed both in motor and generator mode of operation. It may be noted that the direction of conductor currents reverses as we move from one pole to the other. This ensures unidirectional torque to be produced. The derivation of the torque expression is shown below. Thus we see that the above equation is once again applicable both for motor and generator mode of operation. The direction of the electromagnetic torque, Te will be along the direction of rotation in case of motor operation and opposite to the direction of rotation in case of generator operation. When the machine runs steadily at a constant rpm then Te = Tload and Te = Tpm, respectively for motor and generator mode. The emf and torque equations are extremely useful and should be remembered by heart.

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Hysteresis Loss: Unidirectional time varying exciting current: Consider a magnetic circuit with constant (d.c) excitation current I₀. Flux established will have fixed value with a fixed direction. Suppose this final current I₀ has been attained from zero current slowly by energizing the coil from a potential divider arrangement as depicted in Figure (A). Let us also assume that initially the core was not magnetized. The exciting current therefore becomes a function of time till it reached the desired current I and we stopped further increasing it. The flux too naturally will be function of time and cause induced voltage e₁₂ in the coil with a polarity to oppose the increase of inflow of current as shown. The coil becomes a source of emf with terminal-1, ve and terminal-2, -ve. Recall that a source in which current enters through its ve terminal absorbs power or energy while it delivers power or energy when current comes out of the ve terminal. Therefore during the interval when i(t) is increasing the coil absorbs energy. Is it possible to know how much energy does the coil absorb when current is increased from 0 to I₀? This is possible if we have the B-H curve of the material with us. fig(A)

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Single-phase Induction Motor: The winding used normally in the stator (Fig. A) of the single-phase induction motor (IM) is a distributed one. The rotor is of squirrel cage type, which is a cheap one, as the rating of this type of motor is low, unlike that for a three-phase IM. As the stator winding is fed from a single-phase supply, the flux in the air gap is alternating only, not a synchronously rotating one produced by a poly-phase (may be two- or three-) winding in the stator of IM. This type of alternating field cannot produce a torque ((T_o)_st=0.0), if the rotor is stationery (ωr=0.0). So, a single-phase IM is not self-starting, unlike a three-phase one. However, as shown later, if the rotor is initially given some torque in either direction (0.0≠rω), then immediately a torque is produced in the motor. The motor then accelerates to its final speed, which is lower than its synchronous speed. fig(A).Single-phase Induction Motor A three phase motor may be run from a single phase power source. (Figure below) However, it will not self-start. It may be hand started in either direction, coming up to speed in a few seconds. It will only develop 2/3 of the 3-φ power rating because one winding is not used.

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Different laws for calculating magnetic field-Biot-Savart law; We know that any current carrying conductor produces a magnetic field. A magnetic field ℜ is characterized either byH, the magnetic field intensity or by B, the magnetic flux density vector.These two vectors are connected by a rather simple relation:B=μ_rμ_oH; where μ=4π*10^-7 H/M is called the absolute permeability of free space and μ_r, a dimensionless quantity called the relative permeability of a medium (or a material). For example the value of μ_r is 1 for free space or could be several thousands in case of ferromagnetic materials.

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Synchronous Motors:Single phase synchronous motors are available in small sizes for applications requiring precise timing such as time keeping, (clocks) and tape players. Though battery powered quartz regulated clocks are widely available, the AC line operated variety has better long term accuracy-- over a period of months. This is due to power plant operators purposely maintaining the long term accuracy of the frequency of the AC distribution system. If it falls behind by a few cycles, they will make up the lost cycles of AC so that clocks lose no time. Above 10 Horsepower (10 kW) the higher efficiency and leading powerfactor make large synchronous motors useful in industry. Large synchronous motors are a few percent more efficient than the more common induction motors. Though, the synchronous motor is more complex. Since motors and generators are similar in construction, it should be possible to use a generator as a motor, conversely, use a motor as a generator. A synchronous motor is similar to an alternator with a rotating field. The figure below shows small alternators with a permanent magnet rotating field. This figure below could either be two paralleled and synchronized alternators driven by a mechanical energy sources, or an alternator driving a synchronous motor. Or, it could be two motors, if an external power source were connected. The point is that in either case the rotors must run at the same nominal frequency, and be in phase with each other. That is, they must be synchronized. The procedure for synchronizing two alternators is to (1) open the switch, (2) drive both alternators at the same rotational rate, (3) advance or retard the phase of one unit until both AC outputs are in phase, (4) close the switch before they drift out of phase. Once synchronized, the alternators will be locked to each other, requiring considerable torque to break one unit loose (out of synchronization) from the other.

