Welcome to this module focused on solving numerical problems related to the time response of control systems, a crucial area for the GATE Electrical Engineering exam, particularly for Power Systems and Machines specialization. Understanding how systems behave over time when subjected to inputs is fundamental to designing and analyzing electrical systems.

The time response of a control system is its output as a function of time, given a specific input. It's typically divided into two parts: the transient response and the steady-state response. The transient response is the part of the response that changes with time and eventually becomes zero. The steady-state response is the part of the response that remains after the transient response has died out.

For second-order systems, which are commonly used to approximate higher-order systems, several key specifications define the transient response. These are essential for solving numerical problems.

The standard form of a second-order system's transfer function is given by: $G(s) = \frac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2}$ where $\omega_n$ is the undamped natural frequency and $\zeta$ is the damping ratio. The damping ratio ($\zeta$) is critical in determining the nature of the response.

To solve numerical problems, you'll need to know the formulas relating the time response specifications to the system parameters ($\omega_n$ and $\zeta$). These are derived from the system's response to a unit step input.

When faced with a numerical problem, follow these steps:

1. Identify System Type: Determine if the system is first-order, second-order, or can be approximated as such. For GATE, second-order systems are most common.
2. Extract System Parameters: From the given transfer function, identify $\omega_n$ (undamped natural frequency) and $\zeta$ (damping ratio).
3. Determine Response Specifications: Understand what is being asked – rise time, settling time, overshoot, etc.
4. Apply Relevant Formulas: Use the formulas provided earlier, ensuring you use the correct ones based on the specification and the system's damping ratio.
5. Calculate Unknowns: Perform the necessary calculations.
6. Verify Solution: Check if the calculated values are reasonable given the system parameters.

Consider a system with the transfer function $G(s) = \frac{16}{s^2 + 8s + 16}$. Find the time response specifications for a unit step input.

Comparing $G(s) = \frac{16}{s^2 + 8s + 16}$ with the standard form, we have $\omega_n^2 = 16$, so $\omega_n = 4$ rad/s. Also, $2\zeta\omega_n = 8$. Substituting $\omega_n = 4$, we get $2\zeta(4) = 8$, which means $8\zeta = 8$, so $\zeta = 1$. This is a critically damped system.

For a critically damped system ($\zeta = 1$):

• Delay Time ($t_d$): The formula $t_d = \frac{1 + 0.7\zeta}{\omega_n}$ is an approximation for underdamped systems. For critically damped systems, the response is non-oscillatory and reaches the final value directly. A precise calculation for $t_d$ in critically damped systems is more complex, but often approximated or considered differently.
• Rise Time ($t_r$): For $\zeta = 1$, $t_r = \frac{1}{\omega_n} = \frac{1}{4} = 0.25$ seconds.
• Peak Time ($t_p$): For $\zeta = 1$, $t_p$ is undefined as there is no overshoot.
• Maximum Overshoot ($M_p$): For $\zeta = 1$, $M_p = 0\%$.
• Settling Time ($t_s$): For 2% tolerance, $t_s = \frac{4}{\zeta\omega_n} = \frac{4}{1 \times 4} = 1$ second. For 5% tolerance, $t_s = \frac{3}{\zeta\omega_n} = \frac{3}{1 imes 4} = 0.75$ seconds.

Consistent practice with various numerical problems is key to mastering this topic. Focus on understanding the relationship between the system's characteristic equation, its parameters ($\omega_n$, $\zeta$), and the resulting time response specifications. Pay close attention to the units and the specific requirements of each question.