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68-430: In a star (wye) connected topology, with rotation sequence L1 - L2 - L3, the time-varying instantaneous voltages can be calculated for each phase A,C,B respectively by: where: The below images demonstrate how a system of six wires delivering three phases from an alternator may be replaced by just three. A three-phase transformer is also shown. Generally, in electric power systems, the loads are distributed as evenly as

136-548: A linear alternator or a rotating armature with a stationary magnetic field is used. In principle, any AC electrical generator can be called an alternator, but usually, the term refers to small rotating machines driven by automotive and other internal combustion engines. An alternator that uses a permanent magnet for its magnetic field is called a magneto . Alternators in power stations driven by steam turbines are called turbo-alternators. Large 50 or 60 Hz three-phase alternators in power plants generate most of

204-430: A constant instantaneous power since, as long as it is balanced or the same for all phases, it may be written as so that the peak current is for all phases and the instantaneous currents are Now the instantaneous powers in the phases are Using angle subtraction formulae : which add up for a total instantaneous power Since the three terms enclosed in square brackets are a three-phase system, they add up to zero and

272-470: A cycle in phase. Unbalanced operation results in undesirable effects on motors and generators. Provided two voltage waveforms have at least some relative displacement on the time axis, other than a multiple of a half-cycle, any other polyphase set of voltages can be obtained by an array of passive transformers . Such arrays will evenly balance the polyphase load between the phases of the source system. For example, balanced two-phase power can be obtained from

340-415: A rotating magnet, called the rotor , turns within a stationary set of conductors, called the stator , wound in coils on an iron core. The field cuts across the conductors, generating an induced EMF (electromotive force), as the mechanical input causes the rotor to turn. The rotating magnetic field induces an AC voltage in the stator windings. Since the currents in the stator windings vary in step with

408-416: A small amount of electricity, just enough to excite the field coils of the connected alternator to generate electricity. A variation of this system is a type of alternator that uses direct current from a battery for initial excitation upon start-up, after which the alternator becomes self-excited. This method of excitation consists of a smaller alternating-current (AC) generator fixed on the same shaft as

476-428: A three-phase network by using two specially constructed transformers, with taps at 50% and 86.6% of the primary voltage. This Scott T connection produces a true two-phase system with 90° time difference between the phases. Another example is the generation of higher-phase-order systems for large rectifier systems, to produce a smoother DC output and to reduce the harmonic currents in the supply. When three-phase

544-545: A wire is not an ideal conductor. Unlike an ideal conductor, wires can inductively and capacitively couple to each other (and to themselves), and have a finite propagation delay. Real conductors can be modeled in terms of lumped elements by considering parasitic capacitances distributed between the conductors to model capacitive coupling, or parasitic (mutual) inductances to model inductive coupling. Wires also have some self-inductance. Assume an electric network consisting of two voltage sources and three resistors. According to

612-490: Is N = 120 f / P {\displaystyle N=120f/P} , where f {\displaystyle f} is the frequency in Hz (cycles per second). P {\displaystyle P} is the number of poles (2, 4, 6, …), and N {\displaystyle N} is the rotational speed in revolutions per minute (r/min). Old descriptions of alternating current systems sometimes give

680-442: Is a type of alternator that uses direct current from a battery for initial excitation upon start-up, after which the alternator becomes self-excited. This method depends on residual magnetism retained in the iron core to generate a weak magnetic field, which would allow a weak voltage to be generated. This voltage is used to excite the field coils so the alternator can generate stronger voltage as part of its build up process. After

748-451: Is constant at all times. Indeed, let To simplify the mathematics, we define a nondimensionalized power for intermediate calculations, p = 1 V P 2 P T O T R {\displaystyle \scriptstyle p\,=\,{\frac {1}{V_{P}^{2}}}P_{TOT}R} Hence (substituting back): Since we have eliminated θ {\displaystyle \theta } we can see that

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816-415: Is constant. Whenever the electric field between parts of the circuit is non-negligible, such as when two wires are capacitively coupled , this may not be the case. This occurs in high-frequency AC circuits, where the lumped element model is no longer applicable. For example, in a transmission line , the charge density in the conductor may be constantly changing. On the other hand, the voltage law relies on

