electrical dircuits

1. SJTU 1
Chapter 13
Magnetically coupled circuits

2. SJTU 2
φ
Mutual inductance
A single inductor:
dt
di
di
d
N
dt
d
Nv
N
dt
d
v
φφ
φφλ
λ
λ
==∴
=
=
flux:turns;ofnumber:N
linkageflux:
dt
d
NLwhile
dt
di
Lv
φ
=
=∴

3. SJTU 3
φ φ
12111 φφφ +=
dt
di
L
dt
d
Nv 1
1
1
11 ==
φ
dt
di
di
d
N
dt
d
Nv 1
1
12
2
12
22
122
φφ
φφ
==
=
dt
d
NMwhile
dt
di
Mv 12
221212
φ
==∴
Mutual inductance of M21 of coil 2 with respect to coil 1

4. SJTU 4
φ22
φ21
N2N1
v2
v1
i2(t)
22212 φφφ +=
dt
di
L
dt
d
Nv 2
2
2
22 ==
φ
dt
di
di
d
N
dt
d
Nv 2
2
21
1
21
11
211
φφ
φφ
==
=
dt
d
NMwhile
dt
di
Mv 21
112
2
121
φ
==∴
MMM == 2112 (for nonmagnetic cores)

5. SJTU 5
dt
di
L
dt
di
Mv
dt
di
M
dt
di
Lv
2
2
1
2
21
11
+=
+=
dt
d
Nvand
dt
d
Nvand
coil
coil
coil
coil
2
221222
1
112111
φ
φφφ
φ
φφφ
=+=
=+=

6. SJTU 6
dt
d
Nvand
dt
d
Nvand
coil
coil
coil
coil
2
221222
1
112111
φ
φφφ
φ
φφφ
=−=
=−=
dt
di
L
dt
di
Mv
dt
di
M
dt
di
Lv
2
2
1
2
21
11
+−=
−=

7. SJTU 7
1L 2L
M
• •
1v
+
−
2v
+
−
1i 2i
1L 2L
M
•
•
1v
+
−
2v
+
−
1i 2i
dt
di
L
dt
di
Mv
dt
di
M
dt
di
Lv
2
2
1
2
21
11
+−=
−=
dt
di
L
dt
di
Mv
dt
di
M
dt
di
Lv
2
2
1
2
21
11
+=
+=
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

8. SJTU 8
1L 2L
M
• •
1v
+
−
2v
+
−
1i
2i
Examples
1L 2L
M
•
•
1v
+
−
2v
+
−
1i 2i
dt
di
L
dt
di
Mv
dt
di
M
dt
di
Lv
2
2
1
2
21
11
+−=
−=
dt
di
L
dt
di
Mv
dt
di
M
dt
di
Lv
2
2
1
2
21
11
−+=
−=
How could we determine dot markings if we don’t know?

9. 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

10. 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
MLL
Le
221
2
21
−+
−
=
Equivalent inductance
MLL
MLL
Le
221
2
21
++
−
=

11. SJTU 11
Coefficient of coupling
21LL
M
k =
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.

12. SJTU 12
Tee model
1L 2L
M
• •
1v
+
−
2v
+
−
1i 2i
1L M− 2L M−
M1v
+
−
2v
+
−
1i 2i
1L 2L
M
•
•
1v
+
−
2v
+
−
1i 2i
1L M− 2L M−
M1v
+
−
2v
+
−
1i 2i

13. SJTU 13
TEE MODEL
1L 2L
M 1L M− 2L M−
M
• •
Transformer-like Model Tee Model
If Dots on Opposite Sides M M=> → −

14. SJTU 14
Examples of the mutual coupled circuits

15. SJTU 15
Linear transformers
V
R1 R2
ZLL1
L2
M
I1 I2
Primary
winding
Secondary
winding
V
R1 R2
RL+jXLjwL1
jwL2
jwM
I1 I2
Model in frequency field

16. 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
22
11
2
22
2
11
1
1
Z
X
Z
Z
VZ
I
Z
X
Z
V
Ithen
M
M
M
+
=
+
=





17. SJTU 17
V
R1 jwL1
I1
Zr (reflected
impedance)22
2
11
1
Z
X
Z
V
I
M
+
=


Zr
reflected impedance
Equivalent primary winding circuit
222
22
2
22
2
222
22
2
22
2
X
XR
X
Xr
R
XR
X
Rrthen
jXrRrZrlet
M
M
+
−
=
+
=
+=
(reflected resistance)
(reflected reactance)

18. SJTU 18
Z22
I2
11
2
Z
X M
11Z
VZ SM

Equivalent secondary winding circuit
22
2
22
11
2
1
Z
X
Z
Z
VZ
I
M
M
+
=



19. SJTU 19
Ideal transformer
+
-
+
-
1V 2V
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
n
N
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
==

20. SJTU 20
+
-
+
-
1V 2V
1I 2I
1: n nN
N
ti
ti
I
I
n
N
N
tv
tv
V
V
1
)(
)(
)(
)(
2
1
1
2
1
2
1
2
1
2
1
2
−=−==
−=−==




+
-
+
-
1V 2V
1I 2I
1: n
nN
N
ti
ti
I
I
n
N
N
tv
tv
V
V
1
)(
)(
)(
)(
2
1
1
2
1
2
1
2
1
2
1
2
===
===




+
-
+
-
1V 2V
1I 2I
1: n nN
N
ti
ti
I
I
n
N
N
tv
tv
V
V
1
)(
)(
)(
)(
2
1
1
2
1
2
1
2
1
2
1
2
===
−=−==





21. SJTU 21
Transformer as a matching device
+
-
+
-
1V 2V
1I 2I
1: n
RL
-
+
-
+
1V 2V
1I 2I
1: n
RL/n2
+
-
+
1V 2V
1I 2I
1: n
R
-
+
-
+
1V 2V
1I 2I
1: n
R
-
n2
R

22. SJTU 22
Transformer as a matching device
+
-
+
-
1V 2V
1I 2I
1: n
RL
Zin
2
n
Z
Z L
in =
Vs1
Z1 Z2/n2
Vs2/n
Vs2Vs1
Z1
Z2
1: n
I1 I2
Thevenin
equivalent

23. SJTU 23
Vs2Vs1
Z1
Z2
1: n
I1 I2
nVs1
n2
Z1 Z2
Vs2

24. 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
2
1eq n=Z Z 1eq sn=V V
2
2eq
n
=
Z
Z
2s
eq
n
=
V
V
Vs1
Vs2
Z1
Z21: n

25. SJTU 25
The Ideal Transformer

26. SJTU 26
General transformer model
1. Lossless, k=1, but L1,L2,M are not infinite
+
-
+
-
1V 2V
1I 2I
L1 L2
M
+
-
+
-
1V 2V
1I 2I
1: n
L1
1
2
L
L
n =

27. SJTU 27
General transformer model
2. Lossless, k≠1, L1,L2,M are not infinite
+
-
+
-
1V 2V
1I 2I
L1 L2
M +
-
+
-
1V
2V
1I 2I
1: n
LM
LS1 LS2
nMLL
n
M
L
n
M
LLthen
L
L
nlet
S
M
S
−=
=
−==
22
11
2
1

28. SJTU 28
General transformer model
3. No restriction
+
-
+
-
1V 2V
1I 2I
L1 L2
M
+
-
+
-
1V 2V
1I 2I
1: n
LM
LS1 LS2/n2R1 R2/n2

29. 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 =

30. 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.