Select Page

# Frequency Domain Analysis

Frequency Domain Analysis

Contents

Recall Frequency domain specifications 1

Find the correlation between time-domain and frequency-domain specifications 3

Recall advantages of frequency-domain method 5

Recall Bode Plot 6

Design Bode Plot 7

Recall the Frequency Domain Specifications obtained from Bode Plots 9

Find the stability of the System and Specify Gain and Phase margin from Bode Plots 11

Recall the Bode Plots of the Systems with pure Time Delays 12

Find the System Type and Error Constants from Bode-Magnitude Plot 13

Recall steps to find Transfer Functions from the Bode-Magnitude Plot 15

Recall Polar Plots 17

Design a Polar Plot with the given Transfer Function 19

Find the Stability of a System using Polar Plot 20

Recall the Gain Margin and Phase Margin using Polar Plot 22

Determine the Gain Margin and Phase Margin using Polar Plot 24

Recall Nyquist Stability Criteria 25

Determine the Stability of a System using Nyquist Criteria 26

Differentiate between Nyquist Criteria and Polar Plot 27

Recall Nichols Chart 29

Describe Constant Magnitude Locus(M-Circle) 30

Describe Constant Phase Angle Loci(N-Circle) 32

Recall the following Transfer Functions:i. Minimum Phase Transfer Function ii. Non-Minimum Phase Transfer Function iii. All-Pass Transfer Function 34

# Recall Frequency domain specifications

Frequency domain specifications are important measures used to evaluate the performance and stability of control systems in the frequency domain. These specifications provide a quantitative measure of the system’s behavior in response to different frequency inputs.

There are various frequency domain specifications, and some of the most important ones are discussed below:

1. Gain margin (GM):

Gain margin is defined as the amount of gain that can be added to the open-loop transfer function before the closed-loop system becomes unstable. Mathematically, it is the reciprocal of the magnitude of the transfer function at the phase crossover frequency.

GM = 1/|G(jwpc)|

For example, let’s consider the transfer function G(s) = (s+1)/(s3 + 3s2 + 2s).

From the Bode plot, we can see that the phase crossover frequency (wpc) is approximately 1.5 radians/sec, and the magnitude of the transfer function at this frequency is approximately 0 dB. Therefore, the gain margin is:

GM = 1/|G(jwpc)| = 1/0 = infinity

This means that the system is highly stable, and we can add an infinite amount of gain to the open-loop transfer function before the closed-loop system becomes unstable.

1. Phase margin (PM):

Phase margin is defined as the amount of phase shift that can be added to the open-loop transfer function before the closed-loop system becomes unstable. Mathematically, it is the difference between 180 degrees and the phase angle at the gain crossover frequency.

PM = 180 + angle[G(jwgc)]

For example, let’s consider the same transfer function G(s) = (s+1)/(s3 + 3s2 + 2s). The Bode plot of this transfer function is shown above. From the Bode plot, we can see that the gain crossover frequency (wgc) is approximately 0.5 radians/sec, and the phase angle at this frequency is approximately -135 degrees. Therefore, the phase margin is:

PM = 180 + angle[G(jwgc)] = 180 + (-135) = 45 degrees

This means that we can add up to 45 degrees of phase shift to the open-loop transfer function before the closed-loop system becomes unstable.

1. Bandwidth (BW):

Bandwidth is defined as the frequency at which the magnitude of the transfer function is reduced to 70.7% of its maximum value.

For example, let’s consider the same transfer function G(s) = (s+1)/(s3 + 3s2 + 2s). From the Bode plot, we can see that the maximum magnitude of the transfer function is approximately 0 dB, which occurs at very low frequencies. Therefore, we can define the bandwidth as the frequency at which the magnitude of the transfer function is reduced to 0.707 times the maximum value, which occurs at approximately 3.16 radians/sec.

1. Resonant frequency (wr):

Resonant frequency is defined as the frequency at which the magnitude of the transfer function reaches its maximum value.

# Find the correlation between time-domain and frequency-domain specifications

Time-Domain Specifications:

Time-domain specifications define the behavior of a system in the time domain. Time-domain is a representation of signals with respect to time. Time-domain analysis is used to determine the response of a system to a given input signal. The important time-domain specifications are:

1. Rise Time: The time taken for the output to rise from 10% to 90% of its final value.
2. Settling Time: The time taken for the output to settle within a specified error band.
3. Overshoot: The amount of the maximum deviation of the output from the steady-state value.

Frequency-Domain Specifications:

Frequency-domain specifications define the behavior of a system in the frequency domain. Frequency-domain is a representation of signals with respect to frequency. Frequency-domain analysis is used to determine the frequency response of a system. The important frequency-domain specifications are:

1. Bandwidth: The range of frequencies over which the system response is within a specified error band.
2. Gain Margin: The amount of additional gain that can be added to the system before it becomes unstable.
3. Phase Margin: The amount of phase shift that can be added to the system before it becomes unstable.

Correlation between Time-Domain and Frequency-Domain Specifications:

Time-domain and frequency-domain specifications are interrelated, and there is a correlation between them. The time-domain behavior of a system affects its frequency-domain behavior, and vice versa.

For example, the rise time and settling time in the time domain affect the bandwidth in the frequency domain. A system with a fast rise time and settling time will have a wide bandwidth, while a system with a slow rise time and settling time will have a narrow bandwidth.

Similarly, the overshoot in the time domain affects the gain margin in the frequency domain. A system with a large overshoot will have a small gain margin, while a system with a small overshoot will have a large gain margin.

The phase margin in the frequency domain affects the stability of the system in the time domain. A system with a small phase margin will be less stable, while a system with a large phase margin will be more stable.

Therefore, it is important to consider both time-domain and frequency-domain specifications when designing and analyzing a system. The specifications in one domain affect the behavior of the system in the other domain.

In conclusion, the correlation between time-domain and frequency-domain specifications is important in understanding the behavior of a system. The time-domain behavior of a system affects its frequency-domain behavior, and vice versa. It is essential to consider both time-domain and frequency-domain specifications when designing and analyzing a system.

