Chapter II AC fundamentals BEE ` ajanikant A. Metri

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2/18/2013 RIT AC – Alternating Current Voltages of ac sources alternate in polarity and vary in magnitude Voltages produce currents that vary in magnitude and alternate in direction

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2/18/2013 RIT AC – Alternating Current

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2/18/2013 RIT Generating AC Voltages

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2/18/2013 RIT Generating AC Voltages

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The sinusoidal waveform (sine wave) is the fundamental alternating current (ac) and alternating voltage waveform. Sine waves Electrical sine waves are named from the mathematical function with the same shape. 2/18/2013 RIT

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A wave is a disturbance. Unlike water waves, electrical waves cannot be seen directly but they have similar characteristics. All periodic waves can be constructed from sine waves , which is why sine waves are fundamental. 2/18/2013 RIT

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Sine waves are characterized by the amplitude and period. The amplitude is the maximum value of a voltage or current; the period is the time interval for one complete cycle. Sine waves Example The amplitude ( A ) of this sine wave is 20 V The period is 50.0 m s A T 2/18/2013 RIT

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The period of a sine wave can be measured between any two corresponding points on the waveform. Sine waves T T T T T T By contrast, the amplitude of a sine wave is only measured from the center to the maximum point. A 2/18/2013 RIT

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3.0 Hz Frequency Frequency ( f ) is the number of cycles that a sine wave completes in one second. Frequency is measured in hertz (Hz). Example If 3 cycles of a wave occur in one second, the frequency is 1.0 s 2/18/2013 RIT

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The period and frequency are reciprocals of each other. Period and frequency and Thus, if you know one, you can easily find the other. Example If the period is 50 m s, the frequency is 0.02 MHz = 20 kHz. (The 1/ x key on your calculator is handy for converting between f and T .) 2/18/2013 RIT

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Sinusoidal voltages are produced by ac generators and electronic oscillators. Sinusoidal voltage sources Generation of a sine wave A B C D When a conductor rotates in a constant magnetic field, a sinusoidal wave is generated. When the conductor is moving parallel with the lines of flux, no voltage is induced. When the loop is moving perpendicular to the lines of flux, the maximum voltage is induced. 2/18/2013 RIT

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Generators convert rotational energy to electrical energy. A stationary field alternator with a rotating armature is shown. The armature has an induced voltage, which is connected through slip rings and brushes to a load. The armature loops are wound on a magnetic core (not shown for simplicity). AC generator (alternator) Small alternators may use a permanent magnet as shown here; other use field coils to produce the magnetic flux. 2/18/2013 RIT

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AC generator (alternator) By increasing the number of poles, the number of cycles per revolution is increased. A four-pole generator will produce two complete cycles in each revolution. 2/18/2013 RIT

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Function generators Function selection Frequency Output level (amplitude) DC offset CMOS output Range Adjust Duty cycle Typical controls: Outputs Readout 2/18/2013 RIT

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Sine wave voltage and current values There are several ways to specify the voltage of a sinusoidal voltage waveform. The amplitude of a sine wave is also called the peak value, abbreviated as V P for a voltage waveform. Example The peak voltage of this waveform is 20 V. V P 2/18/2013 RIT

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The voltage of a sine wave can also be specified as either the peak-to-peak or the rms value. The peak-to-peak is twice the peak value. The rms value is 0.707 times the peak value. Sine wave voltage and current values Example The peak-to-peak voltage is 40 V. The rms voltage is 14.1 V. V PP V rms 2/18/2013 RIT

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For some purposes, the average value (actually the half-wave average) is used to specify the voltage or current. By definition, the average value is as 0.637 times the peak value. Sine wave voltage and current values Example The average value for the sinusoidal voltage is 12.7 V. V avg 2/18/2013 RIT

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Angular measurements can be made in degrees ( o ) or radians. The radian (rad) is the angle that is formed when the arc is equal to the radius of a circle. There are 360 o or 2 p radians in one complete revolution. Angular measurement 2/18/2013 RIT

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There are 2 p radians in one complete revolution and 360 o in one revolution. To find the number of radians, given the number of degrees: To find the number of degrees, given the number of radians: Angular measurement This can be simplified to:

