HALL EFFECT

                 HALL  EFFECT

Hall effect, named after its discoverer Edwin Herbert Hall (U.S Physicist) in 1879, is a conduction phenomenon which is defined as the development of a transverse electric field in a solid conducting material when it is placed under the influence of mutually perpendicular electric and magnetic fields.

                             Hall effect is observed a the result of force due to magnetic field in a moving electric charge carrier. No matter we consider current as the movement of electrons, holes or as a combination of both ( in the case of semi conductors) a perpendicular magnetic field displaces the moving electric charge in the same direction side ways at right angles to it.

Thus, the accumulation of a specific polarity in one side results in the development of a measureable potential difference within the conductor which is indicated as positive or negative by the detector.

                               Besides solids Hall effect is observed in plasma ( ionised gases) also, but it is significantly different from what we observe in solids.

Then,

Fnet = q (E + v x B )

The net force experienced by a hole moving along y- direction is the sum of forces due to electric field and Magnetic field along y- direction

Force due to electric field     = qEy  

Force due to magnetic field  = q x y- component of v x B

                                                  = q [ - vx Bz - Bz vz]

                                                  = q [ -vx Bz] ( since vz = 0 )

               

               Fnet = Fy = q( Ey – vxBz)

In order to maintain steady state condition, Ey must balance the product vx Bz. Then, Fnet = 0. This condition occurs when the holes are shifted slightly in the negative y direction due to B. As the forces are now balanced, the lateral force becomes zero as they drift along the bar.

This establishment of Ey is known as Hall effect and the resulting voltage is known as Hall voltage Vhall = Ey w

Drift velocity           Vx= Jx  / q po

So, Ey= Jx Bz /q po= RH  Jx  Bz

Po = JxBz/qEy

     =  (Ix/wt  )Bz /  q (v/w   )                                                        

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INTER-RELATIONSHIP BETWEEN VARIOUS MATRICES

In this section,we will be discussing about the relationship between different matrices like incidence matrix(A),Tie-set matrix or Fundamental loop matrix(B) and Cut-set matrix(Q).

A=[A_t:A_l]  where, A_t means matrix of twigs and A_l  means matrix of links

B=[B_t:B_l] where , B_t means matrix of twigs and B_l means matrix of links

                 But for a fundamental loop matrix B_l =U,where U is the identity matrix.

Q=[Q_t:Q_l] where they have the usual meaning as stated above

RELATIONSHIP BETWEEN INCIDENCE MATRIX A AND FUNDAMENTAL LOOP MATRIX B:

We know that A and B are orthogonal.So

                                                   AB^T=0

[A_t:A_l]*B_t^T =0

              U

So A_tB_t^T+A_l=0   or    B_t^T=A_t^-1A_l

B_t= [ -A_t^-1 A_l ]^T

RELATIONSHIP BETWEEN A AND Q:

AI_b = 0     or      QI_b = 0

From this relation,we will get

Q_l = A_t^-1 A_l

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ARMATURE REACTION:

We know that an emf is induced in the coils of armature when there is a change in magnetic flux associated with it.So when there is no load connected to the armature, the current flowing through the armature winding is zero.Therefore the net magnetic flux of the system is created by the field system.

When there is a load connected to the armature winding the current flowing through the armature winding is non-zero value.So since there is a current flow through the winding,it causes generation of magnetic field and hence magnetic flux.Now the net magnetic flux get reduced since the magnetic flux created by armature winding either opposes or crosses perpendicularly to that produced by field system.

Any dc machine has two types of planes known as Magnetic Neutral Plane(MNP) and Geometric Neutral Plane(GNP).

The position of MNP and GNP are important on case of dc machine.GNP is always peerpendicular to the line connecting the centres of magnetic poles.Whereas MNP is always perpendicular to the resultant magnetic flux or mmf.

So when the load connected to armature winding is zero, resultant mmf is from north pole to south pole.So the MNP and GNP coincides.

But when a load is connected to armature winding,the net mmf rotates in a direction along the rotation of armature and hence there is a shift in MNP.Now MNP and GNP never coincides.

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Active Elements and Passive Elements

ACTIVE ELEMENTS

An independent source which can  deliver or absorb energy  continuously is called an active element . The voltage of an ideal source is assumed to be independent of the current in the circuit.

If the current is entering into the positive terminal of battery then power is absorbed by the battery and if the current is entering into the negative terminal of battery then power is delivered by the battery.                              

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Reciprocity Theorem or Reciprocal Theorem

RECIPROCAL THEOREM

A network is said to be reciprocal if the network remains invariant due to the interchange of position of cause and effect.

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Superposition Theorem

SUPERPOSITION  THEOREM

Superposition theorem states that for any linear network consisting of a number of sources,the overall response is equal to the algebraic sum of individual responses considering only one source at a time keeping all other sources inoperative.

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