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Force between two opposite faces of the core across an air gap: In a magnetic circuit involving air gap, magnetic force will exist between the parallel faces across the air gap of length x. The situation is shown for a magnetic circuit in figure A (i). Direction of the lines of forces will be in the clockwise direction and the left face will become a north pole and the right face will become a south pole. Naturally there will be force of attraction Fa between the faces. Except for the fact that this force will develop stress in the core, no physical movement is possible as the structure is rigid. fig(A) Force between parallel faces

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Three Phase synchronous motor: A 3-phase synchronous motor as shown in Figure below generates an electrically rotating field in the stator. Such motors are not self starting if started from a fixed frequency power source such as 50 or 60 Hz as found in an industrial setting. Furthermore, the rotor is not a permanent magnet as shown below for the multi-horsepower (multi-kilowatt) motors used in industry, but an electromagnet. Large industrial synchronous motors are more efficient than induction motors. They are used when constant speed is required. Having a leading power factor, they can correct the AC line for a lagging power factor. The three phases of stator excitation add vectorially to produce a single resultant magnetic field which rotates f/2n times per second, where f is the power line frequency, 50 or 60 Hz for industrial power line operated motors. The number of poles is n. For rotor speed in rpm, multiply by 60. The 3-phase 4-pole (per phase) synchronous motor (Figure below) will rotate at 1800 rpm with 60 Hz power or 1500 rpm with 50 Hz power. If the coils are energized one at a time in the sequence φ-1, φ-2, φ-3, the rotor should point to the corresponding poles in turn. Since the sine waves actually overlap, the resultant field will rotate, not in steps, but smoothly. For example, when the φ-1 and φ-2 sinewaves coincide, the field will be at a peak pointing between these poles. The bar magnet rotor shown is only appropriate for small motors. The rotor with multiple magnet poles (below right) is used in any efficient motor driving a substantial load. These will be slip ring fed electromagnets in large industrial motors. Large industrial synchronous motors are self started by embedded squirrel cage conductors in the armature, acting like an induction motor. The electromagnetic armature is only energized after the rotor is brought up to near synchronous speed.

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Hysteresis loss & loop area: In other words the operating point trace the perimeter of the closed area QFMNKPRTSEQ. This area is called the B-H loop of the material. We will now show that the area enclosed by the loop is the hysteresis loss per unit volume per cycle variation of the current. In the interval 0≤ωt ≤π/2 i is ve and di/dt is also ve, moving the operating point from M to P along the path MNKP. Energy absorbed during this interval is given by the shaded area MNKPLTM shown in fig.(A) (i). In the interval π/2≤ωt≤π, is ve but di/dt is –ve, moving the operating point from P to T along the path PRT. Energy returned during this interval is given by the shaded area PLTRP shown in fig.(A) (ii). Thus during the ve half cycle of current variation net amount of energy absorbed is given by the shaded area MNKPRTM which is nothing but half the area of the loop. In the interval π≤ωt≤3π/2 , i is –ve and di/dt is also –ve, moving the operating point from T to Q along the path TSEQ. Energy absorbed during this interval is given by the shaded area QJMTSEQ shown in fig.(A) (iii).

