Quick Quiz (POLL 1) Identify the correct statement(s): A. πππ needs load as open circuit and πΌπ needs load to be shorted. B. πππ needs load as short circuit and πΌπ needs load to be opened. C. π ππ needs load as open circuit and π π needs load to be shorted. D. π ππ needs load as open circuit and π π also needs same. a. A and B b. B and C c. A and D d. A, B and D Maximum Power Transfer Theorem In many practical situations, a circuit is designed to provide power to a load. There are applications in areas such as communications where it is desirable to maximize the power delivered to a load. Maximum Power Transfer Theorem STATEMENT: Maximum power is transferred to the load when the load resistance equals the Thevenin resistance as seen from the load (π ππ =π πΏ ). Derivation Practice Problem 1 Find the value of resistor π πΏ for transferring maximum power? Also find maximum power? UNIT 2: AC CIRCUITS Lecture 10 Prepared By: Pawandeep Kaur Assistant Professor and Head Syllabus Have an Overview! Todays Topic οΌDifference between DC and AC οΌ AC Fundamentals οΌ Generating ac Voltages οΌAlternating voltage and current οΌConcept of Notations ( i, v, I, V) οΌDefinitions of amplitude, phase, phase difference οΌAverage and RMS value of an AC signal Introduction ο± Historically, dc sources were the main means of providing electric power up until the late 1800s. ο±At the end of that century, the battle of direct current versus alternating current began. Both had their advocates among the electrical engineers of the time. ο±Because ac is more efficient and economical to transmit over long distances, ac systems ended up the winner. The war of the current DC AC Cannot be transmitted to longer distances because of the losses. Safe to transfer over longer city distances It flows in one direction in the circuit. It reverses its direction while flowing in a circuit. Magnitude of current or voltage does not vary with time Magnitude of current or voltage does not vary with time Electrons move steadily in one direction only. Electrons keep switching directions - forward and backward. Power Factor is always 1 Power Factor lies between 0 & 1. The frequency of direct current is zero. The frequency of alternating current is 50Hz or 60Hz depending upon the country. Example: Cell, Battery. Example: Generator. Terminologies used in AC Circuits 1. 2. 3. 4. 5. 6. 7. 8. 9. Peak value Peak to Peak value Instantaneous value Average (Mean) value Cycle Frequency Timeperiod Phase RMS value AC Fundamentals ο Previously you learned that DC sources have ο¬xed polarities and constant magnitudes and thus produce currents with constant value and unchanging direction ο In contrast, the voltages of ac sources alternate in polarity and vary in magnitude and thus produce currents that vary in magnitude and alternate in direction. Sinusoidal ac Voltage ο Sinusoidal ac Voltage One complete variation is referred to as a cycle. Starting at zero, the voltage increases to a positive peak amplitude, decreases to zero, changes polarity, increases to a negative peak amplitude, then returns again to zero. ο Since the waveform repeats itself at regular intervals, it is called a periodic signal. ο Symbol for an ac Voltage Source Lowercase letter e is used to indicate that the voltage varies with time. 17 Sinusoidal ac Current ο During the ο¬rst half-cycle, the source voltage is positive ο Therefore, the current is in the clockwise direction. ο During the second half-cycle, the voltage polarity reverses ο Therefore, the current is in the counterclockwise direction. ο Since current is proportional to voltage, its shape is also sinusoidal 18 Quick Quiz (Poll 1) The frequency of domestic power supply in India is (A) 200 Hz (B) 100 Hz (C) 60 Hz (D) 50 Hz Notations GENERATION OF AC VOLTAGE An alternator is an electrical generator that converts mechanical energy to electrical energy in the form of alternating current Principle: 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. Generating AC Voltages 22 Generating AC Voltages 23 Generating ac Voltages (Method A) ο One way to generate an ac voltage is to rotate a coil of wire at constant angular velocity in a ο¬xed magnetic ο¬eld ο The magnitude of the resulting voltage is proportional to the rate at which ο¬ux lines are cut ο its polarity is dependent on the direction the coil sides move through the ο¬eld. 24 Generating ac Voltages ο Since the coil rotates continuously, the voltage produced will be a repetitive, Time Scales ο Often we need to scale the output voltage in time. ο The length of time required to generate one cycle depends on the velocity of rotation. 