Transcript Chapter 13 Magnetically coupled circuits

Chapter 13
Magnetically coupled circuits
SJTU
1
Mutual inductance
A single inductor:

d
v
 : flux linkage
dt
  N N : number of turns; : flux
 vN
d
d di
N
dt
di dt
di
 vL
dt
d
while L  N
dt
SJTU
2
Mutual inductance of M21 of coil 2 with respect to coil 1


1  11  12
d1
di1
v1  N1
 L1
dt
dt
 2  12
d12
d12 di1
v2  N 2
 N2
dt
di1 dt
di
 v2  M 21
dt
d12
while M 21  N 2
dt
SJTU
3
21
v1
2  21  22
22
v2
i2(t)
d 2
di2
v2  N 2
 L2
dt
dt
1  21
N1 N2
d 21
d 21 di2
v1  N1
 N1
dt
di2 dt
di2
d 21
 v1  M 12
while M 12  N1
dt
dt
M12  M 21  M (for nonmagnetic cores)
SJTU
4
coil1  1   21
coil 2   2  12
dcoil1
and v1  N1
dt
dcoil 2
and v2  N 2
dt
SJTU
di1
di2
v1  L1
M
dt
dt
di1
di2
v2  M
 L2
dt
dt
5
coil1  1   21
coil 2   2  12
dcoil1
and v1  N1
dt
dcoil 2
and v2  N 2
dt
SJTU
di1
di2
v1  L1
M
dt
dt
di1
di2
v2   M
 L2
dt
dt
6
Dot convention
M
i1


i2


L2
v1 L1


M
i1

i2


L2
v1 L1

v2

v2

di1
di2
v1  L1
M
dt
dt
di1
di2
v2  M
 L2
dt
dt
di1
di2
v1  L1
M
dt
dt
di1
di2
v2   M
 L2
dt
dt
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.
SJTU
7
Examples
M
i1


L2
v1 L1


M
i1
i2


v2
v1 L1



L2


v2

i2
di1
di2
v1  L1
M
dt
dt
di1
di2
v2   M
 L2
dt
dt
di1
di2
v1  L1
M
dt
dt
di1
di2
v2   M
 L2
dt
dt
How could we determine dot markings if we don’t know?
SJTU
8
Series connection
1
2
1
2
M
M
(a)mutually coupled coils in
series-aiding connection
(b)mutually coupled coils in
series–opposing connection
Total inductance
LT=L1+L2+2M
LT=L1+L2-2M
SJTU
9
Parallel connection
+ I
V
M
L1
+ I
L2
(a)mutually coupled coils in
parallel-aiding connection
V
M
L1
L2
(b)mutually coupled coils in
parallel–opposing connection
Equivalent inductance
L1L2  M 2
Le 
L1  L2  2M
L1L2  M 2
Le 
L1  L2  2M
SJTU
10
Coefficient of coupling
The coupling coefficient k is a measure of the magnetic
coupling between two coils
k
M
L1 L2
0  k 1
k < 0.5 loosely coupled;
k > 0.5 tightly coupled.
SJTU
11
Tee model
i2
i1
M
i1


i2


L2
v1 L1


v2

L1  M
v1
L2  M
M

i2





L2
v1 L1

v2

v2
M

i2
i1
i1
v1

SJTU

L1  M
L2  M
M

v2

12
TEE MODEL
L1  M
M

L1
L2  M

M
L2
Transformer-like Model
Tee Model
If Dots on Opposite Sides  M  M
SJTU
13
Examples of the mutual coupled circuits
SJTU
14
Linear transformers
M
R1
V
R2
L1
I1
Primary
winding
ZL
L2
I2
jwM
Secondary
winding
R1
V
R2
jwL1
I1
RL+jXL
jwL2
I2
Model in frequency field
SJTU
15
( R1  jwL1 ) I1  jwMI2  V
 jwMI1  ( R2  RL  jwL2  jX L ) I2  0
Total self-impedance of the mesh
containing the primary winding
let Z11  R1  jwL1
Z 22  R2  RL  jwL2  jX L
 R22  jX 22
then I1 

