Network Theorems

Contents

**Apply Thevenin’s Theorem on different types of Networks** 2

**Apply Norton’s Theorem on different types of Networks** 3

**List limitations of Norton’s Theorem** 4

**Describe Superposition Theorem** 5

**Apply Superposition Theorem on different types of Networks** 6

**List limitations of Superposition Theorem** 6

**Describe Maximum Power Transfer Theorem** 7

**Apply Maximum Power Transfer Theorem on different types of Networks** 8

**Describe Reciprocity Theorem** 9

**Apply Reciprocity Theorem on different types of Networks** 10

**List limitations of Reciprocity Theorem** 11

**Describe Milliman’s Theorem** 12

**Apply Millman’s Theorem on different types of Networks** 12

**Describe Tellegen’s Theorem** 13

**Apply Tellegen’s Theorem on different types of Networks** 14

**Describe Compensation Theorem** 15

**Apply Compensation Theorem on different types of Networks** 16

**List limitations of Compensation Theorem** 17

**Recall Substitution Theorem** 18

**Apply Substitution Theorem on different types of Networks** 19

**Recall Network Theorems in AC circuits** 20

**Calculate the response in AC circuits using Network Theorems** 21

**Describe Thevenin’s Theorem**

Thevenin’s theorem is a fundamental principle in electrical circuit theory that states that any linear circuit can be replaced by an equivalent circuit that has a single voltage source and a series resistor. This equivalent circuit is called the Thevenin equivalent circuit.

The Thevenin equivalent circuit is found by replacing all of the resistors and other components in the original circuit with a single resistor, which is the equivalent resistance of the circuit as seen from the output terminals. The equivalent voltage source is then found by measuring the voltage across the output terminals with the load disconnected, and the equivalent resistor is found by calculating the resistance between the output terminals with all voltage sources in the circuit shorted.

Once the Thevenin equivalent circuit is found, it can be used to analyse the original circuit in a much simpler way. The load resistor can be connected to the output terminals of the Thevenin equivalent circuit, and the current and voltage in the load can be calculated using Ohm’s law.

The Thevenin equivalent circuit is particularly useful in circuit design and analysis, as it allows complex circuits to be simplified and analysed in terms of a single voltage source and a series resistor. This can greatly simplify the process of designing and optimising circuits, as well as the process of troubleshooting and repairing existing circuits.

In summary, Thevenin’s theorem is a powerful principle in electrical circuit theory that states that any linear circuit can be replaced by an equivalent circuit that has a single voltage source and a series resistor. This equivalent circuit, called the Thevenin equivalent circuit, can be used to analyse and simplify the original circuit, and is widely used in circuit design and analysis.

**Apply Thevenin’s Theorem on different types of Networks**

Thevenin’s theorem can be applied to a wide range of networks, including both DC and AC circuits, as well as networks containing dependent sources.

Here are the steps to apply Thevenin’s theorem on a circuit:

- Identify the load that needs to be analysed and isolate it from the rest of the circuit.
- Find the Thevenin voltage by calculating the voltage across the load terminals when the load is removed and the circuit is open.
- Find the Thevenin resistance by calculating the resistance between the load terminals with all independent sources turned off.
- Construct the Thevenin equivalent circuit by connecting a voltage source in series with a resistor, with the voltage equal to the Thevenin voltage and the resistance equal to the Thevenin resistance.
- Connect the load back to the circuit and calculate the current through the load and the voltage across it using Ohm’s law and Kirchhoff’s laws.

**Describe Norton’s Theorem**

Norton’s theorem is a circuit analysis technique that simplifies a linear, bilateral circuit with multiple sources into an equivalent circuit with a single current source and a shunt resistor. This theorem is based on the principle of equivalence of circuits, which states that any two circuits are equivalent if they have the same input-output characteristics.

Norton’s theorem is named after American engineer and scientist Edward Lawry Norton, who first published it in 1926.

Here are the steps to apply Norton’s theorem:

- Identify the load that needs to be analysed and remove it from the circuit.
- Find the Norton current by calculating the short-circuit current that flows through the load terminals.
- Find the Norton resistance by calculating the resistance between the load terminals with all independent sources turned off.
- Construct the Norton equivalent circuit by connecting a current source in parallel with a resistor, with the current equal to the Norton current and the resistance equal to the Norton resistance.
- Connect the load back to the circuit and calculate the current through the load and the voltage across it using Ohm’s law and Kirchhoff’s laws.