600 revolutions in 1 minute = 600 rev / 60 s = 10 revolutions in 1 second. The time for 1 revolution = one-tenth of a second = 100 ms 25 Generating ac Voltages (Method-2) ο AC waveforms may also be created electronically using function (or signal) generators. ο With function generators, you are not limited to sinusoidal ac. gear. ο The unit of Figure can produce a variety of variable-frequency waveforms, including sinusoidal, square wave, triangular, and so on. ο Waveforms such as these are commonly used to test electronic 26 Instantaneous Value ο As the coil voltage changes from instant to instant. The value of voltage at any point on the waveform is referred to as its instantaneous value. ο The voltage has a peak value of 40 volts ο The cycle time of 6 ms. οΌ at t = 0 ms, the voltage is zero. οΌ at t=0.5 ms, the voltage is 20V. 27 Voltage and Current Conventions for ac ο First, we assign reference polarities for the source and a reference direction for the current. ο We then use the convention that, when e has a positive value, its actual polarity is the same as the reference polarity, and when e has a negative value, its actual polarity is opposite to that of the reference. ο For current, we use the convention that when i has a positive value, its actual direction is the same as the reference arrow, ο and when i has a negative value, its actual direction is opposite to that of the reference. 28 Voltage and Current Conventions for ac 29 Attributes of Periodic Waveforms ο Periodic waveforms (i.e., waveforms that repeat at regular intervals), regardless of their wave shape, may be described by a group of attributes such as: οΌ Frequency, Period, Amplitude, Peak value. Frequency: The number of cycles per second of a waveform is defined ο Frequency is denoted by the lower-case letter f. ο In the SI system, its unit is the hertz (Hz, named in honor of pioneer researcher Heinrich Hertz, 1857–1894). 30 Attributes of Periodic Waveforms ο Period: ο The period, T, of a waveform, is the duration of one cycle. ο It is the inverse of frequency. ο The period of a waveform can be measured between any two corresponding points ( Often it is measured between zero points because they are easy to establish on an oscilloscope trace). 32 Attributes of Periodic Waveforms Amplitude , Peak-Value, and Peak-to-Peak Value Amplitude (Em): The amplitude of a sine wave is the distance from its average to its peak. Peak-to-Peak Value (Ep-p): It is measured between minimum and maximum peaks. Peak Value The peak value of a voltage or current is its maximum value with respect to zero. 33 Quick Quiz (Poll 2) Peak to peak value of a sine wave is a. Equal to the maximum or phase value of sine wave b. Twice the maximum or phase value of sine wave c. Half of the maximum or phase value of sine wave d. Four times the maximum or phase value of sine wave Quick Quiz (Poll 3) The most common waveforms of ac is a. Square b. Triangular c. Sinusoidal d. Saw tooth The Basic Sine Wave Equation The voltage produced by the previously described generator is: • Em: the maximum coil voltage and • α : the instantaneous angular position of the coil. ο For a given generator and rotational velocity, Em is constant.) ο Note that a 0° represents the horizontal position of the coil and that one complete cycle corresponds to 360°. 36 Radian Measure ο In practice, q is usually expressed in radians per second, ο Radians and degrees are related by : For Conversion: POLL Convert 45Λ to radian A)π/8 B) π/4 C) π/2 D) 0 38 Relationship between ω, T, and f ο Earlier you learned that one cycle of sine wave may be represented as either: ο Substituting these into: Sinusoidal Voltages and Currents as Functions of Time: ο We could replace the angle α as: 39 Quick Quiz (Poll 1 ) The time period or periodic time T of an alternating quantity is the time taken in seconds to complete a. one cycle b. alternation c. none of the above d. Half cycle Quick Quiz (Poll 2 ) The time period of an alternating