884-1265: Is equivalent to { i 1 + ( − i 2 ) + ( − i 3 ) = 0 R 1 i 1 + R 2 i 2 + 0 i 3 = E 1 0 i 1 + R 2 i 2 − R 3 i 3 = E 1 + E 2 {\displaystyle {\begin{cases}i_{1}+(-i_{2})+(-i_{3})&=0\\R_{1}i_{1}+R_{2}i_{2}+0i_{3}&={\mathcal {E}}_{1}\\0i_{1}+R_{2}i_{2}-R_{3}i_{3}&={\mathcal {E}}_{1}+{\mathcal {E}}_{2}\end{cases}}} Assuming R 1 = 100 Ω , R 2 = 200 Ω , R 3 = 300 Ω , E 1 = 3 V , E 2 = 4 V {\displaystyle {\begin{aligned}R_{1}&=100\Omega ,&R_{2}&=200\Omega ,&R_{3}&=300\Omega ,\\{\mathcal {E}}_{1}&=3{\text{V}},&{\mathcal {E}}_{2}&=4{\text{V}}\end{aligned}}}

952-409: Is fed into the rotating field coils through the voltage regulator (VR). This increases the magnetic field around the field coils, which induces a greater voltage in the armature coils. Thus, the output voltage is brought back up to its original value. Alternators used in central power stations also control the field current to regulate reactive power and to help stabilize the power system against

1020-408: Is greatly simplified by the use of the techniques of symmetrical components . An unbalanced system is analysed as the superposition of three balanced systems, each with the positive, negative or zero sequence of balanced voltages. When specifying wiring sizes in a three-phase system, we only need to know the magnitude of the phase and neutral currents. The neutral current can be determined by adding

1088-431: Is needed but only single-phase is readily available from the electricity supplier, a phase converter can be used to generate three-phase power from the single phase supply. A motor–generator is often used in factory industrial applications. In a three-phase system, at least two transducers are required to measure power when there is no neutral, or three transducers when there is a neutral. Blondel's theorem states that

1156-428: Is one of Maxwell's equations ). This has practical application in situations involving " static electricity ". Kirchhoff's circuit laws are the result of the lumped-element model and both depend on the model being applicable to the circuit in question. When the model is not applicable, the laws do not apply. The current law is dependent on the assumption that the net charge in any wire, junction or lumped component

1224-399: Is possible to still model such circuits using parasitic components . If frequencies are too high, it may be more appropriate to simulate the fields directly using finite element modelling or other techniques . To model circuits so that both laws can still be used, it is important to understand the distinction between physical circuit elements and the ideal lumped elements. For example,

1292-439: Is practical among the phases. It is usual practice to discuss a balanced system first and then describe the effects of unbalanced systems as deviations from the elementary case. An important property of three-phase power is that the instantaneous power available to a resistive load, P = V I = V 2 R {\displaystyle \scriptstyle P\,=\,VI\,=\,{\frac {V^{2}}{R}}} ,

1360-448: Is that a small DC exciter current indirectly controls the output of the main alternator. Another way to classify alternators is by the number of phases of their output voltage. The output can be single phase or polyphase. Three-phase alternators are the most common, but polyphase alternators can be two-phase, six-phase, or more. The revolving part of alternators can be the armature or the magnetic field. The revolving armature type has

1428-404: Is the square root of the sum of the squares of the real and imaginary parts, which reduces to With linear loads, the neutral only carries the current due to imbalance between the phases. Devices that utilize rectifier-capacitor front ends (such as switch-mode power supplies for computers, office equipment and the like) introduce third order harmonics. Third harmonic currents are in-phase on each of

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1496-409: Is the total number of branches with currents flowing towards or away from the node. Kirchhoff's circuit laws were originally obtained from experimental results. However, the current law can be viewed as an extension of the conservation of charge , since charge is the product of current and the time the current has been flowing. If the net charge in a region is constant, the current law will hold on

1564-817: Is the total number of voltages measured. A similar derivation can be found in The Feynman Lectures on Physics, Volume II, Chapter 22: AC Circuits . Consider some arbitrary circuit. Approximate the circuit with lumped elements, so that time-varying magnetic fields are contained to each component and the field in the region exterior to the circuit is negligible. Based on this assumption, the Maxwell–Faraday equation reveals that ∇ × E = − ∂ B ∂ t = 0 {\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}=\mathbf {0} } in

