System Gain K And Phase Margin Analysis For G(s)H(s) = Ke^(0.2s) / (s(s+10)(1+0.5s))

by ADMIN 85 views

This article delves into the analysis of a system defined by the open-loop transfer function G(s)H(s) = Ke^(0.2s) / (s(s+10)(1+0.5s)). Our primary objectives are to determine the system gain K required for a gain crossover frequency of 4 rad/s and subsequently calculate the phase margin for this specific value of K. Understanding these parameters is crucial in assessing the stability and performance of the closed-loop system.

Determining the System Gain K for a Gain Crossover Frequency of 4 rad/s

The gain crossover frequency, denoted as ωgc, is a critical parameter in control systems analysis. It signifies the frequency at which the magnitude of the open-loop transfer function, |G(jω)H(jω)|, equals unity (or 0 dB). To find the system gain K that results in a gain crossover frequency of 4 rad/s, we need to solve the magnitude equation:

|G(jωgc)H(jωgc)| = 1

where ωgc = 4 rad/s. Substituting the given transfer function, we have:

|Ke^(0.2jωgc) / (jωgc(jωgc+10)(1+0.5jωgc))| = 1

This magnitude equation can be further expressed as:

|K| * |e^(0.2jωgc)| / (|jωgc| * |jωgc+10| * |1+0.5jωgc|) = 1

Since |e^(jθ)| = 1 for any real θ, the equation simplifies to:

K / (|jωgc| * |jωgc+10| * |1+0.5jωgc|) = 1

Now, let's plug in ωgc = 4 rad/s:

K / (|j4| * |j4+10| * |1+0.5j4|) = 1

We need to calculate the magnitudes of the complex terms:

  • |j4| = 4
  • |j4+10| = √(4^2 + 10^2) = √116 ≈ 10.77
  • |1+0.5j4| = |1+j2| = √(1^2 + 2^2) = √5 ≈ 2.24

Substituting these values back into the equation, we get:

K / (4 * 10.77 * 2.24) = 1

K / 96.49 ≈ 1

Therefore, the system gain K required for a gain crossover frequency of 4 rad/s is approximately:

K ≈ 96.49

This value of K ensures that the open-loop transfer function's magnitude crosses 0 dB at the specified frequency of 4 rad/s. This is a crucial step in control systems design as the gain crossover frequency is directly related to the system's bandwidth and speed of response. A higher gain crossover frequency generally indicates a faster response but can also lead to reduced stability margins. The accurate determination of K is therefore paramount in achieving the desired system performance while maintaining adequate stability.

Calculating the Phase Margin for K ≈ 96.49

The phase margin (PM) is another essential stability metric in control systems. It quantifies the amount of phase lag at the gain crossover frequency needed to bring the system to the verge of instability. A larger phase margin generally indicates a more stable system. The phase margin is defined as:

PM = 180° + ∠G(jωgc)H(jωgc)

where ∠G(jωgc)H(jωgc) is the phase angle of the open-loop transfer function at the gain crossover frequency ωgc. We have already determined that ωgc = 4 rad/s for K ≈ 96.49. Now, we need to calculate the phase angle of G(jω)H(jω) at this frequency. The open-loop transfer function is:

G(s)H(s) = Ke^(0.2s) / (s(s+10)(1+0.5s))

Substituting s = jω and K ≈ 96.49, we get:

G(jω)H(jω) = 96.49 * e^(0.2jω) / (jω(jω+10)(1+0.5jω))

Now, let's plug in ω = 4 rad/s:

G(j4)H(j4) = 96.49 * e^(0.2j4) / (j4(j4+10)(1+0.5j4))

To find the phase angle, we need to calculate the phase of each term:

  • ∠96.49 = 0° (since 96.49 is a positive real number)
  • ∠e^(0.2j4) = 0.2 * 4 radians = 0.8 radians = 0.8 * (180/Ï€)° ≈ 45.84°
  • ∠(j4) = 90°
  • ∠(j4+10) = arctan(4/10) ≈ 21.80°
  • ∠(1+0.5j4) = ∠(1+j2) = arctan(2/1) ≈ 63.43°

Now, we can calculate the total phase angle of G(j4)H(j4):

∠G(j4)H(j4) = ∠96.49 + ∠e^(0.2j4) - ∠(j4) - ∠(j4+10) - ∠(1+0.5j4)

∠G(j4)H(j4) = 0° + 45.84° - 90° - 21.80° - 63.43°

∠G(j4)H(j4) ≈ -129.39°

Finally, we can calculate the phase margin:

PM = 180° + ∠G(j4)H(j4)

PM = 180° - 129.39°

PM ≈ 50.61°

The phase margin of approximately 50.61° indicates a reasonably stable system. A general rule of thumb is that a phase margin between 30° and 60° is considered acceptable for many applications, providing a good balance between stability and performance. A larger phase margin implies a more robust system that is less susceptible to oscillations and instability due to variations in system parameters or operating conditions.

Conclusion

In summary, we have successfully determined the system gain K required for a gain crossover frequency of 4 rad/s, which was found to be approximately 96.49. Subsequently, we calculated the phase margin for this value of K and obtained a result of approximately 50.61°. This phase margin suggests that the system exhibits a reasonable level of stability. The analysis presented here provides valuable insights into the frequency response characteristics and stability of the given system, which are essential considerations in control systems design and implementation. The relationship between gain crossover frequency and phase margin is a critical one, as they jointly dictate the system's transient response and steady-state behavior. By carefully selecting the gain K, designers can achieve the desired performance characteristics while ensuring adequate stability margins.

Further analysis could involve examining the system's step response, evaluating its sensitivity to parameter variations, and exploring potential compensation techniques to further enhance its performance and robustness. The frequency domain analysis conducted here serves as a solid foundation for these more advanced investigations, providing a clear understanding of the system's open-loop behavior and its implications for closed-loop performance. Understanding gain crossover frequency and phase margin is thus paramount for any control systems engineer seeking to design and implement stable and effective control systems.

Keywords Analysis

Determine the system gain K for the gain cross-over frequency to be 4 rad/s

This question focuses on finding the appropriate gain K for a specific gain crossover frequency. To clarify, we can rephrase this as: "How can we calculate the system gain K such that the gain crossover frequency of the system is 4 rad/s?" This emphasizes the process of calculation and clarifies the relationship between K and the gain crossover frequency.

What is the phase margin for this value of K?

This question asks for the phase margin corresponding to the previously determined gain K. A clearer phrasing would be: "Given the value of K calculated previously, what is the resulting phase margin of the system?" This directly links the question to the preceding calculation and emphasizes the dependency of the phase margin on the gain K.