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Chapter 1 – Three phase circuit and three phase power Describe and explain the characteristic in three phase circuits. Analyze the three phase circuit General Format for the Sinusoidal Voltage or Current The basic mathematical format for the sinusoidal waveform is: where: Am is the peak value of the waveform is the unit of measure for the horizontal axis General Format for the Sinusoidal Voltage or Current equation = t states that the angle through which the rotating vector will pass is determined by the angular velocity of the rotating vector and the length of time the vector rotates. For a particular angular velocity (fixed ), the longer the radius vector is permitted to rotate (that is, the greater the value of t ), the greater will be the number of degrees or radians through which the vector will pass. The general format of a sine wave can also be as: The General Format for the Sinusoidal Voltage or Current For electrical quantities such as current and voltage, the general format is: i I m sin t I m sin e Em sin t Em sin where: the capital letters with the subscript m represent the amplitude, and the lower case letters i and e represent the instantaneous value of current and voltage, respectively, at any time t. General Format for the Sinusoidal Voltage or Current Example 1.1 Given e = 5sin, determine e at = 40 and = 0.8. Solution For = 40, e 5 sin 40 3.21 V For = 0.8 180 144 0.8 e 5 sin 144 2.94 V General Format for the Sinusoidal Voltage or Current Example 1.2 (a) Determine the angle at which the magnitude of the sinusoidal function v = 10 sin 377t is 4 V. (b) Determine the time at which the magnitude is attained. General Format for the Sinusoidal Voltage or Current Example 1.2 - solution Vm 10 V; 377 rad/s v Vm sin t V Hence, v 10 sin 377t V When v = 4 V, 4 10 sin 377t Or; sin 1 0.4 23.58 4 sin 377t sin 0.4 10 General Format for the Sinusoidal Voltage or Current Example 1.2 – solution 377t 23.58 0.412 rad 0.412 t 1.09 ms 377 Sinusoidal ac Voltage Characteristics and Definitions Definitions Waveform: The path traced by a quantity, such as voltage, plotted as a function of some variable such as time, position, degree, radius, temperature and so on. Instantaneous value: The magnitude of a waveform at any instant of time; denoted by the lowercase letters (e1, e2). Peak amplitude: The maximum value of the waveform as measured from its average (or mean) value, denoted by the uppercase letters Em (source of voltage) and Vm (voltage drop across a load). Sinusoidal ac Voltage Characteristics and Definitions Definitions Peak value: The maximum instantaneous value of a function as measured from zero-volt level. Peak-to-peak value: Denoted by Ep-p or Vp-p, the full voltage between positive and negative peaks of the waveform, that is, the sum of the magnitude of the positive and negative peaks. Periodic waveform: A waveform that continually repeats itself after the same time interval. Sinusoidal ac Voltage Characteristics and Definitions Definitions Period (T): The time interval between successive repetitions of a periodic waveform (the period T1 = T2 = T3), as long as successive similar points of the periodic waveform are used in determining T Cycle: The portion of a waveform contained in one period of time Frequency: (Hertz) the number of cycles that occur in 1 s 1 hertz, Hz f T Sinusoidal ac Voltage Characteristics and Definitions Sinusoidal ac Voltage Characteristics and Definitions Sinusoidal ac Voltage Characteristics and Definitions Example 1.3 Determine: (a) peak value (b) instantaneous value at 0.3 s and 0.6 s (c) peak-to-peak value (d) period (e) how many cycles are shown (f) frequency Sinusoidal ac Voltage Characteristics and Definitions Example 1.3 – solution (a) 8 V; (b) -8 V at 3 s and 0 V at 0.6 s; (c) 16 V; (d) 0.4 s; (e) 3.5 cycles; (f) 2.5 Hz Three Phase System An ac generator designed to develop a single sinusoidal voltage for each rotation of the shaft (rotor) is referred to as a single-phase ac generator If the number of coils on the rotor is increased in a specified manner, the result is a polyphase ac generator, which develops more than one ac phase voltage per rotation of the rotor In general, three-phase systems are preferred over single-phase systems for the transmission of power for many reasons, including the following: Three Phase System 1. 2. 3. 4. Thinner conductors can be used to transmit the same kVA at the same voltage, which reduces the amount of copper required (typically about 25% less) and in turn reduces construction and maintenance costs. The lighter lines are easier to install, and the supporting structures can be less massive and farther apart. Three-phase equipment and motors have preferred running and starting characteristics compared to single-phase systems because of a more even flow of power to the transducer than can be delivered with a single-phase supply. In general, most larger motors are three phase because they are essentially self-starting and do not require a special design or additional starting circuitry. Three Phase System The frequency generated is determined by the number of poles on the rotor (the rotating