Root Locus Angle of Departure Calculation

What is Angle of Departure?

The angle of departure is a critical parameter in root locus analysis that determines the direction in which the root locus begins to move from a complex conjugate pole pair when a gain is increased. It provides insight into the system's dynamic response characteristics.

In control systems engineering, the angle of departure helps predict how the system's poles will move as the gain increases, which is essential for designing stable and well-behaved control systems. A well-chosen angle of departure can prevent the system from entering unstable regions.

How to Calculate Angle of Departure

Calculating the angle of departure involves several steps that require knowledge of the system's open-loop transfer function and its poles. Here's a simplified overview of the process:

1. Identify the complex conjugate pole pair from which the root locus departs.
2. Determine the angles contributed by all other poles and zeros to the departure point.
3. Calculate the angle of departure using the formula provided below.

The angle of departure is typically measured in degrees and provides valuable information about the system's behavior as the gain increases.

Formula

The angle of departure (θ) can be calculated using the following formula:

Where:

• θ is the angle of departure in degrees
• s is the complex conjugate pole pair (s = σ + jω)
• zᵢ are the zeros of the open-loop transfer function
• pⱼ are the poles of the open-loop transfer function (excluding s)

This formula accounts for the angles contributed by all other system components to the departure point.

Example Calculation

Consider a system with the following open-loop transfer function:

The complex conjugate poles are at s = -1 ± j1. Using the formula, we can calculate the angle of departure as follows:

1. Identify the complex conjugate pole pair: s = -1 ± j1
2. Calculate the angles contributed by the other pole (s = -1) and zero (if any)
3. Apply the formula to find the angle of departure

The calculated angle of departure for this system is approximately 135 degrees, indicating the direction in which the root locus will begin to move as the gain increases.

Interpretation of Results

The angle of departure provides several important insights into the system's behavior:

• It indicates the direction in which the root locus will begin to move as the gain increases.
• A well-chosen angle of departure helps ensure the system remains stable and well-behaved.
• It can be used to predict the system's dynamic response characteristics.

Understanding the angle of departure is essential for designing control systems that meet specific performance requirements and maintain stability.