5 Steps: How to Graph 2nd Order LTI on Bode Plot ⋆ binaryage.com

Understanding the intricacies of second-order linear time-invariant (LTI) systems is crucial in various engineering disciplines. Bode plots, a graphical representation of a system’s frequency response, offer a comprehensive analysis of these systems, enabling engineers to visualize their behavior and make informed design decisions.

In this context, graphing second-order LTI systems on Bode plots is an essential skill. It allows engineers to study the system’s magnitude and phase response over a range of frequencies, providing valuable insights into the system’s stability, bandwidth, and damping characteristics. By utilizing the principles of Bode analysis, engineers can gain a deeper understanding of how these systems behave in different frequency regimes and make necessary adjustments to optimize performance.

To effectively graph second-order LTI systems on Bode plots, it is important to first understand the underlying mathematical equations governing their behavior. These equations describe the system’s transfer function, which in turn determines its frequency response. By applying logarithmic scales to both the frequency and amplitude axes, Bode plots provide a convenient way to visualize the system’s behavior over a wide range of frequencies. By carefully analyzing the resulting plots, engineers can identify key features such as cutoff frequencies, resonant peaks, and phase shifts, and use this information to design systems that meet specific performance requirements.

Introduction to Bode Plots

Bode plots are graphical representations of the frequency response of a system. They are used to analyze the stability, bandwidth, and resonance of a system. Bode plots can be used to design filters, amplifiers, and other electronic circuits.

The frequency response of a system is the output of the system as a function of the input frequency. The Bode plot is a plot of the magnitude and phase of the frequency response on a logarithmic scale.

The magnitude of the frequency response is typically plotted in decibels (dB). The decibel is a logarithmic unit of measurement that is used to express the ratio of two power levels. The phase of the frequency response is typically plotted in degrees.

Bode plots can be used to determine the following characteristics of a system:

Stability: The stability of a system is determined by the phase margin of the system. The phase margin is the difference between the phase of the system at the crossover frequency and 180 degrees. A stable system has a phase margin of at least 45 degrees.
Bandwidth: The bandwidth of a system is the frequency range over which the system has a gain of at least 3 dB.
Resonance: The resonance frequency of a system is the frequency at which the system has a peak gain.

2nd Order Linear Time-Invariant Systems

A 2nd order linear time-invariant (LTI) system is a system that is described by the following differential equation:

y'' + 2ζωny' + ωny^2 = Ku

where:

y is the output of the system
u is the input to the system
ζ is the damping ratio
ωn is the natural frequency
K is the gain

The damping ratio and natural frequency are two important parameters that determine the behavior of a 2nd order LTI system. The damping ratio determines the amount of damping in the system, while the natural frequency determines the frequency at which the system oscillates.

The following table shows the different types of 2nd order LTI systems, depending on the values of the damping ratio and natural frequency:

Damping Ratio	Natural Frequency	Type of System
ζ > 1	Any	Overdamped
ζ = 1	Any	Critically damped
0 < ζ < 1	Any	Underdamped
ζ = 0	ωn = 0	Marginally stable
ζ = 0	ωn ≠ 0	Unstable

Bode plots can be used to analyze the frequency response of 2nd order LTI systems. The shape of the Bode plot depends on the damping ratio and natural frequency of the system.

Transfer Function of a 2nd Order LTI System

A second-order linear time-invariant (LTI) system is described by a transfer function of the form:

“`
H(s) = K / ((s + a)(s + b))
“`

where:
– K is the system gain
– a and b are the poles of the system (the values of s for which the denominator of H(s) is zero)
– s is the Laplace variable

The poles of a system determine its response to an input signal. A system with complex poles will have an oscillatory response, while a system with real poles will have an exponential response.

The following table summarizes the characteristics of second-order LTI systems with different pole locations:

Pole Location	Response
Real and distinct	Two exponential decays
Real and equal	One exponential decay
Complex	Oscillatory decay

The Bode plot of a second-order LTI system is a plot of the system’s magnitude and phase response as a function of frequency.

Asymptotic Behavior Analysis of the Bode Plot

1. High-Frequency Asymptotes

At high frequencies, the Bode plot exhibits predictable asymptotic behavior. For terms with positive exponents, the asymptote follows the slope of that exponent. For example, a term with an exponent of +2 produces an asymptote with a +2 slope (12 dB/octave). Conversely, terms with negative exponents create asymptotes with negative slopes. A term with an exponent of -1 generates an asymptote with a -1 slope (6 dB/octave).

