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Learning Objectives Compare 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.
DC sources have fixed polarities and magnitudes. DC voltage and current sources are represented by capital E and I.
Direct Current (DC)REVIEW
Alternating Current (AC) A sinusoidal ac waveform starts at zero
Increases to a positive maximumDecreases to zeroChanges polarity Increases to a negative maximumReturns to zero
Variation is called a cycle
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
Period Period of a waveform
Time it takes to complete one cycle Time is measured in seconds The period is the reciprocal of frequency
T = 1/f
Frequency Number of cycles per second of a waveform
FrequencyDenoted by f
Unit of frequency is hertz (Hz) 1 Hz = 1 cycle per second
Amplitude and Peak-to-Peak Value Amplitude of a sine wave
Distance from its average to its peak We use Em for amplitude Peak-to-peak voltage
Measured between minimum and maximum peaks We use Epp or Vpp
The Basic Sine Wave Equation The equation for a sinusoidal source is given
where Em is peak coil voltage and is the angular position
The instantaneous value of the waveform can be determined by solving the equation for a specific value of
sin( ) Vme E
(37 ) 10sin(37 ) V = 6.01 Ve
Angular Velocity The 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)
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)
22 (rad/s)f
T
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)
( ) sin 2m
te t E
T (V)
Instantaneous Value The instantaneous value is the value of the
voltage at a particular instant in time.
Example Problem 3
A 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?
A phase shift occurs when e(t) does not pass through zero at t = 0 sec
If e(t) is shifted left (leading), then
e = Em sin ( t + )
If e(t) is shifted right (lagging), then
e = Em sin ( t - )
Phase Shifts
Phase shift The angle by which the wave LEADS or LAGS
the zero point can be calculated based upon the Δt
The phase angle is written in DEGREES
10 s360 360 36
100 s
t
T
Example Problem 4
Write the equations for the waveform below. Express the phase angle in degrees.
v = Vm sin ( t + )
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 power
They are “Root Mean Squared” (RMS) values: The terms RMS and effective are synonymous.
0.7072
0.7072
mrms m
mrms m
VV V
II I