chapter 25 nonsinusoidal waveforms. 2 waveforms used in electronics except for sinusoidal any...
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Chapter 25
Nonsinusoidal Waveforms
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Waveforms• Used in electronics except for sinusoidal
• Any periodic waveform may be expressed as – Sum of a series of sinusoidal waveforms at
different frequencies and amplitudes
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Waveforms• Each sinusoidal components has a unique
amplitude and frequency
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Waveforms• These components have many different
frequencies– Output may be greatly distorted after passing
through a filter circuit
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Composite Waveforms• Waveform made up of two or more
separate waveforms
• Most signals appearing in electronic circuits– Comprised of complicated combinations of dc
and sinusoidal waves
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Composite Waveforms• Once a periodic waveform is reduced to
the summation of sinusoidal waveforms– Overall response of the circuit can be found
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Composite Waveforms• Circuit containing both an ac source and a
dc source– Voltage across the load is determined by
superposition
• Result is a sine wave with a dc offset
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Composite Waveforms• RMS voltage of composite waveform is
determined as
• Referred to as true RMS voltage
22acdcrms VVV +=
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Composite Waveforms• Waveform containing both dc and ac
components– Power is determined by considering effects of
both signals
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Composite Waveforms• Power delivered to load will be determined
by
load
rmsout R
VP
2
=
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Fourier Series
• Any periodic waveform – Expressed as an infinite series of sinusoidal
waveforms
• Expression simplifies the analysis of many circuits that respond differently
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Fourier Series
• A periodic waveform can be written as:– f(t) = a0 + a1cos t + a2cos 2t + ∙∙∙ + an cos nt + ∙∙∙ + b1sin t + b2 sin 2t + ∙∙∙ + bn sin nt + ∙∙∙
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Fourier Series• Coefficients of terms of Fourier series
– Found by integrating original function over one complete period
∫
∫
∫
+
+
+
=
=
=
Tt
tn
Tt
tn
Tt
t
dttntfT
b
dttntfT
a
dttfT
a
1
1
1
1
1
10
)(sin )(2
)( cos )(2
)(1
ω
ω
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Fourier Series• Individual components combined to give a
single sinusoidal expression as:
⎟⎟⎠
⎞⎜⎜⎝
⎛=
+=
+=+°+=+
−
n
n
nnn
n
nnnn
b
a
bac
nxc
nxbnxanxbnxa
1
22
90
tan
and
where
)( sin
sin)( sin sin cos
θ
θ
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Fourier Series• Fourier equivalent of any periodic waveform
may be simplified to– f(t) = a0 + c1sin(t + 1) + c2sin(2t + 2) + ∙∙∙
• a0 term is a constant that corresponds to average value
• cn coefficients are amplitudes of sinusoidal terms
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Fourier Series• Sinusoidal term with n = 1
– Same frequency as original waveform
• First term – Called fundamental frequency
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Fourier Series• All other frequencies are integer multiples
of fundamental frequency
• These frequencies are harmonic frequencies or simply harmonics
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Fourier Series• Pulse wave which goes from 0 to 1,
then back to 0 for half a cycle, will have a series given by
∞=
+= ∑∞
···,,5,3,1
)sin(25.0)(
n
n
tntv
n
ω
π
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Fourier Series• Average value
– a0 = 0.5
• It has only odd harmonics• Amplitudes become smaller
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Even Symmetry• Symmetrical waveforms
– Around vertical axis have even symmetry
• Cosine waveforms– Symmetrical about this axis
– Also called cosine symmetry
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Even Symmetry• Waveforms having even symmetry will be
of the form f(–t) = f(t)
• A series with even symmetry will have only cosine terms and possibly a constant term
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Odd Symmetry
• Odd symmetry – Waveforms that overlap terms on opposite
sides of vertical axis if rotated 180°
• Sine symmetry – Sine waves that have this symmetry
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Odd Symmetry
• Waveforms having odd symmetry will always have the form f(–t) = –f(t)
• Series will contain only sine terms and possibly a constant term
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Half-Wave Symmetry• Portion of waveform below horizontal axis
is mirror image of portion above axis
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Half-Wave Symmetry• These waveforms will always be of the
form
• Series will have only odd harmonics and possibly a constant term
( )tfTtf −=⎟
⎠⎞
⎜⎝⎛+2
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Shifted Waveforms
• If a waveform is shifted along the time axis– Necessary to include a phase shift with each
of the sinusoidal terms
• To determine the phase shift– Determine period of given waveforms
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Shifted Waveforms
• Select which of the known waveforms best describes the given wave
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Shifted Waveforms
• Determine if given waveform leads or lags a known waveform
• Calculate amount of phase shift from = (t/T)•360°
• Write resulting Fourier expression for given waveform
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Shifted Waveforms• If given waveform leads the known
waveform– Add phase angle– If it lags, subtract phase angle
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Frequency Spectrum
• Waveforms may be shown as a function of frequency– Amplitude of each harmonic is indicated at
that frequency
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Frequency Spectrum
• True RMS voltage of composite waveform is determined by considering RMS value at each frequency
··· 23
22
21
2dcrms ++++= VVVVV
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Frequency Spectrum• If a waveform were applied to a resistive
element– Power would be dissipated as if each
frequency had been applied independently
• Total power is determined as sum of individual powers
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Frequency Spectrum• To calculate power
– Convert all voltages to RMS
• Frequency spectrum may then be represented in terms of power
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Frequency Spectrum• Power levels and frequencies of various
harmonics of a periodic waveform may be measured with a spectrum analyzer
• Some spectrum analyzers display either voltage levels or power levels
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Frequency Spectrum• When displaying power levels
– 50- reference load is used
• Horizontal axis is in hertz– Vertical axis is in dB
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Circuit Response to a Nonsinusoidal Waveform
• When a waveform is applied to input of a filter– Waveform may be greatly modified
• Various frequencies may be blocked by filter
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Circuit Response to a Nonsinusoidal Waveform
• A composite waveform passed through a bandpass filter – May appear as a sine wave at desired
frequency
• Method is used to provide frequency multiplication