ee132 mutual inductance
TRANSCRIPT
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EE 132Electric Circuit Theory II
Magnetically-coupled circuits
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Background So far, we have considered conductively coupled circuits,
i.e. one loop affects the neighboring loop through current conduction.
Definition. When two loops with or without contacts between them affect each other through the magnetic field generated by one of them, they are said to be magnetically coupled.
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Background Study of magnetically-coupled circuits - electrical circuits
linked not by hard electrical connections but by the flux lines of a magnetic field.
Transformers Transformers to step-up/step-down voltages Transformers to electrically isolate parts of a circuit Transformers for impedance matching Inadvertent coupling
Review of self-inductance, mutual inductance
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Mutual Inductance The ability of one inductor to induce a voltage across a
neighboring inductor, measured in Henrys.
Consider a single conductor with, a coil with N turns.
When current i flows through the coil, a magnetic flux is produced around it.
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Mutual Inductance Faraday’s Law. The voltage v induced in the coil is
proportional to the number of turns N and the time rate of change of the magnetic flux ; i.e.
Self inductance, L. The inductance that relates the voltage induced in a coil by a time-varying current in the same coil.
dN
dt
d di
Ndi dt
diL
dt For the inductor,
Thus, dL N
di
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Mutual Inductance Consider two coils with self inductances L1 and L2 in close
proximity with each other with
an open-circuited secondary
For i2 = 0
where
Note: Although the two coils are physically separated, they are said to be magnetically coupled.
1 11 12
11
12
leakage flux
mutual flux
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Mutual Inductance Since the entire flux links only coil 1, the voltage induced
in coil 1 is
Only flux links coil 2, thus
1
11 1
dN
dt
12
122 2
dN
dt
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Mutual Inductance Since the entire flux links only coil 1, the voltage induced
in coil 1 is
Only flux links coil 2, thus
Since the fluxes are cause by the current i1 flowing in coil 1,
1
11 1
dN
dt
12
122 2
dN
dt
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Mutual Inductance Since the entire flux links only coil 1, the voltage induced
in coil 1 is
Only flux links coil 2, thus
Since the fluxes are cause by the current i1 flowing in coil 1,
1
11 1
dN
dt
12
122 2
dN
dt
1 1 11 1 1
1
d di diN L
di dt dt
L1 = self inductance of coil 1
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Mutual Inductance Since the entire flux links only coil 1, the voltage induced
in coil 1 is
Only flux links coil 2, thus
Since the fluxes are cause by the current i1 flowing in coil 1,
1
11 1
dN
dt
12
122 2
dN
dt
12 1 12 2 21
1
d di diN M
di dt dt
M21 = mutual inductance of coil 2 with respect to coil 1
Open-circuit mutual voltage across coil 2 (by current in
coil 1)
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Mutual Inductance Suppose we now let current i2 flow in coil 2, while coil 1 is
open-circuited.
The magnetic flux emanating from coil 2 is
where
2 21 22
22
21
leakage flux
mutual flux
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Mutual Inductance The entire flux links coil 2, so the voltage induced in coil
2 is
2
2 2 2 22 2 2 2
2
d d di diN N L
dt di dt dt
L2 = self inductance of coil 2
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The entire flux links coil 2, so the voltage induced in coil 2 is
Since only flux links coil 1, the voltage induced in coil 1 is
Mutual Inductance
2
2 2 2 22 2 2 2
2
d d di diN N L
dt di dt dt
L2 = self inductance of coil 2
21
21 21 2 21 1 1 12
2
d d di diN N M
dt di dt dt
M12 = mutual inductance of coil 1 with respect to coil 2
Open-circuit mutual voltage across coil 1
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Mutual Inductance Note:
Detailed sketch of the windings determine the algebraic sign of M.
In practice, the DOT CONVENTION designates the polarity of the mutual voltage.
.2112 MMM
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Dot Convention Dot convention. Dots are placed beside each coil
(inductor) so that if the currents are entering (or leaving) both dotted terminals, then the fluxes add.
Right hand rule says that curling the fingers (of the right hand) around the coil in the direction of the current gives the direction of the magnetic flux based on the direction of the thumb.
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Dot ConventionIf current enters the dotted terminal of one coil, the
reference polarity of the mutual voltage in the second coil is positive at the dotted terminal of the second coi.
