The complete response of a first-order circuit is equal to: 

Explanation:

Complete Response of a First-Order Circuit

Definition: In electrical engineering, the complete response of a first-order circuit refers to the total behavior of the circuit when subjected to an input signal. A first-order circuit typically consists of a single energy storage element, such as a capacitor or inductor, along with resistive elements. The complete response is a combination of two components: the natural response and the forced response.

Natural Response: The natural response of a circuit represents the behavior of the system due to its internal energy storage elements. It is the transient response of the circuit that occurs when the initial energy stored in the capacitor or inductor dissipates over time. This response is determined by the circuit's initial conditions and does not depend on external inputs.

Forced Response: The forced response, also known as the steady-state response, is the behavior of the circuit due to external input signals. It describes how the circuit reacts to a sustained input, such as a voltage or current source. The forced response is determined solely by the input signal and the circuit's configuration.

Complete Response: The complete response of a first-order circuit is the summation of the natural response and the forced response. Mathematically, it can be expressed as:

Complete Response = Natural Response + Forced Response

This relationship indicates that the total behavior of the circuit is the combined effect of its internal energy dissipation (natural response) and its reaction to external input signals (forced response).

Correct Option:

The correct answer is:

Option 2: (Natural response) + (Forced response)

This option correctly identifies the complete response of a first-order circuit as the sum of its natural response and forced response. This summation accounts for both the transient and steady-state behaviors of the circuit, providing a comprehensive description of its operation.

Important Information

To further understand the analysis, let’s evaluate the other options:

Option 1: (Natural response) - (Forced response)

This option is incorrect because subtracting the forced response from the natural response does not represent the complete behavior of the circuit. The complete response requires the summation of both components, not their difference.

Option 3: (Natural response)/(Forced response)

This option is also incorrect as dividing the natural response by the forced response does not describe the complete response of a first-order circuit. This operation is mathematically and conceptually unrelated to the principle of summation that defines the complete response.

Option 4: (Natural response) × (Forced response)

This option is incorrect because multiplying the natural response by the forced response does not represent the complete response. The complete response is defined as the addition of the two components, not their product.

Conclusion:

The complete response of a first-order circuit is the sum of its natural response and forced response. This summation captures the transient and steady-state behaviors of the circuit, ensuring a comprehensive understanding of its operation. Understanding the distinction between natural and forced responses is crucial for analyzing and designing first-order circuits effectively.