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Reluctance motor: The variable reluctance motor is based on the principle that an unrestrained piece of iron will move to complete a magnetic flux path with minimum reluctance, the magnetic analog of electrical resistance. (Figure below) If the rotating field of a large synchronous motor with salient poles is de-energized, it will still develop 10 or 15% of synchronous torque. This is due to variable reluctance throughout a rotor revolution. There is no practical application for a large synchronous reluctance motor. However, it is practical in small sizes. If slots are cut into the conductorless rotor of an induction motor, corresponding to the stator slots, a synchronous reluctance motor results. It starts like an induction motor but runs with a small amount of synchronous torque. The synchronous torque is due to changes in reluctance of the magnetic path from the stator through the rotor as the slots align. This motor is an inexpensive means of developing a moderate synchronous torque. Low power factor, low pull-out torque, and low efficiency are characteristics of the direct power line driven variable reluctance motor. Such was the status of the variable reluctance motor for a century before the development of semiconductor power control.

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Steinmetz’s empirical formula for hysteresis loss: Based on results obtained by experiments with different ferromagnetic materials with sinusoidal currents, Charles Steimetz proposed the empirical formula for calculating hysteresis loss analytically. Where, the coefficient kh depends on the material and n, known as Steinmetz exponent, may vary from 1.5 to 2.5. For iron it may be taken as 1.6.

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Derivation of an expression for eddy current loss in a thin plate: From physical consideration we have seen that thin plates each of thickness τ, are to be used to reduce eddy loss. With this in mind we shall try to derive an approximate expression for eddy loss in the following section for a thin plate and try to identify the factors on which it will depend. Section of a thin plate τ << L and h is shown in the plane of the screen in Figure (A). fig(A).Elemental eddy current path fig(B)Section of the elemental eddy current path.

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Synchronous condenser: Synchronous motors load the power line with a leading power factor. This is often usefull in cancelling out the more commonly encountered lagging power factor caused by induction motors and other inductive loads. Originally, large industrial synchronous motors came into wide use because of this ability to correct the lagging power factor of induction motors. This leading power factor can be exaggerated by removing the mechanical load and over exciting the field of the synchronous motor. Such a device is known as a synchronous condenser. Furthermore, the leading power factor can be adjusted by varying the field excitation. This makes it possible to nearly cancel an arbitrary lagging power factor to unity by paralleling the lagging load with a synchronous motor. A synchronous condenser is operated in a borderline condition between a motor and a generator with no mechanical load to fulfill this function. It can compensate either a leading or lagging power factor, by absorbing or supplying reactive power to the line. This enhances power line voltage regulation. Since a synchronous condenser does not supply a torque, the output shaft may be dispensed with and the unit easily enclosed in a gas tight shell. The synchronous condenser may then be filled with hydrogen to aid cooling and reduce windage losses. Since the density of hydrogen is 7% of that of air, the windage loss for a hydrogen filled unit is 7% of that encountered in air. Furthermore, the thermal conductivity of hydrogen is ten times that of air. Thus, heat removal is ten times more efficient. As a result, a hydrogen filled synchronous condenser can be driven harder than an air cooled unit, or it may be physically smaller for a given capacity. There is no explosion hazard as long as the hydrogen concentration is maintained above 70%, typically above 91%.

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Construction of Three-phase Induction Motor: This is a rotating machine, unlike the transformer, described in the previous module, which is a static machine. Both the machines operate on ac supply. This machine mainly works as a motor, but it can also be run as a generator, which is not much used. Like all rotating machines, it consists of two parts − stator and rotor. In the stator (Fig. A), the winding used is a balanced three-phase one, which means that the number of turns in each phase, connected in star/delta, is equal. The windings of the three phases are placed 130^o(electrical) apart, the mechanical angle between the adjacent phases being [2*120^o/P], where p is no. of poles. For a 4-pole (p = 4) stator, the mechanical angle between the winding of the adjacent phases, is [2×120°/4=120°/2=60°]. The conductors, mostly multi-turn, are placed in the slots, which may be closed, or semi-closed, to keep the leakage inductance low. The start and return parts of the winding are placed nearly , 180° or (180°−β)apart. The angle of short chording (β) is nearly equal to 30^o , or close to that value. The short chording results in reducing the amount of copper used for the winding, as the length of the conductor needed for overhang part is reduced. There are also other advantages. The section of the stampings used for both stator and rotor, is shown in Fig. (B). The core is needed below the teeth to reduce the reluctance of the magnetic path, which carries the flux in the motor (machine). The stator is kept normally inside a support fig(A) schematic diagram of the stator windings in a three_phase induction motor