Norton’s theorem can be applied to both DC and AC circuits, as well as circuits with dependent sources. It is particularly useful when a circuit has many sources and it is difficult to find the Thevenin equivalent circuit using Thevenin’s theorem.

**Apply Norton’s Theorem on different types of Networks**

Norton’s theorem can be applied to any linear, bilateral circuit with multiple sources, including circuits with dependent sources. Here are the steps to apply Norton’s theorem on different types of networks:

- Resistive networks: To apply Norton’s theorem on a resistive network, we first remove the load and calculate the short-circuit current that flows through the load terminals. Next, we calculate the resistance between the load terminals with all independent sources turned off. The Norton equivalent circuit is then constructed by connecting a current source in parallel with a resistor, with the current equal to the Norton current and the resistance equal to the Norton resistance. Finally, we connect the load back to the circuit and calculate the current through the load and the voltage across it.
- Networks with capacitors or inductors: To apply Norton’s theorem on a network with capacitors or inductors, we first transform the circuit into its phasor representation. We then follow the same steps as in resistive networks to find the Norton equivalent circuit.
- Networks with dependent sources: To apply Norton’s theorem on a network with dependent sources, we first remove the load and simplify the circuit using source transformation and other techniques. We then follow the same steps as in resistive networks to find the Norton equivalent circuit.
- Networks with multiple sources: Norton’s theorem is particularly useful when a circuit has many sources and it is difficult to find the Thevenin equivalent circuit using Thevenin’s theorem. By reducing the circuit to a single current source and a shunt resistor, Norton’s theorem makes it easier to analyse and design circuits, and it helps to reduce the complexity of circuit diagrams.

In summary, Norton’s theorem can be applied to a wide range of networks, including resistive, capacitive, inductive, and networks with dependent sources. It is a powerful tool for simplifying complex circuits and reducing the complexity of circuit diagrams.

**List limitations of Norton’s Theorem**

Norton’s Theorem is a powerful circuit analysis technique that can be used to simplify complex circuits by replacing them with an equivalent circuit that contains a current source and a parallel resistance. However, there are several limitations of Norton’s Theorem that you should be aware of:

- Limited to linear circuits: Norton’s Theorem can only be used to analyze linear circuits, which means that it cannot be used for circuits that contain nonlinear components such as diodes and transistors.
- Requires open-circuit conditions: To apply Norton’s Theorem, you need to first find the open-circuit voltage across the two terminals of the circuit being analyzed. This requirement means that Norton’s Theorem cannot be used for circuits that do not have an open-circuit voltage.
- Limited to DC circuits: Norton’s Theorem is only valid for DC circuits, which means that it cannot be used for circuits that contain time-varying signals such as AC circuits.
- Limited to single-source circuits: Norton’s Theorem can only be applied to circuits that have a single current source. If a circuit contains multiple current sources, then Norton’s Theorem cannot be used to simplify the circuit.
- Limited to two-terminal circuits: Norton’s Theorem is only valid for circuits that have two terminals. It cannot be used for circuits that have more than two terminals.
- Requires complex calculations: In some cases, the calculations required to apply Norton’s Theorem can be complex and time-consuming. This limitation means that Norton’s Theorem may not be the most efficient method for simplifying some circuits.

**Describe Superposition Theorem**

The Superposition Theorem is a principle used in circuit analysis that allows us to analyse the behaviour of a linear circuit with multiple sources by considering the effect of each source separately. According to the theorem, the response (voltage or current) in any branch of a linear circuit with multiple sources is the algebraic sum of the responses caused by each individual source acting alone, with all other sources turned off.

To apply the Superposition Theorem, we follow the following steps:

- Turn off all but one of the independent sources in the circuit (voltage or current sources) by replacing them with their internal resistances. If the source is a voltage source, it is replaced by a short circuit, and if it is a current source, it is replaced by an open circuit.
- Calculate the response (voltage or current) in the branch of interest caused by the one source that is still active.
- Repeat steps 1 and 2 for each independent source in the circuit.
- Finally, find the total response in the branch of interest by adding the individual responses due to each source.

It’s important to note that the Superposition Theorem is only valid for linear circuits, where the voltage-current relationship follows Ohm’s law and other linear equations. Non-linear components such as diodes, transistors, and nonlinear resistors cannot be analysed using this theorem.

