Download Unit 13 PowerPoint Slides

EGR 2201 Unit 13
AC Power Analysis



Read Alexander & Sadiku, Chapter 11.
Homework #13 and Lab #13 due next
week.
Final Exam and Lab Exam next week.
Review:
Power


Supplies energy
Absorb energy
Recall the following key points about power
from the first week of this course.
An element’s power is the rate at which
the element supplies or absorbs energy:
dw
p
dt


Power’s unit of measure is the watt (W).
By convention, we assign a positive sign to
a power value if the element is absorbing
energy, and we assign a negative sign if
the element is supplying energy.
Review: The Power Law

An element’s power is equal to the product of
its voltage times its current:
p  vi

To get the correct sign (+ or ) on the power
value when we use this equation, we must
obey the passive sign convention,
which says that we regard the positive
direction for current as current into an
element’s positive terminal.
Review: Dissipation versus
Storage

Recall also that resistors always absorb
energy. They never supply energy.



So a resistor’s power is always positive.
The energy a resistor absorbs is lost (or
“dissipated”) as heat.
In contrast, inductors and capacitors are
energy-storage elements. At times they
may absorb energy, but at other times they
may supply this energy back to the circuit.

So an inductor’s or capacitor’s power may
be positive at one time but negative at
another time.
Review: Other Power Formulas for
Resistors


By combining the power law (p = v  i)
with Ohm’s law (v = i  R or i = v  R), we
can easily derive two other useful
formulas for the power dissipated by a
resistor:
p=i2R
p=v2R
There are no similar formulas for
capacitors or inductors in DC circuits.
Average Value of a Sinusoid (1 of 2)


Consider a sinusoid that represents any
quantity (voltage, current, power, …) versus
time.
If the sinusoid is
symmetrical about the
horizontal axis, then its
average value is 0.

In the circuits we’ve
studied, a graph of voltage
or current versus time
looks like this. Therefore the
average voltage or
average current is 0.
Average Value of a Sinusoid (2 of 2)

But if the sinusoid is
“shifted up,” then
its average value
(see blue dashed
line) is a positive
number.

As we’ll see, a graph of power versus time in an AC
circuit typically looks like this. Therefore average
power is usually not 0.
Shifting a Sinusoid Up

Mathematically, we can shift a sinusoid up by
adding a positive constant to the sinusoid.
8
6
4
2
0
-2
-4



Example in MATLAB:
>> fplot('5*cos(200*t)', [0, 0.1])
>> hold on
>> fplot('3 + 5*cos(200*t)', [0, 0.1], 'r')
What is the blue sinusoid’s average value?
What is the red sinusoid’s average value?
-6
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Power in AC Circuits

We’ll just
look at
these.
In AC circuits we distinguish several kinds of
power:
Quantity
Symbol
SI Unit
Symbol for
the Unit
Instantaneous power
p(t)
watt
W
Average power (also
called real power)
P
watt
W
Apparent power
S
voltampere
VA
Complex power
S
voltampere
VA
Q
voltampere
reactive
VAR
Reactive power
Instantaneous Power

To find an element’s or network’s
instantaneous power, use the same
power formula as for DC circuits:
p (t )  v(t )i (t )


The t reminds us that in AC circuits,
voltage and current change with time.
So instantaneous power also changes
with time.
This equation holds whether the source
is sinusoidal, triangle, square, etc. But
we’ll focus on the sinusoidal case.
Multiplying Sinusoids


In a network connected to a sinusoidal source,
v(t) and i(t) are sinusoids with the same
frequency. And p(t) = v(t) i(t), so p(t) is the
product of two sinusoids.
Question: What do you get when you multiply
two sinusoids of the same frequency? Let’s
use MATLAB to get an idea.
>> fplot('5*cos(200*t)', [0, 0.1])
>> hold on
>> fplot('8*cos(200*t+70*pi/180)', [0, 0.1],'r')
>>fplot('5*cos(200*t)*8*cos(200*t+70*pi/180)',[0, .1], 'k')
Multiplying Sinusoids