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How many radians are in 45 o ? Angular measurement Example Solution: Example Solution: How many degrees are in 1.2 radians? 2/18/2013 RIT

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Instantaneous values of a wave are shown as v or i . The equation for the instantaneous voltage ( v ) of a sine wave is Sine wave equation where Example If the peak voltage is 25 V, the instantaneous voltage at 50 degrees is V p = q = Peak voltage Angle in rad or degrees 19.2 V 2/18/2013 RIT

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Sine wave equation A plot of the example in the previous slide (peak at 25 V) is shown. The instantaneous voltage at 50 o is 19.2 V as previously calculated. 2/18/2013 RIT

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The sine wave can be represented as the projection of a vector rotating at a constant rate. This rotating vector is called a phasor . Phasors are useful for showing the phase relationships in ac circuits. Phasors 2/18/2013 RIT

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Phase shift where f = Phase shift The phase of a sine wave is an angular measurement that specifies the position of a sine wave relative to a reference. To show that a sine wave is shifted to the left or right of this reference, a term is added to the equation given previously.

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Phase shift Notice that a lagging sine wave is below the axis at 0 o Example of a wave that lags the reference v = 30 V sin ( q - 45 o ) …and the equation has a negative phase shift 2/18/2013 RIT

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Phase shift Notice that a leading sine wave is above the axis at 0 o Example of a wave that leads the reference v = 30 V sin ( q + 45 o ) …and the equation has a positive phase shift 2/18/2013 RIT

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An important application of phase-shifted sine waves is in electrical power systems. Electrical utilities generate ac with three phases that are separated by 120° as illustrated. Phase shift 120 o Normally, 3-phase power is delivered to the user with three hot lines plus neutral. The voltage of each phase, with respect to neutral is 120 V. 120 o 120 o 2/18/2013 RIT

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The power formulas are: Power in resistive AC circuits The power relationships developed for dc circuits apply to ac circuits except you must use rms values in ac circuits when calculating power. 0 V 0 V For example, the dc and the ac sources produce the same power to the bulb: 120 V dc 170 V p = 120 V rms

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Assume a sine wave with a peak value of 40 V is applied to a 100 W resistive load. What power is dissipated? Power in resistive AC circuits Example Solution V rms = 0.707 x V p = 0.707 x 40 V = 28.3 V 8 W 2/18/2013 RIT

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Frequently dc and ac voltages are together in a waveform. They can be added algebraically, to produce a composite waveform of an ac voltage “riding” on a dc level. Superimposed dc and ac voltages 2/18/2013 RIT

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Alternators are ac generators. Utility companies use 3-phase alternators and deliver all three phases to industrial customers. A simplified 3-phase alternator is illustrated. Alternators Phase 1 Phase 2 Phase 3 Neutral 2/18/2013 RIT

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In vehicles, alternators generate ac, which is converted to dc for operating electrical devices and charging the battery. A basic vehicle alternator is illustrated. AC is more efficient to produce and can be easily regulated, hence it is generated and converted to dc by diodes. Alternators The output is taken from the rotor through the slip rings. 2/18/2013 RIT

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There are two major classifications of ac motors. These are the induction motor and the synchronous motor . Both types use a rotating field in the stator windings. AC Motors Induction motors work because current is induced in the rotor by the changing current in the stator. This current creates a magnetic field that reacts with the moving field of the stator, which develops a torque and causes the rotor to turn. Synchronous motors have a magnet for the rotor. In small motors, this can be a permanent magnet, which keeps up with the rotating field of the stator. Large motors use an electromagnet in the rotor, with external dc supplied to generate the magnetic field. 2/18/2013 RIT

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Pulse definitions Ideal pulses 2/18/2013 RIT

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Pulse definitions Non-ideal pulses Notice that rise and fall times are measured between the 10% and 90% levels whereas pulse width is measured at the 50% level. 2/18/2013 RIT

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Triangular and sawtooth waves Triangular and sawtooth waveforms are formed by voltage or current ramps (linear increase/decrease) Triangular waveforms have positive-going and negative-going ramps of equal duration. The sawtooth waveform consists of two ramps, one of much longer duration than the other. 2/18/2013 RIT

