Unit 4: Frequency Response Analysis

1. Introduction to Frequency Response

1.1. Definition

Frequency Response is the steady-state response of a system to a sinusoidal input. When a linear time-invariant (LTI) system is subjected to a sinusoidal input, the output is also a sinusoid of the same frequency, but its amplitude and phase angle may be different from the input signal.

• Input: r(t) = A sin(ωt)
• Output (steady-state): c(t) = B sin(ωt + φ)

The frequency response analysis studies the variation of the output magnitude B and phase angle φ as the input frequency ω is varied from 0 to ∞.

1.2. Frequency Domain Transfer Function

The frequency response is obtained by substituting s = jω into the system's transfer function G(s).

G(jω) = G(s)|_{s=jω}

G(jω) is a complex quantity and can be expressed in two forms:

1. Rectangular Form: G(jω) = Re[G(jω)] + j Im[G(jω)]
2. Polar Form: G(jω) = |G(jω)| ∠G(jω)

Where:

• Magnitude: |G(jω)| = B/A (Ratio of output amplitude to input amplitude)
• Phase Angle: ∠G(jω) = φ (Phase shift between output and input)

1.3. Advantages of Frequency Response Analysis

• It is applicable to systems for which the transfer function may be difficult to determine experimentally. The frequency response can be measured directly from the physical system.
• It provides a straightforward way to design and tune controllers (e.g., PID controllers).
• It offers insights into a system's robustness and its ability to reject disturbances.
• Analysis and design can be performed graphically, which is intuitive.

2. Relationship between Time and Frequency Response

There is a strong correlation between the time-domain response (characterized by parameters like rise time, settling time, peak overshoot) and the frequency-domain response (characterized by parameters like resonant peak, resonant frequency, bandwidth). This relationship is most clearly illustrated for a standard second-order system.

Consider the closed-loop transfer function of a standard second-order system:

T(s) = C(s) / R(s) = ωn² / (s² + 2ζωn s + ωn²)

The frequency response is obtained by s = jω:

T(jω) = ωn² / ((jω)² + 2ζωn(jω) + ωn²)
      = ωn² / (ωn² - ω² + j(2ζωnω))
      = 1 / (1 - (ω/ωn)² + j(2ζω/ωn))

Let u = ω/ωn (normalized frequency).

T(jω) = 1 / (1 - u² + j2ζu)

2.1. Frequency Domain Specifications

• Magnitude (M):
  M = |T(jω)| = 1 / √((1 - u²)² + (2ζu)²)

• Phase (φ):
  φ = ∠T(jω) = -tan⁻¹(2ζu / (1 - u²))

• Resonant Peak (Mr): The maximum value of the magnitude M. It occurs at the resonant frequency. A large Mr corresponds to a large peak overshoot (Mp) in the time domain.
  Mr = 1 / (2ζ√(1 - ζ²)) for 0 < ζ < 0.707

• Resonant Frequency (ωr): The frequency at which the resonant peak Mr occurs.
  ωr = ωn √(1 - 2ζ²) for 0 < ζ < 0.707

• Bandwidth (ωb): The range of frequencies over which the magnitude M is greater than or equal to 1/√2 (or -3 dB) of its value at ω = 0. It indicates the speed of the system's response. A larger bandwidth corresponds to a faster rise time.
  ωb = ωn √((1 - 2ζ²) + √(4ζ⁴ - 4ζ² + 2))

2.2. Correlation Summary

3. Polar Plot

A Polar Plot is a graphical representation of the frequency response G(jω). It is a plot of the magnitude |G(jω)| versus the phase angle ∠G(jω) on polar coordinates as ω is varied from 0 to ∞.

3.1. Construction Procedure

1. Obtain G(jω): Substitute s = jω into the open-loop transfer function G(s)H(s).
2. Separate Magnitude and Phase: Express G(jω) in polar form:
   • Calculate Magnitude: M(ω) = |G(jω)H(jω)|
   • Calculate Phase Angle: φ(ω) = ∠G(jω)H(jω)
3. Find Starting and Ending Points:
   • Calculate M and φ at ω = 0. This is the starting point of the plot.
   • Calculate M and φ as ω → ∞. This is the ending point of the plot.
4. Determine Intersections: Find the points where the plot intersects the real and imaginary axes by:
   • Real Axis Intersection: Find ω for which Im[G(jω)] = 0.
   • Imaginary Axis Intersection: Find ω for which Re[G(jω)] = 0.
5. Sketch the Plot: Plot the points and connect them smoothly as ω increases from 0 to ∞.

3.2. Example: First-Order System

Let G(s) = 1 / (1 + sT).

