Question

Under which conditions is the superposition theorem applicable?

Answer 1

The superposition theorem is a fundamental principle in linear circuit analysis and is used to simplify the analysis of complex electrical circuits. For the superposition theorem to be applicable, the following conditions must be met:

### 1. **Linearity of the Circuit**
   - **Linear Components**: The theorem is applicable only if the circuit consists of linear components. Linear components are those whose voltage-current relationships are linear, meaning they follow Ohm’s Law (V = IR). Examples of linear components include resistors, capacitors, and inductors. Non-linear components, such as diodes and transistors, do not satisfy this condition.
   - **Linear Combination**: The circuit elements should obey the principle of linearity. In other words, the response of the circuit (voltage or current) to a linear combination of inputs should be the linear combination of the responses to each individual input.

### 2. **Superposition Principle**
   - **Additivity**: The superposition theorem relies on the principle of additivity, which states that the response (voltage or current) in any branch of a linear circuit with multiple independent sources is the sum of the responses caused by each source acting independently.
   - **Homogeneity**: It also relies on homogeneity, which means if the input is scaled, the output scales proportionally. This is particularly relevant in linear systems where the output is directly proportional to the input.

### 3. **Independent Sources**
   - **Voltage and Current Sources**: The theorem applies to circuits with independent sources of voltage and/or current. It requires that each source can be considered separately to analyze its effect on the circuit.
   - **Deactivation of Sources**: To apply the theorem, each independent source is considered individually, with all other independent sources turned off. For voltage sources, this means replacing them with short circuits, and for current sources, this means replacing them with open circuits.

### Practical Steps for Application:
1. **Turn Off All Sources Except One**: Consider one independent source at a time and turn off the others. For voltage sources, replace them with a short circuit. For current sources, replace them with an open circuit.
2. **Analyze the Circuit**: Solve the circuit to find the response (voltage or current) due to the active source.
3. **Repeat for All Sources**: Perform the same analysis for each independent source.
4. **Sum the Responses**: Add up all the individual responses to obtain the total response in the circuit.

### Example:
Consider a circuit with two independent voltage sources \( V_1 \) and \( V_2 \) and several resistors. To find the current through a particular resistor using superposition:
1. **Deactivate \( V_2 \)**: Replace \( V_2 \) with a short circuit and solve for the current through the resistor due to \( V_1 \).
2. **Deactivate \( V_1 \)**: Replace \( V_1 \) with a short circuit and solve for the current through the resistor due to \( V_2 \).
3. **Combine Results**: Add the currents found in the previous steps to get the total current through the resistor.

By adhering to these conditions, the superposition theorem provides a powerful and straightforward method for analyzing complex linear circuits.

Answer 2

The Superposition Theorem is a fundamental principle used in electrical engineering to analyze linear circuits with multiple independent sources. This theorem is particularly useful for simplifying the analysis of circuits that have more than one voltage source or current source. Here’s a detailed explanation of the conditions under which the Superposition Theorem is applicable:

### Conditions for Applicability

1. **Linearity of the Circuit Components**:
   - The Superposition Theorem is applicable only to circuits where all components (resistors, capacitors, inductors, and linear dependent sources) exhibit linear behavior. This means the components must obey Ohm's Law and the relationships of linear circuit theory. Specifically:
     - **Resistors**: The voltage across a resistor is directly proportional to the current through it (Ohm's Law: \( V = IR \)).
     - **Capacitors**: The relationship between current and voltage is linear in a capacitor, described by \( I = C \frac{dV}{dt} \).
     - **Inductors**: The relationship between voltage and current is linear in an inductor, given by \( V = L \frac{dI}{dt} \).

2. **Independent Sources**:
   - The theorem applies to circuits with multiple independent voltage and/or current sources. An independent source is one whose value is not affected by the other sources in the circuit.

3. **Superposition in Analysis**:
   - To use the Superposition Theorem, you analyze the circuit multiple times, once for each independent source. Each source is considered separately, while all other sources are turned off (replaced with their internal resistances):
     - **Voltage Sources**: Replace other independent voltage sources with a short circuit (i.e., a wire).
     - **Current Sources**: Replace other independent current sources with an open circuit (i.e., a break in the wire).

4. **Summing Effects**:
   - After analyzing the circuit for each source individually, you sum the individual responses (voltages or currents) to obtain the total response in the circuit. This step involves adding the contributions from each source to get the overall effect on the circuit.

### Example of Applying the Superposition Theorem

Consider a simple circuit with two independent voltage sources \( V_1 \) and \( V_2 \), and a resistor \( R \). To find the voltage across \( R \) using the Superposition Theorem:

1. **Turn off \( V_2 \)**: Replace \( V_2 \) with a short circuit. Analyze the circuit with only \( V_1 \) active. Calculate the voltage across \( R \) due to \( V_1 \) alone.

2. **Turn off \( V_1 \)**: Replace \( V_1 \) with a short circuit. Analyze the circuit with only \( V_2 \) active. Calculate the voltage across \( R \) due to \( V_2 \) alone.

3. **Combine Results**: Add the voltages obtained from the two separate analyses to get the total voltage across \( R \).

### Limitations

1. **Nonlinear Components**: The Superposition Theorem does not apply to circuits with nonlinear components like diodes or transistors in their nonlinear operating regions. For these components, the response is not proportional to the input, and the theorem cannot be used.

2. **Dependent Sources**: If a circuit has dependent sources, these sources depend on circuit variables, and thus cannot be turned off when analyzing the effects of other independent sources. However, the Superposition Theorem still applies to circuits with dependent sources, provided that you carefully account for their effects in the analysis.

In summary, the Superposition Theorem is a powerful tool for analyzing linear circuits with multiple independent sources. It simplifies the analysis by breaking it down into simpler problems, provided the circuit components are linear and the independent sources can be individually considered.