Network Theorems

Network Theorems

CLASSIFY TYPES OF ELECTRICAL CIRCUIT

• Type of circuit element:

   1. Active and passive element

   2. Linear and non linear element

   3. Bilateral and unilateral element

   4. Lumped and distributed element

ACTIVE AND PASSIVE ELEMENT

• Active element :

      "An active element is that which is capable of delivering power to same."

      Examples- voltage source , current source

• Passive element:

     "Passive element cannot deliver power an the contrary absorbs power from the active element."

     Examples- resistor, inductor, capacitor

LINEAR AND NON LINEAR ELEMENT

• Linear :

    "The linear element is one which the voltage ,current relation involves a constant co-efficient."

     The voltage current characteristic is straight line and passing through the origin.

• Non linear:

    "The voltage current characteristic may be a straight line but does not pass through the origin."

BILATERAL AND UNILATERAL ELEMENT

• Bilateral 

    "An element is said to be bilateral in which the voltage current relationship is same for current flowing in either direction."

• Unilateral

    "Where as an element is said to be unilateral in which the voltage–current relation is not the same for current flowing in either direction."

LUMPED AND DISTRIBUTED ELEMENT

• Lumped 

    "Lumped element are those element which are smaller in size and the cause effect relation is simultaneous."

• Distributed

    "The distributed element have large length so that their parameters can not be separated for analysis."

SUPER POSITION THEOREM

• When there are more than one energy sources. 
• The current through a branch is the algebraic sum of currents flowing when each source is considered separately.

• The other sources replaced by their internal resistances.

THEVENIN’S THEOREM

• It state that any network having a number of energy sources and resistances when viewed from its open circuit element A and B can be replaced by a simple equivalent voltage source 𝑉𝑇ℎ in series with a simple equivalent resistance 𝑅𝑇ℎ.
𝑉𝑇ℎ - Thevenin’s equivalent voltage source 
𝑅𝑇ℎ - Thevenin’s equivalent resistance

NORTON’S THEOREM

• Any network having a number of energy sources and resistance when viewed from its open circuit terminals can be replaced by a single equivalent current source 𝐼𝑠𝑐 with single equivalent resistance 𝑅𝑒𝑞 in parallel to it.

𝑅𝑒𝑞- equivalent resistance sources are replaced by their internal resistances and all the current sources are open circuited

𝐼𝑠𝑐- current flowing through the short circuit when opened terminals are shorted

MAXIMUM POWER TRANSFER THEOREM

• This theorem is particularly useful for analysing communication networks because in communication engineering it is usually desirable to deliver maximum power to a load.

• Maximum power transfer theorem states that “maximum power is transferred from the source to the load when the load resistance is equal to the Thevenin’s equivalent resistance.”

𝑅𝐿 = 𝑅𝑇ℎ

𝑅𝑇ℎ - Thevenin’s equivalent resistance

𝑅𝐿 - Load resistance

RECIPROCITY THEOREM

• In many electric circuit problems we are interested in the relationship between an impressed source in one part of the circuit and a response to it in some other part of the same the circuit.

• This is where the property of reciprocity come in useful.

• The circuits having this property are called reciprocal ones, and obey the reciprocity theorem, which is stated below.

• According to this theorem, In a linear passive network, supply voltage V and current I are mutually transferable.

• This ratio V and I is called the transfer resistance (or transfer impedance in an ac system)

STAR DELTA TRANSFORMATION

Delta to star transformation:

• Three resistors R1, R2, R3 are connected in delta in fig.

• We want to obtain the value of equivalent resistors in star connection.

In delta connection resistors 𝑅12 and 𝑅23 + 𝑅31 between terminal 1 and 2 are in parallel so
[𝑅12(𝑅23+𝑅31)]/[𝑅12+𝑅23+𝑅31]                           ....(1)
Where as the equivalent resistance between terminal 1 and 2 in star connection so,
𝑅1+𝑅2                                                                                        ....(2)
𝑅1+𝑅2 = [𝑅12(𝑅23+𝑅31)]/[𝑅12+𝑅23+𝑅31]      ....(3)
If we take transmits 2-3 and 3-1 in turn
R2+R3 = [R23(R31+R12)]/[R12+R23+R31]        ....(4)
R3+R1 = [R31(R12+R23)]/[R12+R23+R31]        ....(5)
We can find the value of R1, R2, R3 from the equation 3,4,5
R1-R3= [R12R23+R12R31−R23+R31−R23R12]/[12+R23+R31]
R1-R3= [R12R31−R23+R31]/[R12+R23+R31]  ....(6)
Adding eq. 5 and 6
2R1=[R12R23+R12R31−R23+R31−R23R12+R12R31−R23+R31] /                    [R12+R23+R31]
2R1= [2R12R31]/[R12+R23+R31]
R1= [R12R31]/[R12+R23+R31]                                ....(7)
Similarly by subtracting eq. 5 from 4 and adding eq. 3 to it.
R2= [R12R23]/[R12+R23+R31]                                ....(8)
Subtracting eq. 3 from eq. 5 and adding eq. 4 to it, we get 
R3= [R23R31]/[R12+R23+R31]                               ....(9)

• Resistance of an arm of star connection is equal to the ratio of multiplication of two resistors of delta meeting at that terminal and sum of the three resistors of delta connection.

Star to delta transformation:

• Three resistors R1, R2, R3 are connected in star in fig.

• We want to obtain the value of equivalent resistors in delta connection.

R1= [R12R31]/[R12+R23+R31]                    ....(1)
R2= [R12R23]/[R12+R23+R31]                    ....(2)
R3= [R23R31]/[R12+R23+R31]                    ....(3)
Dividing eq. 1 by 3
(R1)/(R3)=(R12)/(R23)
R12=(R23)(R1)/(R3)                                             ....(4)
Dividing eq. 2 by 3
(R2)/(R3)=(R12)/(R31)
R31=(R12)(R3)/(R2)                                             ....(5)
Substituting the values of 4 and 5 eq. in eq.1
R1 =     [(R12)(R3/R2)(R23)(R1/R3)]
           ____________________________________
           R23(R1/R3)+ R23 + R12(R3/R2)
R1 =                          (R12R23R1)/R2
          _____________________________________________
          [(R23R1)/R3]+ R23 + [(R23R1R3)/R3R2]
R1 =           R12R23R1/R2
           _____________________________
           R23[(R1/R3)+ 1 +(R1/R2)]
R1 =                 R12R1/R2
           _______________________________
          (R1R2 + R2R3 + R1R3)/R3R2
R1 =          R12R1R3
            _______________________
           R1R2 + R2R3 + R1R3
R1R2 + R2R3 + R1R3 =R12R3
R12= (R1R2/R3)+(R1R3/R3)+(R1R3/R3)
R12=R1 + R2 +(R1R2/R3)                                 ....(6)
Similarly
R23=R2 + R3 +(R2R3/R1)                                 ....(7)
R31=R3 + R1 +(R3R1/R2)                                 ....(8)