Unit-4

Magnetic Circuits

September 9, 2011 Magnetic Circuits 2

Topics to be DiscussedTopics to be Discussed

Magnetic Coupling.Coefficient of Coupling (k).Sign of Mutual Voltage.Dot Convention.

September 9, 2011 Magnetic Circuits 3

Magnetically Coupled Circuits

• A part of magnetic flux produced by a coil in one circuit interlinks with the coil in other circuit.

• When current in one coil changes, there occurs a change in the flux linking with the other. As a result, there is an induced emf in the other coil,

dt

tdiMtv or

dttdi

tv)(

)()(

)( 12

12 =∝

September 9, 2011 Magnetic Circuits 4

•The constant of proportionality M is called coefficient of mutual inductance, or simply mutual inductance.MUTUAL INDUCTANCE is the ability of one inductor to induce a voltage across a neighbouring inductor, measured in henrys (H).

•A circuit element called “mutual inductor” does not exist.• It is defined with reference to two pairs of terminals(i.e., 4 terminals). • The physical device whose operation is based inherently on mutual inductance is called transformer.

September 9, 2011 Magnetic Circuits 5

Magnetic Coupling

Current i1 flowing in coil 1 establishes a total magnetic flux Φ1. Only a part of this flux, Φ12, links with the coil2. The remaining flux Φ11 is confined to coil 1itself.

Thus, Φ1 = Φ11 + Φ12.

September 9, 2011 Magnetic Circuits 6

The emf induced in coil 2 due to the current i1 is given as

dttdΦ

Ntv)(

)( 1222 = Also,

dttdi

Mtv )(

)( 1212 =

)()()()(

1

12221

122

121 tdi

tdΦNM or

dttdΦ

Ndt

tdiM ==∴

M21 is mutual inductance of coil2 with respect to coil1.

Or M21 indicates voltage response at L2 due to current source at L1.

September 9, 2011 Magnetic Circuits 7

The emf induced in coil 1 due to the current i2 is given as

dttdΦ

Ntv)(

)( 2111 = Also,

dttdi

Mtv )(

)( 2121 =

)()()()(

2

21112

211

212 tdi

tdΦNM or

dttdΦ

Ndt

tdiM ==∴

M12 is mutual inductance of coil1 with respect to coil2.

September 9, 2011 Magnetic Circuits 8

lANkNMMM μ12

1221 ===

Mutual Inductance from Geometrical Viewpoint :

lANkN

AlIIkNN

IkΦN

IΦN

didΦNM

μμ

21

1

112

1

12

1

122

1

12221

)/(==

===

The flux that links with the coil 2 is only a part of Φ1.

That is, where 0 ≤ k ≤ 1. 112 kΦΦ =

September 9, 2011 Magnetic Circuits 9

Coefficient of Coupling (k)It is a measure of how close is the coupling between two coils. It gives an idea of what portion of the flux produced by one coil links with the other coil.The flux that links with the coil 2 is only a part of Φ1. That is, where 0 ≤ k ≤ 1. If k = 1, the coils are tightly coupled. The entire flux produced in one coil links with the other.If k = 0, the coils are magnetically isolated. It can be shown that

112 kΦΦ =

21 LLMk =

September 9, 2011 Magnetic Circuits 10

Note that the voltage due to mutual inductance is present independently of and in addition to any voltage due to self-induction.In other words, the voltage across the terminals of coil 1 is composed of two terms,

dttdi

Mdt

tdiLtv

)()()( 21

11 +=

Similarly,

dttdi

Mdt

tdiLtv

)()()( 12

22 +=

September 9, 2011 Magnetic Circuits 11

Sign of Mutual Voltage

The sign depends not only on the current directions, but also on the way the two coils are wound. The induced voltage may be positive or negative. The choice of polarity is made by examining the way in which both coils are physically wound and applying Lenz’s law in conjunction with the right-hand-rule. The procedure is inconvenient in circuit analysis since it is difficult to show the construction details of the coil in circuit schematics. → use the dot convention (often predetermined) “Dot convention” is a convenient way of determining the sign of mutual voltage, without going into the physical construction of the two coils.

September 9, 2011 Magnetic Circuits 12

THE ‘DOT’ CONVENTIONCOUPLED COILS WITH SAME & DIFFERENT WINDING CONFIGURATION

DOT CONVENTION

A current entering the dotted terminal of one coil produces an open‐circuit voltage which is positively sensed at the dottedterminal of the second coil. 

A current leaving the dotted terminal of one coil produces an open‐circuit voltage which is negatively sensed at the dottedterminal of the second coil. 

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September 9, 2011 Magnetic Circuits 14

(a) (b)

(c) (d)

DOT CONVENTION

Fig. (a) is equivalent to Fig. (d), and Fig. (b) is equivalent to Fig. (c)

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September 9, 2011 Magnetic Circuits 16

Dots mark reference polarity for voltages induced by each flux

September 9, 2011 Magnetic Circuits 17

dtdi

Ldtdi

Mv

dtdi

Mdtdi

Lv

22

12

2111

+−=

−=

)()()(

)()()(

22

12

2111

tdtdiLt

dtdiMtv

tdtdiMt

dtdiLtv

+=

+=

Equivalent to a negative mutual inductance

September 9, 2011 Magnetic Circuits 18

−

+

)()()( 211 t

dtdiMt

dtdiLtv −−=−

)()()( 22

12 t

dtdiLt

dtdiMtv −−=

)()()( 211 t

dtdiMt

dtdiLtv +=

)()()( 22

12 t

dtdiLt

dtdiMtv −−=

Convert tobasic case

)(),( 21 tvtvfor equations the Write