Pertemuan 8 Sinus and phasor

1. Pertemuan 8Sinus and phasor Matakuliah : H0042/Teori Rangkaian Listrik Tahun : 2005 Versi : <<versi/01

2. Learning Outcomes Pada akhir pertemuan ini, diharapkan mahasiswa akan mampu : • Menghitung besaran phasor • Membuat diagram phasor

3. Outline Materi • Materi 1 : analisa koordinat komplek • Materi 2 : analisa koordinat geometri • Materi 3 : analisa koordinat rectangular • Materi 4 : diagram phasor • Materi 5 : aplikasi phasor

4. 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) 

5. When Eq.(8-2) is applied to the general sinusoid we obtain 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 Eq.(8-3) • Eq.(8-4)

6. 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.

7. 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)

8. The complex exponential is sometimes called a rotating phasor, and the phasor V is viewed as a snapshot of the situation at t=0.

9. 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)

10. 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

11. 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)

12. 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).

13. (b) The two sinusoids have the same frequent so the additive property of phasors can be used to obtain their sum: SOLUTION (a) The phasor representations of v(t)=v1(t)+ v2(t) are

14. 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.

15. 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

16. (b) The currents have the same frequency, so the additive property of phasors applies. The phasor representing the sum of these current is

17. EXAMPLE 8-3 Use the derivative property of phasors to find the time derivative of v(t)=15cos(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.

18. 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.

19. Koordinat rectangular : Z = R + jX • Koordinat goneometri : Z = Z (cos Q + jSin Q) • Koordinat Polar = Z = Z Qo

20. RESUME • Proses perkalian dan pembagian bilangan impedansi dianjurkan untuk menggunakan bentuk Polar. • Proses penjumlahan dan pengurangan bilangan impedansi dianjurkan untuk menggunakan bentuk rectangular.