69.0k views
1 vote
Construct the Bode Plot for the below frequency response functions. Then, find the phase crossover frequency, gain crossover frequency, gain margin, & phase margin.

a) G(s) = 2(s+2) / s^2 -1
b) G(s) = 2 / s(2+s)(5+s)
Manual calculations only.

User Osukaa
by
8.0k points

1 Answer

6 votes

Answer:

a) G(s) = 2(s+2) / (s^2 -1)

First, let's rewrite the transfer function in its factored form:

G(s) = 2(s+2) / [(s-1)(s+1)]

Now we can create the Bode Plot.

Magnitude plot:

For s = 0, |G(jω)| = [(2*2)/(-1)] = 4

For s → ∞, |G(jω)| → 0

For ω = 1, |G(jω)| = 2.83 ≈ -9 dB

For ω → ∞, |G(jω)| → 0

We can plot these points and connect them using asymptotes as shown below:

Gain crossover frequency = 1 rad/s (where the magnitude curve intersects 0 dB line).

Phase plot:

For s = 0, ∠G(jω) = 90°

For s → ∞, ∠G(jω) → 0°

For ω = 1, ∠G(jω) = 164°

For ω → ∞, ∠G(jω) → 0°

We can plot these points and connect them using an asymptote as shown below:

Phase margin can be calculated by finding the difference between the phase angle at the gain crossover frequency and -180°:

PM = -16°

b) G(s) = 2 / (s(2+s)(5+s))

First, let's rewrite the transfer function in its factored form:

G(s) = 2 / [s(2+s)(s+5)]

Now we can create the Bode Plot.

Magnitude plot:

For s → ∞, |G(jω)| → 0

For ω << 1, |G(jω)| ≈ 0 dB (since the s term dominates)

For ω = 1, |G(jω)| = 0.18 ≈ -13.95 dB

For ω = 2, |G(jω)| = 0.10 ≈ -19.97 dB

For ω = 5, |G(jω)| = 0.04 ≈ -28 dB

We can plot these points and connect them using asymptotes as shown below:

Gain crossover frequency = 2 rad/s (where the magnitude curve intersects 0 dB line).

Phase plot:

For s → ∞, ∠G(jω) → 0°

For ω << 1, ∠G(jω) ≈ -90° (since the s term dominates)

For ω = 1, ∠G(jω) = -93°

For ω = 2, ∠G(jω) = -128°

For ω = 5, ∠G(jω) = -160°

We can plot these points and connect them using asymptotes as shown below:

Phase crossover frequency = 1.26 rad/s (where the phase curve intersects -180° line).

Phase margin can be calculated by finding the difference between the phase angle at the gain crossover frequency and -180°:

PM = -49°

Gain margin can be calculated by finding the difference between the 0 dB line and the magnitude at the phase crossover frequency:

GM = 24 dB

User Dineshkani
by
8.6k points