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Coupling Element and Coupled circuits Coupled inductor Ideal transformer Controlled sources.
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Transcript of Coupling Element and Coupled circuits Coupled inductor Ideal transformer Controlled sources.
Coupling Element and Coupled circuits Coupled inductor Ideal transformer Controlled sources
Coupling Element and Coupled circuitsCoupled 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
2i1i+ +
- -
1v2v
Fig.1 Coupled coil and reference directions
Coupled inductor
Magnetic flux is produced by each coil by the functions
),( 2111 iif 2 2 1 2( , )f i i
Where and are nonlinear function of and 1f 2f 1i 2i
By Faraday’s law
1 1 1 1 21
1 2
d f di f div
dt i dt i dt
2 2 1 2 2
21 2
d f di f div
dt i dt i dt
Coupled inductor
Linear time-invariant coupled inductor
2 1 22 2( ) ( ) ( )t Mi t L i t
If the flux is a linear function of currents
1 11 1 2( ) ( ) ( )t L i t Mi t
1 21 11
di div L M
dt dt 1 2
2 22di di
v M Ldt dt
and
In sinusoid steady-state
1 11 1 2V j L I j M I 2 1 22 2V j MI j L I
Note that the signs of and are positive but the sign for M can be11L 22L or
Coupled inductorDots are often used in the circuit to indicate the sign of M
2i1i+
-1v
+
-2v
H1
H2
Fig. 2 Positive value of M
Coupled inductorCoefficient of coupling
The coupling coefficient is
11 22
| |Mk
L L
If the coils are distance away k is very small and close to zero and equalto 1 for a very tight coupling such for a transformer.
Coupled inductor
Multi-winding Inductors and inductance Matrix
1 11 1 12 2 13 3 ..L I L I L I For more windings the flux in each coil are
2 21 1 22 2 23 3 ..L I L I L I
3 31 1 32 2 33 3 ..L I L I L I
are self inductances and 11 22 33, ,L L L
12 21 13 31 23 32, ,L L L L L L are mutual inductances
In matrix form
φ Li
Coupled inductor
3
2
1
1
2
3
i
i i
i
333231
232221
131211
LLL
LLL
LLL
L
2i1i
+
-
11
dv
dt
+
-
3i + -
22
dv
dt
33
dv
dt
Fig 3 Three-winding inductor
Coupled inductorInduced voltage
dt
d iLv
The induced voltage in term current vector and the inductance matrix is
Example 1
Fig. 4 shows 3 coils wound on a common core. The reference direction of current and voltage are as shown in the figure. Since and has thesame direction but are not therefore is positive while and
1H 2H
3H 12L 13L
23L are negative. 1i1v
+
-
2i+
-2v
3i +3v
-
1H
2H
3HFig. 4
Coupled inductor
1Li
1 11 1 12 2i
It is useful to define a reciprocal inductance matrix
which makes
2 21 1 22 2i
22 11 1211 22 12 21,
det L det L det L
L L Land
where
Thus the currents are
1 11 1 12 2 1
0 0
( ) ( ') ' ( ') ' (0)t t
i t v t dt v t dt i 2 21 1 22 2 2
0 0
( ) ( ') ' ( ') ' (0)t t
i t v t dt v t dt i
Coupled inductorIn sinusoid steady-state
11 121 1 2I V V
j j
21 222 1 2I V V
j j
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
i 1i
2i
+
-
+
-
1v
2v
v
+
-
1 5L
2 2L
3M
1 11 1 2 1 2
2 1 22 2 1 2
5 3
3 2
L i Mi i i
Mi L i i i
1 2 1 2,i i i v v v
Example 2
Fig. 5 shows two coupled inductors connected in series. Determine the Equivalent inductance between the input terminals.
Fig. 5 1 2d dd
dt dt dt
0)0(
1 2 1 28 5 13i i i
13i
L
H
Coupled inductor
i 1i
2i
+
-
+
-
1v
2v
v+
-
1 5L
2 2L
3M
1 11 1 2 1 2
2 1 22 2 1 2
5 3
3 2
L i Mi i i
Mi L i i i
1 2 1 2,i i i v v v
Example 3
Fig. 6 shows two coupled inductors connected in series. Determine the Equivalent inductance between the input terminals.
Fig. 6 1 2d dd
dt dt dt
0)0(
1 2 1 22i i i
1Li
H
Note 11 22 2 | |L L L M for series inductors
Coupled inductor
i 1i
2i
+
-
+
-
1v
2vv
+
-
1 5L
2 2L
3M
2
23
35det
2
Ldet22
11
L
1212
33
5 3detLdet
3 2
L
Example 4
Two coupled inductors are connected in parallel in Fig 6. Determine the Equivalent inductance.