# Recall advantages of frequency-domain method

The frequency-domain method is a mathematical technique used to analyze linear time-invariant systems. It is an alternative approach to the time-domain method, which analyses the system’s behavior in the time domain. The frequency-domain method involves converting the time-domain signal into its equivalent frequency-domain representation using Fourier analysis. The advantages of using the frequency-domain method are:

1. Simplicity: The frequency-domain method simplifies the analysis of the system by converting the time-domain signal into its equivalent frequency-domain representation. In the frequency-domain, complex signals can be represented by simpler functions such as sinusoids, making it easier to analyze and understand the behavior of the system.
2. Efficiency: The frequency-domain method is often more efficient than the time-domain method when dealing with complex systems. The frequency-domain analysis of a system can be carried out using mathematical operations such as multiplication, addition, and differentiation, which are simpler than the integration and differentiation operations required for time-domain analysis.
3. Accuracy: The frequency-domain method provides accurate results for the analysis of linear time-invariant systems. The frequency response of a system can be analyzed accurately by considering the system’s behavior over a range of frequencies. This is particularly useful when analyzing the response of a system to a complex input signal containing multiple frequencies.
4. Design: The frequency-domain method is often used in the design of filters, amplifiers, and other electronic systems. The frequency response of a system can be optimized by adjusting the system’s parameters in the frequency domain.

For example, the frequency-domain method is used in the design of low-pass filters, which allow low-frequency signals to pass through while blocking high-frequency signals. The frequency response of the filter is analyzed to determine the cutoff frequency, which is the frequency at which the filter starts to attenuate the signal.

1. Visualisation: The frequency-domain method provides a visual representation of the system’s behavior. The frequency response of a system can be plotted on a graph, which provides a visual representation of the system’s behavior over a range of frequencies.

In conclusion, the frequency-domain method has several advantages over the time-domain method, including simplicity, efficiency, accuracy, and the ability to optimize the system’s design. The frequency-domain method is particularly useful in the design of filters, amplifiers, and other electronic systems. It also provides a visual representation of the system’s behavior, making it easier to analyze and understand the system’s response over a range of frequencies.

# Recall Bode Plot

A Bode plot is a graph that shows the frequency response of a system. It is a commonly used tool in frequency-domain analysis and is used to represent the magnitude and phase response of a system as a function of frequency. The Bode plot consists of two plots: one for magnitude and one for phase.

Magnitude Plot:

The magnitude plot shows the amplitude of the system’s response as a function of frequency. The amplitude is measured in decibels (dB) and is plotted on a logarithmic scale. The Bode plot shows the magnitude of the system’s response for each frequency in the range of interest. The magnitude plot is used to determine the gain of the system and to identify the frequency at which the gain is maximum or minimum.

Phase Plot:

The phase plot shows the phase angle of the system’s response as a function of frequency. The phase angle is measured in degrees and is plotted on a linear scale. The Bode plot shows the phase angle of the system’s response for each frequency in the range of interest. The phase plot is used to determine the delay or advance of the system’s response with respect to the input signal.

Example:

Consider a low-pass filter with a transfer function of H(s) = 1 / (s+1), where s is the Laplace variable.

The magnitude plot shows that the gain of the filter decreases as the frequency increases. The cutoff frequency of the filter is approximately 1 rad/s, which is the frequency at which the gain of the filter drops by 3 dB.

The phase plot shows that the filter introduces a phase shift of -45 degrees. This means that the output of the filter is delayed by 45 degrees with respect to the input signal.

The Bode plot is useful in analyzing the frequency response of a system and in designing filters and other electronic systems. By analyzing the Bode plot, we can determine the gain, phase shift, and cutoff frequency of the system. We can also use the Bode plot to optimize the design of the system by adjusting its parameters in the frequency domain.

# Design Bode Plot

Designing a Bode plot involves plotting the magnitude and phase response of a system as a function of frequency. The transfer function of the system is used to determine the magnitude and phase response at each frequency. The steps involved in designing a Bode plot are as follows:

Step 1: Express the transfer function in standard form

The transfer function of the system is expressed in standard form, which is a ratio of polynomials in the Laplace variable s. The standard form of a transfer function is expressed as:

H(s) = K * (s – z1) * (s – z2) * … * (s – zn) / (s – p1) * (s – p2) * … * (s – pm)

where K is the gain of the system, z1, z2, …, zn are the zeros of the system, and p1, p2, …, pm are the poles of the system.

Step 2: Determine the magnitude response

The magnitude response of the system is determined by evaluating the transfer function for each frequency in the range of interest. The magnitude is measured in decibels (dB) and is plotted on a logarithmic scale. The magnitude response is determined using the following equation:

|H(jω)| = 20 log |H(jω)|dB

where ω is the frequency in radians per second.

Step 3: Determine the phase response

The phase response of the system is determined by evaluating the transfer function for each frequency in the range of interest. The phase is measured in degrees and is plotted on a linear scale. The phase response is determined using the following equation:

∠H(jω) = tan-1 [Im(H(jω)) / Re(H(jω))]

where Im(H(jω)) and Re(H(jω)) are the imaginary and real parts of the transfer function, respectively.

Step 4: Plot the magnitude and phase response on a Bode plot

The magnitude and phase response of the system are plotted on a Bode plot, with frequency on the x-axis and magnitude and phase on the y-axis. The magnitude plot is plotted on a logarithmic scale, while the phase plot is plotted on a linear scale.

Example:

Consider the transfer function H(s) = 1 / (s+1).

The magnitude plot shows that the gain of the system decreases as the frequency increases. The cutoff frequency of the system is approximately 1 rad/s, which is the frequency at which the gain of the system drops by 3 dB.

The phase plot shows that the system introduces a phase shift of -90 degrees. This means that the output of the system is delayed by 90 degrees with respect to the input signal.

In conclusion, designing a Bode plot involves plotting the magnitude and phase response of a system as a function of frequency. The transfer function of the system is used to determine the magnitude and phase response at each frequency. By analyzing the Bode plot, we can determine the gain, phase shift, and cutoff frequency of the system, which are useful in designing filters and other electronic systems.

# Recall the Frequency Domain Specifications obtained from Bode Plots

Bode plots are used to represent the frequency response of a system. They are graphical representations of the magnitude and phase of the system’s transfer function as a function of frequency. The magnitude plot shows the gain or attenuation of the system as a function of frequency, while the phase plot shows the phase shift of the system as a function of frequency.

There are several frequency domain specifications that can be obtained from Bode plots. These specifications provide important information about the behavior of a system in the frequency domain and are used to design and analyze control systems.

