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Fields & Effects

 

Electromagnetic Induction 2

 

self induction

mutual induction

transformer action

power

efficiency

 

 

 

Self Induction

 

Induced currents only occur when a magnetic field builds or collapses.

Lenz's law tells us that the induced current is such as to oppose the change producing it.

So the induced current will oppose the primary current when the field is building. Conversely, when the field is collapsing the current will be in the opposite direction to try to prevent the collapse.

In the first instance the induced current field will be in a sense to oppose the primary current building. In the second instance when the field is collapsing, it will be in the same direction to lessen the collapse..

 

 

self induction diagram #1

 

 

The induced current is produced by a 'back EMF' - an induced voltage proportional to minus the rate of primary current change.

 

self induction - equation #0

 

self induction equation #1

 

where L (the inductance) is the constant of proportionality.

 

 

Rearranging to make L the subject of the equation. By making E and dI/dt unity we define the unit of self induction, the henry:

 

definition of the henry

 

definition of the henry

 

 

A coil has a self inductance of 1 henry (H) if the back EMF is 1 volt for a current change of 1 ampere/second.

 

By equating the EMF from Neumann's equation with self induction EMF a simplified expression linking the two can be formed.

 

self induction equation #2

 

Integrating between the limits of φ - 0 and I - 0 ,

 

self induction equation #3

 

 

self induction equation #4

 

 

 

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Mutual Induction

 

Mutual induction concerns a pair of coils. A building current in one coil (the primary) produces a building magnetic field around it. This in turn induces a building current to flow in the other coil. The induced current flow is in such a direction as to produce a magnetic field direction to oppose the primary magnetic field. The magnetic fields are directed towards each other. So their total effect is reduced.

 

 

mutual induction diagram #1

 

 

When the primary current direction is reversed its magnetic field direction is also reversed. This has the effect of making current in the secondary reverse direction together with its magnetic field direction. In this case the magnetic fields are in opposing directions.

As with self inductance, the back EMF is proportional to minus the rate of current change. However, in this case the back EMF is in the secondary coil, not and in the primary(as with self induction).

 

self induction - equation #0

 

mutual induction equation #1

 

Where M (mutual inductance) is the constant of proportionality.

There is another important link between self and mutual inductance. It can be shown, assuming 100% flux linkage that,

 

mutual induction equation #2

 

where Lp and Ls are respectively the self inductances of primary and secondary coils.

So the back EMF Es in the secondary coil relates to the rate of current change dIp/dt in the primary.

 

mutual induction equation #3

 

The secondary back EMF also relates to the rate of change of flux linkage in the secondary.

 

mutual induction equation #4

 

Elimenating Es from these two expressions,

 

mutual induction equation #4b

 

Integrating between the limits of φs - 0 and Ip - 0 ,

 

 

mutual induction equation #4c

 

 

 

mutual induction equation #5

 

 

 

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Transformer Action

 

 

transformer construction diagram #1

 

 

Consider the primary coil. There are two opposing EMF's working here : the applied EMF Ep and the back EMF EB .

If I is the current flowing in the primary and R is its resistance, then from Kirchoff's law for pd's in a circuit:

 

transformer equation #00

 

Neumann's equation states that EB is given by :

 

transformer equation #000

 

Substituting for EB in the Kirchoff relation,

 

transformer equation #1

 

Assuming that the coil resistance R is so low as to be negligible, we have,

 

transformer equation #2                           (i      

 

Both primary and secondary coils have the same flux passing through them. So the rate of flux change dφ/dt will also be the same. It follows that the back EMF Es in the secondary is given by :

 

transformer equation #3                           (ii      

 

Dividing equation (ii by equation (i ,

 

transformer equation #4                               (iii    

 

 

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Power in a Transformer

 

Consider a load resistance R connected to the secondary coil.
Quoting the power equation for a circuit,

 

power = current x EMF - equation #9

 

where P is power, I current and E EMF.

 

 

If we assume that there are no losses (ie that the transformer is 100% efficient) we can write :

 

power input = power output

 

If Ip and Is are the currents flowing in the primary and secondary coils, then :

 

transformer equation #5

 

rearranging,

 

transformer equation #6

 

substituting for Es /Ep from (iii the transformer equation,

 

transformer equation #7

 

 

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Transformer Efficiency

 

In reality the efficiency of a transformer is not 100%. However efficiency is still high, being in the range 95-99%.

 

transformer equation #8

 

Ways that power is lost within a transformer :

 

1.) Coil Heating Energy is lost in the coils by resistive heating. The power loss P is given by P = I2R , where R is coil resistance and I the current. This can be reduced by choosing wire thickness according to current.

 

Between the primary and the secondary, the coil with the smaller number of turns carries the larger current. Therefore this coil is made from thicker wire.

 

Remember the resistance of a wire is inversely proportional to its cross-sectional area. A small cross-sectional area gives a higher resistance. A higher resitance gives a greater power loss.

 

 

2.) Eddy Currents Eddy Currents are unwanted induced currents formed in the body of a metal object. Much heating results from high currents induced from low EMFs.

 

To counteract eddy currents the core is laminated. It is constructed of very thin(approx. 1mm) sheets of soft iron. Each sheet is varnished and insulted from the next.

 

 

3.) Hysteresis The core material offers some resistance to the changing strength and direction of the magnetic field(called hysteresis loss). This resistance manifests itself as heat within the core.

 

The remedy is to make the core of specialist metal (eg permalloy, silicon steel) where hysteresis loss is minimal.

 

 

4.) Flux As a result of imperfections in the windings, not all the flux that passes through the primary coil passes through the secondary. This means that not all the energy is transferred between the coils.

 

 

general notes :

 

There are two types of transformer:


A step-up transformer is when Ns > Np and Es > Ep .


For a step-down transformer, the inequality is reversed and Np > Ns and Ep > Es .

 

The applied EMF in the primary coil must be alternating in nature. A changing magnetic field is a requirement for transformer action.

 

 

 

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