2. Low-Frequency Asymptotes

In the low-frequency region, the Bode plot’s asymptotes depend on the constant term. If the constant term is positive, the asymptote remains at 0 dB. If it is negative, the asymptote has a negative slope equal to the constant’s exponent. For instance, a constant term of -2 produces an asymptote with a -2 slope (12 dB/octave).

3. Combined Asymptotic Behavior Analysis

The asymptotic behavior of a transfer function can be a complex interplay of multiple terms. To analyze it effectively, follow these steps:

Identify individual asymptotic behaviors: Determine the high- and low-frequency asymptotes of each term in the transfer function.
Superimpose asymptotes: Overlap the individual asymptotes to create a composite asymptotic profile. This profile outlines the overall shape of the Bode plot.
Breakpoints: Identify the frequencies where asymptotes change slope. These breakpoints indicate where the transfer function’s dominant terms switch.
Mid-Frequency Region: Analyze the behavior between the breakpoints to determine any deviations from the asymptotic lines.

Term	High-Frequency Asymptote	Low-Frequency Asymptote
s + 2	+1 (20 dB/decade)	0 dB
s – 1	0 dB	-1 (20 dB/decade)
1/(s² + 1)	-2 (40 dB/decade)	0 dB

Identifying the Corner Frequencies

The corner frequencies are the frequencies at which the system’s response changes from one type of behavior to another. For a second-order LTI system, there are two corner frequencies: the natural frequency (ωn) and the damping ratio (ζ).

The Natural Frequency

The natural frequency is the frequency at which the system would oscillate if there were no damping. It is determined by the system’s mass and stiffness.

The natural frequency can be found using the following formula:

$$\omega_n = \sqrt{\frac{k}{m}}$$
where:
* ωn is the natural frequency in radians per second
* k is the spring constant in newtons per meter
* m is the mass in kilograms

The Damping Ratio

The damping ratio is a measure of how quickly the system’s oscillations decay. It ranges from 0 to 1. A damping ratio of 0 indicates that the system will oscillate indefinitely, while a damping ratio of 1 indicates that the system will return to its equilibrium position quickly without overshooting.

The damping ratio can be found using the following formula:

$$\zeta = \frac{c}{2\sqrt{km}}$$
where:
* ζ is the damping ratio
* c is the damping coefficient in newtons-seconds per meter
* k is the spring constant in newtons per meter
* m is the mass in kilograms

Constructing the Magnitude Plot

The magnitude plot shows the gain in decibels (dB) as a function of the frequency. To construct the magnitude plot, follow these steps:

1. **Find the cutoff frequency (ωc)**: This is the frequency at which the gain is down by 3 dB from the DC gain.

2. **Find the slope:** The slope of the magnitude plot is -20 dB/decade for a first-order system and -40 dB/decade for a second-order system.

3. **Draw the asymptotes:** Draw two asymptotes, one with the slope found in step 2 and one with a gain of 0 dB.

4. **Interpolate the asymptotes to find the magnitude at the specified frequencies**:

Find the gain in dB at the cutoff frequency from the asymptotes.
Find the frequency at which the gain is 20 dB below the DC gain.
Find the frequency at which the gain is 40 dB below the DC gain (for second-order systems only).
Draw a line connecting these points to approximate the magnitude plot.

5. **Plot the magnitude response:** Plot the gain in dB on the vertical axis and the frequency on the horizontal axis. The resulting plot is the magnitude plot of the 2nd order LTI system.

The following table summarizes the steps for constructing the magnitude plot:

Step	Action
1	Find the cutoff frequency
2	Find the slope
3	Draw the asymptotes
4	Interpolate the asymptotes
5	Plot the magnitude response

Plotting the Phase Plot

The phase plot provides information about the phase shift of the output signal relative to the input signal. To plot the phase plot, follow these steps:

Plot the imaginary part of the transfer function, $Im(H(j\omega))$, on the vertical axis.
Plot the real part of the transfer function, $Re(H(j\omega))$, on the horizontal axis.
The resulting curve is the phase plot.

The phase plot is typically represented as a graph of phase shift (in degrees) versus frequency ($\omega$). The phase shift is calculated using the formula:
“`
Phase Shift = arctan(Im(H(j\omega))/Re(H(j\omega)))
“`

The phase plot can be used to determine the stability and phase margin of the system. A negative phase shift indicates that the output signal is lagging the input signal, while a positive phase shift indicates that the output signal is leading the input signal.