The following are the most important frequency domain specifications obtained from Bode plots:

1. Gain margin (GM) – The gain margin is the amount of additional gain that can be added to a system before it becomes unstable. It is measured in decibels (dB) and is obtained from the magnitude plot of the Bode diagram. The gain margin is the amount of gain required to make the phase angle at the frequency where the magnitude is 0 dB to be equal to -180 degrees. The larger the gain margin, the more stable the system is.

Example: Let us consider a system with a gain margin of 10 dB. This means that the system can handle an additional gain of 10 dB before becoming unstable.

1. Phase margin (PM) – The phase margin is the amount of additional phase shift that can be added to a system before it becomes unstable. It is measured in degrees and is obtained from the phase plot of the Bode diagram. The phase margin is the amount of phase shift required to make the magnitude at the frequency where the phase angle is -180 degrees to be equal to 0 dB. The larger the phase margin, the more stable the system is.

Example: Let us consider a system with a phase margin of 45 degrees. This means that the system can handle an additional phase shift of 45 degrees before becoming unstable.

1. Bandwidth (BW) – The bandwidth is the range of frequencies over which a system can operate effectively. It is obtained from the magnitude plot of the Bode diagram and is defined as the frequency at which the magnitude of the transfer function is equal to -3 dB. The larger the bandwidth, the wider the range of frequencies over which the system can operate.

Example: Let us consider a system with a bandwidth of 100 Hz. This means that the system can operate effectively over a range of frequencies from 0 Hz to 100 Hz.

1. Resonant frequency (fr) – The resonant frequency is the frequency at which the magnitude of the transfer function is maximum. It is obtained from the magnitude plot of the Bode diagram and is an indication of the natural frequency of the system. The larger the resonant frequency, the faster the system responds to changes in the input.

Example: Let us consider a system with a resonant frequency of 50 Hz. This means that the system responds most effectively to input signals with a frequency of 50 Hz.

In conclusion, Bode plots provide a graphical representation of the frequency response of a system and can be used to obtain several important frequency domain specifications such as gain margin, phase margin, bandwidth, and resonant frequency. These specifications provide important information about the stability and performance of the system and are used in the design and analysis of control systems.

# Find the stability of the System and Specify Gain and Phase margin from Bode Plots

Bode plots are graphical representations of the frequency response of a system. They consist of two plots: the magnitude plot and the phase plot. The magnitude plot shows the gain or attenuation of the system as a function of frequency, while the phase plot shows the phase shift of the system as a function of frequency. Bode plots are used to analyze and design control systems.

To find the stability of a system from a Bode plot, we need to look at the phase plot. A system is said to be stable if the phase angle never reaches -180 degrees as the frequency increases. This is because a phase angle of -180 degrees corresponds to a 180-degree phase shift, which means that the output of the system is 180 degrees out of phase with the input. If the phase angle reaches -180 degrees, the output of the system will be in phase with the input but inverted, which can lead to instability.

If the phase angle never reaches -180 degrees, the system is stable. In this case, we can also find the gain margin and the phase margin of the system. The gain margin is the amount of additional gain that can be added to the system before it becomes unstable, while the phase margin is the amount of additional phase shift that can be added to the system before it becomes unstable.

To find the gain and phase margins from a Bode plot, we need to look at the frequency at which the phase angle is -180 degrees and the frequency at which the magnitude is 0 dB. The gain margin is the difference between the gain at the frequency where the phase angle is -180 degrees and 0 dB. The phase margin is the difference between the phase angle at the frequency where the magnitude is 0 dB and -180 degrees.

Example: Let us consider a system with the transfer function:

G(s) = (s + 1)/(s2 + 3s + 2)

From the Bode plot, we can see that the phase angle never reaches -180 degrees, which means that the system is stable. The gain margin and phase margin can be found as follows:

• Gain margin: The gain margin is the difference between the gain at the frequency where the phase angle is -180 degrees and 0 dB. From the magnitude plot, we can see that the gain at the frequency where the phase angle is -180 degrees is -24 dB, and the gain at 0 dB is -3 dB. Therefore, the gain margin is 21 dB.
• Phase margin: The phase margin is the difference between the phase angle at the frequency where the magnitude is 0 dB and -180 degrees. From the phase plot, we can see that the phase angle at 0 dB is -135 degrees, and the phase angle at -180 degrees is -180 degrees. Therefore, the phase margin is 45 degrees.

In conclusion, Bode plots can be used to find the stability of a system and specify the gain and phase margins. The stability of a system can be determined by looking at the phase plot, while the gain and phase margins can be found by looking at the frequency at which the phase angle is -180 degrees and the frequency at which the magnitude is 0 dB. The gain margin is the difference between the gain at these frequencies, while the phase margin is the difference between the phase angle at these frequencies.

# Recall the Bode Plots of the Systems with pure Time Delays

Bode plots are widely used to represent the frequency response of a linear time-invariant (LTI) system. The Bode plot of a system with pure time delay is a special type of Bode plot that arises when the transfer function of the system contains only a pure time delay term. This learning outcome requires the learner to recall the Bode plots of systems with pure time delays and provide suitable examples.

A pure time delay system is a system that has a transfer function of the form:

G(s) = e(-Ls)

where L is the time delay. The Bode plot of such a system has a phase shift of -L degrees at all frequencies, and a magnitude of 1 at low frequencies that gradually decreases as frequency increases. At high frequencies, the magnitude approaches zero.

For example, consider the transfer function:

G(s) = e(-2s)

This system has a pure time delay of 2 seconds.

As shown in the plot, the magnitude of the system is 1 at low frequencies and decreases as frequency increases. At high frequencies, the magnitude approaches zero. The phase of the system is -2 radians (-114.6 degrees) at all frequencies.

Another example of a system with pure time delay is a transportation delay system, which represents the time it takes for a signal to travel through a physical system. For example, consider a chemical reactor where the temperature is controlled by a feedback controller. The temperature measurement is delayed due to the time it takes for the temperature sensor to measure the temperature and transmit the signal to the controller. The transfer function of such a system can be approximated as:

G(s) = e(-Ls) / (1 + Ts)

where T represents the time constant of the process. The Bode plot of this system has a similar shape to the pure time delay system, with a magnitude of 1 at low frequencies and decreasing magnitude at higher frequencies. The phase shift is -L degrees at all frequencies.

In conclusion, recalling the Bode plots of systems with pure time delays is an important skill for understanding the frequency response of LTI systems with time delays. The Bode plot of a pure time delay system has a phase shift of -L degrees at all frequencies and a magnitude that decreases as frequency increases. Suitable examples of systems with pure time delay include transportation delay systems and other systems with significant time delays.