The product’s average value  0 , and the product’s
frequency is twice the frequency of the other two.
A Typical Graph of
Instantaneous Power


In typical AC circuits, a network absorbs
energy during part of the cycle and
supplies energy back to the source
during part of the cycle.
Therefore its power is sometimes
positive and sometimes negative.
Positive p(t): network
is absorbing energy.
Negative p(t): network
is supplying energy.
Instantaneous Power
with Sinusoidal Source

Suppose a network’s voltage and current are
𝑣 𝑡 = 𝑉𝑚 cos 𝜔𝑡 + 𝜃𝑣
and
𝑖 𝑡 = 𝐼𝑚 cos 𝜔𝑡 + 𝜃𝑖
Then its instantaneous power is p(t) = v(t) i(t).
 Using trig identities, we can show that
1
1
𝑝 𝑡 = 𝑉𝑚 𝐼𝑚 cos 𝜃𝑣 − 𝜃𝑖 + 𝑉𝑚 𝐼𝑚 cos 2𝜔𝑡 + 𝜃𝑣 + 𝜃𝑖
2
2

This term does not
depend on t, and thus
is constant. We call it
the average power P.
This term is a sinusoid
whose frequency is
twice the frequency of
v(t) and i(t).
Graph of
Instantaneous Power

On the previous slide we had
1
1
𝑝 𝑡 = 𝑉𝑚 𝐼𝑚 cos 𝜃𝑣 − 𝜃𝑖 + 𝑉𝑚 𝐼𝑚 cos 2𝜔𝑡 + 𝜃𝑣 + 𝜃𝑖
2
2
Constant term
Sinusoid whose
1
amplitude = 𝑉𝑚 𝐼𝑚 .
2
Average Power

The constant term in our previous
equation is the average power. It is
measured in watts.
1
1
𝑝 𝑡 = 𝑉𝑚 𝐼𝑚 cos 𝜃𝑣 − 𝜃𝑖 + 𝑉𝑚 𝐼𝑚 cos 2𝜔𝑡 + 𝜃𝑣 + 𝜃𝑖
2
2
Average power, P

So, 𝑃 =
1
𝑉𝑚 𝐼𝑚
2
cos 𝜃𝑣 − 𝜃𝑖 .
Average Power is Real, Not
Complex



We have 𝑃 =
1
𝑉 𝐼
2 𝑚 𝑚
cos 𝜃𝑣 − 𝜃𝑖 .
Note that everything on the right-hand
side of this equation is real, not
complex.
Therefore, average power P always has
a real value, not a complex value.

So, for example, it would never be correct to
write something like
P = 4+j7 W
or
P = 830 W
Power Factor

We have 𝑃 =
1
𝑉 𝐼
2 𝑚 𝑚
cos 𝜃𝑣 − 𝜃𝑖 .
Power factor



We call cos 𝜃𝑣 − 𝜃𝑖 the power factor.
The symbol for power factor is pf. Its
value is just a number, with no units.
For any given network, pf is a constant
between 0 and 1, so P is a constant
1
between 0 and 𝑉𝑚 𝐼𝑚 .
2
Special Case #1: Resistors
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

Recall that for a
resistor or a resistive
network, current and
voltage are in phase
with each other:
𝜃𝑖 = 𝜃𝑣
So the power factor is 1: cos 𝜃𝑣 − 𝜃𝑖 = 1
1
𝑉 𝐼
2 𝑚 𝑚
And average power 𝑃 =
simplifies to
1
𝑃 = 𝑉𝑚 𝐼𝑚
2
cos 𝜃𝑣 − 𝜃𝑖
Other Average-Power Formulas for
Resistors


