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Chapter 10 Sinusoidal Steady-State Analysis

Chapter 10 Sinusoidal Steady-State Analysis . Charles P. Steinmetz (1865-1923), the developer of the mathematical analytical tools for studying ac circuits. Courtesy of General Electric Co. Heinrich R. Hertz (1857-1894). Courtesy of the Institution of Electrical Engineers. cycles/second.

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Chapter 10 Sinusoidal Steady-State Analysis

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  1. Chapter 10Sinusoidal Steady-State Analysis

  2. Charles P. Steinmetz (1865-1923), the developer of the mathematical analytical tools for studying ac circuits. Courtesy of General Electric Co.

  3. Heinrich R. Hertz (1857-1894). Courtesy of the Institution of Electrical Engineers. cycles/second Hertz, Hz

  4. Sinusoidal Sources Amplitude Period = 1/f Phase angle Angular or radian frequency = 2pf = 2p/T Sinusoidal voltage source vsVm sin(t ). Sinusoidal current source isIm sin(t ).

  5. Example v i circuit element + v _ i Voltage and current of a circuit element. The current leads the voltage by radians OR The voltage lags the current by radians

  6. Example 10.3-1 Find their phase relationship and Therefore the current leads the voltage by

  7. Recall Triangle for A and B of Eq. 10.3-4, where C .

  8. Example 10.3-2 A B B A

  9. Steady-State Response of an RL circuit An RL circuit. From #8&#9 Substitute the assumed solution into 10.4-1 Coeff. of cos Coeff. of sin Solve for A & B

  10. Steady-State Response of an RL circuit (cont.) Thus the forced (steady-state) response is of the form

  11. Complex Exponential Forcing Function Input Response magnitude phase frequency Exponential Signal Note

  12. Complex Exponential Forcing Function (cont.) try We get where

  13. Complex Exponential Forcing Function (cont.) Substituting for A We expect

  14. Example We replace Substituting ie

  15. Example(cont.) The desired answer for the steady-state current interchangeable Or

  16. Using Complex Exponential Excitation to Determine a Circuit’s SS Response to a Sinusoidal Source • Write the excitation as a cosine waveform with • a phase angle • Introduce complex excitation • Use the assumed response • Determine the constant A

  17. Obtain the solution • The desired response is Example 10.5-1

  18. Example 10.5-1(cont.)

  19. Example 10.5-1(cont.) The solution is The actual response is

  20. The Phasor Concept A sinusoidal current or voltage at a given frequency is characterized by its amplitude and phase angle. Magnitude Phase angle Thus we may write unchanged

  21. The Phasor Concept(cont.) A phasor is a complex number that represents the magnitude and phase of a sinusoid. phasor The Phasor Concept may be used when the circuit is linear , in steady state, and all independent sources are sinusoidal and have the same frequency. A real sinusoidal current phasor notation

  22. The Transformation Time domain Transformation Frequency domain

  23. The Transformation (cont.) Time domain Transformation Frequency domain

  24. Example Substitute into 10.6-2 Suppress

  25. Example (cont.)

  26. Phasor Relationship for R, L, and C Elements Time domain Resistor Frequency domain Voltage and current are in phase

  27. Inductor Time domain Frequency domain Voltage leads current by

  28. Capacitor Time domain Frequency domain Voltage lags current by

  29. Impedance and Admittance Impedance is defined as the ratio of thephasor voltage to the phasor current. Ohm’s law in phasor notation phase magnitude or polar exponential rectangular

  30. Graphical representation of impedance R Resistor wL Inductor Capacitor 1/wC

  31. Admittance is defined as the reciprocal ofimpedance. conductance In rectangular form susceptance G Resistor 1/wL Inductor wC Capacitor

  32. Kirchhoff’s Law using Phasors KVL KCL Both Kirchhoff’s Laws hold in the frequency domain. and so all the techniques developed for resistive circuits hold • Superposition • Thevenin &Norton Equivalent Circuits • Source Transformation • Node & Mesh Analysis • etc.

  33. Impedances in series Admittances in parallel

  34. Example 10.9-1 R = 9 W, L = 10 mH, C = 1 mF i = ? KVL

  35. Example 10.9-2 v = ? KCL

  36. Node Voltage & Mesh Current using Phasors va = ? vb = ?

  37. KCL at node a KCL at node b Rearranging Admittance matrix

  38. If Im = 10 A and Using Cramer’s rule to solve for Va Therefore the steady state voltage va is

  39. Example 10.10-1 v = ? use supernode concept as in #4

  40. Example 10.10-1 (cont.) KCL at supernode Rearranging

  41. Example 10.10-1 (cont.) Therefore the steady state voltage v is

  42. Example 10.10-2 i1 = ?

  43. Example 10.10-2 (cont.) KVL at mesh 1 & 2 Using Cramer’s rule to solve for I1

  44. Superposition, Thevenin & Norton Equivalents and Source Transformations Example 10.11-1 i = ? Consider the response to the voltage source acting alone = i1

  45. Example 10.11-2 (cont.) Substitute

  46. Example 10.11-2 (cont.) Consider the response to the current source acting alone = i2 Using the principle of superposition

  47. Source Transformations

  48. Example 10.11-2 IS = ?

  49. Example 10.11-3 Thevenin’s equivalent circuit ?

  50. Example 10.11-4 Thevenin’s equivalent circuit

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