Matakuliah
Tahun
Versi
: H0042/Teori Rangkaian Listrik
: 2005
: <<versi/01
Pertemuan 8
Sinus and phasor
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Learning Outcomes
Pada akhir pertemuan ini, diharapkan mahasiswa
akan mampu :
• Menghitung besaran phasor
• Membuat diagram phasor
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Outline Materi
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•
•
•
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Materi 1 : analisa koordinat komplek
Materi 2 : analisa koordinat geometri
Materi 3 : analisa koordinat rectangular
Materi 4 : diagram phasor
Materi 5 : aplikasi phasor
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•
In electrical engineering the forced sinusoidal component
remaining after the natural component disappears is called the
sinusoidal steady-state response. The sinusoidal steady-state
response is also called the ac response since the driving force
is an alternating current signal. The phasor concept is the
foundation for the analysis of linear circuits in the sinusoidal
steady state. Simply put, a phasor is a complex number
representing the amplitude and phase angle of a sinusoidal
voltage or current. The connection between sinewaves and
complex numbers is provided by Euler's relationship (eq.8-1,2):
Eq.(8-1)
To develop the phasor concept, it is necessary to adopt the point
of view that the cosine and sine functions
Eq.(8-2)
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When Eq.(8-2) is applied to the general sinusoid we
obtain
Eq.(8-3)
• Eq.(8-4)
In the last line of Eq.(8-4), moving the amplitude VA
inside the real part operation does not change the final
result because it is a real constant.
By definition, the quantity in the last line of Eq.(8-4) is the
phasor representation of the sinusoid v(t). The
phasor V is written as
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Eq.(8-5)
Note that V is a complex number determined
by the amplitude and phase angle of the
sinusoid. Fig. 8-1 shows a graphical
representation commonly called a phasor
diagram.
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Two features of the phasor concept need emphasis:
1. Phasors are written in boldface type like V or I1 to
distinguish them from signal waveforms such as v(t)
and i1(t).
2. A phasor is determined by amplitude and phase
angle and does not contain any information about the
frequency of the sinusoid.
In summary, given a sinusoidal signal , the
corresponding phasor representation is . Conversely,
given the phasor , the corresponding sinusoid is found
by multiplying the phasor by and reversing the steps
in Eq.(8-4) as follows:
Eq.(8-6)
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The complex exponential is sometimes called a
rotating phasor, and the phasor V is viewed as a
snapshot of the situation at t=0.
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Properties of Phasors
First, the additive property states that the phasor
representing a sum of sinusoids of the same frequency
is obtained by adding the phasor representations of the
component sinusoids. To establish this property we
write the expression (Eq.(8-7))
where V1(t), V2(t), ... and VN(t) are sinusoids of the same
frequency whose phasor representations are V1, V2... and VN.
The real part operation is additive, so the sum of real parts
equals the real part of the sum. Consequently, Eq.(8-7) can be
written in the form (Eq(8-8)
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Comparing the last line in Eq.(8-8) with the definition of a
phasor, we conclude that the phasor V representing v(t) is
The result in Eq.(8-9) applies only if the component
sinusoids all have the same frequency so that can be
factored out as shown in the last line in Eq.(8-8).
The derivative property of phasors allows us easily to
relate the phasor representing a sinusoid to the phasor
representing its derivative.
Equation (8-6) relates a sinusoid function and its phasor
representation as
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Differentiating this equation with respect time t yields
Eq.(8-10)
From the definition of a phasor, we see that the quantity
(
V) on the right side of this equation is the phasor
representation of the time derivative of the sinusoidal
waveform. This phasor can be written in the form
Eq.(8-11)
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EXAMPLE 8-1
(a) Construct the phasors for the following
signals:
(b) Use the additive property of phasors
and the phasors found in (a) to find
v(t)=v1(t)+v2(t).
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SOLUTION
(a) The phasor representations of v(t)=v1(t)+ v2(t) are
(b) The two sinusoids have the same frequent so the
additive property of phasors can be used to obtain their
sum:
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The waveform corresponding to this phasor sum is
The phasor diagram in Fig. 8-3 shows that summing
sinusoids can be viewed geometrically in terms of phasors.
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EXAMPLE 8-2
(a) Construct the phasors representing the
following signals:
(b) Use the additive property of phasors and the phasors found
in (a) to find the sum of these waveforms.
SOLUTION:
(a) The phasor representation of the three sinusoidal currents are
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(b) The currents have the same frequency, so the additive property of
phasors applies. The phasor representing the sum of these current is
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EXAMPLE 8-3
Use the derivative property of phasors to find the time
derivative of v(t)=15 cos(200t-30).
SOLUTION:
The phasor for the sinusoid is V=15 -30 °. According to
the derivative property, the phasor representing the
dv/dt is found by multiplying V by j.
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The sinusoid corresponding to the
phasor j V is
Finding the derivative of a sinusoid is
easily carried out in phasor form, since it
only involves manipulating complex
numbers.
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• Koordinat rectangular : Z = R + jX
• Koordinat goneometri : Z = Z (cos Q + jSin Q)
• Koordinat Polar = Z = Z Qo
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RESUME
• Proses perkalian dan pembagian bilangan
impedansi dianjurkan untuk menggunakan
bentuk Polar.
• Proses penjumlahan dan pengurangan
bilangan impedansi dianjurkan untuk
menggunakan bentuk rectangular.
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