Coupling Element and Coupled circuits

1
Coupling Element and Coupled circuits

• Coupled inductor
• Ideal transformer
• Controlled sources

2
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
3
Coupled inductor

• Magnetic flux is produced by each coil by the
  functions

Where and are nonlinear function
of and
By Faradays law
4
Coupled inductor

• Linear time-invariant coupled inductor

If the flux is a linear function of currents
and
In sinusoid steady-state
Note that the signs of and are
positive but the sign for M can be
5
Coupled inductor
Dots are often used in the circuit to indicate
the sign of M
Fig. 2 Positive value of M
6
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.
7
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
8
Coupled inductor
Fig 3 Three-winding inductor
9
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
10
Coupled inductor
It is useful to define a reciprocal inductance
matrix
which makes
where
Thus the currents are
11
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.
12
Coupled inductor
Example 2
Fig. 5 shows two coupled inductors connected in
series. Determine the Equivalent inductance
between the input terminals.
Fig. 5
H
13
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
14
Coupled inductor
Example 4
Two coupled inductors are connected in parallel
in Fig 6. Determine the Equivalent inductance.
Fig 6
15
Coupled inductor
The currents are
KVL
By integration of voltage
Therefore
H
Note
for parallel inductors
16
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
Fig. 7
17
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
18
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
19
Ideal transformer

• For multiple windings

(equal volt/ turn)
Fig. 8
20
Ideal transformer

• Impedance transformation

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Impedance transformation
In sinusoid stead state
Fig. 9
22
Controlled sources
Controlled sources are used in electronic device
modeling. There four kinds of controlled source .

• Current controlled current source
• Voltage controlled current source
• Voltage controlled voltage source
• Current controlled voltage source

Fig. 10
23
Controlled sources
Current controlled current source
Current ratio
Voltage controlled current source
Transconductance
Voltage controlled voltage source
Voltage ratio
Current controlled voltage source
Transresistance
24
Controlled sources
Example1
Determine the output voltage from the circuit of
Fig.11
Mesh 1
Fig.11
Mesh 2
25
Controlled sources
Example 2
Determine the node voltage from the circuit of
Fig.12
Fig.12
KCL
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Controlled sources
Diff. (3)
from (1)
then
27
Controlled sources
The initial conditions
From (3)
From (5) and (6) and
can be solved
28
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
29
Controlled sources
Example 3
Consider the circuit of Fig. 13 in sinusoid
steady-state. Find the input impedance of the
circuit.
Fig. 13
30
Controlled sources
Note if the input impedance can be
negative and this two port Network becomes a
negative impedance converter.