Post on 06-Jun-2018
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.
September 9, 2011 Magnetic Circuits 13
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)
September 9, 2011 Magnetic Circuits 15
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