# Bode plots solve frequency domain problems

05/01/1997

Every child who has ever held a spring upright knows that tugging on the top end causes the bottom end to start bouncing and that repeated tugging keeps those oscillations going. Some may notice that even though both ends always oscillate at the same frequency, the bottom end bounces higher at some frequencies than at others. Truly gifted children might even notice that the bottom end oscillates out of sync with the top end and lagsfurther and further behind as the frequency increases.

Engineers know that many mechanical, electrical, and chemical processes with energy-storing components behave the same way. A step change at the input end causes decaying oscillations at the output. A sinusoidal input, on the other hand, causes a sinusoidal output with an amplitude and a lag time that depend on the frequency of the oscillations. Not coincidentally, these two phenomena are related and form the basis of the frequency domain techniques that are fundamental to the analysis of feed-back controllers.

Two ways to calculate
There are basically two ways to analyze the behavior of a process in the frequency domain. The direct method is to drive it with a series of sinusoidal inputs, each with the same amplitude but different frequency. The amplitude and lag time of the sinusoidal outputs that result in each case can be plotted against frequency to produce a Bode plot for the process. The sample Bode plot in the figure shows how high the bottom end of the spring will bounce and how much it will lag the top end when the top end is set oscillating at various frequencies.

The second frequency domain analysis method uses Fourier's Theorem to compute the process' Bode plot indirectly. Fourier's Theorem states that any signal which is not itself a sine wave can be expressed as a sum of sine waves. A step input, for example, results when a few high-amplitude, low-frequency sine waves are added to a larger collection of low-amplitude, high-frequency sine waves. Fourier's Theorem also gives a formula for computing the amplitude of each component sine wave plus its lag time relative to the lowest frequency compo-nent. Note that plotting the amplitude and lag time for each component against its frequency yields a Bode plot of the signal just like theBode plot for an entire process.

In fact, the Bode plot for a process can be derived from the Bode plots of its input and output signals. Simply divide each amplitudein the output's Bode plot by the corresponding amplitude in the input's Bode plot. The result-ing quotient is the amplitude for the process' Bode plot at that frequency. To get the lagtimes for the process' Bode plot, simply subtract each input lag time from the correspondingoutput lag time.

Conversely, multiplying the amplitudes of an arbitrary input signal with the amplitudes of the process will give the amplitudes resulting in the output when that input is actually applied to that process. The output's lag times are computed by adding the lag times of the input to the lag times of the process. With this mathematical 'trick,' control engineers can predict the effects of a controller's actions on any process with a known Bode plot.

Consulting Editor Vance J. VanDoren, Ph.D, P.E., president of VanDoren Industries, West Lafayette, Ind.

