Superposition Theorem

Superposition Theorem Statement: According to the Superposition Theorem, the response (voltage or current) at any point in a linear electrical network with multiple independent sources can be calculated by calculating the individual contributions of each source while assuming the other sources are “turned off” or replaced by their internal resistances.

The Superposition Theorem allows you to analyze a circuit with multiple voltage or current sources by considering the effect of each source separately while treating the others as inactive.

In this article, we will learn all about the Superposition theorem along with its statement, steps to be followed, advantages and disadvantages, applications, numerical problems and frequently asked questions based on it.

What is the Superposition Theorem?

It is used in the electrical circuit analysis, which indicates that in a linear network – any one branch of voltage and current can be found by inspecting the effect of the independent source respectively.

The Superposition Theorem only applies to linear circuits in which the relationship between voltage and current is constant and does not include nonlinear components such as diodes or transistors.

The Superposition Theorem can significantly simplify the analysis of complex circuits, especially when there are many independent sources. However, this theorem can be time-consuming for circuits with a large number of sources, as you need to perform separate calculations for each source. In such cases, other circuit analysis techniques like nodal analysis or mesh analysis might be more efficient.

Guidelines While Using the Superposition Theorem

Here are some Guidelines to keep in mind while using the superposition theorem:

• All the components must be linear with the circuit voltage and current, whenever using the superposition theorem.
• Since power is not a linear quantity, it should not be used with power.
• Ideal voltage source should be removed.
• Ideal current source should be removed so that it can be replaced with an open circuit.
• Real voltage source should be removed , to replace with the internal resistance.
• Consider one source at a time from the various multiple sources.
• Replace all the other (except the selected source) impedances with their internal resistance.
• Calculate the current flowing through the source and the voltage drop across it.
• Repeat the above steps by taking the individual sources one by one.
• Take the algebraic sum once all the current sources and voltage drops have been calculated.

How to Apply Superposition Theorem?

• Turn Off All but One Source: Consider only one independent source (voltage or current source) active, while all other independent sources are turned off (replaced by their internal resistances, which are typically zero for ideal voltage sources and infinite for ideal current sources).
• Analysis of the Circuit: With only one source active, analyze the circuit using circuit analysis techniques such as Ohm’s law, Kirchhoff’s law, and other relevant methods.
• Calculation of the Response: Determine the voltage, current, or any other required parameters in the circuit due to the single active source.
• Repeat for Each Source: Repeat steps 1 to 3 for each independent source in the circuit.
• Combine Responses: After calculating the response for each individual source, you may determine the total response at the required  by adding or superimposing the individual response from each source.

Prerequisites for Using Superposition Theorem

• It is used for the balanced bridge circuits , which are reducible to series/parallel combinations
• Al the components of the circuit must be linear.
• The waves which are being superposed should be of the same type and behaves linearly.
• Two or more sources should be included in the circuit.

Solved Example on Superposition Theorem

Example 1. Find the current through 3 Ω resistor using superposition theorem.

Electric Circuit

Answer:

• Step 1: To find I1.

Consider the 20 V voltage source alone. Short circuit the other voltage source.

To find the current through 3 Ω resistor, it is necessary to determine the total current supplied by the source (I_T).

If we observe the circuit, 3 Ω and 6 Ω resistors are in parallel with each other. This parallel combination is connected in series with a 5 Ω resistor. The equivalent or total resistance is obtained as below,

⇒R_T = 5 + (3*6/9) = 7 Ω

By applying Ohm’s law,

⇒I_T = V/R_T = 20/7 = 2.857 A

Now, the current through 3 Ω resistor is determined by using current division rule. It is given by,

⇒I₁ = I_T*(6/6+3) = 2.857*0.667 = 1.904 A

• Step 2: To find I₂.

Consider the 40 V voltage source alone. Short circuit the other voltage source.

Now, to find the current through 3 Ω resistor, it is necessary to determine the total current supplied by the source (IT).

If we observe the circuit, 3 Ω and 5 Ω resistors are in parallel with each other. This parallel combination is connected in series with a 6 Ω resistor. Hence the equivalent or total resistance is obtained as below,

⇒R_T = 6 + [(3*5)/(3+5)] = 7.875 Ω

By applying Ohm’s law,

⇒I_T = V/R_T = 40/7.875 = 5.079 A

Now, the current through 3 Ω resistor is determined by using current division rule. It is given by,

⇒I₂ = I_T*(5/5+3) = 5.079*0.625 = 3.174 A

The below figure shows the resultant circuit, which depicts the currents produced because of two voltage sources 20 V and 40 V acting individually.

