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Magnetically Coupled Circuits Instructor: Chia-Ming Tsai Electronics Engineering National Chiao Tung Universit y Hsinchu, Taiwan, R.O.C.
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Magnetically Coupled Circuits
Instructor: Chia-Ming TsaiElectronics Engineering
National Chiao Tung UniversityHsinchu, Taiwan, R.O.C.
Contents• Introduction
• Mutual Inductance
• Energy in a Coupled Circuit
• Linear Transformers
• Ideal Transformers
• Applications
Introduction• Conductively coupled circuit means that one loop
affects the neighboring loop through current conduction.
• Magnetically coupled circuit means that two loops, with or without contacts between them, affect each other through the magnetic field generated by one of them.
• Based on the concept of magnetic coupling, the transformer is designed for stepping up or down ac voltages or currents.
Self Inductance
)inductance-(self
is volatgeinduced the turns,For
is volatgeinduced theeach turn,For
1T
di
dNL
dt
diL
dt
di
di
dN
dt
dNv
Ndt
dv
turns
inductance
:inductorAn
N
L
Mutual Inductance (1/5)
dt
diM
dt
di
di
dN
dt
dNv
di
dNL
dt
diL
dt
di
di
dN
dt
dNv
N
L
N
L
121
1
1
122
1222
1
111
11
1
1
11
111
12111
2
2
1
1
where
is 1 coilby generatedflux The
2, coilin current no Assuming
turns
sinductance-self:2 Coil
turns
sinductance-self:1 Coil
dt
diMv
di
dNM
1212
1
12221
is voltagemutualcircuit -open The
is 1 coil respect to2with coil
of inductance-mutual The
Mutual Inductance (2/5)
dt
diM
dt
di
di
dN
dt
dNv
di
dNL
dt
diL
dt
di
di
dN
dt
dNv
N
L
N
L
212
2
2
211
2111
2
222
22
2
2
22
222
21222
2
2
1
1
where
is 2 coilby generatedflux The
1, coilin current no Assuming
turns
sinductance-self:2 Coil
turns
sinductance-self:1 Coil
dt
diMv
Mdi
dNM
2121
212
21112
is voltagemutualcircuit -open The
)(
is 2 coil respect to1with coil
of inductance-mutual The
Mutual Inductance (3/5)• We will see that M12=M21=M.
• Mutual coupling only exists when the inductors or coils are in close proximity, and the circuits are driven by time-varying sources.
• Mutual inductance is the ability of one inductor to induce a voltage across a neighboring inductor, measured in henrys (H).
i1
+
_dt
diMv 1
2
• The dot convention states that a current entering the dotted terminal induces a positive polarity of the mutual voltage at the dotted terminal of the second coil.
Mutual Inductance (4/5)
)(
)(
. and induces
, and induces
2221122
2112111
22212
12111
i
i
dt
diM
dt
diL
dt
dN
dt
dN
dt
dNv
dt
diM
dt
diL
dt
dN
dt
dN
dt
dNv
121
22
122
22212
222
212
11
211
12111
111
)(
)(
Mutual Inductance (5/5)
i1
+
_dt
diMv 1
2
i1
+
_dt
diMv 1
2
i2
+
_dt
diMv 2
1
i2
+
_dt
diMv 2
1
Series-Aiding Connection
dt
diMMLL
dt
diM
dt
diL
dt
diM
dt
diL
vvvdt
diM