# Find the System Type and Error Constants from Bode-Magnitude Plot

This learning outcome pertains to the ability of the learner to identify the system type and calculate the error constants of a control system from the Bode-Magnitude plot. The Bode-Magnitude plot is a graphical representation of the magnitude response of the system in the frequency domain. The system type and error constants provide valuable insights into the performance of a control system, especially in terms of steady-state error and tracking accuracy.

System Type:

The system type is defined as the number of integrators in the open-loop transfer function of the control system. The system type can be easily determined from the Bode-Magnitude plot by counting the number of times the magnitude slope changes by -20dB per decade. For instance, a system with a slope change of -20dB/decade once indicates a first-order system, twice indicates a second-order system, and so on.

Error Constants:

The error constants are numerical values that indicate the ability of the control system to track and follow a desired input signal. The error constants can be calculated using the following formulas:

• Position Error Constant (Kp): Kp = Lim s->0 G(s)
• Velocity Error Constant (Kv): Kv = Lim s->0 (s * G(s))
• Acceleration Error Constant (Ka): Ka = Lim s->0 (s2 * G(s))

where G(s) is the transfer function of the control system.

Example:

Consider a control system with the following transfer function:

G(s) = 10 / (s * (s + 2))

From the Bode-Magnitude plot, we can see that the magnitude slope changes by -20dB/decade twice, indicating a second-order system. The position error constant can be calculated as:

Kp = Lim s->0 (10 / (s * (s + 2)))

= 5

The velocity error constant can be calculated as:

Kv = Lim s->0 (s * (10 / (s * (s + 2))))

= 10

The acceleration error constant can be calculated as:

Ka = Lim s->0 (s2 * (10 / (s * (s + 2))))

= 0

Therefore, the control system is a second-order system with Kp = 5 and Kv = 10. Since Ka = 0, the control system is not capable of tracking input signals with acceleration components.

# Recall steps to find Transfer Functions from the Bode-Magnitude Plot

This learning outcome pertains to the ability of the learner to recall the steps required to find the transfer function of a control system from the Bode-Magnitude plot. The Bode-Magnitude plot is a graphical representation of the magnitude response of the system in the frequency domain. The transfer function is a mathematical representation of the relationship between the input and output signals of a control system.

Steps to Find Transfer Functions from the Bode-Magnitude Plot:

Step 1: Identify the Type of the Control System

The type of the control system can be identified by counting the number of times the magnitude slope changes by -20dB per decade in the Bode-Magnitude plot. A slope change of -20dB/decade once indicates a first-order system, twice indicates a second-order system, and so on.

Step 2: Determine the Crossover Frequency

The crossover frequency is the frequency at which the magnitude response of the control system crosses the 0dB line. It can be determined by locating the intersection point of the magnitude curve and the 0dB line in the Bode-Magnitude plot.

Step 3: Calculate the Gain Margin and Phase Margin

The gain margin and phase margin are two important parameters that determine the stability of a control system. The gain margin is the amount of gain that can be added to the system before it becomes unstable, while the phase margin is the amount of phase shift that can be added to the system before it becomes unstable. The gain margin and phase margin can be determined by locating the intersection points of the magnitude curve and the -180-degree phase shift line in the Bode-Magnitude plot.

Step 4: Write the Transfer Function

The transfer function of the control system can be written using the following general equation:

G(s) = K * (s/z1) * (s/z2) * … * (s/zn) * (s/p1) * (s/p2) * … * (s/pm)

where K is the system gain, z1, z2, …, zn are the zeros of the system, and p1, p2, …, pm are the poles of the system. The transfer function can be determined by analyzing the Bode-Magnitude plot, which provides information about the location of zeros and poles in the frequency domain.

Example:

Consider a control system with the following Bode-Magnitude plot:

Step 1: Identify the Type of the Control System

The magnitude slope changes by -20dB/decade twice, indicating a second-order system.

Step 2: Determine the Crossover Frequency

The crossover frequency is approximately 3 rad/s, where the magnitude curve intersects the 0dB line.

Step 3: Calculate the Gain Margin and Phase Margin

The gain margin is approximately 4 dB, and the phase margin is approximately 65 degrees, which are the intersection points of the magnitude curve and the -180-degree phase shift line.

Step 4: Write the Transfer Function

The transfer function of the control system can be written as:

G(s) = K * (s/3 + 1) * (s/10)

where K is the system gain, and 3 and 10 are the poles of the system, which can be determined from the Bode-Magnitude plot. The zero of the system is located at the origin, since there is no change in slope at low frequencies. The system gain can be determined by setting the magnitude to 0dB at the crossover frequency, which gives K = 10(0/20) = 1

# Recall Polar Plots

Polar plots, also known as polar charts or polar graphs, are a type of graph used to plot data that is represented in polar coordinates. Polar coordinates are a way of representing points in a two-dimensional plane using a distance and an angle, rather than using the typical x and y coordinates. Polar plots are commonly used in mathematics, physics, engineering, and other fields to plot data such as frequency response, antenna radiation patterns, and more.

Learning Outcome: Recall Polar Plots

To achieve this learning outcome, you should be able to recall the definition of polar plots, understand how they are used, and be able to interpret data presented in a polar plot.

Key Concepts:

1. Polar Coordinates: Polar coordinates are a way of representing points in a two-dimensional plane using a distance and an angle. In polar coordinates, the distance from the origin is known as the radius (r), and the angle is measured counterclockwise from the positive x-axis.
2. Polar Plots: Polar plots are graphs that use polar coordinates to plot data. They consist of a circular grid, with the distance from the center representing the radius, and angles marked around the circumference of the circle. Data is plotted by specifying a radius and an angle for each point on the graph.
3. Applications: Polar plots are used in a wide range of applications, including antenna radiation patterns, frequency response plots, and more. They are particularly useful for visualizing data that has circular symmetry or is represented in polar coordinates.

Examples:

1. Antenna Radiation Pattern: An antenna radiation pattern is a plot that shows the strength and direction of radiation from an antenna in three dimensions. A polar plot can be used to represent the radiation pattern in two dimensions, with the distance from the center representing the radiation intensity and the angle representing the direction of radiation. For example, a polar plot of a dipole antenna radiation pattern might show a figure-eight pattern, with maximum radiation in two opposite directions perpendicular to the axis of the dipole.
2. Frequency Response Plot: A frequency response plot is a graph that shows how the output of a system responds to different frequencies of input. A polar plot can be used to represent the frequency response of a system in polar coordinates, with the magnitude of the response represented by the distance from the center of the plot and the phase shift represented by the angle. For example, a polar plot of the frequency response of a low-pass filter might show a straight line at low frequencies, representing little attenuation, and a curve at higher frequencies, representing increasing attenuation.

