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Chapter 5

Chapter 5. Steady-State Sinusoidal Analysis. Where are we going?. A look ahead: High Pass Filter Low Pass Filter Band Pass Filter Three phase Source. 5-1 Sinusoidal Currents and Voltages. v(t) = V m sin( w t + q ) V m = peak value w = angular frequency

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Chapter 5

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  1. Chapter 5 Steady-State Sinusoidal Analysis

  2. Where are we going? A look ahead: • High Pass Filter • Low Pass Filter • Band Pass Filter • Three phase Source

  3. 5-1 Sinusoidal Currents and Voltages v(t) = Vmsin(wt + q) Vm = peak value w = angular frequency q = phase angle

  4. T = period f = Frequency f = 1/T w = 2pf = 2p/T

  5. Definitions • Vrms = root mean square voltage • Vpp = peak-to-peak voltage • Vave = average voltage • Pave = average power

  6. Example: • Find v(t) and i(t) • Vrms = root mean square voltage • Vpp = peak-to-peak voltage • Vave = average voltage • Pave = average power R=50W

  7. Plot v versus t • Label T • Plot v versus wt • Label q

  8. Chapter 5 Steady-State Sinusoidal Analysis

  9. Useful Trig Identities sin(z) = cos(z - 90°) Example: v(t) = 10sin(200t + 30°) (a) Write v(t) as a cosine function. (b) Find Vm, w, f, T and q.

  10. What are the following for a wall outlet? f Vrms = VDCEquivalent Vm Vave Vpp

  11. Notation Summary Sinusoidal v1(t) = V1 cos(wt + q1) Polar Phasor V1 = V1Lq1 Complex Phasor V1 = V1cos(q1) + j V1sin(q1)

  12. Math Imaginary C=A + jB q = phase angle Real What is the magnitude of C? What is the phase (or direction) of C?

  13. Example Problems • Convert the following voltages to phasors in polar form and complex form. • v1(t) = 20 cos(wt - 45°) • v2(t) = 20 sin(wt + 60°)

  14. Example Problems • Convert the following from complex phasors to polar form. • V1 = 30 + j40 • V2 = 4 - j20 • V1 + V2

  15. Phasor Math in Polar Form • (C1q1) (C2q2) = C1 C2 (q1 + q2) • (C1q1) /(C2q2) = C1/C2 (q1 - q2)

  16. Ohm’s Law for AC Circuits V = IZ Impedance

  17. Resistors • Suppose that v(t) = Vmcos(wt) • What is i(t) ? • Hint: v = i R • What is the phase relationship between i and v?

  18. Capacitors • Suppose that v(t) = Vmcos(wt) • What is i(t) ? • Hint: v = q/C • What is the phase relationship between i and v?

  19. Inductors • Suppose that i(t) = Imcos(wt) • What is v(t) ? • Hint: v(t) = L di/dt • What is the phase relationship between i and v?

  20. Reactance, Impedance and Phasors • Go to notes… • Phasor Diagrams for V and I • Impedance Diagrams for R, L and C circuits

  21. The R, L, and C Elements • Ohm’s Law for Peak Values Resistors: Vp=IpR Capacitors: Vp=IpXC Inductors: Vp=IpXL • Go to notes...

  22. Example Problems • Example 3.15: Convert the following from polar to rectangular form. 1053.13 16-30 25120

  23. Example Problems • Example 3.15: Convert the following from rectangular to polar form. 30 + j40 4 - j20 -3 - 4j

  24. Impedance Diagrams Resistor ZR = R0 Capacitor ZC = XC-90 Inductor ZL = XL90

  25. RL Circuit Example Connect at AC power supply in series with an inductor and a resistor. How does VR vary with the input frequency?

  26. RC Circuit Example Connect at AC power supply in series with an capacitor and a resistor. How does VR vary with the input frequency?

  27. RLC Circuit Example Connect at AC power supply in series with an inductor, capacitor and a resistor. How does VR vary with the input frequency?

  28. 3.18 Tuned Resonant Networks • RLC Series circuits are used in radios. • Series RLC networks have a resonant frequency that depends on C and L only. • What capacitance do you need to listen to 107.7 MHz on a radio with a 1mH inductor?

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