dt
diLv
dt
diM
dt
diLv
211221
212121
21
2122
1211
MLLLdt
diMLLv
MMM
2
2
,But
21eq
21
2112
+ _v1 + _v2
Series-Opposing Connection
dt
diMMLL
dt
diM
dt
diL
dt
diM
dt
diL
vvvdt
diM
dt
diLv
dt
diM
dt
diLv
211221
212121
21
2122
1211
MLLLdt
diMLLv
MMM
2
2
,But
21eq
21
2112
+ _v1 + _v2
Example 1
(1b)
gives 2mesh toKVL Applying
(1a)
gives 1mesh toKVL Applying
122222
211111
dt
diM
dt
diLRiv
dt
diM
dt
diLRiv
(2b) )(
(2a) )(
asdomain
phasorin (1) Eq can write We
22212
21111
IIV
IIV
LjRMj
MjLjR
Circuit Model for Coupled Inductors
dt
diL
dt
diMv
dt
diM
dt
diLv
22
12
2111
2212
2111
IIV
IIV
LjMj
MjLj
Example 2
(1b) )42(3
612
0)612(3
gives 2mesh toKVL Applying
(1a) 03)54(12
gives 1mesh toKVL Applying
221
21
21
III
II
II
jj
j
jj
jjj
39.4901.13)42(
04.1491.24
12
gives (1a) into (1b)
21
2
II
I
j
j
Example 3
(1b) 0)185(8
0)52286(26
gives 2mesh toKVL Applying
(1a) 1008)34(100
026)634(100
gives 1mesh toKVL Applying
21
211
21
221
II
III
II
III
jj
jjjjj
jj
jjjj
19693.8
5.33.20
0
100
1858
834
get we(1b) and (1a) From
2
1
2
1
I
I
I
I
jj
jj
Energy in a Coupled Circuit (1/4)
2112222
21121
2222112
0 2220 211222
222
212122112
2211
2110 11111
111111
112
2
1
2
12
1
)(
. to0 from increases , :II Step2
1
)(
. to0 from increases ,0 :I Step
22
1
IIMILILwww
ILIIM
diiLdiIMdtpw
dt
diLi
dt
diMIvivitp
IiIi
ILdiiLdtpw
dt
diLivitp
Iii
II
I
i1
i2
I1
I2
I II
t
Energy in a Coupled Circuit (2/4)
MMM
IIMILILwww
ILIIMw
IiIi
ILw
Iii
2112
2121222
21121
21121212
1122
2221
221
case.former
the toequalmust energy totalBut the
2
1
2
12
1
. to0 from increases , :II Step2
1
. to0 from increases ,0 :I Step
as changed becan process analysis The
i1
i2
I1
I2
I II
t
Energy in a Coupled Circuit (3/4)21
222
211 2
1
2
1iMiiLiLw 21
222
211 2
1
2
1iMiiLiLw
Energy in a Coupled Circuit (4/4)
21
21
2121
2
2211
21222
211
21222
211
0
02
1
02
1
2
1
case,any for 0But
2
1
2
1
asgiven is storedenergy ousinstantane
thes,assignmentcurrent different For
LLM
MLL
MLLiiLiLi
iMiiLiL
w
iMiiLiLw
)10( or
as defined is
The
21
21
kLLkM
LL
Mk
koefficientcoupling c
More about k
0
coupling.perfect means 1
or
asflux of
in terms expressed becan
2211
2221
21
1211
12
k
k
k
k
Coupling vs. Winding Style
Loosly coupled k < 0.5
Tightly coupled k > 0.5
Example
J 73.202
1
2
1
824.2)1( ,389.3)1(
)6.1604cos(254.3
)4.194cos(905.3
6.160254.3
4.19905.3
(1b) 0)416(10
2,mesh For
(1a) 306010)2010(
1,mesh For
56.020
5.2 :Sol
s. 1at inductors coupled
in the storedenergy theand Find
21222
211
21
2
1
2
1
21
21
21
iMiiLiLw
ii
ti
ti
jj
jj
LL
Mk
t
k
I
I
II
II
rad/s 4
V )304cos(60 tv
Linear Transformers
Zin
impedancereflected
pedanceprimary im
ZLjR
MLjR
R
P
RP
L
:
: where
22
22
11in
Z
Z
ZZ
Z
1in
2221
2111
But
0)(
)(
givesmesh two the toKVL Applying
I
VZ
II
IIV
LZRLjMj
MjLjR
R1 and R2
are winding resistances.
