Circuit Theory - Network Theorems Complete Study Notes

NETWORK THEOREM

The fundamental theory on which many branches of electrical engineering, such as electric power, electric machines, control, electronics, computers, communications, and instrumentation are built is the Electric circuit theory. So here the network theorem helps us to solve any complex network for a given condition.

Note: All the theorems are only applicable to Linear Network only, according to the theory of Linear Network they follow the condition of Homogeneity & Additivity.

Homogeneity Principle:

 A system is said to be homogeneous, for any input signal x(t)

If input  x(t) gives → response y(t)

then, it must follow ⇔ k x(t) →k y(t)

i.e scaling in any input signal scales the output signal by same factor.

Additivity Principle:

A system is said to be homogeneous, for any input signal x(t)

If two input x₁(t)+  x₂ (t) ⇔ y₁(t) +  y₂(t)

then, k₁x₁(t) + k₂x₂ (t) ⇔  k₁ y₁(t) + k₂ y₂(t)

i.e the output corresponding to the sum of any two inputs is the sum of of there respective outputs.

1. SUPERPOSITION THEOREM

Superposition theorem finds use in solving a network where two or more sources are present and connected not in series or in parallel.

Superposition theorem states that if a number of voltage or current sources are acting simultaneously in a linear bidirectional network, the resultant response in any branch is the algebraic sum of the responses that would be produced in it, when each source acts alone replacing all other independent sources by their internal resistances.

Procedure for using the superposition theorem:
Step-1: Retain one source at a time in the circuit and replace all other sources with their internal resistances.
Step-2: Determine the output (current or voltage) due to the single source acting alone.
Step-3: Repeat steps 1 and 2 for each of the other independent sources.
Step-4: Find the total contribution by adding algebraically all the contributions due to the independent sources.

So for above given circuit the total response or say current I through resistor R₂ will be equal to the sum of individual response obtained by each source. 

Removing of Active Element in Superposition Theorem:

1. Ideal voltage source is replaced by short circuit.

2. Ideal current source is replaced by open circuit.

Limitation: Superposition cannot be applied to power calculation because the power is related to the square of the voltage across a resistor or the current through a resistor. The squared term results in a non-linear (a curve, not a straight line) relationship between the power and the determining current or voltage.

Example:  Determine the value of current I in 17 ohm resistor by using super-position theorem.

Solution: 

Firstly, only 12 V source is active.

Now, only 10 V source is active.

2. THEVENIN'S THEOREM

Thevenin’s theorem states that any two output terminals of an active linear network containing independent sources (it includes voltage and current sources) can be replaced by a simple voltage source of magnitude V_TH in series with a single resistor R_TH where R_TH  is the equivalent resistance of the network when looking from the output terminals A & B with all sources (voltage and current) removed and replaced by their internal resistances and the magnitude of V_TH is equal to the open circuit voltage across the  A & B terminals. 

The procedure for applying Thevenin’s theorem

To find a current I_L through the load resistance R_L using Thevenin’s theorem, the following steps are followed:

Step-1: Disconnect the load resistance ( R_L) from the circuit,

Step-2: Calculate the open-circuit voltage V_TH at the load terminals (A & B) after disconnecting the load resistance ( R_L ).

Step-3: Redraw the circuit with each independent source replaced by its internal resistance.

Note: Voltage sources should be short-circuited and current sources should be open-circuited.

Step-4: Look backward into the resulting circuit from the load terminals (A & B). Calculate the resistance that would exist between the
load terminals.

Step-5: Place R_TH in series with V_TH to form the Thevenin’s equivalent circuit.

Step-6: Reconnect the original load to the Thevenin equivalent circuit as shown in the load’s voltage, current and power may be calculated by a simple arithmetic operation only.

 3. NORTON'S THEOREM 

Norton’s theorem states that any two output terminals of an active linear network containing independent sources (it includes voltage and current sources) can be replaced by a current source and a parallel resistor R_N. Where, R_N which is the equivalent resistance of the network when looking from the output terminals A & B with all sources (voltage and current) removed and replaced by their internal resistances and the magnitude of I_N is equal to the short-circuit current across the A & B load terminals.

Norton Equivalent Circuit can be shown as:

Example: Find the Norton equivalent circuit of the following circuit at terminal a-b.

Solution: Circuit can be redrawn as:

4. MAXIMUM POWER TRANSFER THEOREM 

Maximum Power Transfer Theorem states a resistive load, being connected to a DC network, consumes maximum power when the load resistance is equal to the thevenin’s equivalent resistance of the source network as seen from the load terminals.

