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SJTU 1
Chapter 13
Magnetically coupled circuits
SJTU 2
Mutual inductance
A single inductor:
dt
di
di
dN
dt
dNv
Ndt
dv
flux : turns;ofnumber :N
linkageflux :
dt
dNLwhile
dt
diLv
SJTU 3
12111
dt
diL
dt
dNv 1
11
11
dt
di
di
dN
dt
dNv 1
1
122
1222
122
dt
dNMwhile
dt
diMv 12
221212
Mutual inductance of M21 of coil 2 with respect to coil 1
SJTU 4
2221
N2N1
v2v1
i2(t)
22212
dt
diL
dt
dNv 2
22
22
dt
di
di
dN
dt
dNv 2
2
211
2111
211
dt
dNMwhile
dt
diMv 21
1122
121
MMM 2112 (for nonmagnetic cores)
SJTU 5
dt
diL
dt
diMv
dt
diM
dt
diLv
22
12
2111
dt
dNvand
dt
dNvand
coilcoil
coilcoil
2221222
1112111
SJTU 6
dt
dNvand
dt
dNvand
coilcoil
coilcoil
2221222
1112111
dt
diL
dt
diMv
dt
diM
dt
diLv
22
12
2111
SJTU 7
1L 2L
M
1v
2v
1i 2i
1L 2L
M
1v
2v
1i 2i
dt
diL
dt
diMv
dt
diM
dt
diLv
22
12
2111
dt
diL
dt
diMv
dt
diM
dt
diLv
22
12
2111
When the reference direction for a current enters the dotted terminal of a coil, the reference polarity of the voltage that it induces in the other coil is positive at its dotted terminal.
Dot convention
SJTU 8
1L 2L
M
1v
2v
1i
2i
Examples
1L 2L
M
1v
2v
1i 2i
dt
diL
dt
diMv
dt
diM
dt
diLv
22
12
2111
dt
diL
dt
diMv
dt
diM
dt
diLv
22
12
2111
How could we determine dot markings if we don’t know?
SJTU 9
Series connection
1 2
M
(a)mutually coupled coils in series-aiding connection
LT=L1+L2+2M
1 2
M
(b)mutually coupled coils in series–opposing connection
LT=L1+L2-2M
Total inductance
SJTU 10
Parallel connection
L1 L2
I+
V
M
L1 L2
I+
V
M
(a)mutually coupled coils in parallel-aiding connection
(b)mutually coupled coils in parallel–opposing connection
MLL
MLLLe 221
221
Equivalent inductance
MLL
MLLLe 221
221
SJTU 11
Coefficient of coupling
21LL
Mk
10 k
The coupling coefficient k is a measure of the magnetic coupling between two coils
k < 0.5 loosely coupled;
k > 0.5 tightly coupled.
SJTU 12
Tee model
1L 2L
M
1v
2v
1i 2i1L M 2L M
M1v
2v
1i 2i
1L 2L
M
1v
2v
1i 2i1L M 2L M
M1v
2v
1i 2i
SJTU 13
TEE MODEL
1L 2L
M1L M 2L M
M
Transformer-like Model Tee Model
If Dots on Opposite Sides M M
SJTU 14
Examples of the mutual coupled circuits
SJTU 15
Linear transformers
V
R1 R2
ZLL1 L2
M
I1 I2
Primary winding
Secondary winding
V
R1 R2
RL+jXLjwL1jwL2
jwM
I1 I2
Model in frequency field
SJTU 16
0)(
)(
2221
2111
IjXjwLRRIjwM
VIjwMIjwLR
LL
2222
2222
1111
jXR
jXjwLRRZ
jwLRZlet
LL
Total self-impedance of the mesh containing the primary winding
Total self-impedance of the mesh containing the secodary winding
22
2
2211
2
22
2
11
1
1
ZX
ZZ
VZI
ZX
Z
VIthen
M
M
M
SJTU 17
V
R1 jwL1
I1
Zr (reflected impedance) 22
2
11
1
ZX
Z
VI
M
Zr
reflected impedance
Equivalent primary winding circuit
22222
222
2
22222
222
2
XXR
XXr
RXR
XRrthen
jXrRrZrlet
M
M
(reflected resistance)
(reflected reactance)
SJTU 18
Z22
I2
11
2
Z
XM
11Z
VZ SM
Equivalent secondary winding circuit
22
2
2211
2
1
ZX
ZZ
VZI
M
M
SJTU 19
Ideal transformer
+
-
+
-
1V2V
1I 2I
1: n
three properties:
