2Learning 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.
3Direct 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.
4Alternating 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
5Alternating 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
6Generating AC VoltageRotating a coil in fixed magnetic field generates sinusoidal voltage.
7Sinusoidal AC CurrentAC current changes direction each cycle with the source voltage.
8Time ScalesHorizontal scale can represent degrees or time.
9Period 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
10Frequency Number of cycles per second of a waveform Denoted by fUnit of frequency is hertz (Hz)1 Hz = 1 cycle per second
11Amplitude 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
12Example Problem 1What is the waveform’s period, frequency, Vm and VPP?
13The 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
14Example Problem 2A sine wave has a value of 50V at =150˚. What is the value of Em?
15Radian MeasureConversion for radians to degrees.2 radians = 360º
16Angular 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)
17Relationship 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)
18Sinusoids 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)
19Instantaneous ValueThe instantaneous value is the value of the voltage at a particular instant in time.
20Example 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?
21Phase 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 - )
22Phase 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
23PHASE RELATIONS i leads v by 110°. i leads v by 80°. V and i are in phase.
24Example Problem 4Write the equations for the waveform below. Express the phase angle in degrees.v = Vm sin ( t + )
25Effective (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.