Transcript Sinusoidal Sources

Sinusoidal Sources
• Voltage or Current
• Sin or Cos
• In our case, choose cos and write in
general
v  Vm cos(t   )
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Graphically
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v  Vm cos(t   )
• Vm = amplitude, or
maximum value of the
signal (volts)
• T = period of the
signal (seconds)
• f = 1/T = frequency of
the signal (Hertz)
• ω = 2πf = angular
frequency of the
signal (radians/sec)
• Φ = phase angle
(radians)
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Average value of a sinusoid
Vaverage
1
 Vdc 
T
t0  T
V
m
cos(t   )dt
t0
Vdc  0
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rms value of a sinusoid
Vrms
1

T
t0  T

Vm2 cos 2 (t   )dt
t0
...
Vrms
Vm

 0.707Vm
2
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Detailed calculation of rms
Vrms
1

T
Vrms 
1

Vrms  Vm
Vrms
Vrms
t0  T

v2
t0

2
2
V
cos
 m td (t )
0




Vm 1 
1 1
(1  cos 2t )d (t ) 
  d (t )   cos 2td (t ) 

 02
2  0
0

Vm  1
1




0

sin
2


sin
0







2

2

V
 m  0.707Vm
2
1
2
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What does rms mean?
• Determine the average power delivered to
the resistor by the sinusoidal voltage
source
+
Vmcos(ωt+θv)
R
1kOhm
-
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1
P
T
1
P
R
t0 T

t0
t0  T

Vm2 cos 2 (t   v )
dt
R
Vm2 cos 2 (t   v )dt
t0
2
rms
V
P
R
If Vrms = 100V, it delivers the same amount of power
to the resistor as a 100V DC source, or it is effectively
equivalent to a DC source of the same magnitude.
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Rms and “Effective Value”
• rms value and “effective value” are used
interchangeably
2
rms
Veff2
V
P

R
R
2
2
P  I rms R  I eff R
P  Vrms I rms  Veff I eff
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Practical Example
• Consider the voltage at a standard
household outlet
– The rms voltage is 120 Volts
– The peak voltage Vm is determined as
Vrms
Vm

 0.707Vm
2
Vm  2Vrms  1.414(120V )  169.7Volts
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