1632-499: Is usually distributed in homes), the motor must contain some mechanism to produce a revolving field, otherwise the motor cannot generate any stand-still torque and will not start. The field produced by a single-phase winding can provide energy to a motor already rotating, but without auxiliary mechanisms the motor will not accelerate from a stop. A rotating magnetic field of steady amplitude requires that all three phase currents be equal in magnitude, and accurately displaced one-third of

1700-480: The Alexanderson alternator were developed as longwave radio transmitters around World War 1 and used in a few high power wireless telegraphy stations before vacuum tube transmitters replaced them. A conductor moving relative to a magnetic field develops an electromotive force (EMF) in it ( Faraday's Law ). This EMF reverses its polarity when it moves under magnetic poles of opposite polarity. Typically,

1768-465: The current and potential difference (commonly known as voltage) in the lumped element model of electrical circuits . They were first described in 1845 by German physicist Gustav Kirchhoff . This generalized the work of Georg Ohm and preceded the work of James Clerk Maxwell . Widely used in electrical engineering , they are also called Kirchhoff's rules or simply Kirchhoff's laws . These laws can be applied in time and frequency domains and form

1836-469: The electric potential (and thus voltage) can be defined in other ways, such as via the Helmholtz decomposition . In the low-frequency limit, the voltage drop around any loop is zero. This includes imaginary loops arranged arbitrarily in space – not limited to the loops delineated by the circuit elements and conductors. In the low-frequency limit, this is a corollary of Faraday's law of induction (which

1904-546: The exterior of each of the components, from one terminal to another. Note that this derivation uses the following definition for the voltage rise from a {\displaystyle a} to b {\displaystyle b} : V a → b = − ∫ P a → b E ⋅ d l {\displaystyle V_{a\to b}=-\int _{{\mathcal {P}}_{a\to b}}\mathbf {E} \cdot \mathrm {d} \mathbf {l} } However,

1972-401: The prime mover turns an alternator which provides electricity for the traction motors (AC or DC). The traction alternator usually incorporates integral silicon diode rectifiers to provide the traction motors with up to 1,200 volts DC. The first diesel electric locomotives, and many of those still in service, use DC generators as, before silicon power electronics, it was easier to control

2040-442: The alternator. The AC stator generates a small amount of field coil excitation current, which is induced in the rotor and rectified to DC by a bridge rectifier built in to the windings where it excites the field coils of the larger connected alternator to generate electricity. This system has the advantage of not requiring brushes, which increases service life, although with a slightly lower overall efficiency. A variation of this system

2108-438: The armature wound on the rotor, where the winding moves through a stationary magnetic field. The revolving armature type is not often used. The revolving field type has a magnetic field on the rotor to rotate through a stationary armature winding. The advantage is that then the rotor circuit carries much less power than the armature circuit, making the slip ring connections smaller and less costly; only two contacts are needed for

Mathematics of three-phase electric power - Misplaced Pages Continue

2176-454: The basis for network analysis . Both of Kirchhoff's laws can be understood as corollaries of Maxwell's equations in the low-frequency limit. They are accurate for DC circuits, and for AC circuits at frequencies where the wavelengths of electromagnetic radiation are very large compared to the circuits. This law, also called Kirchhoff's first law , or Kirchhoff's junction rule , states that, for any node (junction) in an electrical circuit ,

2244-410: The battery to adjust the charging voltage and an over-temperature sensor on the actual alternator to protect it from overheating. High-frequency alternators of the variable-reluctance type were applied commercially to radio transmission in low-frequency radio bands. These were used for transmitting Morse code and, experimentally, for transmitting voice and music. In the Alexanderson alternator , both

2312-400: The boundaries of the region. This means that the current law relies on the fact that the net charge in the wires and components is constant. A matrix version of Kirchhoff's current law is the basis of most circuit simulation software , such as SPICE . The current law is used with Ohm's law to perform nodal analysis . The current law is applicable to any lumped network irrespective of

2380-434: The circuit, provided the system is balanced. Such connections are generally used only when the load on the three phases is part of the same piece of equipment (for example a three-phase motor), as otherwise switching loads and slight imbalances would cause large voltage fluctuations. In practice, systems rarely have perfectly balanced loads, currents, voltages and impedances in all three phases. The analysis of unbalanced cases

