Coupling Element and Coupled circuits
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Transcript Coupling Element and Coupled circuits
Coupling Element and Coupled circuits
Coupled inductor
Ideal transformer
Controlled sources
Coupling Element and Coupled circuits
Coupled elements have more that one branch and branch voltages or branch
currents depend on other branches. The characteristics and properties of
coupling element will be considered.
Coupled inductor
Two coils in a close proximity is shown in Fig.1
i1
+
v1
-
i2
+
v2
-
Fig.1 Coupled coil and reference directions
Coupled inductor
Magnetic flux is produced by each coil by the functions
1 f1 (i1 , i2 )
Where
f1
and
f2
2 f 2 (i1 , i2 )
are nonlinear function of
i1
and i2
By Faraday’s law
d1 f1 di1 f1 di2
v1
dt
i1 dt i2 dt
d2 f 2 di1 f 2 di2
v2
dt
i1 dt i2 dt
Coupled inductor
Linear time-invariant coupled inductor
If the flux is a linear function of currents
and
1 (t ) L11i1 (t ) Mi2 (t )
2 (t ) Mi1 (t ) L22i2 (t )
di1
di2
v1 L11
M
dt
dt
di1
di2
v2 M
L22
dt
dt
In sinusoid steady-state
V1 j L11I1 j M I 2
Note that the signs of
L11 and L22
V2 j MI1 j L22 I 2
are positive but the sign for M can be
or
Coupled inductor
Dots are often used in the circuit to indicate the sign of M
i1
H1
i2
+
+
v1
v2
-
-
H2
Fig. 2 Positive value of M
Coupled inductor
Coefficient of coupling
The coupling coefficient is
k
|M |
L11L22
If the coils are distance away k is very small and close to zero and equal
to 1 for a very tight coupling such for a transformer.
Coupled inductor
Multi-winding Inductors and inductance Matrix
For more windings the flux in each coil are
1 L11I1 L12 I 2 L13 I3 ..
2 L21I1 L22 I 2 L23 I3 ..
3 L31I1 L32 I 2 L33 I3 ..
L11 , L22 , L33
are self inductances and
L12 L21 , L13 L31 , L23 L32
In matrix form
φ Li
are mutual inductances
Coupled inductor
1
2
3
i1
i i2
i3
L11
L L21
L31
L12
L22
L32
L13
L23
L33
i2
i
d1 + 1
v1
dt
+ v d2
2
dt
-
-
Fig 3 Three-winding inductor
i3
+
-
d3
v3
dt
Coupled inductor
Induced voltage
The induced voltage in term current vector and the inductance matrix is
Example 1
di
vL
dt
Fig. 4 shows 3 coils wound on a common core. The reference direction of
current and voltage are as shown in the figure. Since H1and H 2 has the
same direction but H 3 are not therefore L12 is positive while L13 and
i2
i1
L23 are negative.
-
v1
+
+
H1
-
H2
Fig. 4
H3
i3 + v 3
v2
Coupled inductor
It is useful to define a reciprocal inductance matrix
L1
i
which makes
i1 111 122
i2 211 222
where
11
L22
L
L
, 22 11 and 12 21 12
det L
det L
det L
Thus the currents are
t
t
t0
0t
i1 (t ) 11 v1 (t ')dt ' 12 v2 (t ')dt ' i1 (0)
0
0
i2 (t ) 21 v1 (t ')dt ' 22 v2 (t ')dt ' i2 (0)
Coupled inductor
In sinusoid steady-state
11
V1 12 V2
j
j
I 2 21 V1 22 V2
j
j
I1
Series and parallel connections of coupled inductors
Equivalent inductance of series and parallel connections of coupled
inductors can be determined as shown in the example 2.
Coupled inductor
Example 2
Fig. 5 shows two coupled inductors connected in series. Determine the
Equivalent inductance between the input terminals.
i
+
Fig. 5
v
i1
+
v1
L1 5
i2
M 3
-
+
-
v2
-
L2 2
1 L11i1 Mi2 5i1 3i2
2 Mi1 L22i2 3i1 2i2
i i1 i2 , v v1 v2
d d1 d2
(0) 0
dt
dt
dt
1 2 8i1 5i2 13i
L
i
13 H
Coupled inductor
Example 3
Fig. 6 shows two coupled inductors connected in series. Determine the
Equivalent inductance between the input terminals.
i
+
Fig. 6
i1
+
v
v1
L1 5
-
i2
M 3
-
+
v2
-
Note
L2 2
L L11 L22 2 | M |
1 L11i1 Mi2 5i1 3i2
2 Mi1 L22i2 3i1 2i2
i i1 i2 , v v1 v2
d d1 d2
(0) 0
dt
dt
dt
1 2 2i1 i2 i
L
i
1 H
for series inductors
Coupled inductor
Example 4
Two coupled inductors are connected in parallel in Fig 6. Determine the
Equivalent inductance.
i
+
i1
+
v1
L1 5
-
Fig 6
i2
v
-
M 3
+
v2
-
L2 2
L22
det L
2
5
det
3
L11
5
22
detL
5
det
3
L
3
12 12
detL
5
det
3
11
2
3
2
3
2
3
2
5
3
Coupled inductor
The currents are
i1 111 122 21 32
i2 211 222 31 52
KVL
v1 (t ) v2 (t ) and 1 (0) 1 (0) 0
By integration of voltage
Therefore
1 (t ) 2 (t )
i i1 i2 1 22
L
Note
i
1
H
11 22 2 | 12 |
for parallel inductors
Ideal transformer
Ideal transformer is very useful for circuit calculation. Ideal transformer
Is a coupled inductor with the properties
dissipate no energy
No leakage flux and the coupling coefficient is unity
Infinite self inductances
Two-winding ideal transformer
i1
Fig. 7
i2
+
+
v1
v2
-
-
Ideal transformer
Figure 7 shows an ideal two-winding transformer. Coils are wound on ideal
Magnetic core to produce flux. Voltages is Induced on each winding.
