Bode Plot - Interactive Pole/Zero Placement

Simulation

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Zoom: 210.00 rad/s

Load scenarios (up to 3):

The scope is clickable & draggable. More here.

How to run the simulation + useful info

Click "Add pole" or "Add zero".
Move the pole/zero around the plane. Observe the change in the magnitude and phase Bode plots.
Info: Only the first (green) transfer function is configurable. Blue and red transfer functions are cleared when moving poles/zeroes in the plane.
Scenario: 1 pole/zero: can be on real-axis only
Scenario: 2 poles/zeros: can be on real-axis or complex conjugate
Scenario: 3 poles/zeros: the first two can be on real-axis or complex conjugate, the third must be on real-axis
Up to three plots can be shown at the same time.
The transfer function of the pre-loaded high-pass and low-pass filters is scaled to achieve 0 dB attenuation at 0 / infinity, respectively. Once the zeroes/poles are moved/added/deleted, the original calculation will not hold true anymore.
The pole/zero S-place plot can be zoomed in and out using a slider.

MatLab(©) Code

% MatLab(©) Script to generate Bode plots of custom zero/pole location.

clc; clear all; format compact; close all

poles = [-71 + 71j, -71 + -71j];
zeroes = [];
z = poly(zeroes);
p = poly(poles);
Hs = tf(z,p);
x = bodeoptions;
x.XLim = [0.01 100];
figure(1);
bode(Hs,x);

Bode Plot Theory

Transfer Function

See the System Modeling with Transfer Functions article for more details.

Magnitude Bode Plot & Phase Bode Plot

There is so much great material online, please follow these links for excellent lectures and slides:

CT Frequency Response and Bode Plots from MIT
Chapter 6: Frequency response of dynamical systems from the University of Leuven
Bode Plots @ Wiki

Examples

Low-pass Filter A low-pass filter decreases the magnitude of high frequency components. Three examples are provided: single-pole, complex-pole, and three-pole. Higher order results in more aggressive filtering (-20 dB per decade per pole) and phase lag.

The corner frequency of all three filters is 100 rad/s.

The Bode plots of the example three low-pass filters:
High-pass Filter
A high-pass filter decreases the magnitude of low frequency components. Three examples are provided: single-pole, complex-pole, and three-pole. Higher order results in more aggressive filtering (-20 dB per decade per pole) and phase lag.

The corner frequency of all three filters is 100 rad/s.

The Bode plots of the example three high-pass filters:
Notch Filter From Reference 2:

Notch filter could in theory be realized with two zeros placed at +/-(j omega_0). However, such a filter would not have unity gain at zero frequency, and the notch will not be sharp.

To obtain a good notch filter, put two poles close to the two zeros on the semicircle as possible. Since both pole/zero pairs are equal distances to the origin, the gain at zero frequency is exactly one. Same for omega = +/- inf.

The Bode plots of the example notch filter:

The pole-zero map of the example notch filter:
Lead Controller
- Pushes the poles of the closed-loop system to the left.
- Increases response speed and bandwidth.
- Increases the phase margin: the phase of the lead compensator is positive for every frequency, hence the phase will only increase.
The Bode plots of the example lead compensator:

The pole/zero plot of the example lead compensator:
Lag Controller
- The primary function of a lag compensator is to provide attenuation in the high-frequency range to give a system sufficient phase margin.
- The phase-lag characteristic is of no consequence in lag compensation.
- A lag compensator decreases the bandwidth/speed of response:
  - good to reduce the impact of high-frequency noise
  - bad if you want the system to react fast -> use lead compensator
The Bode plots of the example lag compensator:

The pole/zero plot of the example lag compensator:
Lead-Lag Controller
The text below is copied from a public PDF provided by the University of Leuven. It is very well written.

Lead compensation achieves the desired result through the merits of its phase lead contribution.

Lag compensation accomplishes the result through the merits of its attenuation property at high frequencies.

Lag compensation reduces the system gain at higher frequencies without reducing the system gain at lower frequencies. Since the system bandwidth is reduced, the system has a slower speed to response. Thanks to this, the total system gain can be increased, as well as the low-frequency gain and the steady state accuracy can be improved. Also, any high-frequency noise involved in the system is attenuated.

Lag compensation will introduce a pole-zero combination near the origin that will generate a long tail with small amplitude in the transient response.

If both fast responses and good static accuracy are desired, a lag-lead compensator may be employed. By use of the lag-lead compensator, the low-frequency gain can be increased (which means an improvement in steady state accuracy), while at the same time the system bandwidth and stability margins can be increased.

The Bode plots of lead-lag compensator:

The pole/zero plot of the example lead-lag compensator:
PI Controller
See the PI Controller : THEORY + DEMO article for more details.

The Bode plots of the PI controller:

The pole/zero plot of the example PI controller:
PI Controller with Low-pass Filter
A filter is typically applied to the measured signal - voltage, current, speed to remove undesired noise.

The Bode plots of the example PI controller with LPF:

The pole/zero plot of the example PI controller with LPF:

References

See the Textbooks and Journals for Power Electronics and Motor Controls article.
Lecture 9, Poles, Zeros & Filters from Imperial College, London
Chapter 12: Lead and Lag Compensators from the University of Leuven
CT Frequency Response and Bode Plots from MIT