1 / 62

Chapter 8. Sinusoidal Sources and Phasors

Chapter 8. Sinusoidal Sources and Phasors. Chapter Contents. 8.1 Properties of Sinosoids 8.2 RLC Circuit Example 8.3 Complex Sources 8.4 Phasors 8.5 I-V Laws for Phasors 8.6 Impedance and Admittance 8.7 Kirchhoff's Laws and Impedance Equivalents 8.8 Phasor Circuits.

serina-stephenson
Download Presentation

Chapter 8. Sinusoidal Sources and Phasors

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Chapter 8. Sinusoidal Sources and Phasors

  2. Chapter Contents • 8.1 Properties of Sinosoids • 8.2 RLC Circuit Example • 8.3 Complex Sources • 8.4 Phasors • 8.5 I-V Laws for Phasors • 8.6 Impedance and Admittance • 8.7 Kirchhoff's Laws and Impedance Equivalents • 8.8 Phasor Circuits

  3. 8.1 PROPERTIES OF SINUSOIDS • The sine wave :

  4. 8.1 PROPERTIES OF SINUSOIDS • The sinusoid is a periodic function. There is a smallest number s.t.

  5. 8.1 PROPERTIES OF SINUSOIDS • Period and frequency : - Period : - Frequency : German physicist Heinrich R. Hertz (1857-1894)

  6. 8.1 PROPERTIES OF SINUSOIDS • Relation between frequency and radian frequency : • A more general sinusoidal expression : Ф : phase angle, or phase

  7. 8.1 PROPERTIES OF SINUSOIDS - Phase angle Фshould be expressed in radians, but degrees are a very familiar measure for angle. Therefore, we may write

  8. 8.1 PROPERTIES OF SINUSOIDS • Phase leading and lagging : Ex.)

  9. 8.1 PROPERTIES OF SINUSOIDS • Sine and cosine functions : • Half-period shifts and full-period shifts :

  10. 8.1 PROPERTIES OF SINUSOIDS Ex. 8.1)

  11. 8.1 PROPERTIES OF SINUSOIDS • Linear combination of a sine wave and a cosine wave :

  12. 8.1 PROPERTIES OF SINUSOIDS • Quadrature representation of the sinusoid

  13. 8.2 RLC CIRCUIT EXAMPLE • Ex.) Let's find the forced mesh current in the series cct. * By KVL,

  14. 8.2 RLC CIRCUIT EXAMPLE • Trial forced solution : → eq.(1) 

  15. 8.2 RLC CIRCUIT EXAMPLE • Combining the two quadrature terms, 

  16. 8.3 COMPLEX SOURCES • An alternative method for treating ccts with sinusoidal sources = replacing the given sources by complex sources. • A complex number is a point in the complex plane. (1) Rectancluar form :

  17. 8.3 COMPLEX SOURCES (2) Polar form : (3) Trigonometric form :

  18. 8.3 COMPLEX SOURCES (4) Exponential polar form : • Euler's identity : • Therefore, the exponential form is • Polar form of ejθ

  19. 8.3 COMPLEX SOURCES Ex. 8.3) A= 4+j3 <Solution> • Polar form : • Since and , • Exponential polar form :

  20. 8.3 COMPLEX SOURCES • Complex exponential function : ejωt(rotating phasor) • cosωt : projection of this point onto the horizontal axis real part • sinωt : projection of this point onto the vertical axis imaginary part

  21. 8.3 COMPLEX SOURCES • General scaled and phase-shifted complex exponential : Vmej(ωt+ф) • Euler's identity form :

  22. 8.3 COMPLEX SOURCES • Application of complex numbers to electric ccts : superposition * The forced response to will be a complex exponential of the same frequency , .

  23. 8.3 COMPLEX SOURCES Ex. 8.4) <Solution> Replacing the sinusoidal source by the complex source , by KVL, • Substituting the trial solution , .

  24. 8.3 COMPLEX SOURCES Ex. 8.5) <Solution> • First we replace the real excitation by the complex excitation. • Complex response i1 satisfies • Substituting the trial forced solution :

  25. 8.4 PHASORS • The preceding results may be put in much more compact form by the use of quantities called phasors. • The phasor method is credited to Charles Proteus Steinmetz (1865-1923), a famous electrical engineer with the General Electric Company. • Consider the forced response of a cct to sinusoidal excitation at frequency ω. Each sinusoidal source may be expressed as a cosine. • We replace by a complex exponential source.