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Harmonics All repetitive non-sinusoidal waveforms are composed of a fundamental frequency (repetition rate of the waveform) and harmonic frequencies . Odd harmonics are frequencies that are odd multiples of the fundamental frequency. Even harmonics are frequencies that are even multiples of the fundamental frequency. 2/18/2013 RIT

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Harmonics A square wave is composed only of the fundamental frequency and odd harmonics (of the proper amplitude). 2/18/2013 RIT

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Oscilloscopes The oscilloscope is divided into four main sections. 2/18/2013 RIT

Oscilloscopes Vertical Horizontal Trigger Display 2/18/2013 RIT

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Sine wave Alternating current Period (T) Frequency (f) Hertz Current that reverses direction in response to a change in source voltage polarity. The time interval for one complete cycle of a periodic waveform. A type of waveform that follows a cyclic sinusoidal pattern defined by the formula y = A sin q . Selected Key Terms A measure of the rate of change of a periodic function; the number of cycles completed in 1 s. The unit of frequency. One hertz equals one cycle per second. 2/18/2013 RIT

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Instantaneous value Peak value Peak-to-peak value rms value The voltage or current value of a waveform at its maximum positive or negative points. The voltage or current value of a waveform measured from its minimum to its maximum points. The voltage or current value of a waveform at a given instant in time. Selected Key Terms The value of a sinusoidal voltage that indicates its heating effect, also known as effective value. It is equal to 0.707 times the peak value. rms stands for root mean square. 2/18/2013 RIT

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Radian Phase Amplitude Pulse Harmonics The maximum value of a voltage or current. A type of waveform that consists of two equal and opposite steps in voltage or current separated by a time interval. A unit of angular measurement. There are 2 p radians in one complete 360 o revolution. Selected Key Terms The frequencies contained in a composite waveform, which are integer multiples of the pulse repetition frequency. The relative angular displacement of a time-varying waveform in terms of its occurrence with respect to a reference. 2/18/2013 RIT

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Quiz 1. In North America, the frequency of ac utility voltage is 60 Hz. The period is a. 8.3 ms b. 16.7 ms c. 60 ms d. 60 s 2/18/2013 RIT

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Quiz 2. The amplitude of a sine wave is measured a. at the maximum point b. between the minimum and maximum points c. at the midpoint d. anywhere on the wave 2/18/2013 RIT

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Quiz 3. An example of an equation for a waveform that lags the reference is a. v = - 40 V sin ( q ) b. v = 100 V sin ( q + 35 o ) c. v = 5.0 V sin ( q - 27 o ) d. v = 27 V 2/18/2013 RIT

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4. In the equation v = V p sin q , the letter v stands for the a. peak value b. average value c. rms value d. instantaneous value Quiz 2/18/2013 RIT

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5. The time base of an oscilloscope is determined by the setting of the a. vertical controls b. horizontal controls c. trigger controls d. none of the above Quiz 2/18/2013 RIT

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6. A sawtooth waveform has a. equal positive and negative going ramps b. two ramps - one much longer than the other c. two equal pulses d. two unequal pulses Quiz 2/18/2013 RIT

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7. The number of radians in 90 o are a. p /2 b. p c. 2 p /3 d. 2 p Quiz 2/18/2013 RIT

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8. For the waveform shown, the same power would be delivered to a load with a dc voltage of a. 21.2 V b. 37.8 V c. 42.4 V d. 60.0 V Quiz 2/18/2013 RIT

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9. A square wave consists of a. the fundamental and odd harmonics b. the fundamental and even harmonics c. the fundamental and all harmonics d. only the fundamental Quiz 2/18/2013 RIT

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Quiz 10. A control on the oscilloscope that is used to set the desired number of cycles of a wave on the display is a. volts per division control b. time per division control c. trigger level control d. horizontal position control 2/18/2013 RIT

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Quiz Answers: 1. b 2. a 3. c 4. d 5. b 6. b 7. a 8. c 9. a 10. b 2/18/2013 RIT

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By: rbswati (25 month(s) ago)

Metri sir I request you to please send the copy of PPT on my email as unable to down load it. rbswati@gmail.com