# 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

Coupled inductor
Linear time-invariant coupled inductor

If the flux is a linear function of currents

and

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

where

Thus the currents are
Coupled inductor

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

H
Coupled inductor
Example 3

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

Fig. 6

H

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

KVL

By integration of voltage

Therefore

H

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

and

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

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
Example1
Determine the output voltage from the circuit of Fig.11

Mesh 1

Fig.11
Mesh 2
Controlled sources
Example 2
Determine the node voltage from the circuit of Fig.12

Fig.12

KCL
Controlled sources

Diff. (3)

from (1)
then
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

Therefore

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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