Coupling Element and Coupled circuits

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					Coupling Element and Coupled circuits

n   Coupled inductor
n   Ideal transformer
n   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

             Fig.1 Coupled coil and reference directions
Coupled inductor
Magnetic flux is produced by each coil by the functions

Where       and        are nonlinear function of     and

By Faraday’s law
Coupled inductor
Linear time-invariant coupled inductor

    If the flux is a linear function of currents


 In sinusoid steady-state

 Note that the signs of      and         are positive but the sign for M can be
Coupled inductor
Dots are often used in the circuit to indicate the sign of M

                   Fig. 2 Positive value of M
Coupled inductor
Coefficient of coupling

  The coupling coefficient is

 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

                   are self inductances and

                                      are mutual inductances
 In matrix form
Coupled inductor

                   Fig 3 Three-winding inductor
Coupled inductor
Induced voltage
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 the
  same direction but     are not therefore      is positive while   and
      are negative.

      Fig. 4
Coupled inductor
It is useful to define a reciprocal inductance matrix

which makes


 Thus the currents are
Coupled inductor
In sinusoid steady-state

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.

Fig. 5

Coupled inductor
Example 3

Fig. 6 shows two coupled inductors connected in series. Determine the
Equivalent inductance between the input terminals.

Fig. 6


 Note                            for series inductors
 Coupled inductor
Example 4

 Two coupled inductors are connected in parallel in Fig 6. Determine the
 Equivalent inductance.

   Fig 6
Coupled inductor
The currents are


By integration of voltage



 Note                           for parallel inductors
 Ideal transformer
 Ideal transformer is very useful for circuit calculation. Ideal transformer
 Is a coupled inductor with the properties

        q dissipate no energy
        q No leakage flux and the coupling coefficient is unity
        q Infinite self inductances

Two-winding ideal transformer

    Fig. 7
  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                    and                   we have

      In terms of magnetomotive force (mmf) and magnetic reluctance
Ideal transformer
If the permeability     is infinite     becomes zero then


From (1) and (2)

The voltage        does not depend on    or    but it depends only on
Ideal transformer
For multiple windings

                                 (equal volt/ turn)

                        Fig. 8
Ideal transformer
Impedance transformation
Impedance transformation
In sinusoid stead state

                           Fig. 9
Controlled sources
Controlled sources are used in electronic device modeling. There four kinds
of controlled source .
              q Current controlled current source
              q Voltage controlled current source
              q Voltage controlled voltage source
              q Current controlled voltage source

                                                                Fig. 10
Controlled sources

Current controlled current source : Current ratio

Voltage controlled current source :    Transconductance

Voltage controlled voltage source :     Voltage ratio

 Current controlled voltage source :    Transresistance
Controlled sources
Determine the output voltage from the circuit of Fig.11

Mesh 1

Mesh 2
Controlled sources
Example 2
Determine the node voltage from the circuit of Fig.12


   Controlled sources

       Diff. (3)

       from (1)
Controlled sources
 The initial conditions

 From (3)

 From (5) and (6)         and   can be solved
Controlled sources
Other properties

 The instantaneous power entering the two port is

  Since either      or       is zero thus

  If    is connected at port 2


  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.

                          Fig. 13
Controlled sources

Note if      the input impedance can be negative and this two port
Network becomes a negative impedance converter.

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