**Apply Superposition Theorem on different types of Networks**

The Superposition Theorem can be applied to different types of networks, including:

- DC Circuits: In a DC circuit with multiple sources, the Superposition Theorem can be used to find the voltage or current at any point in the circuit. For example, consider a DC circuit with two voltage sources connected in series, and we want to find the voltage across a resistor in the circuit. To apply the Superposition Theorem, we turn off one voltage source and calculate the voltage caused by the other source, and then repeat the process for the other source. Finally, we add the two individual voltages to get the total voltage across the resistor.
- AC Circuits: The Superposition Theorem can also be applied to AC circuits with multiple sources, where the voltage and current vary with time. In this case, we can use the theorem to find the response at any frequency in the circuit. To apply the Superposition Theorem, we follow the same steps as in a DC circuit, but we need to use phasors to represent the AC sources and the circuit components. A phasor is a complex number that represents the amplitude and phase angle of an AC voltage or current.
- Network Theorems: The Superposition Theorem can be used in conjunction with other network theorems, such as the Thevenin’s Theorem and Norton’s Theorem. For example, we can use the Superposition Theorem to find the Thevenin equivalent circuit of a network with multiple sources, by finding the open-circuit voltage caused by one source and the short-circuit current caused by another source. We can then use Thevenin’s Theorem to simplify the network to a single voltage source and a series resistor.
- Amplifier Circuits: The Superposition Theorem can be used to analyse amplifier circuits with multiple input sources. By using the theorem, we can find the output response caused by each input signal separately, and then add them to get the total output response. This is useful for designing and testing amplifier circuits with multiple inputs.

**List limitations of Superposition Theorem**

Although the Superposition Theorem is a useful tool for analysing linear circuits with multiple sources, there are some limitations to its application. Some of the limitations of the Superposition Theorem are:

- Non-Linear Components: The Superposition Theorem is only applicable to linear circuits, where the voltage-current relationship follows Ohm’s law and other linear equations. It cannot be applied to circuits with non-linear components, such as diodes, transistors, and nonlinear resistors.
- Time-Varying Signals: The Superposition Theorem can be applied to circuits with time-varying signals, but it assumes that the circuit is in a steady state. It cannot be used to analyse circuits with transient behaviour or time-varying phenomena, such as switching circuits or circuits with AC signals.
- Mutual Effects: The Superposition Theorem assumes that the different sources in the circuit do not interact with each other. In reality, sources can affect each other, leading to mutual effects that cannot be analysed using the theorem. This is especially true for circuits with feedback or with sources that are physically close to each other.
- Limited Practicality: The Superposition Theorem requires multiple calculations to analyse a circuit with multiple sources, which can be time-consuming and tedious. In addition, turning off sources and replacing them with their internal resistance may not be practical in some circuits, especially if the sources have complex internal structures.
- Assumption of Linearity: The Superposition Theorem assumes that the circuit is linear, which is often an approximation for real-world circuits. In reality, many circuits exhibit nonlinear behaviour due to the presence of components such as diodes, transistors, and nonlinear resistors. The Superposition Theorem cannot be used to analyse such circuits.

**Describe Maximum Power Transfer Theorem**

The Maximum Power Transfer Theorem states that maximum power is transferred from a power source to a load when the impedance of the load matches the complex conjugate of the impedance of the source.

In other words, if we have a voltage source with an internal impedance, and we want to transfer the maximum amount of power to a load, we should make the impedance of the load equal to the complex conjugate of the internal impedance of the source.

Mathematically, the theorem can be expressed as follows:

If a source with a Thevenin equivalent voltage Vth and Thevenin equivalent impedance Zth is connected to a load impedance ZL, then the maximum power is transferred to the load when ZL is equal to the complex conjugate of Zth:

ZL = Zth*

where * denotes complex conjugation.

The maximum power transferred to the load is given by:

Pmax = (|Vth|^{2} / 4Re[Zth])

where Re[Zth] is the real part of the Thevenin equivalent impedance.

The Maximum Power Transfer Theorem is useful in designing circuits to ensure that the maximum amount of power is transferred from the source to the load. However, it should be noted that this theorem is only applicable in situations where the load impedance is variable, and it may not always be the most efficient way to design a circuit for maximum power transfer in all cases.

**Apply Maximum Power Transfer Theorem on different types of Networks**

The Maximum Power Transfer Theorem is a useful tool for determining the load resistance that maximises the power delivered to it from a voltage or current source. The theorem can be applied to different types of circuits, including DC and AC circuits.