quantity is 0.02 second. Its frequency will be a. 25 Hz b. 50 Hz c. 100 Hz d. 0.02 Hz Quick Quiz (Poll 3 ) The angular frequency of an alternating quantity is a mathematical quantity obtained by multiplying the frequency f of the alternating quantity by a factor a. π/2 b. π c. 2π d. 4π Shifted Sine Waves Phasor Representation of Current Voltages with Phase Shifts ο If a sine wave does not pass through zero at t =0 s, it has a phase shift. ο Waveforms may be shifted to the left or to the right Train eg 47 Practice Problem 1 Given the sinusoid 30 sin(4ππ‘ − 75°)calculate its amplitude, phase, angular frequency, period, and frequency. Practice Problem 1 Given the sinusoid 30 sin(4ππ‘ − 75°)calculate its amplitude, phase, angular frequency, period, and frequency. Practice Problem 2 Practice Problem 2 Average Value Algebraic sum of all the values divided by the total number of values. Same concept is applicable for a waveform that varies with time. Thus, the average value over full cycle is ZERO Quick Quiz (Poll ) The average value of current in a sinusoidal signal over full cycle is: A. πΌπ 2 B. πΌπ C. 1 D. 0 2 However, an average value can be defined for the half-cycle (positive or negative) for a sinusoidal signal. Average Value Half Wave Rectifier Full Wave Rectifier Average Value Half Wave Rectifier Full Wave Rectifier Quick Quiz (Poll) Based on the previous results, we can say that the average value of half wave rectifier is_________ than full wave rectifier. A. Double B. Half C. Same D. None of these RMS (Effective) Value ο±The r.m.s value is defined in terms of heating effect. ο±The r.m.s. value of an alternating current is given by that steady (d.c.) current which when flowing through a given circuit for a given time produces the same heat as produced by the alternating current when flowing through the same circuit for the same time. RMS (Effective) Value RMS (Effective) Value RMS Value Half Wave Rectifier Full Wave Rectifier RMS Value Half Wave Rectifier Full Wave Rectifier Quick Quiz (Poll) • What is the effective value of current ? • A)RMS current • B)Average current • C)Instantaneous current • D)total current Quick Quiz (Poll) • The voltage of domestic supply is 230V.This figure represents • A)Mean vale • B)RMS value • C)Peak Value • D) Average value Form Factor and Peak Factor ο±Form Factor, πΎπ = ππππ πππ£π ο±Peak Factor or Crest Factor, πΎπ = ππ ππππ Quick Quiz (Poll) For a pure sinusoidal waveform the Form Factor will always be equal to: 1 A. 2 B. 0.637 C. 1.11 D. 1.414 Importance of Form Factor and Peak Factor ο±Actually some of our meters are designed to measure the RMS values but that is of pure sinusoidal waveforms, if there comes any distortion in the waveform, the meter won't give the correct RMS value. For meter the waveform is still a sinusoidal but it doesn't detect the distortion that's why we use form factor to get accurate value of RMS by just multiplying form factor with the average value of that distorted waveform. It is helpful in finding the RMS values of waveforms other than pure sinusoidal. ο±Similarly, Some loads, such as switching power supplies or lamp ballasts, have current waveforms that are not sinusoidal. They draw a high current for a short period of time, and their crest factors, therefore, can be quite a bit higher than 1.414. Power and Power Factor • In a general ac circuit, we assume: • π£ = π£π sin ππ‘ πππ π = ππ sin(ππ‘ − π) INSTANTANEOUS POWER: π = π£π = ππππ πΌπππ πππ π − cos 2ππ‘ − π ] Average Power • Also called as True Power or Real Power. • It represents the actual amount of power being used, or dissipated, in a circuit. • Recall that : π = π£π = ππππ πΌπππ πππ π − 2 cos ππ‘ − π • Average value of cos ππ‘ − π is ZERO for a full cycle. • Therefore, average power: πππ£π = ππππ πΌπππ πππ π • Units: Watts Apparent Power • Recall that : π = π£π = ππππ πΌπππ πππ π − 2 cos ππ‘ − π • Average value of cos ππ‘ − π is ZERO for a full cycle. • Therefore, Apparent power: ππππ = ππππ πΌπππ • Units: Volt-Amperes (VA) Power Factor • Recall that : π = π£π = ππππ πΌπππ πππ π − 2 cos ππ‘ − π • Average Power: πππ£π = ππππ πΌπππ πππ π • And, Apparent power:ππππ = ππππ πΌπππ Therefore, Average Power = Apparent Power × πππ π Implies, π΄π£πππππ πππ€ππ πππ π= π΄πππππππ‘ πππ€ππ = πππ€ππ πΉπππ‘ππ Numerical Practice