I  Z M V
2
Z11
Total self-impedance of the mesh
containing the secodary winding
V
X M2
Z11 
Z 22
1
X M2
Z 22 
Z 22
SJTU
16
reflected impedance
I1 
R1
V
X M2
Z11 
Z 22
jwL1
V
I1
Zr (reflected
impedance)
Zr
Equivalent primary winding circuit
let Zr  Rr  jXr
X M2
then Rr  2
R22
2
R22  X 22
(reflected resistance)
 X M2
Xr  2
X 22
2
R22  X 22
(reflected reactance)
SJTU
17

Z
V
I2  M
Z11
2
XM
Z11
1
2
M
X
Z 22 
Z 22
Z M VS
Z11
Z22
I2
Equivalent secondary winding circuit
SJTU
18
Ideal transformer
+
I2
I1
V1
-
three properties:
+
V2
1: n
-
1. The coefficient of coupling is unity
(k=1)
2. The self- and mutual inductance of
each coil is infinite (L1=L2=M=∞),
but L1  N1  1 is definite.
L2
V2 v2 (t ) N 2


n

V1 v1 (t ) N1
I2 i2 (t )
N1
1



I
i1 (t )
N2
n
1
N2
n
3. Primary and secondary coils are
lossless.
SJTU
19
+
I2
I1
V1
V2
+
-
1: n
I2
I1
V1
-
1: n
I2
I1
V1
-
+
V2
+
+
+
V2
1: n
-
V2 v2 (t )
N2


 n

V1 v1 (t )
N1
I2 i2 (t ) N1 1



I
i1 (t ) N 2 n
1
V2 v2 (t ) N 2


n

V1 v1 (t ) N1
I2 i2 (t ) N1 1



I
i1 (t ) N 2 n
1
V2 v2 (t )
N

  2  n
V1 v1 (t )
N1
I2 i2 (t )
N1
1



I
i1 (t )
N2
n
1
SJTU
20
Transformer as a matching device
I1
+
I2
V1
RL
-
+
+
V1
I2
R
V1
-
+
V2
1: n
I1
+
+
I2
R
n2R
V1
-
SJTU
-
1: n
I1
+
V2
RL/n2
-
V2
1: n
I2
I1
+
V2
1: n
21
Transformer as a matching device
+
I2
I1
V1
+
RL
-
ZL
Z in  2
n
V2
-
1: n
Thevenin
equivalent
Zin
1: n
Z1
Z2
Z1
Vs1
Vs1
Vs2
I1
Z2/n2
Vs2/n
I2
SJTU
22
1: n
n2 Z1
Z2
Z1
Vs1
Vs2
I1
nVs1
Z2
Vs2
I2
SJTU
23
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
Zeq  n2Z1
Veq  nVs1
Vs1
– Reflecting into primary
Vs 2
Z2
Veq 
Zeq  2
n
n
SJTU
Z1
1: n
Z2
Vs2
24
The Ideal Transformer
SJTU
25
General transformer model
1. Lossless, k=1, but L1,L2,M are not infinite
I2
I1
I2
I1
M
+
V1
-
L1
+
L2 V2
-
+
V1 L1
-
+
V2
1: n
-
L2
n
L1
SJTU
26
General transformer model
2. Lossless, k≠1, L1,L2,M are not infinite
+
V1
I1
I2
M
L1
L2 V2
let n 
+
-
L1
L2
+
V1
I2
I1
LS1
+
LS2
V2
LM
-
-
1: n
M
then LS1  L1 
n
M
LM 
n
LS 2SJTU
 L2  nM
27
General transformer model
3. No restriction
+
V1
I1
M
I2
L2 V2
L1
-
+
V1
+
I2
I1
LS1
R1
LM
LS2
/n2
R2/n2
-
V2
1: n
SJTU
+
-
28
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 M  k L1L2
•
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.
SJTU
29
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.
SJTU
30