The Engineers' Choice Awards highlight some of the best new control, instrumentation and automation products as chosen by Control Engineering subscribers. Vote now (if qualified)!
The System Integrator Giants program lists the top 100 system integrators among companies listed in CFE Media's Global System Integrator Database.
Each year, a panel of Control Engineering and Plant Engineering editors and industry expert judges select the System Integrator of the Year Award winners in three categories.
This eGuide illustrates solutions, applications and benefits of machine vision systems.
Learn how to increase device reliability in harsh environments and decrease unplanned system downtime.
This eGuide contains a series of articles and videos that considers theoretical and practical; immediate needs and a look into the future.
Maximize ROI with integrated control system approach; Microcontrollers vs. PLCs; Power quality; Accelerate and rewire IIoT; Traits for excellent engineers
HMI effectiveness; Distributed I/O; Engineers' Choice Award finalists; System Integrator advice; Inside Machines
Women in engineering; Engineering Leaders Under 40; PID benefits and drawbacks; Ladder logic; Cloud computing
Programmable logic controllers (PLCs) represent the logic (decision) part of the control loop of sense, decide, and actuate. As we know, PLCs aren’t the only option for making decisions in a control loop, but they are likely why you’re here.
This digital report explains how plant engineers and subject matter experts (SME) need support for time series data and its many challenges.
This article collection contains several articles on how advancements in vision system designs, computing power, algorithms, optics, and communications are making machine vision more cost effective than ever before.
Control room technology innovation; Practical approaches to corrosion protection; Pipeline regulator revises quality programs
Cloud, mobility, and remote operations; SCADA and contextual mobility; Custom UPS empowering a secure pipeline
Infrastructure for natural gas expansion; Artificial lift methods; Disruptive technology and fugitive gas emissions
Automation Engineer; Wood Group
System Integrator; Cross Integrated Systems Group
Jose S. Vasquez, Jr.
Fire & Life Safety Engineer; Technip USA Inc.
This course focuses on climate analysis, appropriateness of cooling system selection, and combining cooling systems.
This course will help identify and reveal electrical hazards and identify the solutions to implementing and maintaining a safe work environment.
This course explains how maintaining power and communication systems through emergency power-generation systems is critical.
The Engineers' Choice Awards highlight some of the best new control, instrumentation and automation products as chosen by Control Engineering subscribers. Vote now (if qualified)!
The System Integrator Giants program lists the top 100 system integrators among companies listed in CFE Media's Global System Integrator Database.
Each year, a panel of Control Engineering and Plant Engineering editors and industry expert judges select the System Integrator of the Year Award winners in three categories.
This eGuide illustrates solutions, applications and benefits of machine vision systems.
Learn how to increase device reliability in harsh environments and decrease unplanned system downtime.
This eGuide contains a series of articles and videos that considers theoretical and practical; immediate needs and a look into the future.
Maximize ROI with integrated control system approach; Microcontrollers vs. PLCs; Power quality; Accelerate and rewire IIoT; Traits for excellent engineers
HMI effectiveness; Distributed I/O; Engineers' Choice Award finalists; System Integrator advice; Inside Machines
Women in engineering; Engineering Leaders Under 40; PID benefits and drawbacks; Ladder logic; Cloud computing
Programmable logic controllers (PLCs) represent the logic (decision) part of the control loop of sense, decide, and actuate. As we know, PLCs aren’t the only option for making decisions in a control loop, but they are likely why you’re here.
This digital report explains how plant engineers and subject matter experts (SME) need support for time series data and its many challenges.
This article collection contains several articles on how advancements in vision system designs, computing power, algorithms, optics, and communications are making machine vision more cost effective than ever before.
Control room technology innovation; Practical approaches to corrosion protection; Pipeline regulator revises quality programs
Cloud, mobility, and remote operations; SCADA and contextual mobility; Custom UPS empowering a secure pipeline
Infrastructure for natural gas expansion; Artificial lift methods; Disruptive technology and fugitive gas emissions
Automation Engineer; Wood Group
System Integrator; Cross Integrated Systems Group
Jose S. Vasquez, Jr.
Fire & Life Safety Engineer; Technip USA Inc.
This course focuses on climate analysis, appropriateness of cooling system selection, and combining cooling systems.
This course will help identify and reveal electrical hazards and identify the solutions to implementing and maintaining a safe work environment.
This course explains how maintaining power and communication systems through emergency power-generation systems is critical.
The Engineers' Choice Awards highlight some of the best new control, instrumentation and automation products as chosen by Control Engineering subscribers. Vote now (if qualified)!
The System Integrator Giants program lists the top 100 system integrators among companies listed in CFE Media's Global System Integrator Database.
Each year, a panel of Control Engineering and Plant Engineering editors and industry expert judges select the System Integrator of the Year Award winners in three categories.
This eGuide illustrates solutions, applications and benefits of machine vision systems.
Learn how to increase device reliability in harsh environments and decrease unplanned system downtime.
This eGuide contains a series of articles and videos that considers theoretical and practical; immediate needs and a look into the future.
Maximize ROI with integrated control system approach; Microcontrollers vs. PLCs; Power quality; Accelerate and rewire IIoT; Traits for excellent engineers
HMI effectiveness; Distributed I/O; Engineers' Choice Award finalists; System Integrator advice; Inside Machines
Women in engineering; Engineering Leaders Under 40; PID benefits and drawbacks; Ladder logic; Cloud computing
Programmable logic controllers (PLCs) represent the logic (decision) part of the control loop of sense, decide, and actuate. As we know, PLCs aren’t the only option for making decisions in a control loop, but they are likely why you’re here.
This digital report explains how plant engineers and subject matter experts (SME) need support for time series data and its many challenges.
This article collection contains several articles on how advancements in vision system designs, computing power, algorithms, optics, and communications are making machine vision more cost effective than ever before.
Control room technology innovation; Practical approaches to corrosion protection; Pipeline regulator revises quality programs
Cloud, mobility, and remote operations; SCADA and contextual mobility; Custom UPS empowering a secure pipeline
Infrastructure for natural gas expansion; Artificial lift methods; Disruptive technology and fugitive gas emissions
Automation Engineer; Wood Group
System Integrator; Cross Integrated Systems Group
Jose S. Vasquez, Jr.
Fire & Life Safety Engineer; Technip USA Inc.
This course focuses on climate analysis, appropriateness of cooling system selection, and combining cooling systems.
This course will help identify and reveal electrical hazards and identify the solutions to implementing and maintaining a safe work environment.
This course explains how maintaining power and communication systems through emergency power-generation systems is critical.