Fig 611
225
55 3detL
det3 2
L
Coupled inductor
1 2 1 1( ) ( ) (0) (0) 0v t v t and
1 2( ) ( )t t
1 2 1 22i i i
The currents are
1 11 1 12 2 1 22 3i
2 21 1 22 2 1 23 5i
KVL
By integration of voltage
Therefore
1Li
H
Note 11 22 122 | | for parallel inductors
Ideal transformerIdeal transformer is very useful for circuit calculation. Ideal transformerIs 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
2i1i+
-1v
+
-2vFig. 7
Ideal transformer
1 1 2 2,n and n
11
dv
dt
Figure 7 shows an ideal two-winding transformer. Coils are wound on idealMagnetic core to produce flux. Voltages is Induced on each winding.
If is the flux of a one-turn coil then
Since and we have2
2d
vdt
1 1
2 2
( )(1)
( )
v t n
v t n
In terms of magnetomotive force (mmf) and magnetic reluctance
1 1 2 2
mmf
n i n i
Ideal transformer
1 1 1 1 0n i n i
If the permeability is infinite becomes zero then
1 2
2 1
( )(2)
( )
i t n
i t nand
From (1) and (2)
1 1 2 2( ) ( ) ( ) ( ) 0 (3)v t i t v t i t
The voltage does not depend on or but it depends only on 1v 1i 2i 2v
Ideal transformer
For multiple windings
Ideal
+
-
+ +
--1v
2v
3v
1i
2i
3i
1n
2n
3n
0332211 ininin
31 2
1 2 3
vv v
n n n (equal volt/ turn)
Fig. 8
Ideal transformer
Impedance transformation
Ideal
+ +
--1v
1i 2i
1n 2n
2v
inR
LR
2
22
2
2
1
1
2
1
1
2
2
1
)(
)(
i
v
i
v
i
vR
nn
nn
nn
in 22 iRv L
12
2nin LnR R
Impedance transformation
1 1
2 2
2 21 2
1 2
( ) ( )n nin Ln n
V VZ j Z j
I I
In sinusoid stead state
Ideal
+ +
--1v
1i 2i
1 :n 2n
2v
inZ
LZFig. 9
Controlled sources Controlled sources are used in electronic device modeling. There four kindsof controlled source .
Current controlled current source Voltage controlled current source Voltage controlled voltage source Current controlled voltage source
1i
1 0v +
-
2i
2v+
-1i
1 0i
1v
2i
2v+
-1mg v
+
-
1 0i
1v
2i
2v+
-1v
1 0i
1 0v
2i
2v+
-
+
-
+-
1mr i+- Fig. 10
Controlled sources
Current controlled current source :1
2
i
iCurrent ratio
Voltage controlled current source :2
1m
ig
vTransconductance
Voltage controlled voltage source : 2
1
v
v Voltage ratio
Current controlled voltage source : 2
1m
vr
iTransresistance
Controlled sources
+_sv
1
1'
sR
1R
2
2'
2R
LR+
-2 1v v
+
-
Lv1i
2i-1v
+ss viRR 11)(
11 1 1
1s
s
Rv i R v
R R
Example1
Determine the output voltage from the circuit of Fig.11
Fig.11
Mesh 1
Mesh 22 2 1
2 2
1
2 1
L LL L
L L
Ls
L s
R Rv i R v v
R R R R
R Rv
R R R R
Controlled sources
1
1'
1G
2
2'
2G
+
-
1v+
-
2vsi1C
2 1mi g v
2C
1 1 21 1 1 2
( )(1)s
dv d v vG v C C i
dt dt
Example 2
Determine the node voltage from the circuit of Fig.12
Fig.12
KCL
2 12 2 2 2( )d v v
C G v idt
Controlled sources2 1
2 2 2 1( )
0 (2)md v v
C G v g vdt
)2()1( 1
1 1 1 2 2( ) (3)m sdv
G g v C i G vdt
Diff. (3)2
1 1 21 1 22
( ) (4)sm
didv d v dvG g C G
dt dt dtdt
from (1) 2 11 2 1 1
2
1( ) s
dv dvC C G v i
dt C dt
21 21 1 2 1 1 2 2
121 1 2 1 2 1 1 2
1(5)m s
sG g G did v G G dv G G G
v iC C C dt C C C dt C Cdt
then
Controlled sources
1( )v t
2211 )0(,)0( VvVv
12 2 1 1
1
1(0) (0) ( ) (6)s m
dvi G V g G V
dt C
The initial conditions
From (3)
From (5) and (6) and can be solved2 ( )v t
Controlled sources
Other properties
)()()()()( 2211 titvtitvtp The instantaneous power entering the two port is
Since either or is zero thus)(1 tv 1( )i t
2 2( ) ( ) ( )p t v t i tIf is connected at port 22R 2 2 2v i R
Therefore22 2( )p t i R
Power entering a two port is always negative
Controlled sourcesExample 3
Consider the circuit of Fig. 13 in sinusoid steady-state. Find the inputimpedance of the circuit.
1 1' 2
2'
+
-
sI2 1I I
inZ
LZV
1I LI
Fig. 13
Controlled sources
11
11
1
L
L
s
II
III
II
LLL
sin Z
I
IZ
I
VZ )1(
1
Note if the input impedance can be negative and this two portNetwork becomes a negative impedance converter.
1