A. Source of elect energy:

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A. Source of elect energy

• There are two possible types of power sources
  used in network studies a voltage source and a
  current source (fig 1.1) and they are represented
  as shown

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(a) Voltage Source
(b) Current Source
-
i(t)
?(t)
(b)
(a)
Fig 1.1
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• The sources can again be classified as
  independent and dependent types
• Independent voltage source An ideal
  independent voltage source is one which

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• maintains a constant (specified) voltage across
  the terminals irrespective of the current it
  supplies or receives.

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• In reality even the best possible source will
  have some voltage drop with the increasing
  current due to its internal resistance for an
  ideal source the v-i characteristics will be as
  shown in Fig 1.2.

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• We can see that an ideal voltage source with a
  perfect short circuit at the terminals is an
  impossibility and is a contradiction of terms
  (Fig. 1.3).

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a
_
Shorted
b
Fig 1.3
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• An ideal current source is an element that will
  maintain a specific rate of flow of charge
  regardless of the voltage that appears across its
  terminals.

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• The v-i characteristics is as shown in Fig. 1.4.

a
i(t)I
?
i ?
b
? ?
Fig.1.4
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• A current source can never be an open circuit
  it must have a path for the current to return.

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• Thus a real voltage source will be represented
  by a source voltage V along with an internal
  resistance Ri (? 0, for network computation in
  many situations)

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• and a real current source by a source current
  along with a shunted internal resistance (of
  appreciable value)

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• Dependent Sources

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• If the voltage of a voltage source and the
  current through a current source depend on some
  other variable (say a voltage or current) in some
  part of the network, then such sources are
  dependent sources.

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• There are four types of dependent sources (fig
  1.5) as shown.

-
?K(t)?ij(t)
?K(t)??j(t)
-
ijCurrent through jth element in the network
(VCVS)
(CCVS)
iK(t)A ij(t)
iK(t)??j(t)
VjVoltage across an element j of the network
(VCCS)
(CCCS)
Fig. 1.5
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• (i) Current controlled voltage source (CCVS)
• (ii) Voltage controlled voltage source (VCVS)
• (iii) Current controlled current source (CCCS)
  and
• (iv) Voltage controlled current source (VCCS)

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• Ex 1.1 If Vc -10v and I1 3A Calculate I2
  through R2 and voltage of the VCVS

Solution
50 I1 50 ? 3A 150A I2
?/VCVS 0.1Vc -1V
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• Series and parallel connections of sources
• i) Series connections of voltage sources

Fig1.6
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• Parallel connections of ideal voltage sources, is
  not defined, except when they have identical
  voltages.

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• ii) Parallel connection of current sources

i(t)
i1(t)
i2(t)
i3(t)
in(t)
Fig. 1.7
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• iii) Replacing parallel / series connections of
  a set of ideal voltage/current sources.
• ?1(t) ?2(t) --- ?n(t) ?(t)

-
-
-
-
-
?(t) ?
?3(t)
?1(t)
?2(t)
?n(t)
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• Fig. 1.8
• i1(t) i2(t) i3(t) i4(t) i(t)

?
i(t) i1(t)
i2(t) i3(t)
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• iv) Voltage and current sources in series and
  parallel

-
?(t)
-
i(t)
?
?(t)
-
i(t)
?
i(t)
?(t)
Fig. 1.9a
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• Ex 1.2 Calculate ?x, iy, iz and the power
  supplied by the two sources in Fig. 1.10.
• ?x -2 ? 3V -6V
• iy 2/4 A 0.5A
• iz -(2-0.5) A -1.5A

?x
iz
3?
-
2V
2A
4??
iy
Fig. 1.10
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• Power supplied by voltage source
• 2 ? -1.5 W -3 W
• Voltage across the current source
• (2 6) V 8 V
• Power supplied by current source 8 ? 2 W
  16 W

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BR3
BR3
BR2
v(t)
a
b
c
Net Work
?
BR2
Net Work
BR1
-
-
-
-
Branch
Fig. 1.9b
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• Ex 1.3 It is desired to find voltage ?0 defined
  at terminals a-b in the circuit shown in Fig.
  1.11 assuming that all resistors have a value of
  unity.

