EE 1101 Circ N 4'2

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EE 1101 Circ N 4.2

• Voltage-Ampere Relationships
• Harmonics

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Fundamental freq harmonics

• A sine wave is usually represented by
• v(t) Vsin(wt)
• It has w or f as its frequency and the period T
  given by

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Fundamental freq harmonics

• However, if for some reason (e.g. saturation) the
  sine wave is distorted its equation will change
  to

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Fundamental freq harmonics

• has the fundamental freq w AND other freq
• known as harmonics.
• It is noted that the harmonics are multiples of
  the fundamental freq.
• It is well known that the magnitudes

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Sine wave (clean)
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Sine wave (distorted)
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Other Periodic Functions

• Using Fourier Functions (or otherwise) other
  periodic functions can be expressed in terms of
  sine/cosine functions. These expressions clearly
  bring out the fundamental freq harmonics.
• Examples are (i) square wave (ii) triangular
  wave.

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Square Wave
A
wt
-A
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Triangular Wave
V
wt
-V
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Square wave (Fourier Analysis)
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Square wave (Fourier Analysis)
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Square wave (Fourier Analysis)

• The square wave analysis is

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Triangular wave (Fourier Analysis)
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Triangular wave (Fourier Analysis)
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Triangular wave (Fourier Analysis)
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Fundamental freq harmonics

• In conclusion
• Any wave form can be represented by a sum of
  sinusoidal functions (sine waves) one wave as a
  fundamental the rest as harmonics. The
  harmonics progressively decrease in magnitude
  in practice higher ones are neglected. If we draw
  a freq spectrum the obvious is seen.

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Freq Spectrum of Square Wave
Magnitudes (peak values)
3
5
1
freq
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Phase Phase Difference

• These are best understood when we consider
  voltages currents in a circuit.
• A purely resistive circuit is

R
i
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Phase Phase Difference
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Phase Phase Difference R-L Circuit
i
R
L
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Phase Phase Difference R-L Circuit
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Phasor Diagram for R-L Circuit
Im
I
Re
i
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Phase Phase Difference R-C Circuit

• A capacitive circuit is

R
i
C
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Phase Phase Difference R-C Circuit
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Phasor Diagram for R-C Circuit
Im
i
Re
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Leading lagging sinusoids

• A sinusoidal signal that leads another reaches
  its peak BEFORE the one it leads does so.
• The opposite is indeed true.
• Exercise
• Use the above information to sketch v, i as
  sinusoidal signals clearly indicate