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electronics fundamentals circuits, devices, and applications THOMAS L. FLOYD DAVID M. BUCHLA Chapter 8 – AC Circuits 1 of 77 AC Circuits Alternating Voltage is a voltage that: 1. Continuously varies in magnitude 2. Periodically reverses in polarity 2 of 77 AC Circuits Symbol for a sinusoidal voltage source. 3 of 77 AC Circuits Sine waves The sinusoidal waveform (sine wave) is the fundamental alternating current (ac) and alternating voltage waveform. Electrical sine waves are named from the mathematical function with the same shape. 5 of 77 AC Circuits Generation of a sine wave Sinusoidal voltage sources Sinusoidal voltages are produced by ac generators and electronic oscillators. When a conductor rotates in a constant magnetic field, a sinusoidal wave is generated. C N D B S A B C D A Motion of conductor Conduc tor the conductor is moving parallel with to When When the loop is moving perpendicular the of linesflux, of flux, voltage is induced. the lines thenomaximum voltage is induced. 6 of 77 AC Circuits Sine waves Sine waves are characterized by the amplitude and period. 1. The amplitude is the maximum value of a voltage or current 2. The period is the time interval for one complete cycle. 20 V 15 V The amplitude (A) of this sine wave is 20 V The period is 50.0 s A 10 V 0V t (s) 25 0 37.5 50.0 -10 V -15 V -20 V T 7 of 77 AC Circuits Sine waves The period (T) of a sine wave can be measured between any two corresponding points on the waveform. TT T T A T T A By contrast, the amplitude of a sine wave is only measured from the center to the maximum point. 8 of 77 AC Circuits Frequency Frequency ( f ) is the number of cycles that a sine wave completes in one second. Frequency is measured in hertz (Hz). If 3 cycles of a wave occur in one second, the frequency is 3.0 Hz 1.0 s 10 of 77 AC Circuits 11 of 77 AC Circuits Period and frequency The period and frequency are reciprocals of each other. 1 f T and 1 T f If the period is 50 s, the frequency is 0.02 MHz = 20 kHz. (The 1/x key on your calculator is handy for converting between f and T.) 12 of 77 AC Circuits Sine wave voltage and current values • Instantaneous value (v): Voltage or current at any point on the curve. • Peak value (VP for voltage): The amplitude of a sine wave. 20 V 15 V 10 V The peak voltage of this waveform is 20 V. 0V v VP t (s) 0 25 v 37.5 50.0 -10 V -15 V -20 V 13 of 77 AC Circuits Sine wave voltage and current values Peak to peak value: Value from positive peak to negative peak. Equation = 20 V 15 V V 2V I 2 I PP P PP 10 V 0V P t (s) 0 25 37.5 50.0 -10 V -15 V RMS (root mean squared) value: Is the sinusoidal wave with the same heat value as a DC voltage source (known as the effective value) -20 V V 0.707V rm s P V 1.414V p vms I 0.707 I rm s P I 1.414 I p vms 15 of 77 AC Circuits Sine wave voltage and current values VP = 20 volts 20 V 15 V The peak-to-peak voltage is 40 V. 10 V Vrms 0V 0 VPP t (s) 25 37.5 50.0 -10 V The rms voltage is 14.1 V. -15 V -20 V VP 1.414Vrms VPP 2.828Vrms This is magnitude Vpp 16 of 77 AC Circuits Phase of a Sine Wave Phase: Angular measurement that specifies the position on the sine wave relative to a reference point. 23 of 77 AC Circuits Phase shifts Occurs when a sine wave is shifted right or left in relation to the base/reference sine wave. 25 of 77 AC Circuits Phase shift – Lead/Lag Occurs when a sine wave is shifted right or left in relation to the base/reference sine wave. A - Leads B – Lags by 450 A - Lags B – Leads by 300 26 of 77 AC Circuits Phase shift Example of a wave that lags the reference (not on guided notes) …and the equation has a negative phase shift Referenc e 40 Peak voltage 30 v = 30 V sin (q - 45o) Voltage (V) 20 10 0 0 45 90 135 180 225 270 315 360 405 -20 -30 - 40 Notice that a lagging sine wave is below the axis at 0o Angle () 27 of 77 AC Circuits Phase shift Example of a wave that leads the reference (not on guided notes) Notice that a leading sine Referenc e o wave is above the axis at 0 40 Peak voltage 30 Voltage (V) 20 v = 30 V sin (q + 45o) 10 -45 0 0 -10 -20 -30 -40 45 90 135 180 225 270 315 360 …and the equation has a positive phase shift Angle () 28 of 77 AC Circuits PolyPhase power An important application of phase-shifted sine waves is in electrical power systems. • Electrical utilities generate ac with three phases that are separated by 120o. • 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. 120o 120o 120o 0o 29 of 77 AC Circuits Sine wave equation Instantaneous values of a wave are shown as v or i. The equation for the instantaneous voltage (v) of a sine wave is v V sin q p where Vp = Peak voltage q (theta) = Angle in rad or degrees If the peak voltage is 25 V, the instantaneous voltage at 50 degrees is 19.2 V 30 of 77 AC Circuits Sine wave equation A certain sine wave has a positive-going zero crossing at 0° and an peak value of 40V. Calculate its instantaneous voltage for the degrees listed below for the sine wave below. 