T (or Y) Equivalent Circuit
2
1
2
1
2
1
I
I
V
V
LjMj
MjLj
2
1
2
1
)(
)(
I
I
V
V
cbc
cca
LLjLj
LjLLj
)(
)(2
1
2
1
ML
MLL
MLL
LLjLj
LjLLj
LjMj
MjLj
c
b
a
cbc
cca
П (or ) Equivalent Circuit
221
2
1
1
2
2
1
where MLLK
Kj
L
Kj
MKj
M
Kj
L
V
V
I
I
2
1
2
1
111
111
V
V
I
I
CBC
CCA
LjLjLj
LjLjLj
111
111
1
2
1
2
M
KL
ML
KL
ML
KL
LjLjLj
LjLjLj
Kj
L
Kj
MKj
M
Kj
L
C
B
A
CBC
CCA
Ideal Transformers (1/3)
1. Coils have very large reactance (L1, L2, M ~ )
2. Coupling coefficient is equal to unity (k = 1)
3. Primary and secondary are lossless (series resistances R1= R2= 0)
21 dt
dNv
dt
dNv
2211
Ideal Transformers (2/3)
. thecalled is where
.or 1
coupling,perfect For
gives (1b) into 1(c) ngSubstituti
(1c)
(1a), From
(1b)
(1a)
111
21
1
212
21
21
2
211
2
1211
2212
2111
oturns ratin
nL
L
L
LL
LLMk
jL
ML
L
M
LjMj
LjMj
MjLj
VVVV
IVV
IVI
IIV
IIV
Ideal Transformers (3/3)
nN
N
nN
N
v
vdt
dNv
dt
dNv
1
2
1
2
1
2
1
2
22
11
V
V
nN
N
iviv
1
domain,phasor In
lossless, iser transformidealAn
2
1
1
2
2211
2211
I
I
IVIV
Types of Transformers
• When n = 1, we generally call the transformer an isolation transformer.
• If n > 1 , we have a step-up transformer (V2 > V1).
• If n < 1 , we have a step-down transformer (V2 < V1).
Impedance Transformation
lossless! iser transformThe
loss.without
secondary the todelivered isprimary
the tosuppliedpower complex The
isprimary in thepower complex The
1
2*22
*2
2*111
21
21
2
1
1
2
1
2
1
2
SIVIV
IVS
II
VV
I
IV
V
nn
nn
nN
N
nN
N
matching! impedancefor Useful
) (
1
is source by the
seen as impedanceinput The
2in
2
22
2
2
1
1in
impedancereflected n
nnn
LZZ
I
V
I
V
I
VZ
Zin
How to make a transformer ideal?
Zin
L
L
L
L
L
L
Lj
Lj
Lj
LLLLLj
Lj
MLj
LLMRR
LjR
MLjR
Z
Z
Z
Z
ZZ
ZZ
2
1
2
212
212
1
2
22
1in
2121
22
22
11in
coupling)(perfect and 0 If
oturns ratiL
Ln
nL
L
Lj
Lj
L
LLL
L
the: where
If
1
2
22
1
2
1in
2
ZZZZ
Z
The linear transformer model
Impedance Matching
Linear network
:
complex :
issfer power tran
maximumfor condition The
Th2
*Th2
LLL
LL
Rn
Rn
ZZ
ZZZ
Ideal Transformer Circuit (1/3)
Linear network 1 Linear network 2
Ideal Transformer Circuit (2/3)
nns22
1Th
21 0
VVVV
II
22
2
22
2
2
1
1Th
21
21
1
nnn
nn
n
Z
I
V
I
V
I
VZ
VV
II
1
Ideal Transformer Circuit (3/3)
c c
Applications of Transformers• To step up or step down voltage and current (useful
for power transmission and distribution)
• To isolate one portion of a circuit from another
• As an impedance matching device for maximum power transfer
• Frequency-selective circuits
Applications: Circuit Isolation
When the relationship betweenthe two networks is unknown,any improper direct connectionmay lead to circuit failure.
This connection style canprevent circuit failure.
Applications: DC Isolation
Only ac signal can pass, dc signal is blocked.
Applications: Load Matching
Applications: Power Distribution
