Transcript AC - UniMAP Portal

COVERAGE TOPICS
1.
AC Fundamentals
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2.
AC Analysis
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3.
RL, RC, RLC circuit analysis
Mesh and Nodal analysis
AC power
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4.
AC sinusoids
AC response (reactance, impedance)
Phasors and complex numbers
Average power, Reactive power, Complex
power
Power triangle
Three phase circuit
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Y and Delta connection
Line and Phase voltages
AC SINUSOIDS
SCOPE
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Explain the difference between AC and 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
Define cycle, period, and frequency
Given the analytical expression, sketch and explain the
graph of a sinusoid.
Determine the relative phase of a sinusoidal waveform.
OBJECTIVES (cont)

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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.

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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
V(3.75s)
= -2v
I
1 ohm
time
(sec)
-2v
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
t
1
f 
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

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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.

Figure shows an
oscilloscope trace of
a square wave. Each
horizontal division
represents 50 μs.
Determine the
frequency.
Solution

Since the wave repeats itself
every 200 μs, its period (T) is 200
μs and,
1
f 
 5 kHz
6
200  10 s
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)
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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)
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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 (lab DMM)

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
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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
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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:
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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:
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Wave A peaks at 30ms; Wave B at 60ms
T=120ms
 /360º = Dt/T = (60ms-30ms)/120ms.
 = 90º
LEADING & LAGGING
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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
V1(t)
V2(t)
t = 1 ms/div
Ex:
V2 is the reference. Write the equations.
t = 1 ms/div
V1(t)
V2(t)
SUPERIMPOSED DC & AC
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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?