sketch the underdamped time response of a typical second order feedback control system subjected to a unit step input
I'm sorry, but as a text-based AI, I cannot directly sketch images. However, I can describe the typical shape of the underdamped time response of a second-order feedback control system subjected to a unit step input.
In an underdamped system, the response starts from the initial state and initially overshoots the steady-state value before settling down. The response consists of a series of oscillations known as damped oscillations. The number of oscillations and their damping ratio depend on the system's parameters.
The typical shape of the underdamped time response can be described as follows:
1. The response starts at zero and rapidly rises initially in the positive direction.
2. It overshoots the steady-state value (typically around 1 for a unit step input) and reaches a maximum value known as the peak overshoot.
3. After reaching the peak overshoot, the response starts oscillating around the steady-state value.
- The amplitude of the oscillations gradually decreases over time.
- The oscillation frequency is determined by the system's natural frequency.
- The damping ratio determines how quickly the oscillations decay.
4. Eventually, the response settles down and approaches the steady-state value from below.
- The settling time depends on the system's damping ratio.
- The response may never completely reach the steady-state value due to the presence of damping.
To visualize the underdamped time response, you can search online for "underdamped second-order system step response" to find plots and diagrams that illustrate the behavior.
To sketch the underdamped time response of a typical second-order feedback control system subjected to a unit step input, you can follow these steps:
Step 1: Determine the system's transfer function
The transfer function of a typical second-order system can be written as:
G(s) = (K * ωn^2) / (s^2 + 2ζωn * s + ωn^2)
where:
- K is the system gain.
- ωn is the natural frequency.
- ζ is the damping ratio.
- s is the Laplace variable.
Step 2: Identify the damping ratio ζ
To determine if the system is underdamped, critically damped, or overdamped, you need to examine the value of the damping ratio ζ. In the case of an underdamped system, ζ is less than 1.
Step 3: Calculate the natural frequency ωn
The natural frequency ωn is determined by the system parameters and can be calculated using the formula:
ωn = sqrt(1 / (T * ζ))
where T is the settling time desired for the system response.
Step 4: Sketch the response
For an underdamped system, the time response will exhibit oscillations. The number of oscillations and their shape depends on the value of ζ. The following steps can guide you in sketching the response:
a. Determine the settling time (Ts): The settling time is the time for the response to settle within a specified percentage of the final value. This percentage is typically 2% or 5%.
b. Identify the overshoot (Mp): The overshoot is the maximum deviation of the response from the steady-state value.
c. Identify the peak time (Tp): The peak time is the time at which the maximum response occurs.
d. Sketch the response: Plot the response on a time versus amplitude graph. The response will start from zero and gradually approach the final value with oscillations.
e. Label the key time points: Label the peak time (Tp), the settling time (Ts), the overshoot (Mp), and the 2% or 5% settling bands on the graph.
Keep in mind that the actual shape and values of these response parameters depend on the specific system parameters such as K, ζ, and ωn.
Remember, this is a general procedure for sketching the underdamped response. The actual shapes and values may vary depending on the specific system parameters.