2448-1077: The closed circuit s 2 gives: − R 3 i 3 − E 2 − E 1 + R 2 i 2 = 0 {\displaystyle -R_{3}i_{3}-{\mathcal {E}}_{2}-{\mathcal {E}}_{1}+R_{2}i_{2}=0} This yields a system of linear equations in i 1 , i 2 , i 3 : { i 1 − i 2 − i 3 = 0 − R 2 i 2 + E 1 − R 1 i 1 = 0 − R 3 i 3 − E 2 − E 1 + R 2 i 2 = 0 {\displaystyle {\begin{cases}i_{1}-i_{2}-i_{3}&=0\\-R_{2}i_{2}+{\mathcal {E}}_{1}-R_{1}i_{1}&=0\\-R_{3}i_{3}-{\mathcal {E}}_{2}-{\mathcal {E}}_{1}+R_{2}i_{2}&=0\end{cases}}} which

2516-446: The cost of the magnet material. Since the permanent magnet field is constant, the terminal voltage varies directly with the speed of the generator. Brushless AC generators are usually larger than those used in automotive applications. An automatic voltage control device controls the field current to keep the output voltage constant. If the output voltage from the stationary armature coils drops due to an increase in demand, more current

2584-418: The direct-current rotor, whereas often a rotor winding has three phases, and multiple sections which would each require a slip-ring connection. The stationary armature can be wound for any convenient medium voltage level, up to tens of thousands of volts; manufacture of slip ring connections for more than a few thousand volts is costly and inconvenient. Many alternators are cooled by ambient air, forced through

2652-401: The effects of momentary faults . Often, there are three sets of stator windings, physically offset so that the rotating magnetic field produces a three phase current, displaced by one-third of a period with respect to each other. One cycle of alternating current is produced each time a pair of field poles passes over a point on the stationary winding. The relation between speed and frequency

2720-463: The enclosure by an attached fan on the shaft that drives the alternator. In vehicles such as transit buses, a heavy demand on the electrical system may require a large alternator to be oil-cooled. In marine applications water-cooling is also used. Expensive automobiles may use water-cooled alternators to meet high electrical system demands. Most power generation stations use synchronous machines as their generators. The connection of these generators to

2788-761: The exterior region. If each of the components has a finite volume, then the exterior region is simply connected , and thus the electric field is conservative in that region. Therefore, for any loop in the circuit, we find that ∑ i V i = − ∑ i ∫ P i E ⋅ d l = ∮ E ⋅ d l = 0 {\displaystyle \sum _{i}V_{i}=-\sum _{i}\int _{{\mathcal {P}}_{i}}\mathbf {E} \cdot \mathrm {d} \mathbf {l} =\oint \mathbf {E} \cdot \mathrm {d} \mathbf {l} =0} where P i {\textstyle {\mathcal {P}}_{i}} are paths around

Mathematics of three-phase electric power - Misplaced Pages Continue

2856-437: The fact that the actions of time-varying magnetic fields are confined to individual components, such as inductors. In reality, the induced electric field produced by an inductor is not confined, but the leaked fields are often negligible. The lumped element approximation for a circuit is accurate at low frequencies. At higher frequencies, leaked fluxes and varying charge densities in conductors become significant. To an extent, it

2924-412: The field winding and armature winding are stationary, and current is induced in the armature by the changing magnetic reluctance of the rotor (which has no windings or current-carrying parts). Such machines were made to produce radio frequency current for radio transmissions, although the efficiency was low. Kirchhoff%27s circuit laws Kirchhoff's circuit laws are two equalities that deal with

2992-551: The first law: i 1 − i 2 − i 3 = 0 {\displaystyle i_{1}-i_{2}-i_{3}=0} Applying the second law to the closed circuit s 1 , and substituting for voltage using Ohm's law gives: − R 2 i 2 + E 1 − R 1 i 1 = 0 {\displaystyle -R_{2}i_{2}+{\mathcal {E}}_{1}-R_{1}i_{1}=0} The second law, again combined with Ohm's law, applied to

