Lecture Notes on Network Theory, Study Guides, Projects, Research of Network Theory

Download Lecture Notes on Network Theory and more Study Guides, Projects, Research Network Theory in PDF only on Docsity!

GOVERNMENT COLLEGE OF ENGINEERING,KALAHANDI

DEPARTMENT OF ELECTRICAL ENGINEERING

Lecture notes on Network Theory

Submitted By

Soudamini Behera

BEES2211 Network Theory

MODULE- I

1. NETWORK TOPOLOGY: Graph of a network, Concept of tree, Incidence matrix, Tie-set matrix, Cut-set matrix, Formulation and solution of network equilibrium equations on loop and node basis.
2. NETWORK THEOREMS & COUPLED CIRCUITS: Substitution theorem, Reciprocity theorem, Maximum power transfer theorem, Tellegen’s theorem, Millman’s theorem, Compensation theorem, Coupled Circuits, Dot Convention for representing coupled circuits, Coefficient of coupling, Band Width and Q-factor for series and parallel resonant circuits.

MODULE- II

3. LAPLACE TRANSFORM & ITS APPLICATION: Introduction to Laplace Transform, Laplace transform of some basic functions, Laplace transform of periodic functions, Inverse Laplace transform, Application of Laplace transform: Circuit Analysis (Steady State and Transient).
4. TWO PORT NETWORK FUNCTIONS & RESPONSES : z, y, ABCD and h -parameters, Reciprocity and Symmetry, Interrelation of two-port parameters, Interconnection of two-port networks, Network Functions, Significance of Poles and Zeros, Restriction on location of Poles and Zeros, Time domain behaviour from PoleZero plots.

MODULE- III

5. FOURIER SERIES & ITS APPLICATION: Fourier series, Fourier analysis and evaluation of coefficients, Steady state response of network to periodic signals, Fourier transform and convergence, Fourier transform of some functions, Brief idea about network filters (Low pass, High pass, Band pass and Band elimination) and their frequency response.
6. NETWORK SYNTHESIS : Hurwitz polynomial, Properties of Hurwitz polynomial, Positive real functions and their properties, Concepts of network synthesis, Realization of simple R-L, R-C and L-C functions in Cauer-I,Cauer-II, Foster-I and Foster-II forms.

The theorem can’t be used to solve the network containing two or more sources that are not in series or parallel.

• First obtain the concerned branch voltage and through current given by Vxy Ixy
• The branch may be substituted by independent voltage source or current source shown in fig.respectively.

RECIPROCTY THEORUM

In a linear bilateral network if current flowing through any branch is I due to voltage source E, then the same current will flow when the position of voltage and ammeter are interchanged.in other case E and I are mutually transferable. Now transfer resistance =E/I

Y 11 Y 12 Y 13

Y = Y 21 Y 22 Y 23

Y 31 Y 32 Y 33

Y 12 = Y 21

Y 13 = Y 31

Y 23 = Y 32

Fig. (a) In this case E produces the current I 1 in resistance R 2.

Fig. (b) Now we interchange the position of voltage source E and ammeter that means voltage source E produce the same current (I) in resistance R 1.

Linear circuit – it is a one in which parameters remain constant i.e. they do not change w.r.t. current or voltage

Non-linear circuit – its properties change w.r.t. to current and voltage.

Bilateral circuit – it is one whose properties or characteristics are same in either direction.

Unilateral circuit – it is the circuit whose properties change w.r.t. direction of operation.

Passive network – it is a network which contains no source of emf or voltage source.

Active network – it is a network which contains one or more source of emf.

Dependent or Controlled source – in these type of sources voltage or current source is not fixed but is dependent on a voltage or current fixed at some other part of the circuit.

Ideal voltage source – it is a circuit element where the voltage across it is independent of current. In analysis a voltage source supplies a constant AC or DC voltage b/w its terminal where any current flow through it.

An ideal voltage source has internal resistance 0.

It is able to supply or absorb any amount of current.

Voltage drop in the source is 0.