The following table shows the relationship between the phase shift and the stability of the system:

Phase Shift	Stability
0°	Stable
-90° to 0°	Marginally stable
-90° to -180°	Unstable

The phase margin is the difference between the phase shift at the crossover frequency (where the magnitude of the transfer function is 0 dB) and -180°. A phase margin of at least 45° is generally considered to be acceptable for stability.

Slopes and Breakpoints in the Bode Plot

Slope of the Bode Plot

The slope of the Bode plot indicates the rate of change in the magnitude or phase response of a system with respect to frequency. A positive slope indicates an increase in magnitude or phase with increasing frequency, while a negative slope indicates a decrease. The slope of the Bode plot can be determined by the order of the system and the type of filter it is. For example, a first-order low-pass filter will have a slope of -20 dB/decade in the magnitude plot and -90 degrees/decade in the phase plot.

Breakpoints of the Bode Plot

The breakpoints of the Bode plot are the frequencies at which the slope of the plot changes. These breakpoints occur at the natural frequencies of the system, which are the frequencies at which the system oscillates when it is excited by an impulse. The breakpoints of the Bode plot can be used to determine the resonant frequencies and damping ratios of the system.

Magnitude and Phase Breakpoints of 2nd Order LTI System

Magnitude Breakpoint

Phase Breakpoint

$\omega_n$

$0.707 \omega_n$

$\omega_n \sqrt{1+2\zeta^2}$

$\omega_n$

$\omega_n \sqrt{1-2\zeta^2}$

Overdamped Cases

In the overdamped case, the system’s response to a step input is slow and gradual, without any oscillations. This occurs when the damping ratio (ζ) is greater than 1. The Bode plot for an overdamped system has the following characteristics:

The magnitude response (20 log|H(f)|) is a horizontal line at -6 dB/octave, indicating a roll-off of 6 dB per octave.
The phase response is a straight line with a slope of -90 degrees/decade, indicating a phase lag of 90 degrees at all frequencies.

Underdamped Cases

In the underdamped case, the system’s response to a step input is oscillatory, with the oscillations gradually decreasing in amplitude over time. This occurs when the damping ratio (ζ) is less than 1. The Bode plot for an underdamped system has the following characteristics:

The magnitude response has a peak at the resonant frequency (ωn), with the peak magnitude depending on the damping ratio.
The phase response starts at -90 degrees at low frequencies and approaches -180 degrees at high frequencies, passing through -135 degrees at the resonant frequency.

Critically Damped Cases

In the critically damped case, the system’s response to a step input is the fastest possible without any oscillations. This occurs when the damping ratio (ζ) is equal to 1. The Bode plot for a critically damped system has the following characteristics:

The magnitude response is a horizontal line at -6 dB/octave, indicating a roll-off of 6 dB per octave.
The phase response is a straight line with a slope of -180 degrees/decade, indicating a phase lag of 180 degrees at all frequencies.

Bode Plot Characteristics for Different Damping Cases

Damping Case	Magnitude Response	Phase Response
Overdamped (ζ > 1)	-6 dB/octave	-90 degrees/decade
Underdamped (ζ < 1)	Peak at resonant frequency (ωn)	-90 degrees at low frequencies, -180 degrees at high frequencies
Critically Damped (ζ = 1)	-6 dB/octave	-180 degrees/decade

Impact of Pole and Zero Locations on the Bode Plot

Poles and Zeros at Origin

A pole at the origin gives a -20 dB/decade slope in the magnitude response. A zero at the origin will give a +20 dB/decade slope.

Poles and Zeros at Infinity

A pole at infinity has no effect on the magnitude response. A zero at infinity gives a -20 dB/decade slope.

Poles and Zeros on Real Axis

A pole on the real axis gives a -20 dB/decade slope with a corner frequency equal to the absolute value of the pole location. A zero on the real axis gives a +20 dB/decade slope, also with a corner frequency equal to the absolute value of the zero location.

Poles and Zeros on Imaginary Axis

A pole on the imaginary axis gives a -90 degree phase shift. A zero on the imaginary axis gives a +90 degree phase shift. The corner frequency is equal to the imaginary part of the pole or zero location.

