Assignment Instructions

Exploring and Analyzing Second$-$Order Transfer Functions Across Stability Regions

Part $1$: Exploring Transfer Function Cases

Task $1$: Define General Transfer Functions

Define second$-$order transfer functions and understand different types of roots based on $\backslash$zeta $($damping ratio$)$ variations.

Task $2$: Explore Different Root Cases

Overdamped: $\backslash$zeta $>1$

Underdamped: $0<\backslash$zeta $<1$

Critically Damped: $\backslash$zeta $=1$

Undamped: $\backslash$zeta $=0$

Task $3$: Stability Exploration

Explore both Stable and Unstable regions by adjusting the sign and magnitude of the coefficients in the transfer function.

Part $2$: Poles and Zeros Analysis

Task $4$: Poles and Zeros Visualization

Plot poles and zeros for both stable and unstable scenarios, ensuring that the distinctions and implications of pole locations are discussed.

Part $3$: System Response Analysis

Task $5$: Analyze and Plot System Responses

Investigate and plot system responses for different input signals, ensuring that both stable and unstable scenarios are represented:

Impulse Input

Step Input

Ramp Input

Parabolic Input

Part $3$: Report Writing

Compile findings and analyses into a report, with clear discussion for stable vs$.$ unstable cases.

Depth of analysis regarding stable and unstable cases.