  26. 8.4 PHASORS • Phasor representation of • In general, each current and each voltage will be of the form  We defineandas phasors, i.e., the complex numbers that multiply in the expression for currents and voltages.

  27. 8.4 PHASORS • Relation between the sinusoidal source and the phasor : • Phasor transformation : • Since , by using the polar exponential form,  The amplitude of the sinusoid is the magnitude of its phasor, and the phase angle of the sinusoid is the angle of its phasor.

  28. 8.4 PHASORS Ex. 8.6) <Solution> • Since , we use as the complex exponential source. • Substituting the trial forced solution into eq.(1) : eq.(1) • Therefore, the forced solution :

  29. 8.4 PHASORS • If the sinusoidal source is given in the sine form , we will first convert to the cosine form. • Complex sources and source phasors for cosine and sine sources

  30. 8.4 PHASORS Ex. 8.7) <Solution> • Replacing the sources by exponential sources,

  31. 8.4 PHASORS • KVL equation on mesh-1, where

  32. 8.4 PHASORS • Substituting the trial forced solution :

  33. 8.5 I-V LAWS FOR PHASORS • Relationships between phasor voltage and phasor current for resistors, inductors, and capacitors are very similar to Ohm's law for resistors. • Consider a cct in which all currents and voltages are of the form . We are interested in the forced response only. (1) I-V relationship for resistor cct :

  34. 8.5 I-V LAWS FOR PHASORS • Canceling the eωtfactors, • Since and ,

  35. 8.5 I-V LAWS FOR PHASORS Ex. 8.8) <Solution> • The phasor voltage and current : • Current in time-domain :

  36. 8.5 I-V LAWS FOR PHASORS (2) I-V relationship for inductor cct : • Substituting the complex current and voltage into the time-domain relation, • Therefore, we obtain the phasor relation :

  37. 8.5 I-V LAWS FOR PHASORS • If the current in the inductor is given by , then the phasor current is • therefore, the phasor voltage :

  38. 8.5 I-V LAWS FOR PHASORS • Therefore, the voltage in the time domain : : The current lags the voltage by 90°.

  39. 8.5 I-V LAWS FOR PHASORS (3) I-V relationship for capacitor cct : • Substituting the complex current and voltage into the time-domain relation, • Therefore, we obtain the phasor relation : Since

  40. 8.5 I-V LAWS FOR PHASORS • If the capacitor voltage is given by , then the phasor voltage is • therefore, the phase current :

  41. 8.5 I-V LAWS FOR PHASORS • Therefore, the current in the time domain : : The current leads the voltage by 90°.

  42. 8.5 I-V LAWS FOR PHASORS Ex. 8.9) Determine the current i(t)through a 1 [uF] capacitor when is applied. <Solution>

  43. 8.6 IMPEDANCE AND ADMITTANCE • In general cct with 2 accessible terminals, if the time-domain voltage and current are • then

  44. 8.6 IMPEDANCE AND ADMITTANCE • Impedance : • Therefore, the I-V relation : ( Ohm's law ) • Magnitude and phase of impedance : • Impedance is a complex #, being the ratio of two complex #'s, but it is not a phasor because of no corresponding sinusoidal time-domain function.

  45. 8.6 IMPEDANCE AND ADMITTANCE • Impedance in rectangular form :

  46. 8.6 IMPEDANCE AND ADMITTANCE Ex. 8.10) Determine the impedance in the polar and the rectangular forms. <Solution>

  47. 8.6 IMPEDANCE AND ADMITTANCE • The impedances of resistors, inductors, and capacitors : • Inductive and capacitive reactances: XL, XC

  48. 8.6 IMPEDANCE AND ADMITTANCE • Admittance : • Relation between components of Y and Z :

  49. 8.6 IMPEDANCE AND ADMITTANCE Ex. 8.11) <Solution> • Admittances of a resistor, an inductor, and a capacitor :

  50. 8.7 KIRCHHOFF'S LAWS AND IMPEDANCE EQUIVALENTS • Kirchhoff's Voltage Law • If a complex excitation is applied to a cct, complex voltages appear across the elements in the cct. • KVL around a typical loop : KVL holds for phasors where

More Related