R3
R4

a
R1
-
10v
v0
R5
R2
(a)
b
-
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• To do this we first represent the voltage sources
  as the parallel connection of two identical
  voltage sources and then break the connection as
  shown is successive steps, replacing the
  non-ideal voltage sources by non-ideal current
  sources and vice-versa.

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?
1
a
v1
-
a
1
10
?
1
-
v0
1
1
10
-
b
b
(c)
(b)
?
a
a

0.5
1
?
10/3
v0
-
v0
1
5
1
1
1
1.5
10
10
-
-
b
b
(d)
(e)
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Finally the parallel connected sources and
resistors are combined to obtain the last
equivalent circuit. From which v0 13.333 ?
0.375 V 5V
a

v0
13.33
0.375
_
( f )
b
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• Exercise 1.1
• For the circuit shown in Fig 1.11a, value of
  resistors are defined in the left columns of the
  table below. For each set of values, find the
  voltage v0

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• Replacing a single ideal current source with
  several ideal current sources.

i(t)
a
a
i(t)
?
i(t)
b
N/W
N/W
b
i(t)
(b)
(a)
Fig 1.12
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• Ex 1.4 Find the current i0 defined in the
  circuit shown in Fig.1.13a, assuming that all
  resistors have a value of unity and Is 5A.

R3
R1
R5
R4
Is
R2
R6
i0
Fig. 1.13a
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• Here we represent the current source as a series
  connection of two current sources and add a
  connection from the node between them to node a
  as shown in Fig. 1.13b.

1
1
R1
1
5
a
?
1
1
1
5
i0
Fig. 1.13b
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• Each of the non-ideal current sources may be
  converted to equivalent non-ideal voltage source.

1
1
5
-
1
?
a
1
1
5
-
1
i0
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• By repeated conversion we finally obtain
  Fig.1.13e,whence i02.273 A.

5/4
a
5/3
-
3/4
1
1
3
a
?
1
1
1
i0
5
-
-
1
i0
5
(e)
(d)
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• Ex 1.2
• For the circuit shown in Fig. 1.13a, values of
  is and the resistors (?) are given in the table.
  For each set of values, find i0.

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• B. Network Elements
• There are two types of elements in electrical
  networks active and passive-
• Passive elements are resistances, inductances,
  capacitances and coupled inductances
  (transformers)

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• Active elements are gyrators, dependent
  sources, transistors, op-amps, GIC (general
  impedance converters), NIC (negative impedance
  converters), FDNR (frequency dependent negative
  resistance) etc. There are separate energy
  sources in these elements, without which they can
  not function.

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• Linear and non-linear elements
• A system is linear if superposition theorem
  holds good for the input/ output relationship

output
Input x(t)
Linear System
y(t)
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• If x1? y1
• and x2? y2
• then ax1 bx2 ? ay1 by2
• For a linear resistor, Ohms law is an
  over-simplified observation made on commonly
  available metal conductors over a certain range
  of temperature,

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• V-I relationship is assumed linear over a range
  of operation (Fig. 1.14)

Linear in a small range
v
v/2
v2
Non Linear Resistors
v/1
v1
i0
i
Fig. 1.14
? i
(b)
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• Inductors
• A linear inductor will have a constant
  inductance L, such that flux linkage ?Li.
  Sometimes ? approaches approximately a constant
  value and this leads to a decline in the flux
  building process when the current is increased
  (Fig. 1.15).

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Nonlinear inductor
?
Linear inductor
i
Fig. 1.15
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• Sometimes we are interested in the relation
  between incremental voltages and currents and
  define the effective or incremental resistance
  /inductance around any operating point i0 (say)
  as

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• 
  
• Similarly for an inductance
• There may be different types of nonlinearities
  that may be present in a system element.

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• They are for example
• Relay type
• Fixed dead band only
• Relay with dead band
• Hysteresis type
• Hysteresis with delay type
• Diode type,
• Saturation type

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• (a) Relay type

F(x)?
F0
x?
-F0
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Relay with dead band
F0
-x0
x0
x?
-F0
Fixed dead band only
F(x)
0
-x0
x0
x?
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?f(x)

(d) Hysteresis type
-x1
x1
0
x?
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(e) Hysteresis with dead band

f(x)?
-x2 -x1
x
x2
x1
0
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f(x)

• (f) Diode type
• (g) Saturation type

x?
0
f(x)
x?
0