Here are a few examples of how the Maximum Power Transfer Theorem can be applied to different types of networks:

- DC Circuit: Consider a simple DC circuit with a voltage source and a resistor connected in series. To find the load resistance that maximises the power delivered to it, we need to calculate the internal resistance of the source and set it equal to the load resistance. For example, if the voltage source has an internal resistance of 10 ohms, the load resistance that maximises the power delivered to it would also be 10 ohms.
- AC Circuit: The Maximum Power Transfer Theorem can also be applied to AC circuits. Consider an AC circuit with a voltage source, a resistor, and a capacitor connected in series. To find the load resistance that maximises the power delivered to it, we need to calculate the complex impedance of the source and the load. We can then set the load impedance equal to the complex conjugate of the source impedance to achieve maximum power transfer.
- Amplifier Circuit: In an amplifier circuit, the Maximum Power Transfer Theorem can be used to determine the load resistance that maximises the output power of the amplifier. The internal resistance of the amplifier can be modelled as the output impedance, and the load resistance can be varied to find the value that maximises the power delivered to it.
- Transmission Line: The Maximum Power Transfer Theorem can also be applied to transmission lines. The load impedance that maximises power transfer in a transmission line is the complex conjugate of the characteristic impedance of the line. This is because the characteristic impedance represents the resistance that would maximise the power transfer if the line were terminated in a resistance.

In summary, the Maximum Power Transfer Theorem can be applied to various types of networks, including DC and AC circuits, amplifiers, and transmission lines. The theorem helps to determine the load resistance that maximises power transfer in a given circuit, and is particularly useful in situations where maximum power transfer is desired.

**Describe Reciprocity Theorem**

The Reciprocity Theorem states that in a linear, time-invariant network, the ratio of the voltage at one point to the current at another point is the same as the ratio of the voltage at the second point to the current at the first point, provided that all the sources in the network remain the same.

In simpler terms, the Reciprocity Theorem states that if we have a network with two terminals A and B, and we apply a current to terminal A and measure the resulting voltage at terminal B, the ratio of the voltage to the current is the same as if we had applied the same current to terminal B and measured the resulting voltage at terminal A.

Mathematically, the Reciprocity Theorem can be expressed as follows:

Let i1 be the current flowing into terminal A, and let v2 be the voltage measured at terminal B in response to i1. Then, the ratio v2/i1 is equal to the ratio of the voltage measured at terminal A in response to a current i2 applied to terminal B, divided by i2:

v2/i1 = v1/i2

where v1 is the voltage measured at terminal A in response to i2.

The Reciprocity Theorem is a powerful tool for analyzing complex networks, as it allows us to simplify the analysis by using the measured values at one set of terminals to calculate the values at another set of terminals. However, it should be noted that this theorem only applies to linear, time-invariant networks, and may not be applicable in all situations.

**Apply Reciprocity Theorem on different types of Networks**

The Reciprocity Theorem is a fundamental principle in electrical engineering that can be applied to a wide variety of linear circuits, including passive circuits, active circuits, and transmission lines.

Passive Circuits: Consider a simple passive circuit with two input terminals and two output terminals.

Active Circuits: The Reciprocity Theorem can also be applied to active circuits, such as amplifiers. In this case, the theorem relates the input and output impedances of the amplifier.

Transmission Lines: The Reciprocity Theorem is also applicable to transmission lines, which are used to transmit electrical signals over long distances.

In summary, the Reciprocity Theorem is a powerful tool for analysing and designing a wide variety of linear circuits and systems. It relates the input and output properties of a circuit or system, and can be used to simplify analysis and optimise performance.

**List limitations of Reciprocity Theorem**

While the Reciprocity Theorem is a fundamental principle in electrical engineering, there are some limitations to its applicability. Here are some of the main limitations of the Reciprocity Theorem:

- Nonlinear Circuits: The Reciprocity Theorem is only applicable to linear circuits, where the output response is proportional to the input signal. In nonlinear circuits, the output response may not be proportional to the input signal, and the Reciprocity Theorem may not hold.
- Time-Varying Circuits: The Reciprocity Theorem is only applicable to time-invariant circuits, where the properties of the circuit do not change with time. In time-varying circuits, the properties of the circuit can change with time, and the Reciprocity Theorem may not hold.
- Frequency-Dependent Circuits: The Reciprocity Theorem is only applicable to circuits with frequency-independent parameters, where the circuit properties do not depend on the frequency of the input signal. In frequency-dependent circuits, such as those containing inductors, capacitors, and/or resonant circuits, the properties of the circuit can change with frequency, and the Reciprocity Theorem may not hold.
- Unilateral Circuits: The Reciprocity Theorem is only applicable to bilateral circuits, where the properties of the circuit are the same for signals propagating in both directions. In unilateral circuits, such as diodes and transistors, the properties of the circuit can be different for signals propagating in different directions, and the Reciprocity Theorem may not hold.
- Magnetic Fields: The Reciprocity Theorem is not applicable to circuits involving magnetic fields, since the magnetic field is a vector field that is not invariant under the interchange of the source and load. In circuits involving magnetic fields, other principles, such as Faraday’s Law and Ampere’s Law, must be used to analyse and design the circuit.

Despite these limitations, the Reciprocity Theorem is still a powerful tool for analysing and designing a wide range of linear circuits and systems. Its limitations must be taken into account when applying the theorem, and other principles must be used in cases where the Reciprocity Theorem does not hold.

**Describe Milliman’s Theorem**

Millman’s Theorem is a circuit analysis technique that allows for the quick and efficient analysis of parallel circuits with multiple voltage sources. It is particularly useful for circuits with more than two voltage sources.

The theorem states that for a parallel combination of voltage sources, the voltage at the common junction (or node) of the sources can be found by taking the weighted average of the individual voltage sources, where the weights are the conductances (the reciprocal of resistance) of the branches connected to each source.

Mathematically, Millman’s Theorem can be expressed as follows:

V = (Σ(G_{i} * V_{i})) / Σ(G_{i})

where V is the voltage at the common junction of the voltage sources, G_{i} is the conductance of the ith branch connected to the voltage source V_{i}, and the summation is taken over all the branches connected to the voltage sources.

In practical terms, Millman’s Theorem allows one to find the voltage at a common node in a parallel circuit without having to calculate the currents flowing in each branch of the circuit. This can save time and effort in circuit analysis, especially for circuits with many voltage sources.

It is important to note that Millman’s Theorem is only applicable to circuits that are linear and time-invariant, and that it assumes that the internal resistance of each voltage source is much smaller than the load resistance. If these conditions are not met, the theorem may not provide accurate results.

**Apply Millman’s Theorem on different types of Networks**

Milliman’s theorem is a powerful tool in circuit analysis that can be applied to various types of networks. The theorem states that the total current flowing into a junction or node in a circuit equals the sum of the currents flowing out of that junction or node.

To apply Milliman’s theorem, we can follow the following steps:

- Identify the junction or node in the circuit that we want to analyse.
- Identify all the current sources that are connected to the junction or node and determine the current flowing through each source.
- Identify all the resistors that are connected to the junction or node and determine the voltage drop across each resistor.
- Apply Milliman’s theorem by equating the sum of the current flowing into the junction or node to the sum of the current flowing out of the junction or node.

Here are some examples of how Milliman’s theorem can be applied to different types of networks:

- Series-parallel circuits: In a series-parallel circuit, we can use Milliman’s theorem to analyse the current flowing through each branch. We can apply the theorem at each junction or node in the circuit to determine the current flowing through each resistor and the voltage drop across each resistor.
- Mesh networks: In a mesh network, we can use Milliman’s theorem to analyse the currents flowing in each mesh. We can apply the theorem to each mesh to determine the current flowing through each resistor and the voltage drop across each resistor.
- Delta and star networks: In a delta or star network, we can use Milliman’s theorem to analyse the current flowing through each branch. We can apply the theorem at each junction or node in the circuit to determine the current flowing through each resistor and the voltage drop across each resistor.

**Describe Tellegen’s Theorem**

Tellegen’s theorem, also known as the reciprocity theorem, is a fundamental principle in electrical engineering that relates the voltages and currents in a network to the energy stored in the network. The theorem states that the sum of the instantaneous powers in a network at any instant of time is equal to the total derivative of the energy stored in the network with respect to time.

Mathematically, Tellegen’s theorem can be expressed as follows:

Σiᵢ(t)viᵢ(t) = dW(t)/dt

where Σiᵢ(t)viᵢ(t) is the sum of the instantaneous powers in the network at time t, W(t) is the energy stored in the network at time t, and dW(t)/dt is the total derivative of the energy stored in the network with respect to time.

