C Distributed And Lumped Parameter Systems'

1

• C) Distributed And Lumped Parameter
  Systems.
• Many network elements consist of R, L and C in
  a distributed manner in space (say, a
  transmission line or a rheostat or an inductor
  at high frequencies).

2

• A transmission line between a generating station
  and the load has inductance (due to the magnetic
  field set up by the current) and capacitance
  (between two conductors and between conductor and
  ground).

3

• When a source of energy is connected to the
  transmission line, energy is transported through
  all the parts. However, points along the line do
  not have the same electrical conditions at the
  same instant of time. This means that the
  parameters are distributed along the line.

4

• Fig.1.20a shows a typical transmission line

Fig. 1. 20 (a)
5

• Similarly in a rheostat the inter-turn
  capacitance comes into play (may be along with
  the ground capacitance at high frequencies). The
  inductance, if considered, will come in series
  with the elemental resistance as shown
  (Fig.1.20b).

6
Fig 1.20 (b)
7

• In all our analysis and synthesis work at this
  stage, we shall consider parameters to be
  approximated as lumped.

8

• D) Resistive Ladder Network
• If we have a ladder network as shown in Fig. 1.21

R7
R5
R3
R1
v? v0
-
vs
R4
R6
R8
R2
9

• We compute the branch currents and node voltages
  in successive steps starting from the furthest
  end.
• Let the voltage across R8 be the output voltage
  ?0

10
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11

• For a given voltage Vs/ therefore, all the
  currents and voltages are to be scaled up by a
  factor Vs/ / Vs.. To start with the computation
  one may assume an initial current of 1A to flow
  through R8 or an initial voltage of V01V.

12

• Single and multiport network
• A terminal pair across which the network
  variables are either measured or a source is
  connected or a network element of interest is
  connected is called a port.

13

• A network and its port

14

• Here 1-1/ is the port through which excitation is
  given (Fig.1.22)

R1
1
C

L
R2
1/
Fig. 1.22
15

• Sometimes, we may be interested in injecting
  current into the circuit through some terminal
  pair and measuring the output at some other
  terminal pair under various conditions (Fig.
  1.23).

16
I2
Z1
Z3
Z5
V2
I1
2
1


Z4
Z2
1/
V1
2/
Fig. 1.23
17

• Then we consider the network to be a 2-port
  system and determine the following parameters
  under different conditions.

18

• Open circuit parameters or z- parameters
• b) Short-circuit parameters or y
  parameters
• c) ABCD parameters

19

• Open circuit parameters
• If we apply a voltage v1 with 2-2/ open or apply
  v2 with 1-1/ open we get

2
1
2'
1'
20

• 
  driving point impedance at
  1-1/
• Z11
• 
  driving point impedance at
  2-2/ Z22

21

• If we inject current I1 and keep2-2' open or I2
  and keep 1-1' open and measure the open end
  voltage, we get

22

• transfer impedance
  (at 2-2/) z12
• transfer impedance
  z21

23

• For linear lumped passive bilateral network z12
  z21 .
• One may therefore represent, applying
  superposition theorem, when both the voltage
  sources are active,

24

• i.e, when V1,V2, I1 and I2 are present
• V1 Z11 I1 Z12 I2
• V2 Z12 I1 Z22 I2 -------- (1.1)

25

• b) Short Circuit Parameters
• If we apply a voltage V1, with terminals
  2-2/ shorted or apply a voltage V2 with 1-1/
  shorted, we get,

26

• driving point
  admittance at 1-1/
• driving point
  admittance at 2-2/

27

• transfer
  impedance

28

• Ex. 1.6 Determine the open circuit and short
  circuit parameters of the given network
  (Fig.1.24)

R1
1
2
20
5
10
20
10
1/
2/
29

• All resistance are in ohms
• 
  
• 
  
• 
  
  

30

• If we generalize the case of a 2-port network
  to a multiport network, then eqn (1.1) can be
  written as

31

• 
  
• 
  

(1.2)
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• 
  
• 
  
  
• where n is the number of ports.

33

• For a given network, we could also write the
  equations in terms of the admittance parameters
  (or short circuit or y parameters) as
• I1 y11V1 y12V2 ----- y1nVn
• I2 y21V1 y22V2 ---- y2nVn
• In yn1V1 yn2V2 ---- y nnVn (1.4)

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• or in matrix form
• 
  
  
  
  

35

• If we solve eqn (1.3), we may write
• If we compare this equation with (1.5) , we get
• 
  
• 
  
• 
  
  

36

• Thus we find,
• Please note that
• 
  
• 
  
  

37

• Now in eqn (1.6) if all Vs are zero except Vj,
  then
• 
  
• 
  
• 
  
  

38

• 
  
  
• Similarly if Vk is the only source in the network
  the current in loop j is
• 
  
  

39

• These equations apply to the network as shown
  Fig. 1.25
• 
  
• 
  
• 
  
  

-
-
Ik
Vj
N
Ij
Vk
N
Fig. 1.25 (a)
Fig. 1.25 (b)
40

• In network (a) Vj produces current Ij if Vj is
  moved to loop K, so that VkVj as in Fig.(b),
  What will be the value of Ij under this condition
  ?

