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