A variable resistance R_L is connected to a dc source network as shown in figure above and the Thevenin’s voltage V_Th and Thevenin’s equivalent resistance R_Th of the source network. The aim is to determine the value of R_L such that it consumes maximum power from the DC source.

Steps for Solution of a Network using Maximum Power Transfer Theorem:

Step 1: Remove the load resistance and find the Thevenin’s resistance (R_Th) of the source network looking through the open circuited load terminals.

Step 2: As per maximum power transfer theorem, this R_Th is the load resistance of the network i.e. R_L = R_Th that allows maximum power transfer.

Step 3: Find the Thevenin’s voltage (V_Th) across the open circuited load terminals.

Step 4: Maximum Power Transfer is given by:

 P_max = (V_th)²/ 4R_th

Note: Maximum power transfer condition results in 50 percent efficiency in Thevenin equivalent, however much lower efficiency in the original circuit.

5. MILLMAN’S THEOREM

Millman’s theorem helps to reduce 'N' number of parallel voltage sources  to one. It can be observe in the given figure below.  This technique  permits finding the current through or voltage across R_L without  applying  any method such as mesh analysis, nodal analysis, superposition, and so on.For example,the given three voltage sources can be reduced to one voltage source.

Generally, three steps are included in its application

Step 1: Convert all voltage sources into current sources.

 Step 2: Convert the parallel current source into resulting network as shown below

 I_T= I₁+I₂+I₃   &    G_T = G₁+G₂+G₃

Step 3: Convert the resulting current source to a voltage source by transformation, and the desired single-source network will be shown as

The plus-and-minus signs appears in the last equation to include those cases where the sources may not be supplying energy in the same direction.

In terms of resistances values:

and

6. RECIPROCITY THEOREM 

The reciprocity theorem is applicable to the single-source networks only. The theorem states that the current I in any branch of a network due to a single voltage source E present anywhere else in the network will be equal to the current across the branch in which the source was originally placed if the source is placed in the branch in which the current I was originally(initially) measured.

In other words, the location of the voltage source and the resulting current may be interchanged without a change in magnitude of current. The theorem postulates that the polarity of the voltage source have the same adaptation with the direction of the branch current in each position.

• Example: Verify Reciprocity theorem.

• Interchanging (or reciprocating) the location of I and E in the last figure to demonstrate the validity of the reciprocity theorem.

7. TELLEGEN'S THEOREM

• Tellegen’s theorem is based upon two Kirchhoff’s laws and is also applicable for any lumped network having elements which are linear or non-linear, active or passive, time-varying or time-invariant.
• For a lumped network whose element assigned by associate reference direction for branch voltage v_k and branch current j_k.The product v_kj_k is the power delivered at time t by the network to the element k.
• If all branch voltages and branch currents satisfies KVL and KCL then

Application of Tellegen's Theorem: 

As seen from the last equation, the Tellegen’s Theorem implies the law of energy conservation.“The sum of power delivered by the independent sources to the network elements is equal to the sum of the power absorbed by all the branches of the network.” So, the application of Tellegen's theorem can be classified as 

• Conservation of energy
• Conservation of complex power
• The real(or active) part and the phase of driving point impedance
• Driving point impedance

Example: Find all branch voltages and currents for both networks N1, N2, and then verify Tellegen’s theorem.

8. SUBSTITUTION THEOREM

The substitution theorem states that "If the voltage across any branch and the current flowing through that branch of a dc bilateral network are known, then that branch can be substituted by any one of the combinations that can consist of the same voltage and current through that chosen branch.

More simply, the theorem states that for branch equivalence, the voltage across the terminal and current through the terminal must be the same. Consider the circuit in which the voltage across and current through the branch a-b are determined. Through the use of the substitution theorem, few number of equivalent a-a′ branches are shown. Note that for each equivalent circuit, the terminal voltage & the current remains same.

• By the use of the substitution theorem, the number of equivalent branches are:

Note : for each equivalent, the terminal voltage and current are the same and known potential difference and current in a network can be replaced by an ideal voltage source and current source respectively.

• Example: The current source equivalence where a known current is replaced by an ideal current source, permitting the isolation of R₄ and R₅ as shown below

Recall the discussion of bridge networks that V = 0 and I = 0 were replaced by a short-circuit and an open circuit respectively. 

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