Network Theorems - 1 | Network Theory (Electric Circuits) - Electrical Engineering (EE) PDF Download

Network Theorems

  • The fundamental theories on which many branches of electrical engineering, such as Power Systems, Electric Machines, Control Systems, Analog Electronics, and Instrumentation are built on Network theorems. 
  • Network theorems are crucial elements of Network Theory because they provide essential tools for simplifying and analyzing complex Electrical Networks. 

Note: 

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

So, before jumping to the theorems let's first understand the conditions for Linear Networks.

Linear Networks

An element is considered linear if it satisfies the homogeneity (scaling) property and additive (superposition) property.

Network Theorems - 1 | Network Theory (Electric Circuits) - Electrical Engineering (EE)

1. Homogeneity Property

Let x be the input and y be the output of an element.

  • x(t) → y(t)

If kx(t) is applied to the element, the output must be ky(t).

  • kx(t) → ky(t)

Homogeneity PropertyHomogeneity Property

2. Additivity Property

x1(t) → y1(t),  x2(t) → y2(t)

If (x1(t) + x2(t)) is applied to the element, the output must be y1(t) + y2(t).

  • x1(t) + x2(t) → y1(t) + y2(t)
    Superposition Property
    Superposition Property

If k(x1(t) + x2(t)) is applied to the element, the output must be k(y1(t) + y2(t)).

  • kx1(t) + kx2(t) → ky1(t) + ky2(t)

So, for a network to qualify the application of various theorems must follow the conditions given above.

Superposition Theorem

It states that in a linear network with a number of independent sources, the response can be found by summing the responses to each independent source acting alone, with all other independent sources set to zero.

 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.

Network Theorems - 1 | Network Theory (Electric Circuits) - Electrical Engineering (EE)

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.

I' ⇨ due to source E1 alone

I'' ⇨ due to source E2 alone

I''' ⇨ due to source Ialone

current through resistor R I = I' + I'' + I'''

Question for Network Theorems - 1
Try yourself: In superposition theorem, when we consider the effect of one voltage source, all the other voltage sources are ____________.
View Solution

Removing of Active Element in Superposition Theorem

Network Theorems - 1 | Network Theory (Electric Circuits) - Electrical Engineering (EE)

Limitation of Superposition Theorem

Superposition cannot be applied to power effects because the power is related to the square of the voltage across a resistor or the current through a resistor.

Example: 

Here in the following electrical circuit, we will find the current flowing through the 10 Ω resistor using the superposition theorem.

Network Theorems - 1 | Network Theory (Electric Circuits) - Electrical Engineering (EE)

Solution: Here at first let’s consider the 30 A current source. So we will leave the 30 A current source as it is in the circuit and replace the 60 V voltage source with the short circuit as shown below.

Network Theorems - 1 | Network Theory (Electric Circuits) - Electrical Engineering (EE)

Now the current through 10 Ω resistor is calculated as

Network Theorems - 1 | Network Theory (Electric Circuits) - Electrical Engineering (EE)

[The I1 is calculated using the current divider.]

Now let’s consider a 60 V voltage source. So we will leave the 60 V voltage source as it is and replace the 30 A current source with the open circuits shown below.

Network Theorems - 1 | Network Theory (Electric Circuits) - Electrical Engineering (EE)


Then current through 10 Ω resistor is calculated as

Network Theorems - 1 | Network Theory (Electric Circuits) - Electrical Engineering (EE)

Finally, the total current flowing through the 10 Ω resistor is the algebraic sum of I1 and I2.

Network Theorems - 1 | Network Theory (Electric Circuits) - Electrical Engineering (EE)


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 VTH in series with a single resistor RTH 

where.

  • RTH 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
  • VTH is equal to the open circuit voltage across the A & B terminals.

Thevenin CircuitThevenin Circuit

Procedure for applying Thevenin’s theorem 

  • Open the load resistor.
  • Calculate/measure the open circuit voltage. This is the Thevenin Voltage (VTH).
  • Open current sources and short voltage sources.
  • Calculate /measure the Open Circuit Resistance. This is the Thevenin Resistance (RTH).
  • Now, redraw the circuit with measured open circuit Voltage (VTHin Step (2) as a voltage source and measured open circuit resistance (RTH) in Step (4) as series resistance and connect the load resistor which we had removed in Step (1). This is the equivalent Thevenin circuit of that linear electric network or complex circuit which had to be simplified and analyzed by Thevenin’s Theorem. 
  • Now find the Total current flowing through the load resistor by using Ohm’s Law: IT = VTH / (RTH + RL).

Example: Find current flowing through 1 \OmegaΩ resistor. 