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Starting Current and Torque: as slip at starting (n_r= 0.0) is 1.0,which is the same at standstill (or stalling condition). The magnitude of the induced voltage per phase in the stator winding is nearly same as input voltage per phase fed to the stator, if the voltage drop in the stator impedance, being small, is neglected, i.e. V_s≈V_s, the ratio of the induced emfs per phase in the stator and rotor winding can be taken as the ratio of the effective turns in two windings, i.e. E_s/E_r= t_s' =k_wsT_s and T_r' = k_wrT_r. The winding factor for the stator winding is k_ws = k_ds•k_ps. Same formula is used for the above factor in the rotor winding, assuming it to be wound rotor one. Same formula is used for the above factor in the rotor winding, assuming it to be wound rotor one. (I_s)_st = (I_r)_st•(T_r'/T_s'), neglecting the no load current. This current is normally large, much greater than full load current. This current is reduced by using starters in both types (cage and wound rotor) of IM, which will be taken up in the next lesson.

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Characteristics of a separately excited generator: No load or Open circuit characteristic: In this type of generator field winding is excited from a separate source, hence field current is independent of armature terminal voltage as shown on figure (A). The generator is driven by a prime mover at rated speed, say n rps. With switch S in opened condition, field is excited via a potential divider connection from a separate d.c source and field current is gradually increased. The field current will establish the flux per pole φ. The voltmeter V connected across the armature terminals of the machine will record the generated emf (E_G = (pz/a)φn=kφn). Remember pza is a constant (k) of the machine. As field current is increased, E_G will increase. E_G versus If plot at constant speed n is shown in figure (B). fig(A)connection of separately excited generator

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Starting Methods for Single-phase Induction Motor: The single-phase IM has no starting torque, but has resultant torque, when it rotates at any other speed, except synchronous speed. It is also known that, in a balanced two-phase IM having two windings, each having equal number of turns and placed at a space angle of (electrical), and are fed from a balanced two-phase supply, with two voltages equal in magnitude, at an angle of 90° , the rotating magnetic fields are produced, as in a three-phase IM. The torque-speed characteristic is same as that of a three-phase one, having both starting and also running torque as shown earlier. So, in a single-phase IM, if an auxiliary winding is introduced in the stator, in addition to the main winding, but placed at a space angle of 90°(electrical), starting torque is produced. The currents in the two (main and auxiliary) stator windings also must be at an angle of 90°, to produce maximum starting torque, as shown in a balanced two-phase stator. Thus, rotating magnetic field is produced in such motor, giving rise to starting torque. The various starting methods used in a single-phase IM are described here. Resistance Split-phase Motor: fig(A)

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Capacitor-run motor induction motor: A variation of the capacitor-start motor (Figure below) is to start the motor with a relatively large capacitor for high starting torque, but leave a smaller value capacitor in place after starting to improve running characteristics while not drawing excessive current. The additional complexity of the capacitor-run motor is justified for larger size motors. Capacitor-run motor induction motor.

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Characteristics of a shunt generator: We have seen in the previous section that one needs a separate d.c supply to generate d.c voltage. Is it possible to generate d.c voltage without using another d.c source? The answer is yes and for obvious reason such a generator is called self excited generator. Field coil (F1, F2) along with a series external resistance is connected in parallel with the armature terminals (A1, A2) of the machine as shown in figure (A). Let us first qualitatively explain how such connection can produce sufficient voltage. Suppose there exists some residual field. Therefore, if the generator is driven at rated speed, we should expect a small voltage (kφ_resn) to be induced across the armature. But this small voltage will be directly applied across the field circuit since it is connected in parallel with the armature. Hence a small field current flows producing additional flux. If it so happens that this additional flux aids the already existing residual flux, total flux now becomes more generating more voltage. This more voltage will drive more field current generating more voltage. Both field current and armature generated voltage grow cumulatively. This growth of voltage and the final value to which it will settle down can be understood by referring to (B) where two plots have been shown. One corresponds to the O.C.C at rated speed and obtained by connecting the generator in separately excited fashion as detailed in the preceding section. The other one is the V-I characteristic of the field circuit which is a straight line passing through origin and its slope represents the total field circuit resistance. fig(A) shunt generator fig(B) voltage build in shunt generator

