2 Learning ObjectivesCompare AC and DC voltage and current sources as defined by voltage polarity, current direction and magnitude over time.Define the basic sinusoidal wave equations and waveforms, and determine amplitude, peak to peak values, phase, period, frequency, and angular velocity.Determine the instantaneous value of a sinusoidal waveform.Graph sinusoidal wave equations as a function of time and angular velocity using degrees and radians.Define effective / root mean squared values.Define phase shift and determine phase differences between same frequency waveforms.
3 Direct Current (DC) DC sources have fixed polarities and magnitudes. REVIEWDirect Current (DC)DC sources have fixed polarities and magnitudes.DC voltage and current sources are represented by capital E and I.
4 Alternating Current (AC) A sinusoidal ac waveform starts at zeroIncreases to a positive maximumDecreases to zeroChanges polarityIncreases to a negative maximumReturns to zeroVariation is called a cycle
5 Alternating Current (AC) AC sources have a sinusoidal waveform.AC sources are represented by lowercase e(t) or i(t)AC Voltage polarity changes every cycle
6 Generating AC VoltageRotating a coil in fixed magnetic field generates sinusoidal voltage.
7 Sinusoidal AC CurrentAC current changes direction each cycle with the source voltage.
8 Time ScalesHorizontal scale can represent degrees or time.
9 Period Period of a waveform Time is measured in seconds Time it takes to complete one cycleTime is measured in secondsThe period is the reciprocal of frequencyT = 1/f
10 Frequency Number of cycles per second of a waveform Denoted by fUnit of frequency is hertz (Hz)1 Hz = 1 cycle per second
11 Amplitude and Peak-to-Peak Value Amplitude of a sine waveDistance from its average to its peakWe use Em for amplitudePeak-to-peak voltageMeasured between minimum and maximum peaksWe use Epp or Vpp
12 Example Problem 1What is the waveform’s period, frequency, Vm and VPP?
13 The Basic Sine Wave Equation The equation for a sinusoidal source is givenwhere Em is peak coil voltage and is the angular positionThe instantaneous value of the waveform can be determined by solving the equation for a specific value of
14 Example Problem 2A sine wave has a value of 50V at =150˚. What is the value of Em?
15 Radian MeasureConversion for radians to degrees.2 radians = 360º
16 Angular VelocityThe rate that the generator coil rotates is called its angular velocity ().Angular position can be expressed in terms of angular velocity and time.= t (radians)Rewriting the sinusoidal equation:e (t) = Em sin t (V)
17 Relationship between , T and f Conversion from frequency (f) in Hz to angular velocity () in radians per second = 2 f (rad/s)In terms of the period (T)
18 Sinusoids as functions of time Voltages can be expressed as a function of time in terms of angular velocity ()e (t) = Em sin t (V)Or in terms of the frequency (f)e (t) = Em sin 2 f t (V)Or in terms of Period (T)
19 Instantaneous ValueThe instantaneous value is the value of the voltage at a particular instant in time.
20 Example Problem 3A waveform has a frequency of 100 Hz, and has an instantaneous value of 100V at 1.25 msec.Determine the sine wave equation. What is the voltage at 2.5 msec?
21 Phase ShiftsA phase shift occurs when e(t) does not pass through zero at t = 0 secIf e(t) is shifted left (leading), then e = Em sin ( t + )If e(t) is shifted right (lagging), then e = Em sin ( t - )
22 Phase shiftThe angle by which the wave LEADS or LAGS the zero point can be calculated based upon the ΔtThe phase angle is written in DEGREES
23 PHASE RELATIONS i leads v by 110°. i leads v by 80°. V and i are in phase.
24 Example Problem 4Write the equations for the waveform below. Express the phase angle in degrees.v = Vm sin ( t + )
25 Effective (RMS) Values Effective values tell us about a waveform’s ability to do work.An effective value is an equivalent dc value.It tells how many volts or amps of dc that an ac waveform supplies in terms of its ability to produce the same average powerThey are “Root Mean Squared” (RMS) values:The terms RMS and effective are synonymous.