Root Locus Calculate Departure Angle

What is Departure Angle?

The departure angle is the angle at which a root leaves the imaginary axis in a root locus plot. It's a key parameter in understanding the behavior of control systems as the gain is varied. The departure angle provides insight into how the system's poles move in the complex plane.

In root locus analysis, the departure angle is particularly important when designing control systems. It helps engineers understand how the system's stability and performance characteristics change as the gain is adjusted. By calculating the departure angle, engineers can make informed decisions about system design and compensation.

How to Calculate Departure Angle

Calculating the departure angle involves several steps. First, you need to identify the poles and zeros of the system. Then, you can use the formula for the departure angle to compute its value. The departure angle is typically calculated at the point where the root leaves the imaginary axis.

To calculate the departure angle, you'll need to know the locations of the poles and zeros in the complex plane. The formula for the departure angle involves the sum of the angles between the point of departure and each of the poles and zeros. This sum is then divided by the number of poles minus one.

Formula

This formula accounts for the contributions of both poles and zeros to the departure angle. The departure angle is calculated at the point where the root leaves the imaginary axis, and it provides insight into the behavior of the system as the gain is varied.

Example Calculation

Let's consider a system with two poles at s = -1 and s = -2, and one zero at s = -3. We want to calculate the departure angle at the point where the root leaves the imaginary axis.

First, we identify the point of departure. In this case, the root leaves the imaginary axis at s = -1.5. We then calculate the angles between this point and each of the poles and zeros.

The angles between the point of departure and the poles are θ_p1 = 45° and θ_p2 = 90°. The angle between the point of departure and the zero is θ_z = 135°.

Using the formula, we calculate the departure angle as follows:

In this example, the departure angle is 270°, which indicates the direction in which the root leaves the imaginary axis.

Interpreting Results

The departure angle provides valuable information about the behavior of the system. A positive departure angle indicates that the root moves into the right half of the complex plane, which can lead to instability. A negative departure angle indicates that the root moves into the left half of the complex plane, which is desirable for stability.

By analyzing the departure angle, engineers can make informed decisions about system design and compensation. The departure angle helps engineers understand how the system's stability and performance characteristics change as the gain is adjusted. It provides insight into the behavior of the system and helps engineers make informed decisions about system design and compensation.