3Alternating waveforms IntroductionAlternating waveformsThe term alternating indicates only that the waveform alternates between two prescribed levels in a set time sequence.2
4Sinusoidal ac Voltage Characteristics and Definitions GenerationAn ac generator (or alternator) powered by water power, gas, or nuclear fusion is the primary component in the energy-conversion process.The energy source turns a rotor (constructed of alternating magnetic poles) inside a set of windings housed in the stator (the stationary part of the dynamo) and will induce voltage across the windings of the stator.3
5Various sources of ac power: (a) generating plant; (b) portable ac generator; (c) wind-power station; (d) solar panel; (e) function generator.
6Sinusoidal ac Voltage Characteristics and Definitions Generation:Wind power and solar power energy are receiving increased interest from various districts of the world.The turning propellers of the wind-power station are connected directly to the shaft of an ac generator.Light energy in the form of photons can be absorbed by solar cells. Solar cells produce dc, which can be electronically converted to ac with an inverter.A function generator, as used in the lab, can generate and control alternating waveforms.4
8The electromechanical ac generator, cutaway view (courtesy of General Electric).
9AC GeneratorThe ac generator has slip rings that pick up the induced voltage through a complete rotation cycleThe induced voltage is related to the number of lines of flux being cut. When the loop 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
11One revolution of the wire loop generates one cycle of the sinusoidal voltage.
12Frequency is directly proportional to the rate of rotation of the wire loop in an ac generator.
13Multi-pole ac Generator By increasing the number of poles, the number of cycles per revolution can be increased
14Four poles achieve a higher frequency than two poles for the same r p s.
15Electronic Signal Generators In the lab, we usually use a signal generator to produce a variety of waveforms at a wide range of frequenciesAn oscillator in the signal generator produces the repetitive waveWe are able to set the frequency and amplitude of the signal from the signal generator
16Typical signal generators. (Copyright Tektronix, Inc Typical signal generators. (Copyright Tektronix, Inc. Reproduced by permission.)
18Sinusoidal ac Voltage Characteristics and 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).5
19Sinusoidal ac Voltage Characteristics and 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.6
23Sinusoidal ac Voltage Characteristics and 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 TCycle: The portion of a waveform contained in one period of timeFrequency: (Hertz) the number of cycles that occur in 1 s7
24Period of a Sine WaveThe time required for a sine wave to complete one full cycle is called the period (T)A cycle consists of one complete positive, and one complete negative alternationThe period of a sine wave can be measured between any two corresponding points on the waveform
28Frequency of a Sine Wave Frequency ( f ) is the number of cycles that a sine wave completes in one secondThe more cycles completed in one second, the higher the frequencyFrequency is measured in hertz (Hz)Relationship between frequency ( f ) and period (T) is:f = 1/T
38where: Vpp = 2Vp and Ipp = 2Ip Peak-to-Peak ValuesThe peak-to-peak value of a sine wave is the voltage or current from the positive peak to the negative peakThe peak-to-peak values are represented as:Vpp and Ippwhere: Vpp = 2Vp and Ipp = 2Ip
40Instantaneous Values of Sine Waves The instantaneous values of a sine wave voltage (or current) are different at different points along the curve, having negative and positive valuesInstantaneous values are represented as:v and i
41General 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 axis14
42Sine Wave Formula The general expression for a sine wave is: y = A sin Where: y = an instantaneous value (v or i)A = amplitude (maximum value) = angle along the horizontal axis
43Example of instantaneous values of a sinusoidal voltage.
49Average ValueThe algebraic sum of the areas must be determined, since some area contributions will be from below the horizontal axis.Area above the axis is assigned a positive sign and area below the axis is assigned a negative sign.The average value of any current or voltage is the value indicated on a dc meter – over a complete cycle the average value is the equivalent dc value.21
50Average Value Understanding the average value using a sand analogy: The average height of the sand is that height obtained if the distance form one end to the other is maintained while the sand is leveled off.20
51Average Value of a Sine Wave The average value is the total area under the half-cycle curve divided by the distance in radians of the curve along the horizontal axisVavg = 0.637VpIavg = 0.637Ip
53Effective (rms) Values How is it possible for a sinusoidal ac quantity to deliver a net power if, over a full cycle the net current in any one direction is zero (average value = 0).Irrespective of direction, current of any magnitude through a resistor will deliver power to that resistor – during the positive and negative portions of a sinusoidal ac current, power is being delivered at each instant of time to the resistor.The net power flow will equal twice that delivered by either the positive or the negative regions of sinusoidal quantity.22
54RMS Value of a Sine WaveThe rms (root mean square) value, or effective value, of a sinusoidal voltage is equal to the dc voltage that produces the same amount of heat in a resistance as does the sinusoidal voltageVrms = 0.707VpIrms = 0.707Ip
55An experimental setup to establish a relationship between dc and ac quantities.