For reciprocity network reciprocity theorem is applicable

Z = 𝛴𝑌𝑖^1

DUAL MILL MANS THEORUM

If several ideal current sources (I1 I 2 …. ), in parallel with several impedance z 1 z 2 z 3 Connected in series then the circuit can be replaced by a single ideal current IIn parallel with an impedance (z) such that

𝛴𝐼𝑖𝑍𝑖 𝛴𝑍𝑖 Z=𝛴𝑍𝑖

TELLEGEN’S THEORUM

In any network the sum of instantaneous power absorbed by various elements is always equal to 0. Therefore the total power delivered by various active sources is equal to total power absorbed by various passive element by branches of the network.

.Proof:-

Let Vs is the supply voltage Zs=Internal impedance/source impedance of the circuit =Rs+jXs Zl=Load impedance of the circuit =RL+jXL PL+ Power consumed by load

In the above circuit

I = 𝑉𝑠 𝑍𝑠+𝑍𝑙 = (^) (𝑅𝑠+𝑗𝑋𝑠)+(𝑅𝑙+𝑗𝑋𝑙)𝑉𝑠

= 𝑉𝑠 (𝑅𝑠+𝑅𝑙)+𝑗(𝑋𝑠+𝑋𝑙) PL = I^2 ZL

= 𝑉𝑠^2 {(𝑅𝑠+𝑅𝑙)+𝑗(𝑋𝑠=𝑋𝑙)}^2

According to AC Analysis:-

For maximum power transformation XL and XS are two variables which vary with frequencies where RL and Rs remain constant.

Now PL=PLmax when denominator is minimum. For denominator to be minimum Xs+XL= =>XL=-XS

According to DC analysis:-

If complex part is reduced to 0

PL = 𝑉𝑠

2 (𝑅𝑠+𝑅𝑙)^2 *RL

In this case load resistance RL is variable in nature so that PL=PLMax

𝜕𝑃𝑙 𝜕𝑅𝑙= => (^) 𝜕𝑅𝑙𝜕 [ 𝑉𝑠

(^2) 𝑅𝑙 (𝑅𝑠+𝑅𝑙)^2 ] = 0

=>(𝑅𝑠+𝑅𝑙)

(^2) ∗𝑉 (^2) −𝑉 (^2) 𝑅𝑙(2𝑅𝑠+2𝑅𝑙) (𝑅𝑠+𝑅𝑙)^4 =>Rs=RL

ZL=RL+jXL

ZL=RL-jXS

But, ZS= RS+jXS ZS^ =RS-jXS Hence ZL = ZS

So theorem is proved.

PLMax= 𝑉𝑠

(^2) 𝑅𝑙 (𝑅𝑠+𝑅𝑙)^2 =^

𝑉𝑠^2 𝑅𝑙 4(𝑅𝑙)^2

PLMax= 𝑉𝑠^2 4(𝑅𝑙)^2 /𝑆

Efficiency (ɳ) =𝑜𝑢𝑡𝑝𝑢𝑟 𝑝𝑜𝑤𝑒𝑟𝐼𝑛𝑝𝑢𝑡 𝑝𝑜𝑤𝑒𝑟

= (^) 𝑃𝑠+𝑃𝑙𝑃𝑙 1

= 𝐼

(^2) 𝑅𝑙 𝐼^2 𝑅𝑠+𝐼^2 𝑅𝑙 = (^) 𝑅𝑠+𝑅𝑙𝑅𝑙

In case of maximum power transfer

RS = RL

Step3- calculate the Rth across the open circuit terminal. Ifthere is a source then replace by its internal resistance.

Step4- if there is a current source it can be replaced by an open circuit.

Step5 – connect Rth in series withVth that indicates equivalent circuit

DOT CONVENTION IN TRANSFORMER COILS

Electrical equivalence for magnetic coupled circuit

For same polarity -> effect of mutual inductance

 The induced voltage is additive in nature

V 1 l^ = 𝐿 𝑑𝐼 𝑑𝑡 + 𝐿^

𝑑𝐼 𝑑𝑡

V 1 l^ =V 1 +Vm

V 1 l^ =modified voltage

V 1 =applied voltage

Vm = mutual/induced voltage

Similarly,

V 2 l^ = V 2 +Vm

V 1 (S)=SL 1 .I 1 (S)+SM.I 2 (S)

V 1 (S)=SM.I 1 (S)+SL 2 .I 2 (S)

V 1 l(S) = SL 1 SM I 1 (S)

V 2 l(S) SM SL 2 I 2 (S)

In case of sinusoidal circuit excitation when(s=jw)

V 1 l^ = jWL 1 jWM I 1 V 2 l^ jWM jWL 2 I 2

Case-2:-

When M is – ve The induced voltage is subtractive in nature.

