Download History of AC - Portal UniMAP

SYLLABUS
1.
AC Fundamentals
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2.
AC Analysis
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3.
AC sinusoids
AC response (reactance, impedance)
Phasors and complex numbers
RL, RC, RLC circuit analysis
Mesh and Nodal analysis
AC power
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Average power, Reactive power, Complex power
Power triangle
4. Three phase circuit
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Y and Delta connection
Line and Phase voltages
ALTERNATING CURRENT (AC)
SINUSOIDS
OBJECTIVES
– Explain the difference between alternating (AC) and
direct current (DC).
– Express angular measure in both degrees and radians.
– Compute the peak, peak-peak, and instantaneous values
of a waveform.
– Define and solve for the RMS value of a waveform.
– Define cycle, period, and frequency of a sinusoid.
– Given the analytical expression, sketch and explain the
graph of a sinusoid.
– Determine the relative phase of a sinusoidal waveform.
OBJECTIVES (cont)
– Determine the total voltages and currents that have DC
and AC components.
– Apply Ohm’s Law, KCL, and KVL to analyze a simple
AC circuit.
– Write the time domain equation for any sinusoidal
waveform with a DC component.
SINE WAVES
• Voltage can be produced such that, over time, it
follows the shape of a sine wave
• The magnitude of the voltage continually changes.
• Polarity may or may not change.
– When it does not change, the current does not change
direction.
– When polarity does change, the current changes
direction.
– When graphing a sinusoidal voltage, the polarity
changes only when the magnitude alternates between
“+” and “-” values.
AC SINEWAVE
voltage
Voltage is positive
+
Polarity change
t
0
Voltage is positive
1 cycle
OTHER ACs
SINE WAVE
TRIANGLE WAVE
SQUARE WAVE
HOW IS A SINE WAVE GENERATED ?
• Electromagnetic Induction. (Ship AC generators
produce sine wave voltages through
electromagnetic induction):
– magnetic field
– conductor
– relative motion between the two.
• Electronic Signal Generators
– Function Generators: multi-waveforms.
GENERATING AC VOLTAGES
• One way to generate
ac voltage is to rotate a
coil of wire at constant
angular velocity in a
fixed magnetic field
FARADAY’S LAW
“ Voltage is induced in a circuit whenever the
flux linking (i.e. passing through) the
circuit is changing.. and that the magnitude
of the voltage is proportional to the rate of
change of the flux linkages”
DC vs AC
• DC Source: voltage POLARITY of the source and
current DIRECTION do not change over time.
Voltage
I
V
1 ohm
time
AC SOURCE
– AC source: Voltage polarity changes therefore the current
changes direction.
I
V(1.25s)
= +2v
1 ohm
2v
0
time
(sec)
-2v
V(3.75s)
= -2v
I
1 ohm
1 2 3 4
PERIOD AND FREQUENCY
• Period: Time to complete one complete cycle
– Symbol: T
• Frequency: Number of cycles in one second
– Symbol: f
– Measured in hertz (Hz)
V
1
f 
T
t
FREQUENCY
• Definition: the number of cycles per second
of a waveform
• Denoted by the lower case letter f
• Its unit is the hertz (Hz)
1 hertz  1 cycle per second
Ex.
1 cycle
f=1 Hz
1 second
Ex.
1 cycle
1 cycle
1 second
Ex.
1 cycle
60 cycles
?
1 second
PERIOD
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Definition: the duration of one cycle.
It is the inverse of frequency.
Denoted by the upper case letter T
Measured in second, s
1
1
T  (s) and f  (Hz)
f
T
• The period of a waveform can be measured
between any two corresponding point.
• Often it is measured between zero points
because they are easy to establish on an
oscilloscope trace
T
(between peaks)
t
T
(between zero
points)
T
(Any two
identical points)
Ex.
• Determine the period and frequency of the
waveform of the figure above.
T2 = 10 ms
T1 = 8 ms
Solution
• Time interval T1 does not represent a
period as it is not measured between
corresponding points. Interval T2,
however, is. Thus, T = 10 ms and,
1
f 