41

• Since Z or Y is symmetric, the cofactors
  ??kj?jk and hence
• 
  
• This is stated as the ratio of response to
  excitation is invariant to an interchange of the
  position
• 
  
  

42

• in the network of the excitation and the
  response. Network for which this condition holds
  are said to be reciprocal.
  
• 
  
  

43

• Transmission parameters (A,B,C,D,)
• 
  
  

Fig.1.26
44

• For a two port network these parameters
  (Fig.1.26) relate the voltage and current at one
  port to the voltage and current at the other
  port.
• 
  
• 
  
  

45

• In equation form
• V 1 AV2 BI2
• I1 CV2 DI2 (1.11)
• A,B,C,D are called the transmission parameters.
• 
  
  

46

• These can be defined in terms of ratios of
  voltages and currents under short circuit and
  open circuit conditions
• 
  
• 
  
• 
  
  

47

• If we apply a voltage at 1-1/ and keep the port 2
  open and measure the voltage V2, then we get A
  from the above ratio.
• 
  
• 
  
  

48

• 
  
• i.e., under this open condition we measure the
  ratio of input current and output voltage.
• 
  
  

49

• Similarly
• i.e., under this open condition of port 2, we
  measure the input voltage and output current.
• 
  
• 
  
  

50

• Thus
• 
  
• 
  
• It can be shown from the short circuit
  parameter and eqn (1) by simple manipulation that

51

• 
  
• 
  
  

52

• Exercise 1.4
• i) Prove that BC-AD1
• ii) Find the y, z and A,B,C,D parameters for the
  network shown in Fig. 1.27.
• 
  
  

53

• 
  
• 
  
• 
  
  

Z
1
2
1/
2/
Fig. 1.27(a)
1
2
Y
2/
1/
Fig. 1.27(b)
54

• 
  
• 
  
• 
  
  

1?
1?
2
1
1/2?
2?
1'
2'
Fig. 1.27 (c)
55

• 
  
• 
  
  

Za
1
2
Zb
1/
2/
Fig. 1.27 (d)
56

• 
  
• 
  
• 
  
  

I1
2?
I2
2
1


2?
V2
1?
3I1
V1
-
-
1'
2'
Fig. 1.27 (e)
57

• 
  
  

1?
1/2?


V1
V2
1?
1/2?
2V1
-
-
Fig. 1.27 ( f)
58

• 
  
  

3V1
1?
1
2
-



V1
V2
2?
1?
-
-
2/
1/
Fig. 1.27 (g)
59

• d) The hybrid parameters (h parameters)
• These are used in electronic circuits in modeling
  transistors.
• 
  
  

60

• The two port variables are related as
• V1 h11 I1 h12V2
• I2 h21 I1 h22 V2 (1.13)
• 
  
• 
  
  

61

• It can be see easily from the previous
  relationships that open circuit and short
  circuit parameters
  
• 
  
• 
  
  

62

• These parameters are dimensionally mixed, and
  hence called hybrid parameters.
  
• 
  
• 
  
  

63

• The inverse hybrid parameters, or g parameters
  are given by
• I1 g11 V1 g12 I2 (1.15)
• V2 g21 V1 g22 I2
• 
  
• 
  
  

64

• and

65

• Networks characterized by g and h parameters,
  which are equivalent to two-port network are
  shown in Fig 1.24a and b

66

• 
  
• 
  
  

I1
h11
I2


h21I1
V1

h12V2
h22
V2
-
-
-
Fig. 1.24 (a)
67
g22
I1
I2

V1
g21I2
g11

g21 V 1
V2
-
-
Fig. 1.24(b)
68

• Exercise 1.5
• i) A model for a transistor in the
  common-emitter connection is shown in Fig. 1.25a,
  for which
• V1 (rb re) I1 ?bcV2

69

• 
  
• Fig. 1.25 (a)
• 
  
  

I1
I2


rb
re
re
-
V2
V1
rd
?bcV2
-
-
70

• ii) Write the expression for I2 and determine all
  the h and g parameter for Figures b and c.
  Determine y and z parameters in Fig. 1.25c.
• 
  
  

2V1
1?

-

1?
2? ?
2V1
V2
V1
-
-
Fig. 1.25 (b)
71

• 
  
  

2V3
3I2
I2
1?
I1

-

V3
V1
V2
2?
-
-
-
Fig. 1.25c
72

• e) T and ? networks
• Two-port networks are quite often represented by
  simple equivalent networks of 3-elements as shown
  in Fig. 1.26.

73
Za
Zb
2
1


V2
Zc
-
1'
2 '
Fig. 1.26 (a)
74
Z3


1
2
Z2
V2
V1
Z1
-
-
2'
1'
Fig. 1.26 (b)
75

• We can very easily express y and z parameters in
  terms of the elements za, zb, zc, or z1, z2, z3
  of a T or ? network.
• 
  
• 
  
  

76

• For example, for the T- network
• z11 za zb
• z22 zc zb
• z12 zc
  (1.16)
• 
  
• 
  
  

77

• For a p-network
• 
  

78

• Exercise 1.6
• i) Determine the y-parameters for the
• T- network in Fig. 1.26 a
• ii) Determine the z-parameter for Fig.
  1.26b
• iii) Determine the ABCD parameters
  for Fig. 1.26 a,b
• 
  
• 
  
  

79

• iv) Determine the hybrid parameters (h and g) for
  Fig.1.26 a,b.