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Ch.9 Sinusoids and Phasors

Ch.9 Sinusoids and Phasors. 1. Introduction. AC is more efficient and economical to transmit over long distance Sinusoid is a signal that has the form of the sine or cosine function Sinusoidal current = alternating current (ac) Nature is sinusoidal Easy to generate and transmit

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Ch.9 Sinusoids and Phasors

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  1. Ch.9 Sinusoids and Phasors

  2. 1. Introduction • AC is more efficient and economical to transmit over long distance • Sinusoid is a signal that has the form of the sine or cosine function • Sinusoidal current = alternating current (ac) • Nature is sinusoidal • Easy to generate and transmit • Any practical periodic signal can be represented by a sum of sinusoids • Easy to handle mathematically Electric Circuit, 2007

  3. 2. Sinusoids • Consider the sinusoidal voltage • T: period of the sinusoid Electric Circuit, 2007

  4. Sinusoids (2) • Periodic function • Satisfies f(t) = f(t+nT), for all t and for all integers n • Hence • Cyclic frequency f of the sinusoid Electric Circuit, 2007

  5. Sinusoids (3) • Let us examine the two sinusoids • Trigonometric identities Electric Circuit, 2007

  6. Sinusoids (4) • Graphical approach • Used to add two sinusoids of the same frequency where Electric Circuit, 2007

  7. Example 9.1 • Find the amplitude, phase, period, and frequency of the sinusoid Electric Circuit, 2007

  8. Example 9.2 • Sol) Electric Circuit, 2007

  9. 3. Phasors • Phasor is a complex number that represents the amplitude and phase of a sinusoid • Provides a simple means of analyzing linear circuits excited by sinusoidal sources • Complex number with Electric Circuit, 2007

  10. Phasors (2) • Operations of complex number • Addition: • Subtraction: • Multiplication: • Division: • Reciprocal: • Square Root: • Complex Conjugate: Electric Circuit, 2007

  11. Phasors (3) • Euler’s identity with • Given a sinusoid • Thus, where • Plot of the Electric Circuit, 2007

  12. Phasors (4) • Phasor representation of the sinusoid v(t) Electric Circuit, 2007

  13. Phasors (5) • Derivative & integral of v(t) • Derivative of v(t) • Phasor domain representation of derivative v(t) • Phasor domain rep. of Integral of v(t) Electric Circuit, 2007

  14. Phasors (6) • Summing sinusoids of the same frequency • Differences between v(t) and V • v(t) is time domain representation, while V is phasor domain rep. • v(t) is time dependent, while V is not • v(t) is always real with no complex term, while V is generally complex • Phasor analysis • Applies only when frequency is constant • Applies in manipulating two or more sinusoidal signals only if they are of the same frequency Electric Circuit, 2007

  15. Example 9.3 • Evaluate these complex numbers • Sol) • a) • then • Taking the square root Electric Circuit, 2007

  16. Example • Example 9.4 • Transform these sinusoids to phasors • Example 9.5 • Find the sinusoids represented by these phasors Electric Circuit, 2007

  17. Example • Example 9.6 • Example 9.7 • Using the phasor approach, determine the current i(t) Electric Circuit, 2007

  18. 4. Phasor Relationships for Circuit Elements • Voltage-current relationship • Resistor: ohm’s law • Phasor form • Inductor • Phasor form Electric Circuit, 2007

  19. Phasor Relationships for Circuit Elements(2) • Inductor • The current lags the voltage by 90o. • Capacitor: • Phasor form • The current leads the voltage by 90o. Electric Circuit, 2007

  20. Example 5.6 • The voltage v=12cos(60t+45o) is applied to a 0.1H inductor. Find the steady-state current through the inductor • Sol) • Converting this to the time domain, Electric Circuit, 2007

  21. 5. Impedance and Admittance • Voltage-current relations for three passive elements • Ohm’s law in phasor form • Imdedance Z of a circuit is the ratio of the phasor voltage to the phasor current I, measured in ohms • When , • When , Electric Circuit, 2007

  22. Impedance and Admittance (2) • Impedance = Resistance + j Reactance • where • Adimttance Y is the reciprocal of impedance, measured in siemens (S) • Admittance = Conductance + j Susceptance Electric Circuit, 2007

  23. Example 9.9 • Find v(t) and i(t) in the circuit • Sol) • From the voltage source • The impedance • Hence the current • The voltage across the capacitor Electric Circuit, 2007

  24. 6. Kirchhoff’s law in the frequency domain • For KVL, • Then, • KVL holds for phasors • KCL holds for phasors • Time domain • Phasor domain • KVL & KCL holds in frequency domain Electric Circuit, 2007

  25. 7. Impedance Combinations • Consider the N series-connected impedances • Voltage-division relationship Electric Circuit, 2007

  26. Impedance Combinations (2) • Consider the N parallel-connected impedances • Current-division relationship Electric Circuit, 2007

  27. Example 9.10 • Find the input impedance of the circuit with w=50 rad/s • Sol) • The input impedance is Electric Circuit, 2007

  28. Example 9.11 • Determine vo(t) in the circuit • Sol) • Time domain  frequency domain • Voltage-division principle Electric Circuit, 2007

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