In conclusion, polar plots are a useful tool for representing data in polar coordinates, and can be used to visualize a wide range of phenomena, from antenna radiation patterns to frequency response plots. Understanding polar coordinates and how to interpret polar plots is an important skill in many fields, including mathematics, physics, and engineering.

# Design a Polar Plot with the given Transfer Function

To achieve this learning outcome, you should be able to design a polar plot given a transfer function. This involves understanding how to convert a transfer function into polar coordinates, and how to plot the resulting data in a polar plot.

Key Concepts:

1. Transfer Function: A transfer function is a mathematical representation of a system’s input-output relationship. It is typically represented as a ratio of polynomials in the Laplace variable, s.
2. Polar Coordinates: Polar coordinates are a way of representing points in a two-dimensional plane using a distance and an angle. In polar coordinates, the distance from the origin is known as the radius (r), and the angle is measured counterclockwise from the positive x-axis.
3. Bode Plot: A Bode plot is a graph used to represent the frequency response of a system. It consists of two plots: a magnitude plot, which shows the system’s gain in decibels as a function of frequency, and a phase plot, which shows the phase shift of the system as a function of frequency.
4. Nyquist Plot: A Nyquist plot is a graph used to represent the frequency response of a system in polar coordinates. It plots the magnitude and phase of the system’s transfer function as a function of frequency.

Examples:

1. Transfer Function of a Low-Pass Filter: Consider the transfer function of a simple low-pass filter, given by H(s) = 1/(s+1). To design a polar plot for this transfer function, we can first convert it to polar form by writing H(s) = 1/(1 + s/1). The magnitude of H(s) in polar form is given by |H(jw)| = 1/sqrt(1 + w2), and the phase is given by arg(H(jw)) = -arctan(w). Using these equations, we can plot the magnitude and phase of H(jw) as a function of w to obtain the polar plot.
2. Transfer Function of a Band-Pass Filter: Consider the transfer function of a simple band-pass filter, given by H(s) = (s+1)/(s2+2s+5). To design a polar plot for this transfer function, we can first convert it to polar form by writing H(s) = (1 + s/1)/(1 + 2s/5 + s2/5). The magnitude of H(s) in polar form is given by |H(jw)| = sqrt((1+w2)2/(1+4w2/5+w4/25)), and the phase is given by arg(H(jw)) = arctan((w-1)/(2-w2/5)). Using these equations, we can plot the magnitude and phase of H(jw) as a function of w to obtain the polar plot.

In conclusion, designing a polar plot from a transfer function involves converting the transfer function to polar form and plotting the magnitude and phase in polar coordinates. This technique can be used to visualise the frequency response of a system and to gain insight into its behavior. Polar plots are particularly useful for systems with circular symmetry or systems that are represented in polar coordinates.

# Find the Stability of a System using Polar Plot

To achieve this learning outcome, you should be able to use a polar plot to determine the stability of a system. This involves understanding how to interpret the information provided by a polar plot and how to apply stability criteria.

Key Concepts:

1. Polar Plot: A polar plot is a graph used to represent the frequency response of a system in polar coordinates. It plots the magnitude and phase of the system’s transfer function as a function of frequency.
2. Stability: A system is said to be stable if its output remains bounded for all bounded inputs. In other words, a stable system does not exhibit unbounded oscillations or divergent behavior.
3. Stability Criteria: There are several stability criteria that can be used to determine the stability of a system. One such criterion is the Nyquist stability criterion, which states that a system is stable if and only if the Nyquist plot of its transfer function encircles the point (-1,0) in the complex plane in a counterclockwise direction.

Examples:

1. Stable System: Consider the transfer function of a stable system, given by H(s) = (s+1)/(s+2). To determine the stability of this system using a polar plot, we can first convert the transfer function to polar form as follows:

H(jw) = (jw+1)/(jw+2)

|H(jw)| = sqrt((w2+1)/(w2+4))

arg(H(jw)) = arctan(w) – arctan(2)

Using these equations, we can plot the magnitude and phase of H(jw) as a function of w to obtain the polar plot. The polar plot for this system shows that the magnitude is always less than one and the phase is always negative, indicating that the system is stable.

1. Unstable System: Consider the transfer function of an unstable system, given by H(s) = 1/(s-1). To determine the stability of this system using a polar plot, we can first convert the transfer function to polar form as follows:

H(jw) = 1/(jw-1)

|H(jw)| = 1/sqrt(w2+1)

arg(H(jw)) = -arctan(w)

Using these equations, we can plot the magnitude and phase of H(jw) as a function of w to obtain the polar plot. The polar plot for this system shows that the magnitude is always greater than one and the phase is always positive, indicating that the system is unstable.

In conclusion, the stability of a system can be determined using a polar plot by applying stability criteria. The Nyquist stability criterion is one such criterion that can be used to determine the stability of a system from its polar plot. Polar plots can be particularly useful for identifying unstable modes of a system and for designing controllers that stabilize a system.

# Recall the Gain Margin and Phase Margin using Polar Plot

To achieve this learning outcome, you should be able to understand and recall the concepts of gain margin and phase margin, and how they can be determined from a polar plot.

Key Concepts:

1. Gain Margin: The gain margin of a system is the amount of gain that can be added to the system before it becomes unstable. It is usually expressed in decibels (dB) and can be determined from a polar plot by measuring the distance from the -1 point on the real axis to the intersection of the polar plot with the real axis.
2. Phase Margin: The phase margin of a system is the amount of phase lag that can be added to the system before it becomes unstable. It is usually expressed in degrees and can be determined from a polar plot by measuring the angle between the -1 point on the real axis and the intersection of the polar plot with the real axis.
3. Stability Analysis: Gain margin and phase margin are important indicators of the stability of a system. A system with a large gain margin and phase margin is more stable than a system with a small gain margin and phase margin. In general, a gain margin of 6 dB and a phase margin of 45 degrees are considered good targets for stability.

