Transformers

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Electrical energy is transmitted at high voltage. Which element creates the 120 V that is available in our wall plugs ant home?

Big Ideas: 
  • Transformers are an essential part of any country's electricity distribution system
  • Operating on the principles of electromagnetism, they step up or down the voltage

In this article we’re going to talk about transformers. (Not the robot kind, the electrical kind.) What's so interesting about them and why do we want to learn about them?

There are two main reasons:
1. Transformers have shaped human history. In the late 1800s transformers played a key role in determining whether we would have DC or AC electricity as the national standard for electricity distribution 1
2. In modern times, transformers are an essential part of our electricity distribution system. In order to understand where our electricity comes from, we need to understand transformers.

The key thing that transformers do is they let you change high voltage electrical energy to low voltage or vice versa. This is very important because as we saw in earlier lectures it is extremely inefficient to transmit electricity long distances at low voltages (See article: Transmitting Electricity). In order to make sure that most of the electricity generated makes its way to the customers we need transformers to step the voltage up to high voltage for transmission and then back down again once it gets to our houses.

Here are some examples of different transformers in our everyday world:

Small transformers used in computers
Large transformers on a neighborhood power pole A high voltage transformer in an electrical substation

So what do these three all have in common? For that we want to look inside the transformer :

Image source: http://en.wikipedia.org/wiki/File:Transformer3d_col3.svg

A transformer just consists of an iron core with two coils of wire wrapped around it. These are called the Primary and Secondary coil. By building a different number of turns of wire on each side, the transformer can increase the voltage at the cost of decreased current.

The ratio of Secondary Voltage to Primary Voltage is related to the ratio of the number of turns on each coil, and for a perfectly efficient transformer is given by:

$ \dfrac{{V}_{s}}{{V}_{p}}=\dfrac{{N}_{s}}{{N}_{p}} $

By varying Ns and Np we can create nearly any voltage ratio we want! However, this ONLY works for AC voltage. To understand why that is, we need to figure out what it happening inside that transformer.

The Primary Coil

The primary coil is basically just a coil of wire wrapped around an iron core. This is the essential recipe for an electromagnet2, and in fact the primary coil functions as exactly that. When we run a current through the primary coil, it acts just like a regular magnet.

We can think of the primary coil as creating a magnetic field in the transformer. The direction of the magnetic field lines shows the direction that the magnetic field would push another north magnetic pole, and the density of the magnetic field lines shows the strength of the magnetic field at that location.

The left side of the diagram below shows the magnetic field of a bar magnet. We can see that near the poles (where the magnetic force is strongest) the magnetic field lines are closest together and that the north pole would push another north away from itself. We can also see that the strength of the magnetic field gets weaker when we are far away from the magnet. (You already knew those things, but it is good to review)

Image Source: http://commons.wikimedia.org

The right side shows the magnetic field of an electromagnet, which is identical to a bar magnet.

So if we imagine the primary coil acts as a magnet, then we have a familiar situation: a magnet and a coil of wire are the key ingredients for generating power. However in our earlier discussion we found that we needed motion of the magnet or coil in order to generate power. In a transformer it seems that we can generate power in the secondary coil without moving either!

In order to understand how this is possible we need to expand our earlier understanding of power generation by introducing a new idea: magnetic flux.

Magnetic Flux

Magnetic Flux is basically just the “amount” of the magnetic field that goes through a particular area (such as a loop of wire). Qualitatively speaking, you can imagine calculating flux just by counting the number of magnetic field lines that pierce the loop. However, you also need to look at direction. Field lines in opposite directions tend to cancel. As an example, take a look at the question below:

Which of the following situations has the highest flux through the loop?

We know that the field lines "flow" out of the North pole, so at the location of the loop, the magnetic field is pointing mainly to the right in situations A, B, and D. Since loop D is smaller than A, fewer field lines go through the loop, so the flux in D is less than in A. In B, the fields of two magnets combine. Since the two North poles are near each other, both fields have the same direction and add to the total flux. So the total field and the magnetic flux in B must be twice as large as in A.   
The loop in Setup D is much smaller than the loops in, B or C and thus the flux through the loop will be lower than the other setups. What about situation C? Take a look at the field of a bar magnet again and imagine the loop on top, as shown in C. The field lines are going up through the loop near the North pole and and come back down near the South pole. The number of field lines going up is the same as the number coming back down. Therefore, the magnetic flux (which "measures" the net number of field lines) is zero in C. Thus B has the highest flux through the loop.

The formula for Flux is given by:

 

 \Phi = B\cdot{A}\cdot\cos{\theta}

For this formula, $ \theta $ is measured between the B field and the normal to the coil, so $ \theta $ is zero when the field is going directly through the coil.

