*Notes to a video lecture on http://www.unizor.com*

__Three Phases AC__

*The Basic Principle*

of 3-Phase AC Generation

of 3-Phase AC Generation

Recall the process of generating alternating current (AC) using a pair

of permanent magnets and a wire frame (a coil) rotating around the axis

perpendicular to magnetic field lines.

The pictures below represent the schematic design of such a system and a

graph of an electromotive force (EMF) generated between the ends

*a*and

_{1}*a*of the wire frame

_{2}

**E**_{a1a2}= E_{max}·sin(ωt)where

*E*is the maximum absolute value of EMF,

_{max}*ω*is the angular speed of rotation of the wire frame,

*t*is time.

An obvious improvement to this design is to use the power of rotation

more efficiently by having two wire frames on the same axis positioned

perpendicularly to each other. In this case we can have two independent

sources of EMF with the only difference of one of them to be shifted in

time relatively to another by 1/2 of the time of rotation.

This shift is related to a simple fact that at the moment one wire

frame, aligned along the magnetic line, crosses these magnetic lines

with the highest rate, while another wire frame, that is perpendicular

to magnetic field lines, moves along these lines without actual

crossing. Then the roles are changed, as the coils rotate.

The EMF between the ends

*b*and

_{1}*b*of the second wire frame will then be

_{2}

**E**_{b1b2}= E_{max}·sin(ωt−π/2)Why stop at two wire frames? Let's have three coils positioned at 120°

relative to each other. Now we will have three independent sources of

EMF shifted in time from each other by 1/3 of the time of rotation (

**phase**

shift) - the time needed by one wire frame to take the position between

the magnet poles, previously taken by another wire frame.

Three different EMF, therefore, will be equal to

**E**_{a1a2}= E_{max}·sin(ωt)

**E**_{b1b2}= E_{max}·sin(ωt−2π/3)

**E**_{c1c2}= E_{max}·sin(ωt−4π/3) = E_{max}·sin(ωt+2π/3)*Practical Implementation*

The design of a three phase generator, as depicted above, is just the

first try of an idea. If the magnet is fixed and three wire frames

(coils) are rotating between its poles, it presents a problem to connect

these coils to transmit the generated electricity to consumers, we need

sliding contacts, brushes and other impractical devices.

In real life generators the three coils make up a stator - a fixed part

of a generator, while the magnet is rotating inside a circle of coils by

external power (like steam, water, wind etc.), generating the

alternating current in the coils, which allows to make an electric

connection to coils fixed.

At the first glance, three coils have three pairs of connections with

sinusoidal EMF generated in each pair and, to transfer AC electricity

from all coils to consumers, we seem to need three pairs of wires, two

from each coil - six wires altogether. This, however, can be improved by

using the following technique.

Let's connect ends

*a*,

_{2}*b*and

_{2}*c*

_{2}of three coils together (see a picture below, it's a black wire at the

bottom connected to a black circle going around all coils) and see what

kind of resulting voltage will be observed on each end of the coils.

This is called a

**star**connection of the generator's coils.

We know that the electric potential (EMF) on each of the above contacts

has a sinusoidal magnitude with a time shift by 1/3 of a period relative

to each other. When we connect these three contacts, the potential at

the joint will be

**E**_{0}= E_{max}·sin(ωt) + E_{max}·sin(ωt−2π/3) + E_{max}·sin(ωt+2π/3) = E_{max}·Xwhere

**X = sin(ωt) + sin(ωt−2π/3) + sin(ωt+2π/3) =**

= sin(ωt) + sin(ωt)·cos(2π/3) − cos(ωt)·sin(2π/3) + sin(ωt)·cos(2π/3) + cos(ωt)·sin(2π/3) =

= sin(ωt) + sin(ωt)·(−1/2) − cos(ωt)·(√3/2) + sin(ωt)·(−1/2) +

cos(ωt)·(√3/2) = 0= sin(ωt) + sin(ωt)·cos(2π/3) − cos(ωt)·sin(2π/3) + sin(ωt)·cos(2π/3) + cos(ωt)·sin(2π/3) =