Examples:

1. Determining Gain Margin: Consider the transfer function of a system, given by H(s) = (s+2)/(s+1)(s+3). To determine the gain margin of this system using a polar plot, we can first convert the transfer function to polar form and plot the polar plot as shown below:

|H(jw)| = sqrt((w2+4)/(w2+1)(w2+9))

arg(H(jw)) = arctan(2/w) – arctan(w) – arctan(3/w)

The intersection of the polar plot with the real axis occurs at -0.74 dB, which represents the gain margin of the system.

1. Determining Phase Margin: Consider the transfer function of a system, given by H(s) = (s+1)/(s+2)(s+3). To determine the phase margin of this system using a polar plot, we can first convert the transfer function to polar form and plot the polar plot as shown below:

|H(jw)| = sqrt((w2+1)/(w2+4)(w2+9))

arg(H(jw)) = arctan(w) – arctan(2/w) – arctan(3/w)

The angle between the -1 point on the real axis and the intersection of the polar plot with the real axis is 45.6 degrees, which represents the phase margin of the system.

In conclusion, the gain margin and phase margin of a system can be determined from a polar plot. These measures of stability provide important information for system design and control. A system with large gain margin and phase margin is generally more stable than a system with small gain margin and phase margin.

# Determine the Gain Margin and Phase Margin using Polar Plot

To achieve this learning outcome, you should be able to apply the concepts of gain margin and phase margin to determine these measures of stability from a given polar plot.

Key Concepts:

1. Gain Margin: The gain margin of a system is the amount of gain that can be added to the system before it becomes unstable. It is usually expressed in decibels (dB) and can be determined from a polar plot by measuring the distance from the -1 point on the real axis to the intersection of the polar plot with the real axis.
2. Phase Margin: The phase margin of a system is the amount of phase lag that can be added to the system before it becomes unstable. It is usually expressed in degrees and can be determined from a polar plot by measuring the angle between the -1 point on the real axis and the intersection of the polar plot with the real axis.
3. Stability Analysis: Gain margin and phase margin are important indicators of the stability of a system. A system with a large gain margin and phase margin is more stable than a system with a small gain margin and phase margin. In general, a gain margin of 6 dB and a phase margin of 45 degrees are considered good targets for stability.

Examples:

1. Determining Gain Margin: Consider the polar plot of a system shown below. The intersection of the polar plot with the real axis occurs at -1.2 dB, which represents the gain margin of the system.
2. Determining Phase Margin: Consider the polar plot of a system shown below. The angle between the -1 point on the real axis and the intersection of the polar plot with the real axis is 38 degrees, which represents the phase margin of the system.

In conclusion, the gain margin and phase margin of a system can be determined from a polar plot. These measures of stability provide important information for system design and control. A system with large gain margin and phase margin is generally more stable than a system with small gain margin and phase margin.

# Recall Nyquist Stability Criteria

To achieve this learning outcome, you should be able to recall the Nyquist stability criteria and understand its significance in system stability analysis.

Key Concepts:

1. Nyquist Plot: The Nyquist plot is a graphical representation of the complex frequency response of a system. It is obtained by plotting the imaginary part of the transfer function against the real part while varying the frequency.
2. Nyquist Stability Criteria: The Nyquist stability criteria is a mathematical tool used to determine the stability of a closed-loop control system using the Nyquist plot. The criteria states that a closed-loop control system is stable if and only if the Nyquist plot of the open-loop transfer function does not encircle the -1 point on the complex plane in the clockwise direction.
3. Significance: The Nyquist stability criteria is a powerful tool for analyzing the stability of feedback control systems. It can be used to analyze systems with multiple feedback loops, non-minimum phase systems, and systems with time delays. It is also useful in designing stable control systems by providing insights into the effect of system parameters on stability.

Examples:

1. Stable System: Consider the open-loop transfer function given by G(s) = 1/(s+2). The Nyquist plot of this transfer function is shown below. The Nyquist plot does not encircle the -1 point in the clockwise direction, which indicates that the system is stable.
2. Unstable System: Consider the open-loop transfer function given by G(s) = 1/(s-1). The Nyquist plot of this transfer function is shown below. The Nyquist plot encircles the -1 point in the clockwise direction once, which indicates that the system is unstable.

In conclusion, the Nyquist stability criteria provides a powerful tool for analyzing the stability of feedback control systems. The Nyquist plot can be used to determine if a system is stable or unstable. The criteria is particularly useful for analyzing complex systems with multiple feedback loops and time delays.

# Determine the Stability of a System using Nyquist Criteria

To achieve this learning outcome, you should be able to apply the Nyquist stability criteria to determine the stability of a closed-loop control system.

Key Concepts:

1. Closed-loop Control System: A closed-loop control system is a system in which the output of the system is fed back to the input to adjust the behavior of the system.
2. Nyquist Plot: The Nyquist plot is a graphical representation of the complex frequency response of a system. It is obtained by plotting the imaginary part of the transfer function against the real part while varying the frequency.
3. Nyquist Stability Criteria: The Nyquist stability criteria is a mathematical tool used to determine the stability of a closed-loop control system using the Nyquist plot. The criteria states that a closed-loop control system is stable if and only if the Nyquist plot of the open-loop transfer function does not encircle the -1 point on the complex plane in the clockwise direction.
4. Significance: The Nyquist stability criteria is a powerful tool for analyzing the stability of feedback control systems. It can be used to analyze systems with multiple feedback loops, non-minimum phase systems, and systems with time delays. It is also useful in designing stable control systems by providing insights into the effect of system parameters on stability.

Examples:

1. Stable System: Consider the closed-loop control system shown below, where G(s) and H(s) are the transfer functions of the plant and feedback paths, respectively. The open-loop transfer function of the system is given by G(s)H(s). The Nyquist plot of the open-loop transfer function is shown below. The Nyquist plot does not encircle the -1 point in the clockwise direction, which indicates that the system is stable.
2. Unstable System: Consider the closed-loop control system shown below, where G(s) and H(s) are the transfer functions of the plant and feedback paths, respectively. The open-loop transfer function of the system is given by G(s)H(s). The Nyquist plot of the open-loop transfer function is shown below. The Nyquist plot encircles the -1 point in the clockwise direction once, which indicates that the system is unstable.

In conclusion, the Nyquist stability criteria provides a powerful tool for analyzing the stability of closed-loop control systems. The Nyquist plot can be used to determine if a system is stable or unstable. The criteria is particularly useful for analyzing complex systems with multiple feedback loops and time delays. It is also useful in designing stable control systems by providing insights into the effect of system parameters on stability.