30 20 Voltage (V) v V p sin q 40 10 0 Vp = Peak voltage q (theta) = Angle in rad or degrees -1 0 -2 0 -3 0 - 40 45°, 125°, 180°, 220°,325° 31 of 77 AC Circuits Phasors Phasor (aka Phase Vector): Representation of a sine wave whose amplitude (A) and angular frequency (ω - omega) are a constant rate. 90 180 0 0 90 180 360 33 of 77 AC Circuits 34 of 77 AC Circuits Power in resistive AC circuits A sinusoidal voltage produces a sinusoidal current 35 of 77 AC Circuits Power in resistive AC circuits Kirchhoff’s voltage law applies to AC circuits just like DC circuits 36 of 77 AC Circuits Power in resistive AC circuits Power in AC circuits is calculated using RMS values for voltage and current. The power formulas are: P Vrms I rms 2 Vrms P R 2 P I rms R The dc and the ac sources produce the same power to the bulb: 120 Vdc 0V ac or dc source 170 Vp = 120 Vrms 0V Bulb WHY? 37 of 77 AC Circuits Power in resistive AC circuits Assume a sine wave with a peak value of 40 V is applied to a 100 W resistive load. What power is dissipated? 40 30 Voltage (V) 20 10 0 -1 0 -2 0 -3 0 - 40 Vrms = 0.707 x Vp = 0.707 x 40 V = 28.3 V 2 Vrms 28.3 V 2 P 8W R 100 W 38 of 77 AC Circuits AC generator (alternator) • Generators convert rotational energy to electrical energy. • 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). Small alternators may use a permanent magnet Others use field coils to produce the magnetic flux. 44 of 77 AC Circuits 45 of 77 AC Circuits AC generator (alternator) • Increasing the number of poles increases the number of cycles per revolution. • A four-pole generator will produce two complete cycles in each revolution. 46 of 77 AC Circuits Output Frequency of an AC Generator Ns f 120 f – frequency (Hz) N – number of poles s - speed in RPM 1 4 2 3 47 of 77 AC Circuits Alternators • In vehicles, alternators generate ac, which is converted to dc for operating electrical devices and charging the battery. • AC is more efficient to produce and can be easily regulated, hence it is generated and converted to DC by diodes. Housing The output is taken from the rotor through the slip rings. Stator coils Rotor Diode plate Diodes Slip rings 48 of 77 AC Circuits AC Motors There are two major classifications of ac motors. 1. Induction motor 2. Synchronous motor. 3. Both types use a rotating field in the stator windings. 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. 49 of 77 AC Circuits Rotating the stator produces a net magnetic field 50 of 77 AC Circuits Rotating the stator produces a net magnetic field 51 of 77 AC Circuits Induction vs. Stator Motors Squirrel Cage Rotor 52 of 77 AC Circuits Induction vs. Stator Motors 53 of 77 AC Circuits Pulse definitions Ideal pulses Leading (rising) edge Leading (falling) edge Trailing (falling) edge Trailing (rising) edge Baseline Am plitude Am plitude Baseline Pulse width (a) Positive-going pulse Pulse width (b) Negative-going pulse 55 of 77 AC Circuits Repetitive pulse waveforms • Periodic waveforms repeat at fixed intervals. • Pulse repetition frequency: Rate at which the pulses repeat. • Duty Cycle – Ratio of pulse width (tw) to the period (T) Percent Duty Cycle = tw 100 T Vavg = baseline = (duty cycle)( amplitude) 57 of 77 AC Circuits Nonsinusoidal Wave Forms • Define terms on page 359 • Rise time: • Fall time: • Pulse width: 58 of 77 AC Circuits Voltage ramps Ramp – Linear increase or decrease in voltage or current. Yaxis V I or Slope = Xaxis t t 59 of 77 AC Circuits 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 (same T T slope either increasing or decreasing). T T The sawtooth waveform consists of two ramps, one of much longer duration than the other. (unequal slopes in either direction). 60 of 77 AC Circuits Oscilloscope A device that traces the graph of a measured electrical signal on its screen. 63 of 77 AC Circuits Video on Osciloscope • http://www.cleanvideosearch.com/media/acti on/yt/watch?v=8VEg6L2QG5o • The name of video is: AC vs Dc explain how to use an oscilloscope 65 of 77 AC Circuits Reading an Oscilloscope 66 of 77 AC Circuits Calculate for each wave: Period, Peak, Peak to Peak, RMS T= 3.0 ms Vp = 1250 mv Vpp = 2500 mv RMS = 833.8 mV T= 20 ms Vp = 1.5 v Vpp = 3.0 v RMS = 1.06 V 500 mv 0.5 ms T= 6000 µs; 6 ms Vp = 20.4 v Vpp = 40.8 v RMS = 14.4 V T= 30 µs Vp = 24 v Vpp = 48 v RMS = 16.97 V 6v 300 µs 12 v 15 µs 67 of 77 AC Circuits Selected Key Terms Sine wave A type of waveform that follows a cyclic sinusoidal pattern defined by the formula y = A sin q. Alternating Current that reverses direction in response to a current change in source voltage polarity. Period (T) The time interval for one complete cycle of a periodic waveform. Frequency (f) A measure of the rate of change of a periodic function; the number of cycles completed in 1 s. Hertz The unit of frequency. One hertz equals one cycle per second. 68 of 77 AC Circuits Selected Key Terms Instantaneous The voltage or current value of a waveform at value a given instant in time. Peak value The voltage or current value of a waveform at its maximum positive or negative points. Peak-to-peak The voltage or current value of a waveform value measured from its minimum to its maximum points. rms value 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. 69 of 77 AC Circuits 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 70 of 77 AC Circuits 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 71 of 77 AC Circuits Quiz 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 74 of 77 AC Circuits Quiz 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 75 of 77 AC Circuits Quiz 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 60 V 45 V 30 V 0V t (s) 0 25 37.5 50.0 -30 V -45 V -60 V 77 of 77 AC Circuits 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 79 of 77