If current leaves the dotted terminal of one coil, the reference polarity of the mutual voltage in the second coil is negative at the dotted terminal of the second coil.
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Dot Convention
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Dot Convention
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Dot Convention
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Dot Convention
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Dot Convention
i1(t)
+
–
v1(t) L1
i2(t)
+
–
v2(t)L2
M
dt
diL
dt
diMv
dt
diM
dt
diLv
22
12
2111
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Dot Convention
dt
diL
dt
diMv
dt
diM
dt
diLv
22
12
2111
i1(t)
+
–
v1(t) L1
i2(t)
+
–
v2(t)L2
M
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Dot Conventioni1(t)
+
–
v1(t) L1
i2(t)
+
–
v2(t)L2
M
dt
diL
dt
diMv
dt
diM
dt
diLv
22
12
2111
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Dot Conventioni1(t)
+
–
v1(t) L1
i2(t)
+
–
v2(t)L2
M
dt
diL
dt
diMv
dt
diM
dt
diLv
22
12
2111
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Mutually-coupled AC coils The frequency-domain model of the coupled circuit
is essentially identical to that of the time domain.
I1
+
–
V1 sL1
I2
+
–
V2sL2
sM
ssMIsIsLsV
ssMIsIsLsV
1222
2111
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At complex frequency,I1(s)
+
–
V1(s) sL1
I2(s)
+
–
V2(s)sL2
sM
sIsLssMIsV
ssMIsIsLsV
2212
2111
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Example 1 Calculate the phasor currents I1 and I2.
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Solution
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Example 2 Determine V0 in the circuit below.
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Example Find the phasor voltage V2 in the network below
due to the complex forcing function V1. Ω
V12 Ω
3 Ω2ss
•
•
I1 I2
+
V2
_
+
_
s
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Energy Storage The energy stored in an inductor at time t is
The stored energy is the sum of the energies supplied to the primary & secondary terminals.
tLitw 2
2
1
Mi1(t)
+
–
v1(t) L1
i2(t)
+
–
v2(t)L2
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Energy Storage Instantaneous powers are
Suppose that at t0, i1(t0)= 0 & i2(t0) = 0,
then w(t0) = 0.
22
21
21222
12
121
111
idt
diL
dt
diMivP
idt
diM
dt
diLivP i
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Energy Storage
Assume that beginning at t0, keep i2 = 0 and increase i1 until at time t1,
From
.0, 12111 tiIti
2110 111
111211
2210
1
1
0
1
0
2
1
.00,
ILdiiL
dt
diiLdtppw
dt
ditittt
I
t
t
t
t
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Energy Storage
Then we maintain i1=I1 and increase i2, until at time t2,
Since i1 is held constant at I1,
.222 Iti
2222112
0 222112
222
21122
2
1
2
2
1
ILIIM
diiLIM
dtdt
diiL
dt
diIMw
I
t
t
.0 102 ttt
dt
di for
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Energy Storage
At time t = t2,
If we reverse the order in w/c we increase i1 and i2, i.e. from t0 < t < t1, increase i2, so that i2 (t1) = I2, i1 = 0;
from t1 < t < t2, keep i2 = I2, while increasing i1 so that at t2, i1(t2) = I1.
2222112
211
2102
2
1
2
1ILIIMIL
wwtwtw
2222121
2112 2
1
2
1ILIIMILtw
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Energy StorageFor both cases,
Then w(t2) should be the same.
Equal only if
.222121 ItiIti and
2222121
2112 2
1
2
1ILIIMILtw
2222112
2112 2
1
2
1ILIIMILtw
.2112 MMM
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Energy Storage In general,
where (+) if both currents enter the dotted (undotted) terminal, (-) if otherwise.
The coefficient of coupling, k, between the inductors is given by
22221
211 2
1
2
1ILIMIILw
21LL
Mk
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Coefficient of Coupling If k = 0, no coupling exists, M = 0.
We can write
since then
122
21
11
12
21
2112
21
LL
MM
LL
Mk
.1,122
21
11
12
.10 k
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Coefficient of Coupling If k = 1, all of the flux links all of the turns of both
windings. Thus we have a Unity-Coupled Transformer.
If k < 0.5, loosely coupled. e.g air-core transformers
If k > 0.5, tightly coupled. e.g. iron-core transformers
Note: Value of k and M depends on the physical dimensions, no. of turns of each coil, their relative positions to one another, and the magnetic properties of the core.