# Differentiate between Nyquist Criteria and Polar Plot

To achieve this learning outcome, you should be able to differentiate between Nyquist criteria and polar plot and explain their respective advantages and disadvantages.

Key Concepts:

1. Nyquist Criteria: The Nyquist stability criteria is a mathematical tool used to determine the stability of a closed-loop control system using the Nyquist plot. The criteria states that a closed-loop control system is stable if and only if the Nyquist plot of the open-loop transfer function does not encircle the -1 point on the complex plane in the clockwise direction.
2. Polar Plot: The polar plot is a graphical representation of the magnitude and phase angle of the transfer function of a system as a function of frequency. It is obtained by plotting the magnitude of the transfer function against the phase angle while varying the frequency.
3. Advantages of Nyquist Criteria: The Nyquist criteria is particularly useful for analyzing complex systems with multiple feedback loops and time delays. It is also useful in designing stable control systems by providing insights into the effect of system parameters on stability.
4. Disadvantages of Nyquist Criteria: The Nyquist criteria requires the computation of the entire Nyquist plot, which can be time-consuming and computationally intensive. It also assumes that the closed-loop control system is linear and time-invariant, which may not be the case in practical applications.
5. Advantages of Polar Plot: The polar plot provides a simple and intuitive graphical representation of the frequency response of a system. It is useful for analyzing the gain and phase margins of a closed-loop control system.
6. Disadvantages of Polar Plot: The polar plot does not provide a direct measure of stability, and it cannot be used to analyze complex systems with multiple feedback loops and time delays.

Examples:

1. Nyquist Criteria: Consider the closed-loop control system shown below, where G(s) and H(s) are the transfer functions of the plant and feedback paths, respectively. The open-loop transfer function of the system is given by G(s)H(s). The Nyquist plot of the open-loop transfer function is shown below. The Nyquist plot does not encircle the -1 point in the clockwise direction, which indicates that the system is stable.
2. Polar Plot: Consider the closed-loop control system shown below, where G(s) and H(s) are the transfer functions of the plant and feedback paths, respectively. The open-loop transfer function of the system is given by G(s)H(s). The polar plot of the open-loop transfer function is shown below. The gain margin and phase margin of the system can be read directly from the polar plot.

In conclusion, Nyquist criteria and polar plot are two different tools used to analyze closed-loop control systems. The Nyquist criteria is particularly useful for analyzing complex systems with multiple feedback loops and time delays, while the polar plot provides a simple and intuitive graphical representation of the frequency response of a system. Both tools have their respective advantages and disadvantages, and the choice of tool depends on the specific application and the complexity of the system being analyzed.

# Recall Nichols Chart

The Nichols chart is a graphical tool used in control systems engineering to analyze and design closed-loop systems. It is named after American engineer Nathaniel B. Nichols, who developed it in 1947. The chart provides a graphical representation of the frequency response and stability of a system, and is useful in designing feedback control systems for stable operation.

The Nichols chart is a polar plot of the open-loop transfer function of a system in which the magnitude is represented on the radial axis and the phase is represented on the angular axis. The chart can be used to determine the stability of a system by evaluating the phase margin and gain margin.

Phase margin is the amount of additional phase lag that can be added to the system before it becomes unstable. A system is considered stable if its phase margin is greater than zero. Gain margin is the amount of additional gain that can be added to the system before it becomes unstable. A system is considered stable if its gain margin is greater than one.

The Nichols chart is particularly useful for designing controllers that can stabilize a system. For example, suppose we have a control system with a transfer function of G(s) and a feedback system with transfer function H(s). We want to design a controller C(s) that stabilizes the system. To do this, we need to find a point on the Nichols chart that satisfies the stability requirements. We can then use this point to determine the parameters of the controller C(s) that will stabilize the system.

The following example demonstrates how the Nichols chart can be used to design a controller for a simple control system:

Suppose we have a control system with a transfer function of G(s) = K/(s+1)(s+2), and a feedback system with transfer function H(s) = 1. We want to design a controller C(s) that stabilises the system.

1. Plot the Nichols chart for the open-loop transfer function G(s). The plot will have the magnitude of G(s) on the y-axis and the phase on the x-axis.
2. Determine the stability margins for the system. The phase margin is the angle between the phase curve and the -180° line at the frequency where the magnitude is 0 dB. The gain margin is the amount of gain required to make the phase margin 0 dB. In this case, the phase margin is 58.3° and the gain margin is 0.47.
3. Find a point on the Nichols chart that satisfies the stability requirements. In this case, we need a phase margin of at least 60° and a gain margin of at least 1. We can see that the point (-0.55 dB, 73°) satisfies these requirements.
4. Use the point on the Nichols chart to determine the parameters of the controller C(s) that will stabilize the system. In this case, we can use the formula C(s) = (s+0.87)/s to design a controller that stabilises the system.

In conclusion, the Nichols chart is a useful tool for analyzing and designing closed-loop control systems. It provides a graphical representation of the frequency response and stability of a system, and can be used to determine the parameters of a controller that will stabilize the system. By understanding and recalling the properties of the Nichols chart, engineers can design stable control systems that meet performance requirements.

# Describe Constant Magnitude Locus(M-Circle)

In control systems engineering, the constant magnitude locus, also known as M-circle, is a graphical tool used to analyze the frequency response of a system. It is a polar plot of the magnitude of the open-loop transfer function of a system, where the magnitude is held constant while the phase is varied.

The M-circle is a locus of points on the polar plot that represent the points on the Nyquist plot where the magnitude of the open-loop transfer function is constant. This locus is important in analyzing and designing feedback control systems because it can be used to determine the stability and performance of a system.

The M-circle is a useful tool for analyzing the frequency response of a system. For example, suppose we have a feedback control system with a transfer function of H(s) and an open-loop transfer function of G(s). We want to determine the stability and performance of the system using the M-circle.

To use the M-circle, we first plot the Nyquist plot of the open-loop transfer function G(s). We then draw a circle on the Nyquist plot with a radius equal to the desired constant magnitude. The M-circle is the locus of points on the Nyquist plot where the circle intersects the curve of the Nyquist plot.

The following example demonstrates how the M-circle can be used to analyze the stability and performance of a control system:

Suppose we have a feedback control system with a transfer function of H(s) = 1/(s+1), and an open-loop transfer function of G(s) = K/(s+1)(s+2). We want to determine the stability and performance of the system using the M-circle.

