Definition of Network

A network serves to be an interconnectedness that rests amid nodes or computers apportioning resources inside their group or to additional group of systems. The mature model of a single computer attending to all of the organization’s computational needs has been substituted by one in which a big number of disconnect but co-ordinate computers carry out the job.

These systems are anticipated as computer networks. As per this, a computer network can make available for an authoritative communication middling in the midst of extensively broke up employees. Utilizing a network, it is straightforward for two or more people who exist far apart to inscribe a report jointly.


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Fig 1: Network Diagram


Definition of Lattice

A lattice contributes very well for an intermittent organization of points that shape an efficient structure for a range of geometric problems. Not only this, a lattice can come out to be a place of intersection points of an endless, customary n- dimensional grid. Moreover, a lattice is foreboded a whole lattice if each of its non empty subset has slightest upper bound and most enceinte.

The least and grooviest constituents of the lattice, if they stay alive are called up as the bounds of the lattice. They are announced by 0 and 1 correspondingly. A lattice which has together 0 and 1 is anticipated bounded lattice.

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Figure 2: Example of Lattices

What is Lattice Networks?

These are also foretold as Grid or Mesh network. Lattice networks are merely networks, where nodes are dressed in a rectangular lattice, directed to conquer the most important negative aspect of the ER model. Due to this only, the Lattice networks are an instance for allotted parallel computation, grid computing and wired circuits.

Lattice networks contribute for a rough calculation for networks whose nodes are aimlessly settled, but this does not present for a wonderful usual structure but furnishes for an approximated structure for the nodes. They are moderately inappropriate for hypothetical models in network analysis, for the reason that they appear to be artificial.

A lattice network is a symmetrical and balanced network with four arms. The arms lying in impedance ZA are called up series arms of the lattice network. The arms are found out to be dwelling of impedance ZB are foreboded shunt or diagonal arms. In this manner, the lattice network can be rearranged in the bridge structure as established in the figure which is very appropriate for the lattice network analysis. Lattice networks are practiced in filter sections and are also utilized in attenuators. On certain occasions, Lattice structures are expended in fondness to ladder structures in some extraordinary applications.


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Figure 3: (a) A Lattice Network (b) a lattice network rearranged as a bridge (c) Zd = 0 in a lattice network

Lattice Network Circuit Analysis

As a Lattice Network is a symmetrical network, we draw from expressions for the characteristic impedance (Z0) and propagation constant (γ). It is very expedient to make use of bridge structure of the lattice network for the estimate of propagation constant.

(A)  Characteristic impedance (Z0)

Think about the closed path 1-2-2'-1'-1, in the above fig 3 (b) implementing KVL we find

-ZA (I1) – Z0 (IR) – ZA (IS – I1 + IR) + E =O

-ZA · I1 – Z0 · IR – ZA · IS + ZA · I1 – ZA · IR = -E

(ZA) IS + (Z0 + ZA) IR = E        (1)

Mull over closed path 1-2'-2-1'-1, implementing KVL, we get

-ZB (IS – I1) + Z0 (IR) – ZB (I1 – IR) + E = 0

-ZB · IS + ZB · I1 + Z0 · IR – ZB · I1 + ZB · IR = -E

(ZB) IS – (Z0 + ZB) IR = E         (2)

From equation (1),


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From Equation (2),

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Corresponding equations (3) and (4),

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(E – ZA · IS) (Z0 + ZB) = (ZB · IS – E) (ZO + ZA)

E · Z0 + E · ZB – ZA · Z0 · IS – ZA · ZB · IS = ZB · Z0 IS – E · Z0 + ZA · ZB · IS – E · ZA

E (2 ZO + ZA + ZB) = IS [(Z0 + ZB) · ZA + ZB (Z0 + ZA)]

E (2ZO + ZA + ZB) = (Z0 + ZB) · ZA + (Z0 + ZA) · ZB

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Other than by the property of the symmetrical network, the input impedance of the network displaced in its characteristic impedance is equal to Z0.

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Placing Value  equationIn above equation,

   Z0 (2ZO + ZA + ZB) = Z0 · ZA + Z0 · ZB + 2ZA · ZB

2Z20 + Z0 · ZA + Z0 · ZB = Z0 · ZA + Z0 · ZB + 2ZA · ZB

2ZO2 = 2ZA · ZB

Z0 = √ZA · ZB                          (5)



(B)  In terms of open and short circuit impedances

For the computation of open and short circuit impedances fixing up bridge structure of the lattice network as evidenced in below figure


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Fig 4: open and short circuit impedances of symmetrical lattice network

Consider fig 4 (a), 

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Think about Fig 4 (B),

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Multiplying equations (6) and (7) we can engrave,

ZOC · ZSC = [(ZA + ZB) / 2] [2ZA · ZB / (ZA + ZB)] = ZA · ZB = Z02

Z0 = √ZOC · ZSC                                           (8)

(C) Propagation constant (γ)

For any symmetrical network, propagation constant can be conveyed as,

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Consider equations for present IR given by equation (3) and (4)

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But we be acquainted with that E = IS · Z0

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(D) Impedance ZA and ZB in terms of characteristic impedance (Z0) and propagation constant (γ)

Consider equation,



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eγ (Z0 – ZA) = Z0+ ZA

Z0 (eγ– 1)  = ZA (eγ + 1)

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ZA = Z0 · tanh γ / 2                       (11)


Likewise, image   

eγ(ZB – Z0) = ZB + ZO

(eγ -1) ZB = Z0 (eγ + 1)image

ZB = Z0 · coth γ/2                                (12)

For this reason only, the lattice network with impedance conveyed in terms of characteristic impedance and propagation constant is as demonstrated in the below figure 5


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Fig 5: Symmetrical Lattice Network Impedances Intermsof Z0 and γ


Lattice Network Z Parameters

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When

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Consequently

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If the network is symmetric, then Za = Zd, Zb = Zc

Therefore

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When I2 = 0, V2 is the voltage across 2-2

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Replacing the value of V1from (1),

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Then

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If the network is symmetric, Za = Zd, Zb= Zc

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When the input port is unlock, I1 = 0,

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The network can be redrawn as established in figure 6

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Fig 6: The network can be redrawn

Interchanging the value of V2 into V1, we acquire

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If the network is symmetric, Za = Zd, Zb= Zc,

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If the network is symmetric, Za= Zd, Zb = Zc,

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Therefore

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From the above equations we encompass,

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If the network is symmetric, Za = Z

Synthesis of Lattice Networks

(A) With both ends dismissed in R

Think about a lattice network which gets ceased at both the ends in confrontation R as shown in fig 7.


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Fig 7. Lattice Network


The transfer task for this network are collapsed by,


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But Z21 = Z12 and Z22 = Z11

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But Z112 – Z212 = R2 condition for constant resistance

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Replace with values of Z21 and Z11in terms of Za and Zb,

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But ZaZb = R2 so  image

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At the moment input impedance of the lattice network is R and source impedance is also R. Half the source voltage will come into sight at port 1 which is V1. This half comes out in the equation of V2 / Vg so transfer function V2 / V1 can be published as,


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(B) With 1Ω termination

As deducted earlier,


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This is found out to be appropriate when Vg is not associated and therefore network turns a network with 1Ω termination, with R = 1Ω

The network is demonstrated in fig 8.


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                          Fig 8: The network


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Get hold of the Lattice equivalent of a symmetrical T network shown in below figure



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A two-port network can be realized as a symmetric lattice if it is shared and symmetric. The Z parameters of the network


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