# Activity: Generators and transformers

#### Aims

- Understand how an alternating current is induced in a rotating coil.
- Derive an equation for the EMF in a rotating coil.
- Understand why AC power = I
_{rms}V_{rms}_{ }. - Explain the operation of a transformer.
- Solve ideal transformer problems.
- understand why transformers are used in power transmission.

#### The rotating coil

In this section we will be considering a rectangular coil rotating in a uniform magnetic field, this is best viewed in 3D.

Let us consider the coil in the position shown below.

- Use Flemings RHR to deduce that the current flows from Q -P on side A and S -R on side B.
- How much flux is enclosed by the coil at this position?

If the coil moves a small amount the flux enclosed will change from zero to some small value. This is a large rate of change.

- What can you deduce about the EMF induced in this position?

1/4 of a turn later

- Is any current induced in this position? Try using Flemings RHR.

In this position the flux enclosed is maximum, if the coil rotates by a small angle the change is minimal.

- What can you deduce about the EMF in this position?

Another 1/4 turn

- Use Flemings RHR to show that the the current flows from P - Q on side A and R - S on side B.

Note that the current has now change direction in the coil,1/2 revolution before it was flowing from Q - P.

In the simulation below you can see how the EMF and current are related to coil position.

Observe the positions of the coil for maximum EMF. Remember that Faraday's law says EMF is equal to the "rate of change flux enclosed" not the "flux enclosed", examiners like to catch students out with questions about this.

You can see from the simulation above that the EMF is sinusoidal, we will now show why by deriving the equation for EMF.

To define the position of coil we use will use the angle between the normal to the plane of the coil and the field.

If the angular velocity is ω and the area of the coil A

- Write an expression for the flux enclosed by the coil at angle θ
- Write and expression for the angle θ in terms of ω and t
- Substitute for angle θ in the first equation.

This is the flux enclosed by the coil (φ), but EMF = -dφ/dt

- Show that EMF = BAωsinωt (If the coil has N turns EMF = BANωsinωt)

#### AC generator

To turn the rotating coil into a generator you need to connect it to a circuit. This can be done with slip rings as shown in this simulation. Walter Fendt generator

- Observe the effect of increasing the frequency.

According to Lenz's law the current induced will oppose the change producing it.

- Use Flemmings LHR on the induced current to show that it will cause a force opposing the motion.

To keep the coil moving at constant speed we must apply a balancing force (Newt 1)

- Will work be done by this force?
- What happens to the energy transferred?
- If the circuit was broken so no current could flow what would happen to the motion of the coil?
- If the circuit had no resistance what would happen when you started moving the coil?

#### The electric motor

If a current is passes through a coil placed in a magnetic field then the coil will experience a torque causing it to rotate 1/4 of a revolution

- Use Flemings LHR to deduce the direction of rotation of the coil.

To create continuous rotation the direction of the current would need to be changed, this is done using a commutator as can be seen in this animation Walter Fendt motor.

(note: Walter Fendt uses a different RHR I use the LHR for motors)

You can see that the motor is the same as a generator a fact that is used to increase the efficiency of electric cars. To slow the car down the motor is used as a generator to charge the battery.

Here is a blog post I made some years ago.

#### Alternating current

The output of the rotating coil is alternating current (AC) this means it has changing direction and magnitude, in the case of a rotating coil the signal is also sinusoidal. Below is a simulation showing a sinusoidaly changing Voltage.

Consider this signal connected to a resistance, R.

- Electrical power = V
^{2}/R. Will the power dissipated in a resistor be constant? - If the peak value of V is V
_{o}what is the maximum power generated? - What is the average value of V
^{2}? - Show that average power dissipated is

This is the same as

V_{rms} is the root mean square voltage (the root of the mean of the squares) this is the voltage that would give the same power dissipation as an equal DC voltage.

- Calculate the power dissipated when an AC signal of 5 V peak is connected to a 10 Ω resistor.
- What DC signal would give the same power dissipation?
- Why would a 5V DC signal result in more power?

Exercises 62 - 65 page 256

#### The Transformer

A transformer consists of two coils wrapped around the same iron core. Have a look at the demonstration transformer in the lab. See what happens when you switch it on and off. Note that the neon light bulb needs about 50 V to light up.

Fill in the blanks to get an explanation of how a transformer works

flux current magnetic EMF Faraday's secondary

When the switch is closed flows in the primary.

This induces a field in the core.

According to law the increasing field in the will induce an EMF.

The size of this EMF is proportional to the rate of change of .

So, since the increase in flux is very rapid the induced is enough to light the neon.

- Why doesn't the bulb stay lit after the switch is closed?
- An electric fence can be powered by a 12 V car battery. How do you think a high voltage is produced.

The reason such a large EMF was induced in the demonstration was because the rate of change of flux was so big, If a sinusoidal signal is connected then the PD across the secondary can be calculated using the formula.

If the transformer is 100% efficient we also have

- Explain where the second equation comes from.

Exercise 66 page 256

#### Power Losses

Explain why the following lead to power loss and identify where the energy goes.

- Joule heating in the coils
- eddy currents are induced in the iron core
- flux leaks out of the core and passes through nearby metal objects.
- it requires energy to change the direction of magnetization of the core.

#### Power transmission

When electrical energy is transmitted through cables energy is lost. From the equation I^{2}R we can see that the power loss is proportional to the current squared so reducing the current will reduce power loss. This can be done by stepping up the voltage. This simulation does the calculations and can be used to check the answers to the following questions.

Consider a power station producing 100 MW of power at 30 kV, calculate:

- The current from the power station.
- The current flowing through the cables if the Voltage is stepped up to 115 kV.
- The power loss in the cables if they have resistance = 2.1 Ω
- The power delivered to the town.
- The % power loss

Compare this value with a transmission voltage of 300 kV.

Exercise 67 page 258

If you want to know more about power transmission try Transformer simulation.