1. Plot the Nyquist plot of the open-loop transfer function G(s). The plot will have the real part of G(s) on the x-axis and the imaginary part on the y-axis.
2. Draw a circle on the Nyquist plot with a radius of 1. This represents the constant magnitude locus.
3. Determine the intersection points of the M-circle and the Nyquist curve. These points represent the points on the Nyquist curve where the magnitude of the open-loop transfer function is 1.
4. Determine the stability and performance of the system. The system is stable if the M-circle does not enclose the point (-1,0) on the Nyquist plot. The system has good performance if the intersection point of the M-circle and the Nyquist curve is close to the point (-1,0), indicating a low steady-state error.

In conclusion, the M-circle is a useful tool for analyzing the frequency response of a system. It can be used to determine the stability and performance of a feedback control system by plotting a circle on the Nyquist plot and determining the intersection points with the Nyquist curve. By understanding and describing the properties of the M-circle, engineers can design stable and high-performance control systems.

# Describe Constant Phase Angle Loci(N-Circle)

Constant Phase Angle Loci, also known as N-Circles, are a type of locus that is useful in analyzing the behavior of electrical circuits that have multiple signal sources. The locus is constructed by plotting points that have a constant phase angle with respect to a reference signal. In this context, the phase angle is the relative position of a point in a waveform cycle, expressed in degrees or radians.

1. Definition of Constant Phase Angle Loci:

Constant Phase Angle Loci are curves that represent the locus of points in a complex plane that have a constant phase angle with respect to a reference signal. They are constructed by plotting points on the complex plane that have a constant ratio of imaginary to real components. For example, if we consider a sinusoidal signal with a frequency of 1 Hz, the points that have a phase angle of 45 degrees will lie on a circle with a radius of 1/sqrt(2).

1. Properties of Constant Phase Angle Loci:
• The locus is a circle centered at the origin of the complex plane.
• The radius of the circle depends on the frequency of the signal and the phase angle.
• The locus represents all possible values of a signal that have a constant phase angle with respect to the reference signal.
• The angle between two N-Circles represents the phase difference between the corresponding signals.
1. Construction of Constant Phase Angle Loci:

To construct an N-Circle, we start by defining the reference signal and its phasor representation. We then consider another signal that has a constant phase angle with respect to the reference signal, and determine its phasor representation. The locus of points that have this phase relationship is a circle centered at the origin, with a radius proportional to the magnitude of the phasor.

For example, let us consider two sinusoidal signals with frequencies f1 and f2, and phase angles θ1 and θ2, respectively. We can represent these signals as phasors using Euler’s formula as:

V1 = Vm1 ∠θ1 = Vm1(cosθ1 + j sinθ1)

V2 = Vm2 ∠θ2 = Vm2(cosθ2 + j sinθ2)

The N-Circle corresponding to the phase angle θ1 is a circle with radius Vm1 centred at the origin, while the N-Circle corresponding to the phase angle θ2 is a circle with radius Vm2. The angle between these two circles is equal to (θ1 – θ2).

1. Applications of Constant Phase Angle Loci:
• Analysis of electrical circuits with multiple signal sources: N-Circles are useful in analyzing circuits that have multiple sinusoidal signal sources, by providing a graphical representation of the phase relationships between the signals.
• Design of filters: N-Circles can be used to design filters that have a specific phase response, by selecting the appropriate values of capacitance and inductance in the filter circuit.
• Control systems: N-Circles can be used to analyze the stability of control systems, by determining the phase margin and gain margin of the system.

In conclusion, Constant Phase Angle Loci (N-Circles) in terms of their definition, properties, construction, and applications. By understanding the concept of N-Circles, learners can analyze electrical circuits with multiple signal sources, design filters with specific phase responses.

# Recall the following Transfer Functions:i. Minimum Phase Transfer Function ii. Non-Minimum Phase Transfer Function iii. All-Pass Transfer Function

Transfer functions are a fundamental concept in signal processing that relate the input and output signals of a system. They are widely used in the design and analysis of filters, amplifiers, and control systems. A transfer function is a mathematical representation of the relationship between the input and output signals of a system in the frequency domain.

Here are detailed notes on this learning outcome, along with suitable examples:

1. Minimum Phase Transfer Function:

A minimum phase transfer function is a transfer function that has all its poles and zeros in the left half of the complex plane. This means that the system has a causal and stable impulse response. The phase response of a minimum phase system is such that it reaches its maximum and minimum values at the same frequencies as the magnitude response.

For example, a second-order minimum phase transfer function can be represented as:

H(s) = K (s – z1) (s – z2) / (s – p1) (s – p2)

where K is the gain, z1 and z2 are the zeros, and p1 and p2 are the poles of the transfer function. If we plot the poles and zeros of this transfer function on a complex plane, we will find that they are all in the left half of the plane.

1. Non-Minimum Phase Transfer Function:

A non-minimum phase transfer function is a transfer function that has one or more zeros in the right half of the complex plane. This means that the system has a non-causal or unstable impulse response. The phase response of a non-minimum phase system is such that it reaches its maximum and minimum values at frequencies different from those at which the magnitude response reaches its maximum and minimum values.

For example, a second-order non-minimum phase transfer function can be represented as:

H(s) = K (s – z1) (s – z2) / (s – p1) (s – p2)

where K is the gain, z1 and z2 are the zeros, and p1 and p2 are the poles of the transfer function. If we plot the poles and zeros of this transfer function on a complex plane, we will find that one or both of the zeros are in the right half of the plane.

1. All-Pass Transfer Function:

An all-pass transfer function is a transfer function that has the same magnitude response as a minimum phase transfer function, but with a phase response that varies linearly with frequency. This means that an all-pass system has a stable and causal impulse response, but its phase response introduces a delay without affecting the magnitude response.

For example, a second-order all-pass transfer function can be represented as:

H(s) = (s – z1) / (s – p1)

where z1 and p1 are the zero and pole of the transfer function, respectively. If we plot the pole and zero of this transfer function on a complex plane, we will find that they are complex conjugates, lying on a line that is perpendicular to the imaginary axis. The phase response of this transfer function is given by:

Φ(ω) = -2tan-1[(ω – ω0)/Δω]

where ω0 is the frequency at which the phase shift is zero, and Δω is the bandwidth of the all-pass filter.