Network Theorems - 1 | Network Theory (Electric Circuits) - Electrical Engineering (EE)


Network Theorems - 1 | Network Theory (Electric Circuits) - Electrical Engineering (EE)Solution: Open Load Resistor

Network Theorems - 1 | Network Theory (Electric Circuits) - Electrical Engineering (EE)

Network Theorems - 1 | Network Theory (Electric Circuits) - Electrical Engineering (EE)

  • Voltage Source are shorted

Network Theorems - 1 | Network Theory (Electric Circuits) - Electrical Engineering (EE)

  • Network Theorems - 1 | Network Theory (Electric Circuits) - Electrical Engineering (EE)Equivalent Circuit to find Rth

Network Theorems - 1 | Network Theory (Electric Circuits) - Electrical Engineering (EE)

Network Theorems - 1 | Network Theory (Electric Circuits) - Electrical Engineering (EE)Network Theorems - 1 | Network Theory (Electric Circuits) - Electrical Engineering (EE)

Let us solve a Previous Year's Question that appeared in GATE EE 2020.

Question: 

The Thevenin equivalent voltage, VTH, in V (rounded off to 2 decimal places) of the network shown below, is _______ . [GATE EE 2020]

Network Theorems - 1 | Network Theory (Electric Circuits) - Electrical Engineering (EE)

Solution:

Network Theorems - 1 | Network Theory (Electric Circuits) - Electrical Engineering (EE)

From the figure we can observe that,

V1 = 4V

As the load is open, therefore the current flowing through 5 Ohm resistor is zero.

By applying the nodal analysis, we get

Network Theorems - 1 | Network Theory (Electric Circuits) - Electrical Engineering (EE)

Therefore, Vth = 14 V


Question for Network Theorems - 1
Try yourself:Calculate the Thevenin resistance and Thevenin voltage across the terminal AB for the following circuit.
Find the Thevenin resistance across the terminal AB
View Solution

Norton's Theorem

Norton’s theorem states that any linear network containing can be replaced by a current source and a parallel resistor. 

  • RN = RTH 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 VTH is equal to the open circuit voltage across the A & B terminals.
  • IN is the Load current.

Norton`s CircuitNorton's CircuitNetwork Theorems - 1 | Network Theory (Electric Circuits) - Electrical Engineering (EE)

Example: 

For the given circuit, determine the current flowing through 10 Ω resistor using Norton’s theorem.
Network Theorems - 1 | Network Theory (Electric Circuits) - Electrical Engineering (EE)

Since the question here, is to determine the current through 10 Ω resistor, it is considered as the load.

(a) To find Norton’s current, Remove the load resistor(10 Ω), short it with a wire and the circuit is redrawn as below.

Network Theorems - 1 | Network Theory (Electric Circuits) - Electrical Engineering (EE)

In this circuit, we need to find the current IN, which is Norton’s current flowing from a to b. To find the value of IN, it is necessary to determine the total current in the circuit.

If you observe the circuit, 3 Ω resistor and 2 Ω resistor are in parallel with each other. This parallel combination is in series with 1 Ω resistor. Thus,

  Network Theorems - 1 | Network Theory (Electric Circuits) - Electrical Engineering (EE)

Now, the total current IT is given by,

  Network Theorems - 1 | Network Theory (Electric Circuits) - Electrical Engineering (EE)

The current through the 2 Ω resistor (or Norton’s current IN) is obtained by applying current division rule:

  Network Theorems - 1 | Network Theory (Electric Circuits) - Electrical Engineering (EE)


(b) To find Norton’s resistance, remove the load resistor, short the voltage source and circuit is redrawn as below.

Network Theorems - 1 | Network Theory (Electric Circuits) - Electrical Engineering (EE)

In this circuit, we can observe that the 2 Ω resistor is in series with the parallel combination of 1 Ω and 3 Ω resistors. Thus the equivalent value of resistance is obtained as,

  Network Theorems - 1 | Network Theory (Electric Circuits) - Electrical Engineering (EE)

(c) Norton’s Equivalent Circuit. It is drawn by connecting Norton’s voltage IN, Norton’s resistance RN and load resistor in series, as shown below:

Network Theorems - 1 | Network Theory (Electric Circuits) - Electrical Engineering (EE)


From this circuit, the current through the load RL = 10 Ω resistor is obtained using current division rule. It is given by,

  Network Theorems - 1 | Network Theory (Electric Circuits) - Electrical Engineering (EE)

Maximum Power Transfer Theorem

The maximum power transfer theorem states that, to obtain maximum external power from a source with a finite Internal Impedance (Say Resistance) the resistance of the load must equal to the resistance of the source as viewed from its output terminals.

