Welcome back to Linear Circuits, where, today, we're doing
Part 2 of our Power Factors and Power Triangles lecture.
In the previous lesson, we talked about calculating Complex Power
and identifying what the various aspects of that complex power represented.
Today, we're going to continue on talking about the
power factors and power triangles, most, particularly, about power triangles.
And as we understand those things, we'll then be able to talk more about maximum
power transfer in AC systems.
The objectives of this lesson are to have you use power triangles and to calculate
power angle and power factor, real and
Reactive Power, and Apparent Power for a system.
When calculating the Complex Power we plotted it on the complex plane.
Where we had the real part here, and the imaginary part here.
And we label all of the various values.
What we're going to do is we're going
to take the triangle out of this complex plane,
and draw it like this.
We see that we have all the same basic measurements.
We have the P, the Q, and the absolute value of S, corresponding to
Real Power, Reactive Power, and the Parent
Power, and our angle theta, our power angle.
So to remind us of the equation we use for finding the
Complex Power, I've listed it here and I've put the power triangle.
And, what we're going to do is find out
how to calculate all these things from the equations.
So, here we have the voltage equation, and the current equation would be m cosine
of omega t, plus theta v, and Im cosine of omega t plus theta i.
Now we'll go through, one by one, all of the different
important things that we're going to be defining for these power triangles.
First of all, the power angle.
Power angle theta, is the single right here.
And
that corresponds to theta v minus theta i, which we
can pull immediately from our equations for voltage and current.
We're going to call the cosine of theta the power factor.
And the reason we'll say that is that we can calculate the length of P, given we
know, theta, the Apparent Power, absolute value of S,
as P is equal to modulo S, times cosine
of theta.
And so that's why we called it the power factor because it's how much
we're multiplying by our Apparent Power to
find the average power that's being consumed.
So that means that we can calculate Average Power by taking the RMS voltage.
Multiplying it by the RMS current and multiplying that by cosine theta.
And this just follows immediately.
It consists of our calculation of how we found the Apparent Power.
Then for Reactive Power, Q, we want to find the length of this leg.
And using, again, basic trigonometry, Q is going to be
equal to the Apparent Power times the sine of theta.
Now,
here's where it gets a little bit hairy as far as units are concerned.
Average power is going to be measured in watts, because we're doing work.
And watts is the rate at which energy is being used.
But, when we're talking about Reactive Power, it corresponds to power
that's temporarily being stored up to be used at a later time.
And it's not actually doing any real work.
So to help distinguish it, we're going to be using a different unit.
The AC units are Volt-Amperes Reactive.
Sometimes distinguished as V-A-R-S, or VARS.
It's basically measuring the same thing because we're taking a voltage.
Multiplying it by a current and
then multiplying it by something that's unitless.
which is basically the same thing we have for power.
And so we're just going to call
it something different to help distinguish that
it's not really being used, it's just a temporary storage type of a thing.
And Apparent Power we're again going to use a different unit.
We're going to use Volt-Amperes or VA, and we've
already discussed how to calculate that Apparent Power.
So now we see that we can calculate all these different values, from the,
the voltage and the current equations.
And that they have a relationship that corresponds to
the trigonometric relationship that we see in a triangle.
Now we can look at the relationships between the Apparent Power, the Real
Power, and the Complex Power, or
the, Reactive Power, rather for different devices.
So, first of all, the resistive case, we've already looked at.
The voltage and the currents multiply together
to get something that is non-negative, for power.
Here in the green. And I'm still using the same v is
red, i is blue and green is P. Again, all different units here.
I, here, illustrate the Impedance of this device because it's all real.
It's just a real line out here. Now, this is a triangle, in, the
generative sense, where this phase angle, this angle here in this corner is 0.
And we all
see that the, the Impedance triangles are similar,
in the mathematical sense of similar, to the Complex Power triangles.
All of my power, all of my Parent Power is
Real Power and there is no Reactive Power for a resistor.
Now, as I move this around.
Get more of a proper triangle. Now I have a slightly inductive load,
because I have a resistor, I have an inductor.
Now, you'll notice that, on these plots, my voltages stay the same.
I'm moving currents with respect to the voltages
and then, obviously, the powers are changing as well.
No longer are the voltage and the current in phase.
Now, the voltage, leads the current, a little
bit, or the current lags behind the voltage.
This is my Impedance triangle
for this system, where the real part comes from the resistor.
The imaginary part comes from the inductor.
So the real part and the imaginary part are giving us this.
And this theta, corresponds to this data because these are two similar triangles.
For systems that have, a, an inductive load, my phase
angle is going to be between 0 and pi halves for
my power angle.
And my power is going to be somewhere between the
extremes of 0 and the value of the Apparent Power.
But so is my react, my Reactive Power,
somewhere between 0 and the the Apparent Power.
We're going to say that the system is lagging, because it's inductive.
And we say that it lags, or it's lagging because the current
lags behind the voltage, the voltage hits its maximum point before the current.
We'll go to the extreme case where it is a pure inductor.
Now my triangles are again degenerative because
my angle, my power angles are pi halves.
There's no Real Power, real average power being consumed.
All of the Apparent Power goes to Reactive Power.
I can do the same thing with capacitive loads.
Like this.
Now, we see that the current leads the voltage, so it's going to be leading.
This is my Impedance triangle, which is similar to my power triangle.
My theta, my power angle, is between minus pi halves and 0.
My power is between 0 and the Apparent Power, and
my Reactive Power is now going to be negative because it's
down here in this quadrant.
And it's going to be between the minus value of the Apparent Power and the 0.
We can see all of the relationships here.
And finally, looking at the purely capacitive case,
all of the power is Reactive Power because
the end of the capacitor stores up the
energy temporarily to be used at a later time.
What are the implications of this?
Well, first of all, only Real Power is being transformed as
heat or light or moving a motor or doing real work.
And, typically that's what we're most interested in.
We don't really care about how much power is being temporarily stored up in
the system, we want to know how much is being used to do real work.
But that doesn't mean we can neglect the Reactive Power.
Reactive power causes increased currents.
And so that means that the more Reactive Power
we have in our system, the more currents need
to be used in the lines connecting to your device.
And so you
[INAUDIBLE],
have more loss in the transmission lines, leading to that device.
So if you're a power company, you'll want to give some heat to that Reactive Power.
Now, private customers, for their private homes and residences, typically,
only are charged for the Real Power that they consume.
The power company doesn't really care too much about how much Reactive
Power there is because a resi, residential home doesn't have a lot.
But if you're a big
industrial system, with big motors, there's a lot
of inductive loads in those types of systems.
And so consequently industrial consumers, are often charged at a lower rate
for the Reactive Power because the power companies needs to have more equipment.
They have more losses in their transmission
lines and they need to design their
systems with extra transformers and extra devices,
to be able to handle those increased loads.
So,
[INAUDIBLE]
because of this, big industrial users might want
to give some consideration to their systems and
change their systems a little bit to reduce
the amount of Reactive Power that they have.
To reduce the costs that they pay to the power company.
In summary, we've defined power angle and power
factory, factor, Real and Reactive Power, and Apparent Power.
And showed how they all correlate in a triangle behavior,
using power triangles, we can see
the relationship between all those different values.
In the next lesson, we'll see how we can use reactive elements like capacitors
and inductors to control the amount of Reactive Power that we have in the system.
And we can use, or we can analyze maximum power transfer for AC systems, and look at
how that is different from the maximum power
transfer that was calculated in DC systems, until then.