V 1 l^ = 𝐿 𝑑𝐼 𝑑𝑡 − 𝑀^

𝑑𝐼 𝑑𝑡 V 2 l^ = −𝑀 𝑑𝐼1𝑑𝑡 + 𝐿2 𝑑𝐼2𝑑𝑡

CO-EFFICIENT OF COUPLING

Consider two coils having self-inductance L 1 and L 2 placed very close to each other. Let the number of turns of the two coils be N 1 and N 2 respectively. Let coil 1 carries current i 1 and coil 2 carries current i 2.

Due to current i 1 , the flux produced is Φ 1 which links with both the coils. Then from the previous knowledge mutual inductance between two coils can be written as M = N 1 Φ 21 /i 1 ............. (1)

where Φ 21 is the part of the flux Φ 1 linking with coil 2. Hence we can write, Φ 21 = k 1 Φ 1. ... M = N 1 (k1 Φ 1 )/i 1 ........... (2)

Similarly due to current i 2 , the flux produced is Φ 2 which links with both the coils. Then the mutual inductance between two coils can be written as M = N 2 Φ 21 /i 2 …........ (3)

Where Φ 21 is the part of the flux Φ 2 linking with coil 1. Hence we can write Φ 21 = k 2 Φ 2. ... M = N 2 (k 2 Φ 2 )/i 2 ............ (4)

Multiplying equations (2) and (4),

But N 1 Φ 1 /i 1 = Self-induced of coil 1 = L 1

N 2 Φ 2 /i 2 = Self-induced of coil 2 = L 2

... M^2 = k 1 k 2 L 1 L 2 ... M = √ (k 1 k 2 ) √ (L 1 L 2 ) Let k = √ (k 1 k 2 ) ... M = k √ (L 1 L 2 ) .......... (5) Where ‘k’ is called coefficient of coupling. ... k = M/ (√ (L 1 L 2 ))

a. When k = 1, it is called critical coupling and the network is called coupled network. b. When k > 1, it is called tight coupling. c. When k < 1 ,it is called looser coupling, RESONANCE IN A COUPLED CIRCUIT

Resonance is typical state or condition of a system during which the frequency of oscillation produced by an external forcing function matches with the natural frequency of the system by causing a response of maximum amplitude.

Resonance doesn’t take place on the steady state condition and for its occurrence in a system there must be some disturbances from outside the system that introduces oscillation in the system.

As the system continues with these oscillation interchange of energy takes place b/w two independent energy storing component present within the system (components like inductor and capacitor)

In some cases resonance occurs when the inductive reactance and capacitive reactance of the circuit are equal in magnitude then resonance occurs.

SERIES RESONANCE

Electrical resonance occurs in an AC circuit when the two reactance which are opposite and equal cancel each other out as XL = XC and the point on the graph at which this happens is where the two reactance curves cross each

NETWORK TOPOLOGY

Defination: - Topology refers to the science of place. In mathematics, topology is a branch of geometry in which figure are considered perfectly elastic.

Network topology refers to the properties that relates to the geometry of the network, which remains unaffected, when the graph is twisted or folded; provided that no parts are cut & no new connections are made.

Why to study network topology?

Ans: to suppress the nature of the circuit elements that make up the network

Terminology:-

Node: - it is an equipotential point at which two or more circuit elements are joined.

From the above figure nodes are:- a,b,c,d,e,f,g,h

Junction: - It is that point of a network where three or more circuit elements are joined.

From the above figure junctions are:-c, e, b, g

Essential node:- A node that joins three or more circuit elements.

From the above figure junctions are:-c, e, b, g