Keep in mind this is for a constant B field. A bar magnet doesn’t have any areas where the field is constant, but when we think about the area nearest the poles this formula still gives us the right idea.

Understanding Power Generation with Flux

Rather than thinking about the magnetic forces on the charges, we can just look at the flux through the coil:

Whenever we change the flux of the magnetic field through the coil, we generate a voltage.

Keep in mind the flux does NOT need to increase; only to change from its current value. However increasing the flux will generate one voltage, and decreasing the flux will generate the same voltage in the other direction (For more details, see Article on Faraday's Law).

This is a rather abstract concept. Let’s try to relate it back to something we’ve already discussed: generating power with moving coils.

In this example, the flux through this coil is constant. As the coil moves to the right the field strength, angle, and area all stay constant, and the flux doesn’t change. Therefore, there is no voltage generated around this loop, which matches with our earlier result.
For this situation, the loop is moving from a region of no field to a region of uniform field. As it moves, the amount of field going through the loop is increasing, which means the flux is increasing. Changing flux means there IS a voltage generated around this loop, which matches with our earlier result

In this case the loop is moving from a region of field OUT of the page to a region of field INTO the page. Here it is important to note that the direction of the field is important when calculating flux. We get to choose which side of the loop is the positive side for the purposes of calculating $ \theta $, but once we make a choice we need to stick with it. Therefore, if the second field has $ \theta $ = 0˚, then the first field will have $ \theta $= 180˚.

This means the flux will go from a negative value to a positive value, and the flux will indeed be changing as we move from one region to the next. It is also important to note that this change will be larger than the previous case: going from negative flux to positive flux is a bigger jump than going from zero flux to positive flux (assuming the B field is the same strength.)

Changing flux means there IS a voltage generated around this loop, which matches with our earlier result.

Rate of Change Flux

Does it matter how quickly we change the flux? Yes! We could have guessed this from our earlier knowledge that the force on an electron moving through a magnetic field depends on the electron’s velocity. It matters how quickly you move.

The formula for Voltage generated by changing magnetic flux is:

{V} =\dfrac{-\Delta\Phi}{\Delta{t}} = \dfrac{-\Delta({B}\cdot{A}\cdot\cos{\theta})}{\Delta{t}}(*\*)

If there are N turns in a coil, the flux creates a voltage in each of them an they add in series, so equation (*) becomes:

{V} =-N\dfrac{\Delta\Phi}{\Delta{t}} =-N\dfrac{\Delta({B}\cdot{A}\cdot\cos{\theta})}{\Delta{t}}(*\**\*)

Application to Transformers

So this new idea of flux gives us the same results as our earlier understanding of moving charges and fields. However, now we can make sense of situations where the flux changes without anything moving, such as in a transformer.

By varying the input voltage to the primary coil, we can create a changing magnetic field. When the secondary coil is exposed to this changing magnetic field, it has a changing flux in it which induces a voltage in the secondary coil.

The iron core channels almost all of the flux that is generated by the primary coil into the secondary coil, and so prevents loss of energy. For a basic transformer we often assume that ALL of the flux is channeled into the secondary.

Image Source: http://en.wikipedia.org/wiki/File:Transformer3d_col3.svg

To see how the transformer changes the voltage up and down, we just use the induced voltage equation (**) for both coils.

{V}_{p} = -{N}_{p}\cdot\dfrac{\Delta\Phi}{\Delta{t}}

{V}_{s} = -{N}_{s}\cdot\dfrac{\Delta\Phi}{\Delta{t}}

If we assume the core channels all of the magnetic flux, then $ \dfrac{\Delta\Phi}{\Delta\textnormal{t}} $ is identical for both coils. This gives us:

$ \dfrac{{V}_{p}}{{N}_{p}}=\dfrac{{V}_{s}}{{N}_{s}} $

or

$ \dfrac{{V}_{s}}{{V}_{p}}=\dfrac{{N}_{s}}{{N}_{p}} $

By varying the number of windings on the primary and secondary coil we can easily step the voltage up or down to any value.

Why we need AC power for transformers

We can also see the reason that transformers only work with AC voltage: a voltage is only generated in the secondary coil when the magnetic flux is changing. This means that we need to be constantly changing the voltage in the primary coil. An AC voltage does exactly that: the voltage oscillates between a positive and negative value, and is always changing.

The main reason that we use AC voltage in our homes today is that it enables transformer technology, which allows us to send electricity long distances with low loss.

Now we have a more complete picture of where our electricity comes from. It looks something like this:

Source: USA National Energy Development Project (public domain)

For a more detailed description of the voltages at the different stages shown above, see the article on Heating Efficiency.

In subsequent lectures we will talk about the real-life efficiency of this process.

Click here for simulations that help you better understand magnetic electromagnetism

 

 

 

 

 

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