Transcript Intro to AC and Sinusoids

Lesson 13 Intro to AC &
Sinusoidal Waveforms
Learning Objectives
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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.
REVIEW
Direct Current (DC)
DC sources have fixed polarities and
magnitudes.
 DC voltage and current sources are
represented by capital E and I.
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Alternating Current (AC)
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A sinusoidal ac waveform starts at zero
 Increases
to a positive maximum
 Decreases to zero
 Changes polarity
 Increases to a negative maximum
 Returns to zero
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Variation is called a cycle
Alternating Current (AC)
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AC sources have a sinusoidal waveform.
AC sources are represented by lowercase e(t)
or i(t)
AC Voltage polarity changes every cycle
Generating AC Voltage
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Rotating a coil in fixed magnetic field generates
sinusoidal voltage.
Sinusoidal AC Current
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AC current changes direction each cycle with the
source voltage.
Time Scales
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Horizontal scale can represent degrees or time.
Period
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Period of a waveform
 Time
it takes to complete one cycle
Time is measured in seconds
 The period is the reciprocal of frequency
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T
= 1/f
Frequency
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Number of cycles per second of a
waveform
 Frequency
 Denoted
by f
Unit of frequency is hertz (Hz)
 1 Hz = 1 cycle per second
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Amplitude and Peak-to-Peak Value
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Amplitude of a sine wave
 Distance
from its average to its peak
We use Em for amplitude
 Peak-to-peak voltage
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 Measured
between minimum and
maximum peaks
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We use Epp or Vpp
Example Problem 1
What is the waveform’s period, frequency, Vm and VPP?
The Basic Sine Wave Equation
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The equation for a sinusoidal source is given
e  Em sin( ) V
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 
e(37)  10sin(37) V = 6.01 V
Example Problem 2
A sine wave has a value of 50V at =150˚. What is the
value of Em?

Radian Measure
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Conversion for radians to degrees.
2 radians = 360º
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)
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Rewriting the sinusoidal equation:
e (t) = Em sin  t (V)
Relationship between , T and f
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Conversion from frequency (f) in Hz to angular
velocity () in radians per second
 = 2 f
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(rad/s)
In terms of the period (T)
2
  2 f 
T
(rad/s)
Sinusoids as functions of time
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Voltages can be expressed as a function of time
in terms of angular velocity ()
e (t) = Em sin  t
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(V)
Or in terms of the frequency (f)
e (t) = Em sin 2 f t (V)
 Or in terms of Period (T)
t
e(t )  Em sin 2
(V)
T
Instantaneous Value
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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?
Phase Shifts
A phase shift occurs when e(t) does not pass
through zero at t = 0 sec
 If e(t) is shifted left (leading), then
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e = Em sin ( t + )
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If e(t) is shifted right (lagging), then
e = Em sin ( t - )
Phase shift
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The angle by which the wave LEADS or LAGS
the zero point can be calculated based upon
the Δt
 10  s 
 t 
    360  
 360  36
T 
 100  s 
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The phase angle is written in DEGREES
PHASE RELATIONS
i leads v by 80°.
i leads v by 110°.
V and i are in phase.
Example Problem 4
Write the equations for the waveform below. Express
the phase angle in degrees.
v = Vm sin ( t + )
Effective (RMS) Values
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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.
Vm
Vrms 
 0.707Vm
2
Im
I rms 
 0.707 I m
2