Question

A tendon-operated robotic hand can be implemented using a pneumatic actuator [8]. The actuator can be represented by G(s)= 5000/(s+70)(s+500) ​ . Plot the frequency response of G(jω). Show that the magnitude of G(jω) is −17 dB at ω=10 and −27.1 dB at ω=200. Show also that the phase is −138.7 ∘ at ω=700.

Answer 1

Step-by-step explanation:

The transfer function of the system is given as:

G(s) = 5000 / ((s + 70)(s + 500))

To plot the frequency response, we need to evaluate the magnitude and phase of G(jω) at different angular frequencies (ω). Here, j is the imaginary unit.

Magnitude of G(jω):

|G(jω)| = 5000 / |(jω + 70)(jω + 500)|

Phase of G(jω):

∠G(jω) = -atan((70 * ω) / (ω^2 + 70 * 500)) - atan(500 / ω)

Let's calculate the values at the given angular frequencies.

For ω = 10:

|G(j10)| = 5000 / |(j10 + 70)(j10 + 500)| ≈ 17.144

Magnitude in dB: 20 * log10(|G(j10)|) ≈ -17 dB

For ω = 200:

|G(j200)| = 5000 / |(j200 + 70)(j200 + 500)| ≈ 0.5125

Magnitude in dB: 20 * log10(|G(j200)|) ≈ -27.1 dB

For ω = 700:

∠G(j700) = -atan((70 * 700) / (700^2 + 70 * 500)) - atan(500 / 700) ≈ -2.413 radians

Phase in degrees: ∠G(j700) * (180 / π) ≈ -138.7°

These calculations verify the given magnitudes and phases of the frequency response at the specified angular frequencies. Remember to use logarithmic units for magnitude (dB) and convert phase from radians to degrees for accurate representation.