We’ve just seen that, for a resistor,
1
𝑃 = 𝑉𝑚 𝐼𝑚
2
By combining this with Ohm’s law, we
can easily derive two other useful
formulas for the average power
dissipated by a resistor:
𝑃=
1 2
𝐼𝑚 𝑅
2
and
𝑃=
2
1 𝑉𝑚
2 𝑅
Summary for Resistors

Compare the following formulas for computing
a resistor’s power in a DC circuit and
computing a resistor’s average power in a
sinusoidal AC circuit:
DC
𝑝 = 𝑣𝑖
𝑝 = 𝑖2𝑅
𝑣2
𝑝=
𝑅
AC
1
𝑃 = 𝑉𝑚 𝐼𝑚
2
1 2
𝑃 = 𝐼𝑚 𝑅
2
1 𝑉𝑚2
𝑃=
2 𝑅
Special Case #2: Inductors



Recall that for an
inductor or an
inductive network,
current lags voltage
by 90: 𝜃𝑖 = 𝜃𝑣 − 90°
So the power factor is 0: cos 𝜃𝑣 − 𝜃𝑖 = 0
And average power 𝑃 =
simplifies to
𝑃=0
1
𝑉𝑚 𝐼𝑚
2
cos 𝜃𝑣 − 𝜃𝑖
Special Case #3: Capacitors



Recall that for a
capacitor or a
capacitive network,
current leads voltage
by 90: 𝜃𝑖 = 𝜃𝑣 + 90°
So the power factor is 0: cos 𝜃𝑣 − 𝜃𝑖 = 0
And average power 𝑃 =
simplifies to
𝑃=0
1
𝑉𝑚 𝐼𝑚
2
cos 𝜃𝑣 − 𝜃𝑖
The General Case

We’ve looked at three special cases:




Resistors: pf=1 and 𝑃 =
1
𝑉 𝐼 .
2 𝑚 𝑚
Inductors: pf=0 and P=0.
Capacitors: pf=0 and P=0.
For other components (such as
sources), pf is a number between 0 and
1, and the formula for P does not
simplify as it did in the special cases.
We’re left with:
1
𝑃 = 𝑉𝑚 𝐼𝑚 cos 𝜃𝑣 − 𝜃𝑖
2
The General Case


In a general circuit containing sources,
resistors, capacitors, and inductors, only
the sources and the resistors have nonzero average power.
1
𝑉 𝐼
2 𝑚 𝑚
The general formula 𝑃 =
cos 𝜃𝑣 − 𝜃𝑖
applies to each element, but for the
capacitors and inductors it simplifies to 0,
and for the resistors it simplifies to
1
1 2
𝑃 = 𝑉𝑚 𝐼𝑚 = 𝐼𝑚 𝑅
2
2
Review: Maximizing the Load
Power


In many applications, we wish to
maximize the power transferred from a
source to a load.
Replacing the source with its Theveninequivalent circuit, we have the following
situation:
Thevenin-equivalent
of source
Variable load
resistance
Review: Maximum Power Transfer
Theorem

For DC resistive circuits, the
maximum power
transfer theorem
says that maximum
power is transferred
to a load when the
load resistance equals
the source’s Thevenin
resistance (RL = RTh).
What About for AC Circuits?

For AC circuits we have a similar
situation, except instead of a
Thevenin-equivalent resistance RTh and
a load resistance RL, we have a
Thevenin-equivalent impedance ZTh
and a load impedance ZL.
Maximum Average Power Transfer
Theorem for AC Circuits


The maximum average power transfer
theorem says that
maximum average power
is transferred to a
load when the load
impedance equals the
complex conjugate of the source’s
Thevenin impedance:
𝐙𝐿 = 𝐙 ∗𝑇ℎ
Also, 𝑃𝑚𝑎𝑥 =
part of 𝐙 𝑇ℎ .
𝐕𝑇ℎ 2
,
8𝑅𝑇ℎ
where 𝑅𝑇ℎ is the real
Different Ways to Give AC Values

We’ve seen two ways to specify the
size of an AC current or voltage:




Peak-to-peak value.
Peak (or maximum) value, also called
the amplitude.
A third common way is called the
effective value (or rms value).
To understand this third way, we
have to think about how to compare
AC and DC.
Comparing DC and AC—Like
Apples and Oranges?
Does it make any sense to try to
compare a DC current (or voltage) to
an AC current(or voltage)?
Example, which current shown below
is greater?