1. G(jω) = 1 / (1 + jωT)
2. Magnitude: M(ω) = 1 / √(1 + (ωT)²)
   Phase: φ(ω) = -tan⁻¹(ωT)
3. At ω = 0:
   M(0) = 1
φ(0) = 0°
   Starting point: (1, ∠0°)
4. At ω → ∞:
   M(∞) = 0
φ(∞) = -90°
   Ending point: Origin (0, ∠-90°)

The plot is a semi-circle in the fourth quadrant, starting at (1,0) on the real axis and ending at the origin, approaching along the -90° axis.

3.3. Effect of System Type

The starting point (ω=0) of the polar plot depends on the Type of the system (number of poles at the origin).

• Type 0 System: Starts on the positive real axis. (K, ∠0°)
• Type 1 System: Starts at infinity along the negative imaginary axis. (∞, ∠-90°)
• Type 2 System: Starts at infinity along the negative real axis. (∞, ∠-180°)

The ending point (ω→∞) of the polar plot is always the origin, and the angle at which it approaches depends on n-m (number of poles - number of zeros). Angle = -(n-m) * 90°.

4. Stability in Frequency Domain: Gain and Phase Margins

Gain Margin (GM) and Phase Margin (PM) are two crucial metrics used to assess the relative stability of a closed-loop system from its open-loop frequency response.

• Phase Crossover Frequency (ωpc): The frequency at which the phase angle of G(jω)H(jω) is -180°.
• Gain Crossover Frequency (ωgc): The frequency at which the magnitude of G(jω)H(jω) is 1 (or 0 dB).

4.1. Gain Margin (GM)

GM is the factor by which the system's gain can be increased before it becomes unstable. It is measured at the phase crossover frequency ωpc.

• Definition: GM = 1 / |G(jωpc)H(jωpc)|
• In decibels: GM (dB) = 20 log(GM) = -20 log(|G(jωpc)H(jωpc)|)
• Stability Condition: For a stable system, |G(jωpc)H(jωpc)| < 1, which means GM > 1 (or GM (dB) > 0).

4.2. Phase Margin (PM)

PM is the additional phase lag required at the gain crossover frequency ωgc to make the system unstable.

• Definition: PM = 180° + ∠G(jωgc)H(jωgc)
• Stability Condition: For a stable system, the phase angle at ωgc must be less negative than -180°. This means PM > 0.

4.3. Interpretation

• Stable System: ωgc < ωpc. Positive GM and PM.
• Marginally Stable System: ωgc = ωpc. GM = 1 (0 dB) and PM = 0°. The Nyquist plot passes through the (-1, j0) point.
• Unstable System: ωgc > ωpc. Negative GM and PM.

5. Nyquist Plot and Nyquist Stability Criterion

The Nyquist plot is a powerful graphical technique for determining the stability of a closed-loop system by observing its open-loop frequency response.

5.1. The Nyquist Contour

The Nyquist plot is the mapping of a specific contour from the s-plane, called the Nyquist Contour, onto the G(s)H(s)-plane.

• The Nyquist Contour encloses the entire right-half of the s-plane (RHP).
• It consists of:
  1. The entire imaginary (jω) axis from ω = -∞ to ω = +∞.
  2. A large semi-circle of infinite radius R→∞ that encloses the RHP.
• Modification for Poles on jω-axis: If G(s)H(s) has poles on the jω axis (e.g., at the origin for a Type 1 system), the contour is indented with a small semi-circle of radius ε→0 to bypass these poles.

5.2. Cauchy's Principle of Argument

The Nyquist criterion is based on this mathematical principle. It states:
If a closed contour in the s-plane encloses P poles and Z zeros of a function F(s), then the corresponding contour in the F(s)-plane will encircle the origin Z - P times in the clockwise direction.

Let N be the number of clockwise encirclements of the origin.
N = Z - P

5.3. The Nyquist Stability Criterion

We apply the Principle of Argument to the characteristic equation 1 + G(s)H(s) = 0. Let F(s) = 1 + G(s)H(s).

• Zeros of F(s): The roots of 1 + G(s)H(s) = 0, which are the closed-loop poles.
• Poles of F(s): The poles of G(s)H(s), which are the open-loop poles.

The criterion relates the encirclements of the -1+j0 point by the G(s)H(s) plot to the stability of the closed-loop system.

Let:

• P = Number of open-loop poles of G(s)H(s) in the RHP (this is known).
• Z = Number of closed-loop poles in the RHP (this is what we want to find; Z=0 for stability).
• N = Number of clockwise encirclements of the -1+j0 point by the Nyquist plot of G(s)H(s).

The relationship is:
N = Z - P

For the closed-loop system to be stable, there must be no closed-loop poles in the RHP, i.e., Z = 0.

Therefore, the stability condition becomes:
N = -P

This means the number of clockwise encirclements of the -1+j0 point must equal the number of open-loop poles in the RHP. (Note: A counter-clockwise encirclement is taken as N = -1).

