Question

Use the given transfer function and identify the correct steady-state response yss(t) to the given input function f(t). t(s)=y(s)f(s)=7514s 18, f(t)=10 sin 1.5t

Answer 1

Given transfer function, the identified correct steady-state response ys(t) to the given input function is: y (t) = 27.116 Sin (1.5 t - 49.4°)

What is a transfer function?

A transfer function is a mathematical link between a numerical input to a dynamic system and the subsequent output in statistical time-series analysis, signal processing, and control engineering.

What is the calculation justifying the above result?

Step 1

Recall that we are given T(s) = Ys)/F(s) = 75/(145 + 18)

⇒ T(Jw) = 75/ [18 + J14w]

This is called a steady state response to Sinusoidal input.

f(t) = [T(t) ↔T(jw)] → y(t A Sin (wi + φ)

Where A = A1 x [T(jw)] w= w1

⇒ A = A1 [T(jw1)]

And,

φ = θ + [T(jw)] w = w1

Step 2

Given, f(t) = 10 Sin 1.5t

A1 = 10, W1 = 1.5 red/see, θ = 0°

The Magnitude of T(jw) is

[T(jw)] = 75/√[(18²) + (14w)²

⇒ [T(jw] w = w1 = 1.5

= 75/√[(18²) + (141.5)²

⇒ [T(jw1)] = 2.7116

Hence,

A = A1 x [T(jw1)] =

10 x 2.7116

⇒ A = 2.116

⇒ Phase of T(jw) is,

[T(jw)] = - tan ⁻¹ (14w/18)

Hence, φ = 0° - Tan⁻1 (21/18)

φ = -49.4°

Hence,

y(t) = A sin (w1t + φ)

⇒ y (t) = 27.116 Sin (1.5 t - 49.4°)

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