3060-407: The frequency in terms of alternations per minute, counting each half-cycle as one alternation ; so 12,000 alternations per minute corresponds to 100 Hz. An alternator's output frequency depends on the number of poles and the rotational speed. The speed corresponding to a particular frequency is called the synchronous speed . This table gives some examples: Alternators may be classified by

3128-444: The generator. The early machines were developed by pioneers such as Michael Faraday and Hippolyte Pixii . Faraday developed the "rotating rectangle", whose operation was heteropolar – each active conductor passed successively through regions where the magnetic field was in opposite directions. Lord Kelvin and Sebastian Ferranti also developed early alternators, producing frequencies between 100 and 300 Hz . The late 1870s saw

3196-432: The initial AC voltage buildup, the field is supplied with rectified voltage from the alternator. A brushless alternator is composed of two alternators built end-to-end on one shaft. Until 1966, alternators used brushes with rotating field. With the advancement in semiconductor technology, brushless alternators are possible. Smaller brushless alternators may look like one unit, but the two parts are readily identifiable in

3264-416: The introduction of the first large-scale electrical systems with central generation stations to power Arc lamps , used to light whole streets, factory yards, or the interior of large warehouses. Some, such as Yablochkov arc lamps introduced in 1878, ran better on alternating current, and the development of these early AC generating systems was accompanied by the first use of the word "alternator". Supplying

3332-429: The large versions. The main alternator is the larger of the two sections, and the smaller one is the exciter. The exciter has stationary field coils and a rotating armature (power coils). The main alternator uses the opposite configuration with a rotating field and stationary armature. A bridge rectifier , called the rotating rectifier assembly, is mounted on the rotor. Neither brushes nor slip rings are used, which reduces

3400-446: The method of excitation, number of phases, the type of rotation, cooling method, and their application. There are two main ways to produce the magnetic field used in the alternators: by using permanent magnets , which create their persistent magnetic field, or by using field coils . The alternators that use permanent magnets are specifically called magnetos . In other alternators, wound field coils form an electromagnet to produce

3468-514: The nature of the network; whether unilateral or bilateral, active or passive, linear or non-linear. This law, also called Kirchhoff's second law , or Kirchhoff's loop rule , states the following: The directed sum of the potential differences (voltages) around any closed loop is zero. Similarly to Kirchhoff's current law, the voltage law can be stated as: ∑ i = 1 n V i = 0 {\displaystyle \sum _{i=1}^{n}V_{i}=0} Here, n

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3536-405: The neutral. The neutral current is the inverted vector sum of the line currents. See Kirchhoff's circuit laws . We define a non-dimensionalized current, i = I N R V P {\displaystyle i={\frac {I_{N}R}{V_{P}}}} : Since we have shown that the neutral current is zero we can see that removing the neutral core will have no effect on

3604-400: The number of measurement elements required is one less than the number of current-carrying conductors. Alternator An alternator is an electrical generator that converts mechanical energy to electrical energy in the form of alternating current . For reasons of cost and simplicity, most alternators use a rotating magnetic field with a stationary armature . Occasionally,

3672-435: The number of wearing parts. The main alternator has a rotating field and a stationary armature (power generation windings). Varying the amount of current through the stationary exciter field coils varies the 3-phase output from the exciter. This output is rectified by a rotating rectifier assembly mounted on the rotor, and the resultant DC supplies the rotating field of the main alternator and hence alternator output. The result

3740-440: The position of the rotor, an alternator is a synchronous generator. The rotor's magnetic field may be produced by permanent magnets or by a field coil electromagnet. Automotive alternators use a rotor winding, which allows control of the alternator's generated voltage by varying the current in the rotor field winding. Permanent magnet machines avoid the loss due to magnetizing current in the rotor, but are restricted in size due to

3808-531: The power may be split between the engine starting battery and the domestic or house battery (or batteries) by use of a split-charge diode ( battery isolator ) or a voltage-sensitive relay. Due to the high cost of large house battery banks, Marine alternators generally use external regulators. Multistep regulators control the field current to maximize the charging effectiveness (time to charge) and battery life. Multistep regulators can be programmed for different battery types. Two temperature sensors can be added: one for