Tellegen’s theorem applies to any type of network, including linear and nonlinear circuits, as well as networks that contain energy-storage elements such as capacitors and inductors. The theorem is particularly useful for analysing the behaviour of complex circuits and for determining the energy transfer between different parts of a circuit.

One important application of Tellegen’s theorem is in the analysis of electromagnetic systems, where it is used to relate the voltages and currents in a network to the electromagnetic fields and energy stored in the system. The theorem is also used in the analysis of power systems, where it is used to relate the power flowing in and out of a network to the energy stored in the system.

Overall, Tellegen’s theorem is a fundamental principle in electrical engineering that relates the voltages and currents in a network to the energy stored in the network, and is a powerful tool for analysing the behaviour of complex circuits and systems.

**Apply Tellegen’s Theorem on different types of Networks**

Tellegen’s theorem is a fundamental principle in electrical engineering that applies to any type of network, including linear and nonlinear circuits, as well as networks that contain energy-storage elements such as capacitors and inductors. The theorem relates the voltages and currents in a network to the energy stored in the network, and can be used to analyse the behaviour of complex circuits and systems. Here are some examples of how Tellegen’s theorem can be applied to different types of networks:

- RLC circuits: In an RLC circuit, Tellegen’s theorem can be used to relate the energy stored in the capacitors and inductors to the power flowing through the resistors. The theorem can be applied to the entire circuit or to individual components, depending on the analysis being performed.
- Power systems: In a power system, Tellegen’s theorem can be used to relate the power flowing in and out of the system to the energy stored in the system. The theorem can be applied to individual generators, transformers, and transmission lines, as well as to the overall system.
- Electromagnetic systems: In an electromagnetic system, Tellegen’s theorem can be used to relate the electromagnetic fields and energy stored in the system to the voltages and currents in the system. The theorem can be applied to a variety of systems, including antennas, motors, and generators.
- Nonlinear circuits: In a nonlinear circuit, Tellegen’s theorem can be used to analyse the behaviour of the circuit by relating the energy stored in the circuit to the power flowing through the components. The theorem can be applied to individual components or to the entire circuit.

Overall, Tellegen’s theorem is a powerful tool for analysing the behaviour of a wide variety of circuits and systems, including RLC circuits, power systems, electromagnetic systems, and nonlinear circuits. The theorem provides a fundamental relationship between the voltages, currents, and energy stored in a network, and can be used to gain insights into the behaviour of complex systems.

**Describe Compensation Theorem**

The Compensation theorem, also known as the Norton’s theorem or the Thevenin’s theorem, is a fundamental principle in electrical engineering that allows a complex network to be simplified into an equivalent circuit. The theorem states that any two-terminal network can be replaced by an equivalent circuit consisting of a single voltage source and a series resistor (in the case of Thevenin’s theorem) or a single current source and a parallel resistor (in the case of Norton’s theorem).

The equivalent circuit is determined by calculating the open-circuit voltage or short-circuit current of the original network and then using that value to calculate the equivalent resistance of the network. The open-circuit voltage or short-circuit current represents the response of the network to an external excitation, and the equivalent resistance represents the relationship between the voltage and current in the network.

The Compensation theorem is particularly useful for analysing the behaviour of complex networks and for designing circuits that meet specific performance requirements. The theorem allows a network to be simplified into a more manageable form, without losing any important information about the network’s behaviour.

One important application of the Compensation theorem is in the analysis and design of electronic circuits, where it is used to simplify complex networks into equivalent circuits that can be more easily analysed or modified. The theorem is also used in the design of power systems, where it is used to model the behaviour of complex power networks and to optimise the performance of the system.

Overall, the Compensation theorem is a fundamental principle in electrical engineering that allows complex networks to be simplified into equivalent circuits consisting of a single voltage source and a series resistor (in the case of Thevenin’s theorem) or a single current source and a parallel resistor (in the case of Norton’s theorem). The theorem is a powerful tool for analysing and designing circuits and systems, and is widely used in a variety of applications.