Special Case (Most Common): If the open-loop system is stable (P=0), then for closed-loop stability, we need N=0. The Nyquist plot must not encircle the -1+j0 point.

5.4. Nyquist Plot Construction and Analysis

1. Sketch the Polar Plot: Draw the plot for G(jω) from ω = 0⁺ to ω = +∞.
2. Sketch the Inverse Plot: The plot for ω = -∞ to ω = 0⁻ is the mirror image of the polar plot with respect to the real axis.
3. Complete the Contour:
   • If there are no poles on the jω axis, the plot is closed.
   • If there are poles at the origin, the small semi-circle s = εe^(jθ) from θ=-90° to +90° maps to a large arc of n * 180° in the G(s)H(s)-plane, where n is the number of poles at the origin.
4. Apply the Criterion:
   • Count P, the number of open-loop poles in the RHP.
   • Count N, the number of clockwise encirclements of the -1+j0 point.
   • Calculate Z = N + P. If Z=0, the system is stable.

5.5. GM and PM from the Nyquist Plot

• Gain Margin: Find the intersection of the plot with the negative real axis. Let this point be -a. Then GM = 1/a.
• Phase Margin: Draw a unit circle centered at the origin. Find the point where the Nyquist plot intersects this circle. The angle between the negative real axis and the vector to this intersection point is the Phase Margin.

6. Root Locus Technique

The Root Locus is a graphical method that shows the paths (loci) of the roots of the characteristic equation (i.e., the closed-loop poles) as a single system parameter, typically the gain K, is varied from 0 to ∞.

6.1. Fundamental Concept

The closed-loop transfer function is T(s) = KG(s) / (1 + KG(s)H(s)).
The characteristic equation is 1 + KG(s)H(s) = 0.

Any point s in the s-plane that lies on the root locus must satisfy this equation for some K > 0. This leads to two conditions:

1. Angle Condition: The angle of the open-loop transfer function must be an odd multiple of 180°.
   ∠G(s)H(s) = ±180°(2q + 1) where q = 0, 1, 2, ...

2. Magnitude Condition: The magnitude must be 1. This is used to find the value of K for a specific point on the locus.
   K|G(s)H(s)| = 1 or K = 1 / |G(s)H(s)|

6.2. Rules for Constructing a Root Locus

Let G(s)H(s) have n poles and m zeros.

1. Symmetry: The root locus is always symmetric with respect to the real axis.
2. Number of Branches: The number of branches in the locus is equal to n (the number of open-loop poles).
3. Starting and Ending Points:
   • Branches start (K=0) at the open-loop poles.
   • Branches end (K=∞) at the open-loop zeros. If n > m, then n-m branches will end at infinity.
4. Locus on Real Axis: A point on the real axis is on the root locus if the total number of real poles and real zeros to its right is odd.
5. Asymptotes: For branches going to infinity, their paths approach straight lines called asymptotes.
   • Number of Asymptotes: n - m
   • Angle of Asymptotes (φa): φa = (±180°(2q + 1)) / (n - m) for q = 0, 1, 2, ...
   • Centroid (σa) (point of intersection on real axis): σa = (Σ(real parts of poles) - Σ(real parts of zeros)) / (n - m)
6. Breakaway and Break-in Points: These are points on the real axis where multiple branches leave (breakaway) or meet (break-in). They are found by solving dK/ds = 0 or, equivalently, d(-1/G(s)H(s))/ds = 0.
7. Angle of Departure/Arrival:
   • Angle of Departure (from a complex pole p_j):
     θd = 180° - (sum of angles from other poles top_j) + (sum of angles from zeros top_j)
   • Angle of Arrival (at a complex zero z_j):
     θa = 180° - (sum of angles from other zeros toz_j) + (sum of angles from poles toz_j)
8. Intersection with Imaginary (jω) Axis: This point indicates the onset of instability. It can be found by:
   • Applying the Routh-Hurwitz stability criterion to the characteristic equation.
   • Find the value of K that makes a row of the Routh array zero (the "marginal gain" K_mar).
   • Solve the auxiliary equation (from the row above the zero row) to find the frequencies ±jω of intersection.

6.3. Using the Root Locus for Design

• Stability: The system is stable as long as all branches of the root locus are in the left-half of the s-plane. The value of K at which the locus crosses the jω axis is the maximum gain for stability.
• Transient Response: The location of the dominant closed-loop poles on the locus for a given K determines the transient response. For example, a desired damping ratio ζ corresponds to a line drawn from the origin at an angle θ = cos⁻¹(ζ). The intersection of this line with the root locus gives the pole locations for that ζ, and the magnitude condition can be used to find the required gain K.