Sinusoidal Alternating Waveforms

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Sinusoidal Alternating Waveforms

• Chapter 13
• Sinusoids and Amplitude

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Syllabus

• Review handout
• Any questions??

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Introduction

• Circuits 1 is dc with the assumption of constant
  dc.
• Circuits 2 is alternating current, where the
  signal varies with time.
• Furthermore signal is periodic that is it
  repeats in uniform time.
• Commonly ac refers to a sinusoidal voltage, the
  case for 99 of this class.

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Why Sinusoids

• Sinusoid demonstrate with O-scope.
• simple to generate from rotating machines
• maintains its shape through sums, products,
  integration and differentiation.
• maintains shape through capacitors and inductors
• can be stepped up or down with transformers.

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Non-sinusoidal Waveforms

• With all those properties, sinusoids are not the
  be-all end-all.
• Recent electronics makes use of many non
  sinusoidal waveforms.
• ac periodic waveform could be square, or
  triangle, etc.
• However, can analyze any (well-behaved) periodic
  waveform as a sum of sinusoids.

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Sinusoidal ac Voltage Characteristics and
Definitions

• 13.2

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Definition Waveform

• Typically shown graphically as
• amplitude (or range)
• voltage
• current
• power
• domain
• time
• distance
• degrees
• radians

range
domain
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Sinusoid Example
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The Sine Wave

• A cycle can be measured in time
• or as the distance between successive similar
  points
• Represent one period as a circle.
• Whats the distance around a circle?
• Circumference 2 pi radius
• choose a circle of unit radius (r 1)
• Circumference 2 pi

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Period

• The time required for one repetition of a
  waveform.
• Standard notation T
• Measured from any point until that point is
  repeated.

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Period
T 2 seconds
T
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Cycle

• The portion of a waveform contained in one
  period.
• The waveform has cycled from a starting point
  to where that starting point is repeated.
• The period then measures the time of one cycle
  or seconds per cycle.

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Frequency

• Answers how often do cycles occur?
• Frequency is then the number of cycles per
  second.
• Frequency is the inverse of the period.
• Standard notation lowercase f.
• Units of Hertz.

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Frequency
T 2 seconds gt f 0.5 Hz or 1/2 cycle in one
second.
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Instantaneous Value

• magnitude of the waveform at any instant of
  time.
• give me a time, Ill tell you a magnitude
• one time, one value
• standard notation lower case letters, v, i.

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Instantaneous Value
0.3 sec gt 140 v
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Peak Amplitude

• Maximum instantaneous value of the waveform
  measured from average
• standard notation uppercase letters

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Peak Amplitude
Average 0, peak amplitude 170 v
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Peak Value

• maximum instantaneous value of a waveform
  measured from zero volts.
• highest value reached
• no standard notation. Must indicate or describe.

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Peak Value
220 volts
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Peak to Peak Value

• Difference between highest and lowest values of
  the waveform
• standard notation Vp-p or Ip-p

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Peak to Peak Value
Vp-p 340 volts
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The Sine Wave

• A cycle can be measured in time
• or as the distance between successive similar
  points
• Represent one period as a circle.
• Whats the distance around a circle?
• Circumference 2 pi radius
• choose a circle of unit radius (r 1)
• Circumference 2 pi

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Radians

• Radians measure the distance around a unit
  circle.

2 p radians
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Degrees

• For convenience, the Babylonians chose to
  measure a circle in fractions of 360 degrees.
• 360 is evenly divisible by prime numbers 2, 2,
  2, 3, 3, 5. And thus any combination of those
  primes.
• 2, 3, 4, 5, 6, 8, 9 ...
• This measure is not based on length, but on angle.

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Degrees

• Degrees measure an angular fraction of a circle.

360 degrees
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Unit Circle
p/2
Quadrant II
Quadrant I
90
p
0
2p
Quadrant IV
Quadrant III
3p/2
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Unit Circle
p/2
3p/4
p/4 45
p
0
2p
7p/4
5p/4
3p/2
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Unit Circle
p/2
p/3
p/6
60
p
30
0
2p
3p/2
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Radians Degrees

• Equivalence or conversion between radians and
  degrees is based on

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Rotating Vector

• A sinusoid is the projection of a rotating
  radius vector on a time varying waveform plot.
• The rate of the radius vector motion is
  distance/time in units of radians/second.

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Angular Velocity

• standard notation Greek lower case Omega.

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Frequency

• For once around the circle, 2 pi distance is
  covered in one period of time.

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General Format for the Sinusoidal Voltage or
Current

• Most general form

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Constant Offset

• A sinusoid could be offset from zero by the
  addition of a dc source.

e
C E
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Phase Relations

• Suppose you choose a starting time reference
  when the amplitude is not zero going positive?
• The shift from zero can be mathematically
  accounted in a phase angle.
• Standard notation Greek lowercase theta.

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Phase Angle
This waveform starts about 2/3 of the way to the
maximum.
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Phase Angle Example
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Phase Relationship

• A phase angle is only meaningful as a
  comparison.
• Usually an arbitrary zero phase angle is chosen.
  
• Other waveforms are then compared to that zero
  reference.

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Phase Angle Calculation
This is a ratio and proportion problem. The
phase difference is the ratio of separation of
the sinusoids compared to a complete period.
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Lead-Lag
The pink waveform leads the blue by 60 degrees.
Blue is the reference. Or the blue lags the pink
by 60 degrees.
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Lead - Lag

• If a waveform has an instantaneous value greater
  than zero at the domain 0 reference, then that
  waveform LEADS the reference.
• If instantaneous value less than zero, LAGS.
• 180 degrees is maximum lead.

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Maximum Lead-Lag

• The maximum lead or lag is 180? or ½ cycle.
• Instead of saying a waveform leads by 300?, say
  the waveform lags by 60?.

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Example Phase Relationship
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Example Phase Relationship

• C goes through zero positive 120 degrees after A
  
• C lags A by 120 degrees
• B goes through zero positive 240 degrees after A
• B LEADS A by 120 degrees
• recall maximum lead/lag of 180 degrees
• B lags C by 120 degrees

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Phase Measurements

• Compare distance between waveforms with period.
• Phase relations are only meaningful if the
  frequency is the same.