Poles in the Left Half Plane (LHP)

Poles in the LHP contribute to the stability of the system. They give a -20 dB/decade slope in the magnitude response and a -90 degree phase shift. The corner frequency is equal to the distance from the pole location to the imaginary axis.

Zeros in the Left Half Plane (LHP)

Zeros in the LHP do not contribute to the stability of the system. They give a +20 dB/decade slope in the magnitude response and a +90 degree phase shift. The corner frequency is equal to the distance from the zero location to the imaginary axis.

Complex Poles and Zeros

Complex poles and zeros give a combination of the above effects. The magnitude response will have a slope that is a combination of -20 dB/decade and +20 dB/decade, and the phase response will have a combination of -90 degree shift and +90 degree shift. The corner frequency is equal to the distance from the pole or zero location to the origin.

Pole-Zero Cancellations

If a pole and a zero are located at the same frequency, they will cancel each other out. This will result in a flat (zero slope) magnitude response and a linear phase response in the frequency range around the cancellation frequency.

Pole or Zero Location Magnitude Slope Phase Shift Corner Frequency

Origin -20 dB/decade 0 –

Infinity 0 -20 dB/decade –

Real Axis (positive) -20 dB/decade -90 Pole location

Real Axis (negative) -20 dB/decade -90 -Pole location

Imaginary Axis (positive) 0 +90 Zero location

Imaginary Axis (negative) 0 -90 -Zero location

Left Half Plane (LHP) -20 dB/decade -Phase angle Distance to imaginary axis

Right Half Plane (RHP) +20 dB/decade +Phase angle Distance to imaginary axis

Complex Plane Combination of above Combination of above Distance to origin

Pole-Zero Cancellation 0 Linear Cancellation frequency

Gain and Phase Margin Calculations

Bode plots are indispensable for calculating gain and phase margins, which determine the stability and robustness of a control system. Gain margin measures the amount by which the system’s gain can be increased before instability occurs, while phase margin measures the amount by which the system’s phase can be increased before instability arises. Bode plots provide a straightforward method for determining these margins, ensuring control system stability.

Loop Shaping for Control System Design

Using Bode plots, control engineers can shape the frequency response of a control loop to achieve desired performance characteristics. By adjusting the gain and phase of the system at specific frequencies, they can optimize the loop’s stability, bandwidth, and disturbance rejection capabilities, ensuring optimal system operation.

Stability Analysis of Systems with Multiple Inputs and Outputs

Bode plots are particularly useful for analyzing the stability of MIMO (Multiple-Input Multiple-Output) systems, where interactions between multiple inputs and multiple outputs can complicate stability assessment. By constructing Bode plots for each input-output pair, engineers can identify potential stability issues and design control strategies to ensure system robustness.

Compensation Design for Feedback Control Loops

Bode plots provide a valuable tool for designing compensation networks to improve the performance of feedback control loops. By adding lead or lag compensators, engineers can adjust the system’s frequency response to enhance stability, reduce steady-state errors, and improve dynamic performance.

Analysis of Closed-Loop Systems

Bode plots are essential for analyzing the closed-loop behavior of control systems. They enable engineers to predict the system’s output response to external disturbances and determine system parameters such as rise time, settling time, and frequency response.

Predictive Control and Model-Based Design

Bode plots are increasingly used in predictive control and model-based design approaches, where system models are developed and used for control. By comparing the actual Bode plots with the predicted ones, engineers can validate models and design control systems that meet performance specifications.

How to Graph 2nd Order LTI on Bode Plot

A second-order linear time-invariant (LTI) system can be represented by the following transfer function:

“`
H(s) = K * (s + z1) / (s^2 + 2*zeta*wn*s + wn^2)
“`

where K is the gain, z1 is the zero, wn is the natural frequency, and zeta is the damping ratio.

To graph the Bode plot of a 2nd order LTI system, follow these steps:

Calculate the gain, zero, natural frequency, and damping ratio of the system.
Create a Bode plot template with frequency on the x-axis and magnitude and phase on the y-axis.
Plot the gain as a horizontal line at 20*log10(K) dB.
For the magnitude plot, plot a curve with a slope of -20 dB/decade for frequencies below wn and a slope of -40 dB/decade for frequencies above wn.
For the phase plot, plot a curve with a slope of -90 degrees/decade for frequencies below wn and a slope of -180 degrees/decade for frequencies above wn.
Adjust the magnitude and phase curves based on the zero and damping ratio of the system.