56When the same amount of heat is being produced by the resistor in both setups, the sinusoidal voltage has an rms value equal to the dc voltage.
57Effective (rms) Values The formula for power delivered by the ac supply at any time is:The average power delivered by the ac source is just the first term, since the average value of a cosine wave is zero even though the wave may have twice the frequency of the original input current waveform.23
58Effective (rms) Values The equivalent dc value is called the effective value of the sinusoidal quantityorandWhere: Im and Em are max (peak) values24
59Effective (rms) Values InstrumentationA true rms meter will read the effective value of any waveform and is not limited to only sinusoidal waveforms.You should make sure that your meter is a true rms meter, by checking the manual, if waveforms other than purely sinusoidal are to be encountered.25
60Angular Measurement of a Sine Wave A degree is an angular measurement corresponding to 1/360 of a circle or a complete revolutionA radian (rad) is the angular measure along the circumference of a circle that is equal to the radius of the circleThere are 2 radians or 360 in one complete cycle of a sine wave
61Phase Relations If a sinusoidal expression should appear as e = - Emsin tthe negative sign is associated with the sine portion of the expression, not the peak value Em .Phase MeasurementsWhen determining the phase measurement we first note that each sinusoidal function has the same frequency, permitting the use of either waveform to determine the period.Since the full period represents a cycle of 360°, the following ratio can be formed:19
62The Sine WaveIf we define x as the number of intervals of r (the radius) around the circumference of a circle, thenC = 2r = x • r and we find x = 2Therefore, there are 2 rad around a 360° circle, as shown in the figure.10
63Angular measurement showing relationship of the radian to degrees.
64Angular measurements starting at 0º and going counterclockwise.
66The Sine WaveThe quantity is the ratio of the circumference of a circle to its diameter.For 180° and 360°, the two units of measurement are related as follows:11
67The Sine WaveThe sinusoidal wave form can be derived from the length of the vertical projection of a radius vector rotating in a uniform circular motion about a fixed point.The velocity with which the radius vector rotates about the center, called the angular velocity, can be determined from the following equation:12
71The Sine Wave The angular velocity () is: or Since () is typically provided in radians per second, the angle obtained using = t is usually in radians.The time required to complete one revolution is equal to the period (T) of the sinusoidal waveform. The radians subtended in this time interval are 2.or13
72General Format for the Sinusoidal Voltage or Current The 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:15
73Demonstrating the effect of on the frequency and period.
74Phase of a Sine WaveThe phase of a sine wave is an angular measurement that specifies the position of a sine wave relative to a referenceWhen a sine wave is shifted left or right with respect to this reference, there is a phase shift
75Expressions for Shifted Sine Waves When a sine wave is shifted to the right of the reference by an angle , it is termed laggingWhen a sine wave is shifted to the left of the reference by an angle , it is termed leading
76Ohms’s Law and Kirchhoff’s Laws in AC Circuits When time-varying ac voltages such as a sinusoidal voltage are applied to a circuit, the circuit laws that were studied earlier still applyOhm’s law and Kirchhoff’s laws apply to ac circuits in the same way that they apply to dc circuits
77sinusoidal voltage produces a sinusoidal current
78Illustration of Kirchhoff’s voltage law in an ac circuit
79Superimposed dc and ac Voltages DC and ac voltages will add algebraically, to produce an ac voltage “riding” on a dc level
82Pulse WaveformsA pulse has a rapid transition (leading or rising edge) from a baseline to an amplitude level, then, after a period of time, a rapid transition (trailing or falling edge) back to the baseline levelPulses can be positive-going, or negative-going, depending upon where the baseline isThe distance between rising and falling edge is termed the pulse width
83Non-ideal PulseA non-ideal pulse has a rising and falling time interval, measured between 10% and 90% of its AmplitudePulse width is taken at the half-way point
85Repetitive PulsesAny waveform that repeats itself at fixed intervals is periodicThe time from one pulse to the corresponding point on the next pulse is the period, T ( f =1/T )The duty cycle is the ratio of the pulse width (tw) to the period (T), and is usually expressed as %Duty cycle = (tw/T)100%Square waves have a 50% duty cycle
88Triangular and Sawtooth Waveforms 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 slopeThe sawtooth waveform is a special case of the triangular wave consisting of two ramps, one of much longer duration than the other. A sawtooth voltage is sometimes called a sweep voltage
91HarmonicsA repetitive non-sinusoidal waveform is composed of a fundamental frequency (repetition rate of the waveform) and harmonic frequenciesOdd harmonics are frequencies that are odd multiples of the fundamental frequencyEven harmonics are frequencies that are even multiples of the fundamental frequencyComposite waveforms vary from a pure sine wave, they may contain only even harmonics, only odd harmonics or both