Welcome back. I'm Nathan Parrish.
And again, this is linear circuits. Today, we're going to be talking about
power and energy. So first of all, we're going to be trying
to calculate power and energy. Describing the difference between power
and energy. Using the conservation of energy to find
an unknown energy. And using power to calculate an unknown
current or unknown voltage. So from our previous class, we talked
about voltage, this kind of this electric potential based upon the distribution of
charge inside of a device. And then we looked at the battery.
And how it could be charging or discharging, based upon the direction of
the current flow. But in the same context, now we can start
talking about power and energy. And, in the next class, we will look at
some basic circuit diagrams. But actually today, we're going to be
doing our first bit of circuit analysis. Lesson objectives are to calculate power
from an energy function,and to calculate energy from the power function.
To use the conservation of energy to find a power of an unknown device, calculate
power from voltage and current, and finally to find a voltage or a current
for a device with an unknown, for a known power.
So in talking with, about power, power is measured in units of watts and power is
essentially the rate at which energy is being consumed.
Energy being measured in joules. So power is in joules per second.
And one watt is equal to one joule per second.
Now, we can actually use a little bit of calculus to help us do some calculation
about power. Power is going to be the change in energy
with respect to time. Or we can use the chain rule, this sort
of, the change in energy with respect to charge times the change in charge with
respect to time, is going to be equivalent.
And so, if we have energy divided by charge, you might remember from last
class, that's voltage. And if I have charge with respect to
time, that's going to be current. So it turns out that we can calculate
electric power by taking a voltage and multiplying it by a current.
We can also find an energy by integrating the power with respect to time, since
power is the rate at, at which energy is being consumed or generated.
So all we have to do is integrate with respect to time, and then we can find our
energy. In calculations, we're going to be using
w to represent energy. And we're going to use p to represent
power. One of the things that students sometimes
forget that's actually a very useful tool when you're doing analysis, is
remembering that energy cannot instantaneously change.
The power can, it's not a problem. So we can look at this example.
Here we have an energy curve, where the energy is being consumed, the energy
decreases and decreases and decreases until this point, where there's no energy
left. And now, it is regenerated.
And remembering that we can calculate power by taking the derivative of energy.
But we know that this derivative is basically the slope of the slide and so
we are basically just going down one. So oue power is negative 1 watt.
Over here the slope is now positive 1. So we have 1 watt.
At this point in the middle, it's kind of a mathematical problem.
There is no derivative here because if we come from one side or the other, we get
two different answers. And so it's kind of undefined what the
derivative happens to to be here. But this only happens at one isolated
point, and so it really doesn't matter, as far as we're concerned.
So here we just used some circles to identify that we don't have a defined
amount of power. Now if we take an integral, of the power,
we get the energy. Now suppose that we wanted this energy,
to change, instantaneously in time. Or just do jumps.
What would have to happen? Well, over a tiny little slice, if we
remember the definition of an integral, we have to have an inordinately large
amount of power. So large, in fact, that as we make the
area around it smaller and smaller, the value has to overcome that ever
decreasing window of space. Which basically means that we have to
have infinite power. Well infinite power is not something that
we can have. So it turns out that we cannot have the
energy change instantaneously. But power can change instantaneously as
much as it likes. Now suppose that you run a power company
and you want to charge your customers. Well, how are you going to charge them?
Are you going to charge them by the amount of power that they are using or
the amount of energy? And why?
I want you to think about it for a minute.
In fact, I'm going to even stick up a little sign.
A little pause, to let you know that it's okay to pause the video and take a little
bit of time to think about it before you answer.
So what did you come up with? It turns out, if you look at your power
bill, you're probably going to see something about your power company
charging you in units of kilowatt hours. And this is actually a measurement of
energy. It seems to be a little bit easier to
work with than joules, since a kilowatt hour is equal to about 360 joules.
And so it's just an easier number to work with.
But essentially, it's energy. And why energy?
Well that's because they don't really care about what rate you're using the
power. What they want to know is how what, or
how much energy you're actually consuming.
So it's important for them to keep track of that.