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Permanent-split capacitor motor: This type of motor suffers increased current magnitude and backward time shift as the motor comes up to speed, with torque pulsations at full speed. The solution is to keep the capacitor (impedance) small to minimize losses. The losses are less than for a shaded pole motor. This motor configuration works well up to 1/4 horsepower (200watt), though, usually applied to smaller motors. The direction of the motor is easily reversed by switching the capacitor in series with the other winding. Single phase induction motor with embedded stator coils.

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Capacitor Split-phase Motor: The motor described earlier, is a simple one, requiring only second (auxiliary) winding placed at a space angle of 90° from the main winding, which is there in nearly all such motors as discussed here. It does not need any other thing, except for centrifugal switch, as the auxiliary winding is used as a starting winding. But the main problem is low starting torque in the motor, as this torque is a function of, or related to the phase difference (angle) between the currents in the two windings. To get high starting torque, the phase difference required is 90° (Fig.(A)b), when the starting torque will be proportional to the product of the magnitudes of two currents. As the current in the main winding is lagging by φ_m, the current in the auxiliary winding has to lead the input voltage by φ_a, with (φ_m φ_a=90°). φ_a is taken as negative (-ve), while mφ is positive ( ve). This can be can be achieved by having a capacitor in series with the auxiliary winding, which results in additional cost, with the increase in starting torque, The two types of such motors are described here. Capacitor-start Motor:

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Capacitor-start and Capacitor-run Motor: fig (A)

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Load characteristic of shunt generator: With switch S in open condition, the generator is practically under no load condition as field current is pretty small. The voltmeter reading will be E_o . In other words, E_o and I_a = 0 is the first point in the load characteristic. To load the machine S is closed and the load resistances decreased so that it delivers load current I_L. Unlike separately excited motor, here IL ≠ Ia. In fact, for shunt generator, I_a = I_L - I_f. So increase of I_L will mean increase of I_a as well. The drop in the terminal voltage will be caused by the usual Iara drop, brush voltage drop and armature reaction effect. Apart from these, in shunt generator, as terminal voltage decreases, field current hence φ also decreases causing additional drop in terminal voltage. Remember in shunt generator, field current is decided by the terminal voltage by virtue of its parallel connection with the armature. Figure (A) shows the plot of terminal voltage versus armature current which is called the load characteristic. One can of course translate the V versus I_a characteristic into V versus I_L characteristic by subtracting the correct value of the field current from the armature current. For example, suppose the machine is loaded such that terminal voltage becomes V1 and the armature current is I_a1. The field current at this load can be read from the field resistance line corresponding to the existing voltage V1 across the field as shown in figure (A). Suppose If1 is the noted field current. Therefore, I_Ll = I_a1- I_f1.Thus the point [I_a1, V₁] is translated into [I_Ll, V₁] point. Repeating these step for all the points we can get the V versus I_L characteristic as well. It is interesting to note that the generated voltage at this loading is E_G1 (obtained from OCC corresponding to I_f1). Therefore the length PQ must represents sum of all the voltage drops that has taken place in the armature when it delivers I_a.

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Torque-slip (speed) Characteristics: The torque-slip or torque-speed characteristic, as per the equation derived earlier, is shown in Fig.(A). The slip is s=(w_s-w_r)/ w_s = n_s-n_r/ n_s= 1-(n_r/n_s). The range of speed, n_r is between 0.0 (standstill) and n_s (synchronous speed). The range of slip is between 0.0 (n_r= n_s) and 1.0 (n_r= 1.0) fig.(A)Torque-slip (speed) Characteristics of induction motor

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Gross Torque Developed: The current per phase in the rotor winding is