Question

All problems pertain to the unity negative feedback control system below, a proportional controller with a control gain K:

For this problem, G(s) is defined as:
G(s) = (s + 3)(s² + 2s + 4) / (s + 1)

(a) Write G(jω) in the form G(jω) = u(ω) + jv(ω).

Answer 1

To write G(jω) in the form G(jω) = u(ω) + jv(ω), we substitute jω for s in the given transfer function, G(s). By simplifying the expression step by step, we find that u(ω) = -ω³ + 12 and v(ω) = (15ω² + 10ω) / (jω + 1).

Step-by-step explanation:

To write G(jω) in the form G(jω) = u(ω) + jv(ω), we substitute jω for s in the given transfer function, G(s).

Let's simplify the expression G(jω) step by step:

1. (jω + 3)((jω)² + 2(jω) + 4) / ((jω) + 1)
2. (jω + 3)(-ω² + 2jω + 4) / (jω + 1)
3. (-ω³ + 3jω² + 4jω + 12ω² + 6jω + 12) / (jω + 1)
4. (-ω³ + 15jω² + 10jω + 12) / (jω + 1)
5. (-ω³ + 12) + (15jω² + 10jω) / (jω + 1)

Therefore, u(ω) = -ω³ + 12 and v(ω) = (15ω² + 10ω) / (jω + 1).