part of the generator) and the speed with which the shaft is turned. Throughout the United States the line frequency is 60 Hz, whereas in Europe the chosen standard is 50 Hz. On aircraft and ships the demand levels permit the use of a 400 Hz line frequency. The three-phase system is used by almost all commercial electric generators. Three Phase System Most small emergency generators, such as the gasoline type, are one-phased generating systems. The two-phase system is commonly used in servomechanisms, which are self-correcting control systems capable of detecting and adjusting their own operation. Servomechanisms are used in ships and aircraft to keep them on course automatically, or, in simpler devices such as a thermostatic circuit, to regulate heat output. The number of phase voltages that can be produced by a polyphase generator is not limited to three. Any number of phases can be obtained by spacing the windings for each phase at the proper angular position around the stator. Three-Phase Generator The three-phase generator has three induction coils placed 120° apart on the stator. The three coils have an equal number of turns, the voltage induced across each coil will have the same peak value, shape and frequency. Three-Phase Generator At any instant of time, the algebraic sum of the three phase voltages of a three-phase generator is zero. Three-Phase Generator The sinusoidal expression for each of the induced voltage is: Y-Connected Generator If the three terminals denoted N are connected together, the generator is referred to as a Y-connected three-phase generator. Y-Connected Generator The point at which all the terminals are connected is called the neutral point. If a conductor is not attached from this point to the load, the system is called a Y-connected, three-phase, three-wire generator. If the neutral is connected, the system is a Yconnected three-phase, four-wire generator. The three conductors connected from A, B and C to the load are called lines. Y-Connected Generator The voltage from one line to another is called a line voltage The magnitude of the line voltage of a Y-connected generator is: Phase Sequence (Y-Connected Generator) The phase sequence can be determined by the order in which the phasors representing the phase voltages pass through a fixed point on the phasor diagram if the phasors are rotated in a counterclockwise direction. Y-Connected Generator with a YConnected Load Loads connected with three-phase supplies are of two types: the Y and the ∆. If a Y-connected load is connected to a Y-connected generator, the system is symbolically represented by YY. Y-Connected Generator with a YConnected Load If the load is balanced, the neutral connection can be removed without affecting the circuit in any manner; that is, if Z 1 = Z2 = Z3 then IN will be zero Since IL = V / Z the magnitude of the current in each phase will be equal for a balanced load and unequal for an unbalanced load. In either case, the line voltage is Y-∆ System There is no neutral connection for the Y-∆ system shown below. Any variation in the impedance of a phase that produces an unbalanced system will simply vary the line and phase currents of the system. Y-∆ System For a balanced load, Z1 = Z2 = Z3. The voltage across each phase of the load is equal to the line voltage of the generator for a balanced or an unbalanced load: V = EL. Kirchhoff’s current law is employed instead of Kirchhoff’s voltage law. The results obtained are: Example For the system in figure shown below a. Find the phase angle delta2,3 for phase sequence ABC b. Find the current in each phase of load c. Find the magnetude of the line current ∆-Connected Generator In the figure below, if we rearrange the coils of the generator in (a) as shown in (b), the system is referred to as a three-phase, three-wire. ∆-Connected Generator ∆-connected ac generator In this system, the phase and line voltages are equivalent and equal to the voltage induced across each coil of the generator: E AB E AN and e AN 2 E AN sin t EBC EBN and eBN 2 E BN sin( t 120) or ECA ECN and eCN 2 ECN sin( t 120) EL = Eg Only one voltage (magnitude) is available instead of the two in the Y-Connected system. ∆-Connected Generator Unlike the line current for the Y-connected generator, the line current for the ∆-connected system is not equal to the phase current. The relationship between the two can be found by applying Kirchhoff’s current law at one of the nodes and solving for the line current in terms of the phase current; that is, at node A, IBA = IAa + IAC or IAa = IBA - IAC = IBA + ICA ∆-Connected Generator The phasor diagram is shown below for a balanced load. In general, line current is: Phase Sequence (∆- Connected Generator) Even though the line and phase voltages of a ∆ connected system are the same, it is standard practice to describe the phase sequence in terms of the line voltages In drawing such a diagram, one must take care to have the sequence of the first and second subscripts the same In phasor notation, EAB = EAB 0 EBC = EBC 120 ECA = ECA 120 Example For the system shown below : a. Find the voltage across each phase of load. b. Find the magnetude of the line voltage