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Fundamentals of Electric Circuits by Alexander- Sadiku

Circuit Analysis-II Spring-2015 EE -1112 Instructor : Hafiz Zaheer Hussain Email: zaheer.hussain@ee.uol.edu.pk www.hafizzaheer.pbworks.com Department of Electrical Engineering The University of Lahore Week 1 & 2. Fundamentals of Electric Circuits by Alexander- Sadiku. Chapter 9

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Fundamentals of Electric Circuits by Alexander- Sadiku

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  1. Hafiz Zaheer Hussain

  2. Circuit Analysis-II Spring-2015EE -1112Instructor: Hafiz Zaheer HussainEmail: zaheer.hussain@ee.uol.edu.pkwww.hafizzaheer.pbworks.comDepartment of Electrical EngineeringThe University of Lahore Week 1 & 2 Hafiz Zaheer Hussain

  3. Fundamentals of Electric CircuitsbyAlexander-Sadiku Chapter 9 Sinusoidal Steady-State Analysis Hafiz Zaheer Hussain

  4. Content 9.1 Introduction 9.2 Sinusoids’ features 9.3 Phasors 9.4 Phasorrelationships for circuit elements 9.5 Impedance and admittance 9.6Kirchhoff’s laws in the frequency domain 9.7 Impedance combinations Hafiz Zaheer Hussain

  5. 9.1 Introduction We now begin the analysis of circuits in which the source voltage or current is time-varying. In this chapter, we are particularly interested in sinusoidally time-varying excitation, or simply, excitation by a sinusoid A sinusoid is a signal that has the form of the sine or cosine function. A sinusoidal current is usually referred to as alternating current (AC). Such a current reverses at regular time intervals and has alternately positive and negative values. Circuits driven by sinusoidal current or voltage sources are called ac circuits Hafiz Zaheer Hussain

  6. 9.2 Sinusoids’ features • We are interested in sinusoids for a number of reasons. • nature itself is characteristically sinusoidal. • The motion of a pendulum • The vibration of a string • The ripples on the ocean surface etc. • Sinusoidal signal is easy to generate and transmit. • Sinusoidal play an important role in the analysis of periodic signals. • Sinusoid is easy to handle mathematically. • The derivative and integral of a sinusoid are themselves sinusoids. • For these the sinusoid is an extremely important function in circuit analysis. Hafiz Zaheer Hussain

  7. 9.2 Sinusoids’ (2) • A sinusoid is a signal that has the form of the sine or cosine function. • A general expression for the sinusoid, where Vm = the amplitudeof the sinusoid ω = the angular frequency in radians/s Ф = the phase Hafiz Zaheer Hussain

  8. 9.2 Sinusoids’ (3) A periodic function is one that satisfies v(t) = v(t + nT), for all t and for all integers n. If • Only two sinusoidal values with the same frequency can be compared by their amplitude and phase difference. • If phase difference is zero( ) , they are in phase; • If phase difference is not zero( ) , they are out of phase. Hafiz Zaheer Hussain

  9. 9.2 Sinusoids’ (4) Hafiz Zaheer Hussain

  10. 9.2 Sinusoids’ (5) Q Hafiz Zaheer Hussain

  11. 9.2 Sinusoids’ (6) Hafiz Zaheer Hussain

  12. 9.2 Sinusoids’ Example 9.1 Given a sinusoid, , calculate its amplitude, phase, angular frequency, period, and frequency. Hafiz Zaheer Hussain

  13. 9.2 Sinusoids’ PP 9.2 Hafiz Zaheer Hussain

  14. 9.2 Sinusoids’ (7) Trigonometric Identities (9.9) (9.9) Hafiz Zaheer Hussain

  15. 9.2 Sinusoids’ (8) A graphical approach may be used to relate or compare sinusoids as an alternative to using the trigonometric identities in Eqs. (9.9) and (9.10). Consider the set of axes shown in Fig. 9.3(a). The horizontal axis represents the magnitude of cosine, while the vertical axis (pointing down) denotes the magnitude of sine. Angles are measured positively counterclockwise from the horizontal, as usual in polar coordinates. This graphical technique can be used to relate two sinusoids. Hafiz Zaheer Hussain

  16. 9.2 Sinusoids’ (9) We obtain Hafiz Zaheer Hussain

  17. 9.2 Sinusoids’ Example Given v(t) = Vm sin (ωt +10o). Transform to Cosine Solution v(t) = Vm sin (ωt +10o) v(t) = Vm cos (ωt + 10o - 90o) v(t) = Vm cos (ωt – 80o) Hafiz Zaheer Hussain

  18. 9.2 Sinusoids’ Example 9.2 Solution: Let us calculate the phase in three ways. The first two methods use trigonometric identities, while the third method uses the graphical approach. ■ METHOD 1 In order to compare v1 and v2 we must express them in the same form. If we express them in cosine form with positive amplitudes, Hafiz Zaheer Hussain