But we shouldn't lose complete identity of power because it could become a
problem there as well. Suppose that everybody in the city
simultaneously decided to run their vacuum cleaners and air conditioners all
at once. Well, the power company might have some
problems. They wouldn't be able to generate enough
power that instant to be able to provide for everybody at once.
Even though the energy wasn't too much for it to provide, it's the rate that
becomes a problem. So, in practice, your power company kind
of looks at both. But when you're charged, you're going to
be paying for energy. Another very useful property that we can
use, is something called the conservation of energy.
Just like most closed systems, the total amount of energy that is being provided
or consumed by the system, is a constant value, in time.
No energy is created or destroyed in the system.
But, we actually are going to find it more convenient, generally, to work with
power rather than energy. And so let's take a look at a little bit
of a derivation here. We sum up the total energy in a system.
We're going to get some constant value. Now what we can do then is take the time
derivative of this value and if I take a time derivative of some constant, that's
going to be equal to zero. And it turns out that as long as this sum
is over a finite number of energies, we can interchange the derivative and the
sum to give us this, where we're summing up the derivatives.
Well these derivatives are just powers, individual powers, and we're summing all
of these powers up and we then get zero. So what does this actually mean in
practice? Well, what it really means is that
because energy is conserved, we know that the rate of power which a system is
consuming or generating energy, has to be 0 overall, throughout the system.
So if we sum up all of the powers that are being consumed and all of the power
that's being generated in the system at any point in time, it always has to be 0.
So this is the excellent way to check your work.
It also turns out to be something useful on analysis, if we happen to know
something about the power in the system. Again reference directions are going to
come up. like I said, it's something that students
really have challenges with, especially at first, keeping the reference
directions straight. And because we're taking a product of a
current and a voltage, if we just naively multiply them together, we can get
different answers depending upon how they're related.
And as far as we're concerned, there's only two real configurations to consider.
The first configuration is where our current arrow is pointing from our plus
to our minus. And the other is when our current arrow
is going from our minus to our plus. So, then we'll start off by looking at
the one on the left, where the arrow goes from the plus to the minus.
We're going to calculate power by multiplying the current and the voltage.
What this ends up doing then, is it means that positive power means power's being
consumed, not generated. This might be counterintuitive to what
you might think. Thinking that negative would be that it's
being used and positive would be that it's being generated.
But we're actually going to have positive power being consumed power.
And that's the general trend that we're going to use.
And so consequently, if we flip the references, where the arrow's going the
other way, to keep the values consistent, we have to multiply by a negative 1.
So in this case, power is going to be equal to minus i times v, minus the
current times the voltage. So be very careful when you're
calculating power to look. Is my arrow going from the plus to the
minus? Just multiply them together.
Is it going from the minus to the plus? Remember to switch to the reference
directions so you're multiplying by negative one.
So now we're going to look at a practical, practical problem, and do an
analysis. Now suppose we have a light bulb.
And we have 120 va, volts, of voltage, going to the light bulb to power it.
And this particular light bulb is rated 60 watts.
Now suppose we want to know the current that is flowing through the light bulb.
Well at this point we have all the tools we need to be able to solve this problem.
So, I want you to take a look at the system and try and solve it yourself.
And so I'm going to go ahead and put up a pause button and then we will continue on
after you've had a chance to try to solve it yourself.
Okay, so we have 60 watts of power. And we notice that the arrow for current
is going from the plus to the minus. So in this configuration, power p is
equal to i times v, current times voltage.
So that means the power, 60 watts, is equal to 120 volts times i.
Where i is equal to 60 watts divided by 120 volts.
So the current is equal to one half of an m.
Now, if you were able to solve this problem, it might be a good thing to go
onto the forums and post, because you'll notice that we're not going to actually
show the answer to the solution here on the slide.
To summarize, we've describe the relationship between power and energy and
how to calculate them. We also looked at how voltage and current
interact to give us electric power, and in turn, electric energy.
We presented a derivation for the conservation of power, and how this
property can used in analysis. And we solved our very first simple
analysis problem. Next lesson, we will be taking our first
look at circuit diagrams and seeing how these things come together.
I remind you again, to take a look at the forms.
Answer questions that you know the answers to.
It's great practice, teaching others. As well as ask any questions that you
might have from this lesson. Until then, we will see you next time.
Cheers.