**Apply Compensation Theorem on different types of Networks**

The Compensation theorem (Thevenin’s and Norton’s theorem) is a powerful tool for analysing and simplifying complex electrical networks. Here are some examples of how the theorem can be applied to different types of networks:

- DC circuits: In a DC circuit, the Compensation theorem can be used to simplify a complex network into an equivalent circuit consisting of a single voltage source and a series resistor (Thevenin’s theorem) or a single current source and a parallel resistor (Norton’s theorem). This equivalent circuit can be used to calculate the voltage or current at any point in the original network.
- AC circuits: In an AC circuit, the Compensation theorem can be used to simplify a complex network into an equivalent circuit consisting of a single voltage source and a series impedance (Thevenin’s theorem) or a single current source and a parallel impedance (Norton’s theorem). This equivalent circuit can be used to calculate the voltage or current at any point in the original network.
- Power systems: In a power system, the Compensation theorem can be used to model the behaviour of a complex network of generators, transformers, and transmission lines. The equivalent circuit can be used to calculate the voltage and current at any point in the system, and to optimise the performance of the system.
- Electronic circuits: In an electronic circuit, the Compensation theorem can be used to simplify a complex network of resistors, capacitors, and transistors into an equivalent circuit consisting of a single voltage source and a series resistor (Thevenin’s theorem) or a single current source and a parallel resistor (Norton’s theorem). This equivalent circuit can be used to analyse the behaviour of the circuit and to optimise its performance.

Overall, the Compensation theorem is a powerful tool for analysing and simplifying complex electrical networks in a wide variety of applications. The theorem allows a complex network to be simplified into an equivalent circuit consisting of a single voltage source and a series resistor (Thevenin’s theorem) or a single current source and a parallel resistor (Norton’s theorem), which can be used to calculate the voltage and current at any point in the network. The theorem is widely used in the design and optimization of electrical circuits and systems.

**List limitations of Compensation Theorem**

While the Compensation theorem (Thevenin’s and Norton’s theorem) is a powerful tool for analysing and simplifying complex electrical networks, it does have some limitations. Here are some limitations of the theorem:

- The Compensation theorem applies only to linear, time-invariant networks. It cannot be applied to networks that are nonlinear or time-varying.
- The theorem assumes that the network is bilateral, meaning that the response to a given stimulus is the same in both directions. If the network is non-bilateral, the theorem cannot be applied.
- The theorem assumes that the network is passive, meaning that it does not produce energy. If the network is active, meaning that it produces energy, the theorem cannot be applied.
- The theorem assumes that the network is finite, meaning that it has a finite number of components. If the network is infinite, the theorem cannot be applied.
- The Compensation theorem provides an equivalent circuit for a network, but it does not provide information about the actual behaviour of the network. The equivalent circuit may not accurately represent the behaviour of the original network in all cases.
- The theorem assumes that the network is linear, meaning that the response to a given stimulus is proportional to the stimulus. If the network is highly nonlinear, the theorem may not provide an accurate representation of the network.

Overall, while the Compensation theorem is a powerful tool for analysing and simplifying complex electrical networks, it has limitations and cannot be applied in all cases. Engineers must take into account these limitations and apply the theorem carefully to ensure accurate results.

**Recall Substitution Theorem**

The Substitution theorem is a technique used in circuit analysis that allows a part of a circuit to be replaced by an equivalent circuit, simplifying the original circuit without changing its behaviour. The theorem states that any part of a circuit between two terminals can be replaced by an equivalent circuit consisting of a voltage source in series with a single resistor.

The Substitution theorem can be stated mathematically as follows:

Given a circuit with a two-terminal section A-B, the voltage across the terminals A-B and the current flowing through them can be found by substituting a voltage source V with a series resistance R between the terminals A-B, where V is the open-circuit voltage across terminals A-B, and R is the resistance of the equivalent circuit looking into the terminals A-B with all the sources turned off.

This equivalent circuit can be obtained by calculating the resistance between the terminals A-B with all independent sources turned off and replacing all dependent sources with their equivalent circuit.

The Substitution theorem is useful in circuit analysis because it allows a complex circuit to be simplified by replacing a part of the circuit with an equivalent circuit that is easier to analyse. By using this theorem, engineers can break down a complex circuit into smaller, more manageable parts and then combine the results to obtain the behaviour of the entire circuit.

**Apply Substitution Theorem on different types of Networks**

The Substitution Theorem is a powerful tool used in electrical circuit analysis to simplify complex circuits. It states that any active or passive linear network can be replaced by an equivalent circuit that has the same voltage and current characteristics at the output terminals. In other words, any network can be replaced by a simpler network with the same input/output behavior.