Network Theorems - 1 | Network Theory (Electric Circuits) - Electrical Engineering (EE)

Power delivered to the load resistance:
Network Theorems - 1 | Network Theory (Electric Circuits) - Electrical Engineering (EE)

To find the maximum power, differentiate the above expression with respect to resistance RL and equate it to zero. Thus,
Network Theorems - 1 | Network Theory (Electric Circuits) - Electrical Engineering (EE)
Thus in this case, the maximum power will be transferred to the load when load resistance is just equal to internal resistance of the battery.

Results of Maximum Power Transfer Theorem

  • The maximum power transfer takes place when the load resistance R= Rth 
  • The maximum power transferred to the load Pmax = PL (R= Rth) = (Vth)2/ 4Rth 
  • The efficiency of power transfer η = PL/Ps 
  • where PL- Power delivered to the Load 
  • Ps- Power Generated by Source

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

Example: 

The maximum power drawn by the load RL in the below circuit will be:Network Theorems - 1 | Network Theory (Electric Circuits) - Electrical Engineering (EE)

Calculation:

Here Rth= 5 ohm and Vth= 10 V and RL= 5 ohm.

So Thevenin equivalent would be:

Network Theorems - 1 | Network Theory (Electric Circuits) - Electrical Engineering (EE)

So, power across load can be calculated by calculating current I across RL.

I = Vth/Req

Req = 5 + 5

= 10 ohm

I = 10/10

I = 1 A.

So, power across RL = I­2RL

= 1 × 5

= 5 W

Frequently Asked Questions (FAQs)

Q.1. What is the superposition theorem?

Superposition theorem is a circuit analysis theorem that is used to solve the network where two or more sources are present and connected.

Q.2. Is the superposition theorem valid for AC circuits?

The superposition theorem is valid for AC circuits.

Q.3. Is the superposition theorem applicable to power?

The requisite of linearity indicates that the superposition theorem is only applicable to determine voltage and current, but not power. Power dissipation is a nonlinear function that does not algebraically add to an accurate total when only one source is considered at a time.

Q.4. Can the superposition theorem be applied to non-linear circuits?

No, the superposition theorem can only be applied to non-linear circuits.

Q.5. Why do we use the superposition theorem?

The superposition theorem is very important in circuit analysis because it converts a complex circuit into a Norton or Thevenin equivalent circuit.

The document Network Theorems - 1 | Network Theory (Electric Circuits) - Electrical Engineering (EE) is a part of the Electrical Engineering (EE) Course Network Theory (Electric Circuits).
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FAQs on Network Theorems - 1 - Network Theory (Electric Circuits) - Electrical Engineering (EE)

1. What is the Superposition Theorem in network analysis?
Ans. The Superposition Theorem states that in a linear network with multiple sources, the response at any point can be determined by considering each source separately and adding the individual responses. This means that we can analyze the network by turning off all but one source at a time and calculating the response, and then superimpose these individual responses to obtain the overall response.
2. What is Thevenin's Theorem and how is it applied in electrical networks?
Ans. Thevenin's Theorem states that any linear electrical network can be replaced by an equivalent circuit consisting of a single voltage source in series with a single impedance. The voltage source is called the Thevenin voltage, and the impedance is called the Thevenin impedance. This equivalent circuit can be used to simplify complex networks and analyze the behavior of the network from the perspective of a single source and load.
3. What is Norton's Theorem and how does it differ from Thevenin's Theorem?
Ans. Norton's Theorem is similar to Thevenin's Theorem, but instead of replacing a network with a voltage source and impedance, it replaces it with a current source and impedance. The current source is called the Norton current, and the impedance is called the Norton impedance. Norton's Theorem provides an alternative way to simplify and analyze electrical networks, particularly when the load is better characterized by a current rather than a voltage.
4. What is the Maximum Power Transfer Theorem and how does it help in optimizing network performance?
Ans. The Maximum Power Transfer Theorem states that the maximum power is transferred from a source to a load when the impedance of the load is equal to the complex conjugate of the impedance of the source. By finding the impedance that maximizes the power transfer, we can optimize the performance of a network. This theorem is particularly useful when designing circuits or systems where power efficiency is important, as it helps us determine the ideal load impedance for maximum power transfer.
5. Can these network theorems be applied to non-linear networks?
Ans. No, these network theorems are applicable only to linear networks. Linear networks have the property of proportionality, meaning that the response is directly proportional to the input. Non-linear networks, on the other hand, do not exhibit this proportionality and often have complex and unpredictable behaviors. Therefore, the network theorems discussed, such as the Superposition Theorem, Thevenin's Theorem, Norton's Theorem, and Maximum Power Transfer Theorem, cannot be directly applied to non-linear networks.
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