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4.4 Power Triangles

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Linear Circuits 2: AC Analysis

乔治亚理工学院

4.6（80 个评分） | 6.6K 名学生已注册

This course explains how to analyze circuits that have alternating current (AC) voltage or current sources. Circuits with resistors, capacitors, and inductors are covered, both analytically and experimentally. Some practical applications in sensors are demonstrated.

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4.6（80 个评分）

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AA

Jan 01, 2019

Really beneficial especially with covering Filters and Transformers

MR

Mar 29, 2018

The best course and understandable. Thanks dear teachers

从本节课中

Module 4: Power

The description goes here

4.1 Power Module Overview1:18

4.2 Root Mean Square8:29

4.3 Power Factor12:06

4.4 Power Triangles10:10

4.5 AC Power Transfer13:59

Sample Problem: AC Power 15:12

Sample Problem: AC Power 214:30

Sample Problem: AC Power 37:41

Sample Problem: AC Power 49:38

Sample Problem: AC Power 54:42

教学方

Dr. Bonnie H. Ferri
Professor
Dr. Joyelle Harris
Director of the Engineering for Social Innovation Center

脚本

Welcome back to linear circuits where today we're doing part

two of our power factors and power triangles lecture.

In the previous lecture 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, 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 for

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 we calculated the complex power, we plotted it on a complex plane.

Where we had the real part here and

the imaginary part here and we labeled all the various values.

What we are going to do is 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 the real power,

reactive power, and the parent power, and the hour angle beta.

All right, our power angle, so,

to remind us of the equation, we use proof 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

with the m cosine of omega T, plus state E 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, the 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 are going to call the cosign of theta the power factor.

The reason we are able to say that, is if you calculate the length of P,

given we know theta and the apparent power and absolute value of S, as P is equal to

modular s, cosign of theta, and so that's why we call 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,

because of our calculation of how we found the apparent power.

For reactive power, Q we want to find the length of this leg, and

you think again basic trigonometry.

It was going to be equal to the apparent power times the sine of beta.

Now, here's where it gets a little bit hairy as far as units are concerned.

How much power is going to be measured in watts because we are doing work, and

watts is the rate by which energy is being used.

But, when we are talking about reactive power it corresponds to

power that is temporarily being stored up to be used at a later time, and

it is not actually doing any real work.

So, to help distinguish it, we are going to be using a different unit.

The units are Volt-Amperes Reactive, sometimes distinguished as VARs, 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 thing.

And apparent power we're again going to use a different unit,

we're going use volt amperes or va and

we've all ready discussed how to calculate that apparent power.

So now we can see that we can calculate all these different values from

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, brother for

different devices.

So first of all, the resistance case, we've already looked at.

The voltage and the current multiply together to give something that is not

negative or power, they're in the green, and [INAUDIBLE] is in the same,

B 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 and

this is a triangle in a degenerative sense where this

phase angle, this angel here in this corner is zero.

And we all see that the impedance triangles

are similar in the mathematical sense similar to the complex power triangles.

All my power, all my apparent power is real power, and there is no reactive power

for a resister, 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 in 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 can be imaginary part giving us this and

this theta corresponds to this theta, because these are two similar triangles.

For systems that have an inductive load,

my phasing goal's going to be between 0 and pi 1/2 for my power angle.

And the power is going to be similar between the extremes of zero and

it would be value of the apparent power.

But so is my reactive power, somewhere between zero and the apparent power.

I'm going to say that the system is lagging, because it's inductive and

we say that it lags or is lagging because the current lags behind the voltage.

The voltage hits its maximum point before the current.

If I go to the extreme case where it is a pure inductor,

now my triangles are again degenerative because my power angles are pi halves.

There is no real power being consumed.

All 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, wich is similar to my power triangle.

My theta, my power angle, is between minus pi halves and zero.

My power is between zero, 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

zero, we can see all of the relationships here.

And finally, looking at the purely capacitive case,

all of the power is reactive power because again 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 the motor ro 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.

This slide 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 have more loss in the transmission lines leading to that device.

So if you're a power company, Bill Walton give it 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 the residential home doesn't have a lot.

But if your big industrial system with big motors,

there's a lot of inducting rules in those types of systems.

And so, consequently, industrial consumers are often charged at a lower rate for

the reactive power.

Because the power company needs to have more equipment to have more loses 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 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 defined power angle and power factor, real and

reactive power and apparent power.

And showed how they all correlate in a triangle behaviour using

power triangle we can see the relationship between all those different values.

In the next lesson will see how we can use reactive elements Like capacitors and

conductors to control the amount of reactive power that we have in the system.

And 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.

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