The Substitution Theorem can be applied to different types of networks, including:

- Resistive networks: In a resistive network, the Substitution Theorem can be used to replace a network of resistors with a single equivalent resistor. This can be useful for simplifying complex circuits and making them easier to analyze. The equivalent resistance is calculated by shorting the output terminals of the circuit and finding the total resistance between the input terminals.
- Capacitive networks: In a capacitive network, the Substitution Theorem can be used to replace a network of capacitors with a single equivalent capacitor. This can be useful for simplifying circuits that have multiple capacitors and making them easier to analyze. The equivalent capacitance is calculated by opening the output terminals of the circuit and finding the total capacitance between the input terminals.
- Inductive networks: In an inductive network, the Substitution Theorem can be used to replace a network of inductors with a single equivalent inductor. This can be useful for simplifying circuits that have multiple inductors and making them easier to analyze. The equivalent inductance is calculated by shorting the output terminals of the circuit and finding the total inductance between the input terminals.
- Mixed networks: The Substitution Theorem can also be applied to networks that contain a combination of resistors, capacitors, and inductors. In this case, the equivalent circuit will contain a single equivalent resistor, capacitor, and inductor that together have the same input/output characteristics as the original circuit.

To apply the Substitution Theorem, we first identify the network that we want to simplify and the output terminals that we want to replace. Next, we either short-circuit or open-circuit the output terminals, depending on whether we are dealing with a resistive network or a capacitive/inductive network.

**Recall Network Theorems in AC circuits**

In AC circuits, several network theorems can be used to simplify the analysis and design of circuits. Some of the key network theorems used in AC circuits are:

- Superposition theorem: This theorem states that the total response of a linear circuit with multiple sources is the sum of the responses due to each source acting independently. The theorem applies to circuits with both AC and DC sources.
- Thevenin’s theorem: This theorem states that any linear AC circuit can be replaced by an equivalent circuit consisting of a single voltage source and a series impedance. The equivalent impedance is the impedance seen by the load terminals of the original circuit when all sources are turned off.
- Norton’s theorem: This theorem is similar to Thevenin’s theorem, but instead of a voltage source, it uses a current source and a parallel impedance.
- Maximum Power Transfer theorem: This theorem states that the maximum power is transferred from a source to a load when the load impedance is equal to the complex conjugate of the source impedance.
- Reciprocity theorem: This theorem states that the impedance seen looking into a port of a linear circuit is the same as the impedance seen when the source and load are interchanged.
- Tellegen’s theorem: This theorem states that the sum of the instantaneous power supplied by all sources in a network is equal to the sum of the instantaneous power consumed by all elements in the network.

Overall, these theorems can be used to simplify AC circuits, find equivalent circuits, optimise power transfer, and more. By understanding and applying these theorems, engineers can design and analyse AC circuits more efficiently and effectively.

**Calculate the response in AC circuits using Network Theorems**

Here are some examples of how to calculate the response in AC circuits using network theorems:

- Superposition theorem: Consider a circuit with two AC voltage sources, each with a different frequency. To calculate the response of the circuit, we can apply the superposition theorem by calculating the response due to each source acting independently and then summing the results.
- Thevenin’s theorem: Consider a complex AC circuit with multiple sources and impedances. To simplify the circuit, we can use Thevenin’s theorem to find the equivalent circuit consisting of a single voltage source and impedance. This simplifies the circuit and makes it easier to analyse.
- Norton’s theorem: Similar to Thevenin’s theorem, we can use Norton’s theorem to simplify a complex AC circuit by finding the equivalent circuit consisting of a single current source and parallel impedance.
- Maximum power transfer theorem: Consider a circuit with a source and a load. To maximise the power transfer between the source and the load, we can use the maximum power transfer theorem to determine the load impedance that maximises the power transferred.
- Reciprocity theorem: Consider a circuit with two ports, with a source connected to one port and a load connected to the other. To find the impedance seen looking into one port, we can use the reciprocity theorem to determine the impedance seen looking into the other port when the source and load are interchanged.
- Tellegen’s theorem: Consider a circuit with multiple sources and loads. To verify the conservation of energy in the circuit, we can use Tellegen’s theorem to sum the instantaneous power supplied by all sources and compare it to the sum of the instantaneous power consumed by all elements in the circuit.

Overall, these theorems provide powerful tools for analysing and designing AC circuits, and they can be used in a wide range of applications. By understanding and applying these theorems, engineers can optimise the performance of AC circuits and ensure that they meet design specifications.