Current (mA)

250
200
150
100
50
0
0
2
4
6
8
10
Time (ms)

Well, at some times the DC current is
greater and at other times the AC is
greater.
Can We Compare DC and AC?


DC currents and AC currents are very
different, but we can still draw some
comparisons between them.
For example: if an AC
current flows through
a resistor and a DC
current flows through
a resistor of the same
size, each current will
deliver power to its
resistor.
The Idea Behind Effective Values


For a given AC current, can we say what
size DC current would deliver the same
power to a resistor as the average
power delivered by our AC current?
Example: Suppose that when an AC current
with peak value 2 A flows through R, the
average power is 1 W. What size DC
current would give the same power for a
resistor of the same size?
i(t) = 2 cos t A
I=?
P=1W
P=1W
Effective Value of an AC Current

So, by definition, an AC current’s
effective value is the DC current
that delivers the same power to a
resistor as the AC current delivers.
Effective Value of an AC Voltage

An AC voltage’s effective value is
defined in the same way. An AC
voltage’s effective value is the DC
voltage that delivers the same power
to a resistor as the AC voltage
delivers.
Root-mean-square



We’ve defined what we mean by
effective value, but how can we
compute effective values?
Answer: to compute an AC current’s
effective value, take the square root
of the mean (average) of its square:
1 𝑇2
𝐼𝑒𝑓𝑓 =
𝑖 𝑑𝑡
𝑇 0
Effective values are also called rms
(root-mean-square) values.
Root-mean-square for Sinusoids

For a sinusoidal current,
𝑖 𝑡 = 𝐼𝑚 cos 𝜔𝑡 + 𝜃𝑖 , taking the rootmean-square is equivalent to dividing
the current’s amplitude by 2:
𝐼𝑒𝑓𝑓 =

𝐼𝑚
2
≈ 0.707𝐼𝑚
Similarly for sinusoidal voltages. If
𝑣 𝑡 = 𝑉𝑚 cos 𝜔𝑡 + 𝜃𝑣 , then
𝑉𝑒𝑓𝑓 =
𝑉𝑚
2
≈ 0.707𝑉𝑚
Outlet Voltage in the USA

The voltage at wall outlets in the USA
is a sinusoid, with a frequency of
60 Hz and a voltage of 120 V rms.
Some AC Sources in Multisim

AC Voltage Source, RMS Value
Group=Sources > Family=POWER_SOURCES >
AC_POWER

AC Voltage Source, Peak Value
Group=Sources >
Family=SIGNAL_VOLTAGE_SOURCES > AC_VOLTAGE

AC Current Source, Peak Value
Group=Sources >
Family=SIGNAL_CURRENT_SOURCES > AC_CURRENT
DC Versus AC on Multimeter

Most digital multimeters can measure
DC voltage, DC current, AC voltage,
AC current.
DC Voltage
DC Current
AC Voltage
AC Current
Fluke 45
Voltage
DC or AC?
Current
Fluke 8050A
DC or AC?


When a multimeter is set to measure
DC voltage or current, it actually
displays the average value of the
voltage or current.
When a multimeter is set to measure
AC voltage or current, it actually
displays the rms (or effective)
value of the voltage or current.
AC Instruments in Multisim

By default, voltmeters and ammeters in
Multisim are set to DC mode. You can
change a meter to AC mode by doubleclicking it and selecting AC.
AC Instruments in Multisim

Multisim also has several oscilloscopes.
One of these is modeled on the Tektronix
oscilloscopes that we have in our lab.