Transcript Radians

AC Fundamentals
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Radians
Converting Between Radians and Degrees
Frequency and Period
Angular Velocity
Generated Angle
Instantaneous Current and Voltage
Effective Value of ac Current or Voltage
Capacitance and Inductance
Radians
Key Point:
When the length of an
arc equals the length of
the radius of a circle,
the angle is equal to 1
radian.
One radian = 57.3°.
Converting Between Radians
and Degrees
To convert from
radians to degrees:
To convert from
radians to degrees:
Convert 85o to radians
Convert 3.42 radians to degrees
angle = 37°
Frequency and Period
The frequency (f) is the
number of times per
second that we
generate a sine wave.
The period (T) is the
time it takes to
generate one complete
sine wave.
T = 400 μs. Find f
f = 1/T
T = 1/f
f = 15 kHz. Find T
Angular Velocity
Angular velocity, ω, is
the angular motion or
speed of a rotating
vector or phasor.
Angular velocity is
measured in radians
per second or in
degrees per second.
ω = 2πf
o
ω = 360 f
Generated Angle
The generated angle at any
instant, t, can be calculated as:
angle in radians = 2πft = ωt
angle in degrees = 360° ft = ωt
Given:
f = 2kHz
t = 100 μs
ωt = 1.26 rad
ωt = 72.0°
Self-Test 18-1 #4, 5, 6
Instantaneous Current and
Voltage
In ac circuits,
instantaneous values
of current and voltage
may be found using
the following
equations:
i = Ipk sin ωt
e = Epk sin ωt
Self-Test 18-2 #1, 2
Let Ipk = 40 mA, Epk = 12 V,
and f = 5 kHz
Find i and e after 70 μs
i = 40 mApk sin ωt
= 32.4 mAt
e = 12 Vpk sin ωt
= 9.71 Vt
Effective Value of ac Current or
Voltage
The rms or effective
value of an ac voltage or
current is the value that
converts the same
energy as does a dc
value.
The rms value equals
0.707 of the maximum
or peak of a sine wave.
Irms = Ieff = 0.707 Ipk
Vrms = Veff = 0.707 Vpk
Ipk = 1/ Ieff = 1.414 Irms
Vpk = 1/ Veff = 1.414 Vrms
Capacitance and Inductance
Key Point:
Whenever we have
capacitance or inductance
in an ac circuit, the current
and voltage are out of
phase. When the phase
angle, theta, θ, is known, we
can calculate the
instantaneous values of
current and voltage.
When current is the
reference,
i = Ipk sin ωt
e = Epk sin (ωt  )
When voltage is the
reference,
e = Epk sin ωt
i = Ipk sin (ωt  )
Self-Test 18-2 #5