100
Hz
3
10 10 s
PEAK VALUES (VP, IP)
• Max Voltage (Current)
– Symbol VM ( IM )
– The maximum value of V (I) measured from
the point of inflection (“baseline or DC offset”)
– From the graph: VM - VDC
– Also called “Amplitude”
V
VM or Amplitude
baseline
VDC
t
PEAK TO PEAK VALUES (VPP, IPP)
• Peak to Peak Voltage (Current)
– Symbol VPP ( IPP )
– The difference between the maximum value of V
(I) and the minimum value of V (I)
– From the graph: VMAX – VMIN
– Equals twice peak value VPP = 2VP
V
VMAX
VPP
t
VMIN
ROOT-MEAN-SQUARE (VRMS, IRMS )
• Named for the mathematical process by which the
value is calculated. “Effective Voltage (VEFF)”
• “ The RMS value of a sine wave is equal to the
value of an equivalent DC circuit that would
produce the same heating effect or power in a load
as the given sine wave.”
• Most meters read in RMS
• The voltage accessed at electrical wall sockets is
RMS.
ROOT-MEAN-SQUARE (VRMS, IRMS )
VRMS
2

VP  0.707  VP
2
COMPATIBILITY OF VALUES
Vrms
VM
Vpp
• When Peak voltages are used as source values,
current calculations will also be in Peak values.
• Likewise, an RMS source produces answers in
RMS.
• When solving a problem make sure all values are
expressed ONE way (peak, peak to peak, or
RMS)!
VOLTAGE & CURRENT VALUES
• Ohm’s Law still applies: V=IR
• If current changes with time and R is a
constant, voltage will also change with time
– Voltage will be proportional to current
VOLTAGE & CURRENT VALUES
– A graph of current and voltage in a resistor
produces identical waveforms:
• Peak at the same time
• Cross the same baseline, at the same time
• Differ only in amplitude:
– IP is 1/R of VP
INSTANTANEOUS VALUES
• Instantaneous Values ( v, i )
– value of voltage and current at any:
• instant in time or at
• at any angle
• Mathematically expressed 2 ways:
v(t)  VM sin( 2 ft   )
v(  )  VM sin(    )
ANGULAR DOMAIN
• We can identify points on the sine wave in terms of
an angular measurement (degrees or radians).
– The instantaneous value of the sine wave can be
related to the angular rotation of the generator,
(1 rotation = 360°=2 radians)
 

rad  
 deg
 180 
180 

deg  
rad
  
• Sine Wave Angles: Degrees & Radians
– 2 radians = 360o
1 radian = 57.3o
TIME DOMAIN
• Because the time to complete a cycle is frequency
dependent, we can also identify points on the sine
wave in terms of time.
v(t)  VM sin( 2 ft   )
• To convert between the time domain and angular
domain remember:
2 ft   t  
PHASE ANGLE
• Symbol is  (theta). It is expressed as an angle
• Phase angle specifies the lateral shift in the position
of a sine wave from a reference wave.
• Examine the same event, on each wave:
– Two events occurring at the same angle or at the
same time are in phase.
– Events occurring at different angles or at
different times are out of phase.
PHASE ANGLE (angular domain)
• Wave A is the reference wave:
– Wave B is 90° out of phase.
PHASE ANGLE (Time domain)
• Wave A is the reference wave. Compare the
positive peak events:
–
–
–
–
Wave A peaks at 30ms; Wave B at 60ms
T=120ms
 /360º = Dt/T = (60ms-30ms)/120ms.
 = 90º
LEADING & LAGGING
• Since wave B peaked after the reference wave peaked, we
say it LAGS the reference wave by 90º ;  = - 90º
• If wave B was the reference, wave A would peak before the
reference wave (B). We would say it LEADS the reference
wave;
 = + 90º
• Note: Because
it is the reference
wave,  for ANY
reference wave
is 0 º
Ex:
– Compute the phase angle if:
• V1(t) is the reference wave
• V2 (t) is the reference wave
t = 1 ms/div
V1(t)
V2(t)
Ex:
V2 is the reference. Write the equations.
t = 1 ms/div
V1(t)
V2(t)
SUPERIMPOSED DC & AC
• A circuit can have both a DC voltage source and
an AC
• We say that the “AC rides on the DC”
• The graph of the voltage is displaced vertically
from 0, to the DC voltage level. Algebraically:
v(t)  Vdc  VM sin( 2 ft   )
v(  )  Vdc  VM sin(    )
REVIEW QUIZ
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The difference between DC and AC ?
3 items required for electromagnetic induction.
Frequency is equal to ?
Name 3 different Sine wave values.
How many radians in 360 degrees ?
If the peak value is 170 V, the RMS value = ?
What type of shift does a phase angle represent?