1. The coefficient of coupling is unity (k=1)
2. The self- and mutual inductance of each coil is infinite (L1=L2=M=∞), but is definite.
3. Primary and secondary coils are lossless.
nN
N
ti
ti
I
I
nN
N
tv
tv
V
V
1
)(
)(
)(
)(
2
1
1
2
1
2
1
2
1
2
1
2
nN
N
L
L 1
2
1
2
1
SJTU 20
+
-
+
-
1V2V
1I 2I
1: n nN
N
ti
ti
I
I
nN
N
tv
tv
V
V
1
)(
)(
)(
)(
2
1
1
2
1
2
1
2
1
2
1
2
+
-
+
-
1V2V
1I 2I
1: nnN
N
ti
ti
I
I
nN
N
tv
tv
V
V
1
)(
)(
)(
)(
2
1
1
2
1
2
1
2
1
2
1
2
+
-
+
-
1V2V
1I 2I
1: n nN
N
ti
ti
I
I
nN
N
tv
tv
V
V
1
)(
)(
)(
)(
2
1
1
2
1
2
1
2
1
2
1
2
SJTU 21
Transformer as a matching device
+
-
+
-
1V2V
1I2I
1: n
RL
-
+
-
+
1V2V
1I 2I
1: n
RL/n2
+
-
+
1V2V
1I 2I
1: n
R
-
+
-
+
1V2V
1I 2I
1: n
R
-
n2R
SJTU 22
Transformer as a matching device
+
-
+
-
1V2V
1I 2I
1: n
RL
Zin
2n
ZZ Lin
Vs1
Z1 Z2/n2
Vs2/nVs2Vs1
Z1 Z2
1: n
I1 I2
Thevenin equivalent
SJTU 23
Vs2Vs1
Z1 Z2
1: n
I1 I2
nVs1
n2 Z1 Z2
Vs2
SJTU 24
Solving Ideal Transformer Problem
• Method 1: Write out equations first– Loop equations or Nodal equations– Two more transformer equations
• Method 2 : Form equivalent circuit first– Reflecting into secondary
– Reflecting into primary
21eq nZ Z 1eq snV V
22eq n
Z
Z2s
eq n
VV
Vs1Vs2
Z1Z21: n
SJTU 25
The Ideal Transformer
SJTU 26
General transformer model1. Lossless, k=1, but L1,L2,M are not infinite
+
-
+
-
1V2V
1I 2I
L1 L2
M +
-
+
-
1V2V
1I 2I
1: n
L1
1
2
L
Ln
SJTU 27
General transformer model2. Lossless, k≠1, L1,L2,M are not infinite
+
-
+
-
1V2V
1I 2I
L1 L2
M +
-
+
-
1V2V
1I 2I
1: n
LM
LS1 LS2
nMLLn
ML
n
MLLthen
L
Lnlet
S
M
S
22
112
1
SJTU 28
General transformer model3. No restriction
+
-
+
-
1V2V
1I 2I
L1 L2
M
+
-
+
-
1V2V
1I 2I
1: n
LM
LS1 LS2/n2R1 R2/n2
SJTU 29
SUMMARY• Mutual inductance, M, is the circuit parameter relating the
voltage induced in one circuit to a time-varying current in another circuit.
• The coefficient of coupling, k, is the measure of the degree of magnetic coupling. By definition, 0≤k≤1
• The relationship between the self-inductance of each winding and the mutual inductance between the windings is
• The dot convention establishes the polarity of mutually induced voltage
• Reflected impedance is the impedance of the secondary circuit as seen from the terminals of the primary circuit, or vise versa.
21LLkM
SJTU 30
SUMMARY
• The two-winding linear transformer is a coupling device made up of two coils wound on the same nonmagnetic core.
• An ideal transformer is a lossless transformer with unity coupling coefficient(k=1) and infinite inductance.
• An ideal transformer can be used to match the magnitude of the load impedance, ZL, to the magnitude of the source impedance, ZS, thus maximizing the amount of average power transferred.