By superposition theorem, the total current is determined by adding the individual currents produced by 20 V and 40 V.

Therefore, the current through 3 Ω resistor is :

I₁ + I₂ = 1.904 + 3.174 = 5.078 A

Applications of Superposition Theorem

Some applications of Superposition Theorem are:

• Superposition Theorem can be used to study the effects of each source separately on parameters such as voltage levels, current levels, etc.
• Superposition Theorem can be used to study how each element affects the overall circuit behavior in large electronic circuits.
• Superposition Theorem can be used how a single element like amplifiers, filters, and other electronic elements affect the large analog circuits when any defect arises.
• Superposition Theorem can be used in sensors evaluation where multiple sensors are involved an each sensor has to studied on the basis of their performance.
• Superposition Theorem can be used in Engineering Electromagnetics where the fault has to studied in transmission line analysis related to voltage, current, resistance, capacitance, inductance, etc.
• Superposition Theorem can be used for multiple-frequency application to analyze the effects of each frequency component for the overall filter response.

Advantages of Superposition Theorem

Some Advantages of Superposition Theorem are:

• It makes the analysis of the circuit easy by breaking down the circuits into smaller parts for easy understanding.
• It makes the analysis of an individual source easy by making it easy to study one component of an electronic circuit without impacting the other components.
• Insight into Circuit Behavior: By studying the effect of each source independently, we can gain insight into how different sources interact with each other and the resulting circuit behavior.
• It makes the study of an electronic component about its behavior towards other components and the overall circuits.

Limitations of Superposition Theorem

Some Limitations of Superposition Theorem are:

• Limited to Linear Circuits: The Superposition Theorem only applies to linear circuits when voltage and current have a constant relationship. It cannot be used in circuits including nonlinear components such as diodes and transistors.
• May Not Capture All Effects: In some cases, interactions between sources can lead to effects that the Superposition Theorem doesn’t fully capture. It assumes that the interaction between sources is negligible, which might not always be the case.
• Doesn’t Provide Overall Solution: The Superposition Theorem provides solutions for individual sources but doesn’t directly yield the overall response when all sources are active. Summing up individual solutions may not always result in an accurate representation of the circuit behavior when all sources are present.
• Not Suitable for Complex Networks: For circuits with interconnected components and complex feedback loops, applying the Superposition Theorem might not be practical or accurate in capturing all circuit dynamics.

Conclusion

The Superposition Theorem allows you to evaluate complex circuits with multiple voltage or current sources by evaluating each source individually and setting all other sources to zero (Voltage source is replaced by short circuit, Current source is replaced by open circuit).

This allows you to break down a difficult circuit into simpler sub-circuits and more quickly solve for voltage, current, etc.

Superposition Theorem – FAQs

What is the principle of Superposition Theorem used in circuit analysis ?

The Superposition Theorem is a principle used in circuit analysis to simplify the analysis of circuits with multiple independent sources. It states that the response at any point in a linear circuit can be determined by considering the effect of each source individually while treating the other sources as inactive.

What is the theory of superposition?

The theory of superposition is a principle in physics and engineering stating that in a linear system, the net response at any given time or point due to multiple stimuli is the sum of the responses that each stimulus would produce individually.

When can the Superposition Theorem be applied?

The Superposition Theorem only applies to linear circuits where the voltage-current relationship is constant. It is incompatible with circuits incorporating non linear components such as diodes, transistors, or non linear resistors.

How does the Superposition Theorem work?

The theorem works by analyzing the circuit under the influence of each independent source one at a time while turning off the other sources or replacing them with their internal resistances. The total response is then obtained by superposing the individual responses due to each source.

What are the steps to apply the Superposition Theorem?

The steps involve turning off each independent source except the one being analyzed, calculating the response due to that source, and repeating the process for all sources. The final response is obtained by summing up the individual responses.

Why do we use superposition theorem?

We use the superposition theorem in electrical engineering to simplify the analysis of complex linear circuits by breaking them down into simpler, more manageable components.