= sin(ωt) + sin(ωt)·(−1/2) − cos(ωt)·(√3/2) + sin(ωt)·(−1/2) +

cos(ωt)·(√3/2) = 0

Therefore,

**E**_{0}= 0There will be no difference in electrical potential between the joint and the ground.

This fact enables to transmit all three phases of generated electricity along four wires - one from

*a*(

_{1}**phase 1**) contact, one from

*b*(

_{1}**phase 2**) contact, one from

*c*(

_{1}**phase 3**) contact and one

**neutral**from a joint connection to

*a*,

_{2}*b*and

_{2}*c*.

_{2}The

**neutral**wire is usually grounded since its electric potential is equal to zero.

With this arrangement we still have an advantage of having three

independent phases of alternating current, but we need only four wires

to transmit it - three

**phase**wires and one

**neutral**.

Connecting any device to any phase and a neutral wires, we will get a closed circuit with AC running in it.

*Energy Consideration*

Obviously, putting two or three coils in a stator of a generator doubles

or triples the energy output carried by outgoing wires. The Law of

Energy Conservation must work, so where is the energy is coming from?

Recall the electromagnetic induction experiment described in the lecture

"Faraday's Law" in the "Electromagnetic Induction" chapter of this

course with a wire moving in the uniform magnetic field.

Since we physically move wire's electrons in one direction

perpendicularly to magnetic field lines, the Lorentz force pushes them

perpendicularly to both, the direction of the movement of a wire and the

direction of the magnetic field lines, that is, along the wire, thereby

creating an electric current between wire ends.

Now electrons are moving with a wire in one direction and along the wire in another.

The first movement maintains the electric current in the wire, but the

second, again, is a subject of the Lorentz force that pushes the

electrons perpendicularly to their direction, that is opposite to the

original direction of a wire movement.

This force resists the movement of a wire in its original direction. We

have to perform work against this force to move the wire.

Similar considerations are true in a case of a circular movement of a

wire frame in a magnetic field or, if wire coils are in a stator, the

force is needed to rotate the magnet in a rotor. That is, we have to

spend energy to generated the electricity, the rotor's rotation is

possible only if we apply the force against the Lorentz forces resisting

this rotation. The magnetic field generated by the electric current in a

wire coil of a stator resists the rotation of a magnet in a rotor.

If we have more than one coil in a stator, each one resists the rotation

of a rotor, so we have to spend proportionally more effort to rotate

the rotor.

The Law of Energy Conservation works. The more coils we have in a stator

- the more electricity is generated, but the more resistance to a

rotor's rotation needs to be overcome.

*Three Phase AC Motor*

The lecture "AC Motors" of this chapter described the necessity of having a rotating magnetic field to make an AC motor.

To achieve such a rotating field we had to resort to artificially create

a second AC current with a phase shift by 90° using a capacitor or a

transformer.

Most of household AC motors (like in a fan) work on this principle, they need only two wires, which are, as we can say now, a

**phase**wire and a

**neutral**one.

Powerful industrial level AC motors (like in a water pump that works in a

tall building to pump water to the roof tank) needs more power, and we

can use all three phases to create a rotating magnetic field.

So, all four wires coming from the AC generator, three

**phase**wires and one

**neutral**

one go into an AC motor, whose principal construction very much

resembles the one described in the previous lecture. The only difference

is, we already have three wires with AC phase shifted by 120°

relatively to each other. So, we have to position three wire coils in a

stator at 120° angles to each other, connect one end of each coil to a

corresponding

**phase**wire and another end - to a common

**neutral**

wire, and the rotating field is ready. Then it will work pretty much as

it was described in the "AC Motors" lecture, but smoother because three

phases make a smoother rotation of a magnetic field than two phases.

At the end I would like to say again, that it was Nicola Tesla's genius

that created all the basic principles, based on which all the current AC

motors are working now. His contributions to our industrial development

are grossly underappreciated. Calling an electric car model "Tesla" is a

late but well deserved tribute to his creativity.