3876-838: The proper amount of voltage from generating stations in these early systems was left up to the engineer's skill in "riding the load". In 1883 the Ganz Works invented the constant voltage generator that could produce a stated output voltage, regardless of the value of the actual load. The introduction of transformers in the mid-1880s led to the widespread use of alternating current and the use of alternators needed to produce it. After 1891, polyphase alternators were introduced to supply currents of multiple differing phases. Later alternators were designed for various alternating current frequencies between sixteen and about one hundred hertz for use with arc lighting, incandescent lighting, and electric motors. Specialized radio frequency alternators like

3944-417: The rotating magnetic field. A device that uses permanent magnets to produce alternating current is called a permanent magnet alternator (PMA). A permanent magnet generator (PMG) may produce either alternating current or direct current if it has a commutator . This method of excitation consists of a smaller direct-current (DC) generator fixed on the same shaft as the alternator. The DC generator generates

4012-414: The salt-water environment. Marine alternators are designed to be explosion proof (ignition protected) so that brush sparking will not ignite explosive gas mixtures in an engine room environment. Depending on the type of system installed, they may be 12 or 24 volts. Larger marine diesels may have two or more alternators to cope with the heavy electrical demand of a modern yacht. On single alternator circuits,

4080-436: The solution is { i 1 = 1 1100 A i 2 = 4 275 A i 3 = − 3 220 A {\displaystyle {\begin{cases}i_{1}={\frac {1}{1100}}{\text{A}}\\[6pt]i_{2}={\frac {4}{275}}{\text{A}}\\[6pt]i_{3}=-{\frac {3}{220}}{\text{A}}\end{cases}}} The current i 3 has

4148-506: The speed of DC traction motors. Most of these had two generators: one to generate the excitation current for a larger main generator. Optionally, the generator also supplies head-end power (HEP) or power for electric train heating . The HEP option requires a constant engine speed, typically 900 r/min for a 480 V 60 Hz HEP application, even when the locomotive is not moving. Marine alternators used in yachts are similar to automotive alternators, with appropriate adaptations to

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4216-515: The sum of currents flowing into that node is equal to the sum of currents flowing out of that node; or equivalently: The algebraic sum of currents in a network of conductors meeting at a point is zero. Recalling that current is a signed (positive or negative) quantity reflecting direction towards or away from a node, this principle can be succinctly stated as: ∑ i = 1 n I i = 0 {\displaystyle \sum _{i=1}^{n}I_{i}=0} where n

4284-475: The supply phases and therefore will add together in the neutral which can cause the neutral current in a wye system to exceed the phase currents. Any polyphase system, by virtue of the time displacement of the currents in the phases, makes it possible to easily generate a magnetic field that revolves at the line frequency. Such a revolving magnetic field makes polyphase induction motors possible. Indeed, where induction motors must run on single-phase power (such as

4352-461: The three phase currents together as complex numbers and then converting from rectangular to polar co-ordinates. If the three-phase root mean square (RMS) currents are I L 1 {\displaystyle I_{L1}} , I L 2 {\displaystyle I_{L2}} , and I L 3 {\displaystyle I_{L3}} , the neutral RMS current is: which resolves to The polar magnitude of this

4420-405: The total power becomes or showing the above contention. Again, using the root mean square voltage V = V p 2 {\displaystyle V={\frac {V_{p}}{\sqrt {2}}}} , P T O T {\displaystyle P_{TOT}} can be written in the usual form For the case of equal loads on each of three phases, no net current flows in

4488-470: The total power does not vary with time. This is essential for keeping large generators and motors running smoothly. Notice also that using the root mean square voltage V = V p 2 {\displaystyle V={\frac {V_{p}}{\sqrt {2}}}} , the expression for P T O T {\displaystyle P_{TOT}} above takes the following more classic form: The load need not be resistive for achieving

4556-482: The utility grid requires synchronization conditions to be met. Alternators are used in modern internal combustion engine automobiles to charge the battery and to power the electrical system when its engine is running. Until the 1960s, automobiles used DC dynamo generators with commutators . With the availability of affordable silicon-diode rectifiers, alternators were used instead. In later diesel-electric locomotives and diesel electric multiple units ,

4624-410: The world's electric power, which is distributed by electric power grids . Alternating current generating systems were known in simple forms from the discovery of the magnetic induction of electric current in the 1830s. Rotating generators naturally produced alternating current, but since there was little use for it, it was normally converted into direct current via the addition of a commutator in

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