  19. 9.2 Sinusoids’ Example 9.2(count..) Hafiz Zaheer Hussain

  20. 9.2 Sinusoids’ PP 9.2 Hafiz Zaheer Hussain

  21. Content 9.1 Introduction 9.2 Sinusoids’ features 9.3 Phasors 9.4 Phasorrelationships for circuit elements 9.5 Impedance and admittance 9.6Kirchhoff’s laws in the frequency domain 9.7 Impedance combinations Hafiz Zaheer Hussain

  22. 9.3 Phasors where • A phasor is a complex number that represents the amplitude and phase of a sinusoid. • It can be represented in one of the following three forms: Rectangular Polar Exponential Hafiz Zaheer Hussain

  23. 9.3 Phasors , Example 9.3 Hafiz Zaheer Hussain

  24. 9.3 Phasors , Example 9.3 Hafiz Zaheer Hussain

  25. 9.3 Phasors , Example 9.3 (count..) Hafiz Zaheer Hussain

  26. 9.3 Phasors , PP 9.3 • Evaluate the following complex numbers: a. b. Solution: a. –15.5 + j13.67 b. 8.293 + j2.2 Hafiz Zaheer Hussain

  27. 9.3 Phasors (2) Mathematic operation of complex number: • Addition • Subtraction • Multiplication • Division • Reciprocal • Square root • Complex conjugate • Euler’s identity Hafiz Zaheer Hussain

  28. 9.3 Phasors(3) • Transform a sinusoid to and from the time domain to the phasor domain: (time domain) (phasor domain) • Amplitude and phase difference are two principal concerns in the study of voltage and current sinusoids. • Phasor will be defined from the cosine function in all our proceeding study. If a voltage or current expression is in the form of a sine, it will be changed to a cosine by subtracting from the phase. Hafiz Zaheer Hussain

  29. 9.3 Phasors(3) Table 9.1 Sinusoid-phasor transformation Hafiz Zaheer Hussain

  30. 9.3 Phasors(3) Lead or Lag seen Via Phasors Such a graphical representation ofphasorsis known as a phasor diagram. Hafiz Zaheer Hussain

  31. 9.3 Phasors , Example 9.4 Transform the following sinusoids to phasors: i = 6cos(50t – 40o) A v = –4sin(30t + 50o) V Solution: I A b. v(t) = -4 sin (30t +50o)V v(t) = 4cos (30t + 50 + 90o) v(t) = 4cos (ωt + 140o) Hafiz Zaheer Hussain

  32. 9.3 Phasors , Example 9.5 Hafiz Zaheer Hussain

  33. 9.3 Phasors , Example 9.5 Hafiz Zaheer Hussain

  34. 9.3 Phasors , PP 9.5 Solution: • v(t) = 10cos(wt + 210o) V • Since • i(t) = 13cos(wt + 22.62o) A Transform the sinusoids corresponding to phasors: Hafiz Zaheer Hussain

  35. 9.3 Phasors , Example 9.6 Answer Hafiz Zaheer Hussain

  36. 9.3 Phasors , PP 9.6 Hafiz Zaheer Hussain

  37. 9.3 Phasors (4) The differences between v(t) and V: • v(t) is instantaneous or time-domain representationV is the frequency or phasor-domain representation. • v(t) is time dependent, V is not. • v(t) is always real with no complex term, V is generally complex. Note: Phasor analysis applies only when frequency is constant; when it is applied to two or more sinusoid signals only if they have the same frequency. Hafiz Zaheer Hussain

  38. 9.3 Phasors (5) Relationship between differential, integral operation in phasor listed as follow: Hafiz Zaheer Hussain

  39. 9.3 Phasors , Example 9.7 Use phasor approach, determine the current i(t) in a circuit described by the integrodifferential equation. Hafiz Zaheer Hussain

  40. 9.3 Phasors , PP 9.7 Hafiz Zaheer Hussain

  41. Content 9.1 Introduction 9.2 Sinusoids’ features 9.3 Phasors 9.4 Phasorrelationships for circuit elements 9.5 Impedance and admittance 9.6Kirchhoff’s laws in the frequency domain 9.7 Impedance combinations Hafiz Zaheer Hussain

  42. 9.4 Phasorrelationships for circuit elements Transform the voltage-current relationship from the time domain to the frequency domain for each element. we will assume the passive sign convention. Voltage and Current are in phase in resistance Hafiz Zaheer Hussain

  43. 9.4 Phasorrelationships for circuit elements Hafiz Zaheer Hussain

  44. 9.4 Phasorrelationships for circuit elements Hafiz Zaheer Hussain

  45. 9.4 Phasorrelationships for circuit elements Hafiz Zaheer Hussain

  46. 9.4 Phasorrelationships for circuit elements The V-I Relationships for Capacitor Hafiz Zaheer Hussain

  47. 9.4 Phasorrelationships for circuit elements Hafiz Zaheer Hussain

  48. 9.4 Phasorrelationships for circuit elements Hafiz Zaheer Hussain

  49. 9.4 Phasorrelationships for circuit elements • Summary of voltage-current relationship Resistor: Capacitor: Inductor: Hafiz Zaheer Hussain

  50. 9.4 Phasorrelationships for circuit elements Hafiz Zaheer Hussain

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