If
is the flux of a one-turn coil then
Since
d 1
v1
dt
1 n1 , and 2 n2
d2
and v2
dt
v1 (t ) n1
v2 (t ) n2
we have
(1)
In terms of magnetomotive force (mmf) and magnetic reluctance
mmf
n1i1 n2i2
Ideal transformer
If the permeability is infinite
becomes zero then
n1i1 n1i1 0
i1 (t )
n2
i2 (t )
n1
and
(2)
From (1) and (2)
v1 (t )i1 (t ) v2 (t )i2 (t ) 0
(3)
The voltage v1does not depend on i1 or i2 but it depends only on v2
Ideal transformer
For multiple windings
n2 i2
n1i1 n2i2 n3i3 0
+
v2
v1 v2 v3
n1 n2 n3
-
i1
Ideal i3
+
+
v1
v3
-
n1
n3
Fig. 8
(equal volt/ turn)
Ideal transformer
Impedance transformation
Ideal i
2
i1
+
+
v1
v2
-
n1
Rin
n2
( n1 n2 )v2
-
v1
Rin n
i1 ( 2 n )i2
1
Rin
n1
n2
2
RL
RL
n1 2
n2
v2
i2
v 2 RL i2
Impedance transformation
In sinusoid stead state
Ideal i
2
i1
+
+
v1
v2
-
Z in
n1 : n2
V1
Zin ( j )
I1
n1
n2
2
ZL
Fig. 9
-
V2
I2
n1
n2
2
Z L ( j )
Controlled sources
Controlled sources are used in electronic device modeling. There four kinds
of controlled source .
i1
+
v1
-
0
Current controlled current source
Voltage controlled current source
Voltage controlled voltage source
Current controlled voltage source
i2
i1 0
+
i1
v2
-
+
v1
-
+
v1
+
-
+
v1
-
i2
i1 0
v2
g m v1
-
i2
i1 0
+
v2
-
i2
+
v1 0
-
rm i1
+
-
v2
Fig. 10
Controlled sources
Current controlled current source : Current ratio
i2
i1
Voltage controlled current source :
i2
gm
v1
Voltage controlled voltage source :
Current controlled voltage source :
Transconductance
Voltage ratio
v2
v1
Transresistance
v2
rm
i1
Controlled sources
Example1
Determine the output voltage from the circuit of Fig.11
1
Mesh 1
( Rs R1 )i1 vs
R1
v1 i1 R1
vs
Rs R1
Mesh 2
Rs
R2
i1
+
v_s
+
R1
+
v1
-
v2 v1
1'
RL
RL
vL i2 RL
v2
v1
R2 RL
R2 RL
RL
R1
vs
R2 RL Rs R1
2
+
i2 RL
vL
-
-
2'
Fig.11
Controlled sources
Example 2
Determine the node voltage from the circuit of Fig.12
1
2
+
G1
is
C1
C2
v1
+
G2
v2
-
1'
Fig.12
i2 g m v1
2'
KCL
dv1
d (v1 v2 )
G1v1 C1
C2
is
dt
dt
d (v2 v1 )
C2
G2 v2 i2
dt
(1)
Controlled sources
(1) (2)
Diff. (3)
d (v2 v1 )
G2 v2 g m v1 0
dt
dv
(G1 g m )v1 C1 1 is G2 v2
dt
dv1
d 2 v1 dis
dv
(G1 g m )
C1 2
G2 2
dt
dt
dt
dt
C2
from (1) dv2
then
d 2 v1
dt 2
1
dt
C2
(2)
(3)
(4)
dv1
(
C
C
)
G
v
i
2
1 1
s
1
dt
G1 g m G2 G1 G2 dv1 G1G2
G2
1 dis
v
is (5)
1
C1
C1 C2 dt C1C2
C1 dt C1C2
Controlled sources
The initial conditions
v1 (0) V1 , v2 (0) V2
From (3)
dv1
1
(0) is (0) G2V2 ( g m G1 )V1
dt
C1
From (5) and (6)
v1 (t ) and
v2 (t ) can be solved
(6)
Controlled sources
Other properties
The instantaneous power entering the two port is
p(t ) v1 (t )i1 (t ) v2 (t )i2 (t )
Since either v1 (t ) or
i1 (t ) is zero thus
p(t ) v2 (t )i2 (t )
If
R2 is connected at port 2
Therefore
v2 i2 R2
p(t ) i22 R2
Power entering a two port is always negative
Controlled sources
Example 3
Consider the circuit of Fig. 13 in sinusoid steady-state. Find the input
impedance of the circuit.
1
I1
1'
IL
2
+
Is
V
I 2 I1
2'
Z in
Fig. 13
ZL
Controlled sources
I s I1
I 1 I 1 I L
IL
I1
1
ZLIL
V
Z in
(1 ) Z L
Is
I1
Note if 1 the input impedance can be negative and this two port
Network becomes a negative impedance converter.