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05-Chapter 5 - Sinusoidal

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05-Chapter 5 - Sinusoidal Powered By Docstoc
					Chapter 5
Steady-State
Sinusoidal Analysis
Where are we going?
   A look ahead:
     HighPass Filter
     Low Pass Filter

     Band Pass Filter

     Three phase Source
5-1 Sinusoidal Currents and Voltages

 v(t) = Vmsin(wt + q)
    Vm = peak value
    w = angular frequency
    q = phase angle
T = period
f = Frequency
f = 1/T
w = 2pf = 2p/T
Plot   v versus t
 Label   T


Plot   v versus wt
 Label   q
Chapter 5
Steady-State
Sinusoidal Analysis
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.
Definitions
 Vrms = root mean square voltage
 Vpp = peak-to-peak voltage
 Vave = average voltage
 Pave = average power

             Vm               Vave  0
      Vrms 
               2
      V pp  2Vm            Pave  I rms Vrms
                                      R=50W

 Example:
   Find  v(t) and i(t)
   Vrms = root mean square voltage
   Vpp = peak-to-peak voltage
   Vave = average voltage
   Pave = average power
What are the following for a wall outlet?

  f
  Vrms = VDCEquivalent
  Vm
  Vave
  Vpp
Notation Summary
Sinusoidal
               v1(t) = V1 cos(wt + q1)
Polar Phasor
            V1 = V1 L q1
Complex Phasor
            V1 = V1 cos(q1) + j V1 sin(q1)


                      j  1
Math
       Imaginary
                     C=A + jB

             q = phase angle
                         Real



             What is the magnitude of C?

                      C  A B    2        2


             What is the phase (or direction) of C?
                                 B    1
                         q  tan
                                 A
 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°)
Example Problems
 Convertthe following from complex
 phasors to polar form.
  V1 = 30 + j40
  V2 = 4 - j20

  V1 + V2
Phasor Math in Polar Form
 (C1q1)   (C2q2) = C1 C2 (q1 + q2)

 (C1q1)   /(C2q2) = C1/C2 (q1 - q2)
Ohm’s Law for AC Circuits

      V=IZ

            Impedance
 Resistors
 Suppose that       v(t) = Vmcos(wt)

 What   is i(t) ?
   Hint:   v=iR


 What   is the phase relationship between i and v?
Capacitors
 Suppose that       v(t) = Vmcos(wt)

 What   is i(t) ?
   Hint:   v = q/C


 What   is the phase relationship between i and v?
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?
Reactance, Impedance and Phasors

 Go   to notes…

 Phasor   Diagrams for V and I

 Impedance Diagrams    for R, L and C circuits
The R, L, and C Elements
   Ohm’s Law for Peak Values
      Resistors: Vp=IpR
      Capacitors: Vp=IpXC
      Inductors: Vp=IpXL

   Go to notes...
  Example Problems
 Example   3.15:

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

  Convert the following from rectangular to polar
    form.
  30 + j40
  4 - j20
  -3 - 4j
Impedance Diagrams
Resistor
 ZR = R0

Capacitor
 ZC = XC-90

Inductor
  ZL = XL90
 RL Circuit Example
Connect at AC power supply in series with an
 inductor and a resistor.
How does VR vary with the input frequency?
 RC Circuit Example
Connect at AC power supply in series with an
 capacitor and a resistor.
How does VR vary with the input frequency?
 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?
 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.

                  1
           fs 
                2p LC
 Whatcapacitance do you need to listen to
 107.7 MHz on a radio with a 1mH inductor?

				
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posted:10/30/2011
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