Bode Plot, Stability, Draw Bode Plot, Gain Margin and Phase Margin

What is a Bode Plot?

In order to assess the stability of a control system, a Bode plot is a graph that is frequently used in control system engineering. Two graphs, the Bode magnitude plot (which expresses the magnitude in decibels) and the Bode phase plot (which expresses the phase shift in degrees), are used to map the frequency response of a system.

Hendrik Wade Bode created the first Bode plots in the 1930s while he was employed by Bell Labs in the United States. Contrary to the Nyquist stability criterion, Bode plots cannot handle transfer functions with right half plane singularities, despite the fact that they provide a relatively straightforward method for calculating system stability.

Understanding Bode plots requires knowledge of both gain margins and phase margins. Here is a definition of these terms.

Gain Margin

The stability of the system increases with the Gain Margin (GM) value. The amount of gain that can be raised or lowered without causing the system to become unstable is referred to as the gain margin. Usually, the magnitude is given in decibels (dB).

Usually, the Bode plot (as depicted in the diagram above) allows us to read the gain margin directly. So To do this, find the vertical distance at the frequency where the Bode phase plot is equal to 180° between the magnitude curve on the Bode magnitude plot and the x-axis. The frequency of the phase crossover is at this location.

So It is crucial to understand that the Gain and the Gain Margin are two distinct concepts. In actuality, the gain margin (measured in decibels, or dB) is the opposite of the gain. When we examine the Gain margin formula, this will make sense.

Gain Margin Formula

The formula for Gain Margin (GM) can be expressed as:

G is the gain where. The magnitude (in dB) is taken from the magnitude plot’s vertical axis at the phase crossover frequency.

In our example shown in the graph above, the Gain (G) is 20. Hence using our formula for gain margin, the gain margin is equal to 0 – 20 dB = -20 dB (unstable).

Phase Margin

The stability of the system will increase with the Phase Margin (PM) value. The amount of phase that can be increased or decreased without the system becoming unstable is referred to as the phase margin. Typically, it is described as a phase in degrees.

Usually, the Bode plot (as depicted in the diagram above) allows us to read the phase margin directly. To do this, find the vertical distance at the frequency where the Bode magnitude plot equals 0 dB between the phase curve (on the Bode phase plot) and the x-axis. The frequency at which the gain crossover occurs is this.

It is crucial to understand that the phase lag and the phase margin are two different concepts. When we examine the phase margin formula, this will make sense.

Phase Margin Formula

The formula for Phase Margin (PM) can be expressed as:

Where

is the phase lag (a number less than 0). This is the phase as read from the vertical axis of the phase plot at the gain crossover frequency.

In our example shown in the graph above, the phase lag is -189°. Hence using our formula for phase margin, the phase margin is equal to -189° – (-180°) = -9° (unstable).

As another example, if an amplifier’s open-loop gain crosses 0 dB at a frequency where the phase lag is -120°, then the phase lag -120°. Hence the phase margin of this feedback system is -120° – (-180°) = 60° (stable).

Bode Plot Stability

Below is a list of criterion relevant to drawing Bode plots (and calculating their stability):

1. Gain Margin: The stability of the system will increase as the gain margin does. It refers to the gain amount, which can be changed without causing the system to become unstable. dB is typically used to express it.
2. Phase Margin: The stability of the system will increase as the phase margin increases. It speaks of the phase that can be changed without causing the system to become unstable. It is typically written as phase.
3. Gain Crossover Frequency: In a bode plot, this term describes the frequency at which the magnitude curve crosses the zero dB axis.
4. Phase Crossover Frequency: In this plot, this term describes the frequency at which the phase curve crosses the negative times axis.
5. Corner Frequency: The break frequency or corner frequency is the frequency at which the two asymptotes intersect or cut.
6. Resonant Frequency: The value of frequency at which the modulus of G (jω) has a peak value is known as the resonant frequency.
7. Factors: So Every loop transfer function {i.e G(s) × H(s)} product of various factors like constant term K, Integral factors (jω), first-order factors ( 1 + jωT)(± n) where n is an integer, second-order or quadratic factors.
8. Slope: Each factor has a corresponding slope, and each factor’s slope is expressed in dB per decade.
9. Angle: Each factor has a corresponding angle, which is expressed in degrees for each factor.

Now there are some results that one should remember in order to plot the Bode curve. These results are written below:

• Constant term K: The slope of this factor is 0 dB per decade. This constant term has no corner frequency corresponding to it. This constant term also has zero associated phase angle.
• Integral factor 1/(jω)ⁿ: This factor has a slope of -20 × n (where n is an integer)dB per decade. So There is no corner frequency corresponding to this integral factor. The phase angle associated with this integral factor is -90 × n. Here n is also an integer.
• First-order factor 1/ (1+jωT): This factor has a slope of -20 dB per decade. The corner frequency corresponding to this factor is 1/T radian per second. So The phase angle associated with this first factor is -tan^{– 1}(ωT).
• First order factor (1+jωT): This factor has a slope of 20 dB per decade. The corner frequency corresponding to this factor is 1/T radian per second. So The phase angle associated with this first factor is tan^{– 1}(ωT).
• Second order or quadratic factor : [{1/(1+(2ζ/ω)} × (jω) + {(1/ω²)} × (jω)²)]: This factor has a slope of -40 dB per decade. So The corner frequency corresponding to this factor is ωⁿ radian per second. The phase angle associated with this first factor is

How to Draw Bode Plot

We can create a Bode plot for any type of control system by keeping the aforementioned considerations in mind. So Let us talk about how to draw a Bode plot now:

1. Substitute the s = jω in the open loop transfer function G(s) × H(s).
2. Find the corresponding corner frequencies and tabulate them.
3. Now, we must select a frequency range for each semi-log graph such that the plot start frequency is lower than the lowest corner frequency. So Mark the angular frequencies along the x-axis, the slopes along the left side of the y-axis by placing a zero slope in the middle, and the phase angle along the right side by placing a minus 180 degree in the center.
4. Calculate the gain factor and the type of order of the system.
5. Now calculate the slope corresponding to each factor.

For drawing the Bode magnitude plot:

• On the paper with the semi-log graph, mark the corner frequency.
• In the given order, tally these variables from top to bottom.
• Constant term K.
• Integral factor

• First order factor

• First order factor (1+jωT).
• Second order or quadratic factor:

• Now, using the given factor’s corresponding slope, draw the line. Add the slope of the following factor to each corner frequency to change the slope. The magnitude plot will be provided.
• Calculate the gain margin.

For drawing the Bode phase plot:

1. Add up all of the phase transitions to calculate the phase function.
2. So Plot a curve by substituting different values into the aforementioned function to determine the phase at various points. A phase curve will appear.
3. Calculate the phase margin.

Bode Stability Criterion

A technique used in control system engineering to assess a system’s stability is known as the Bode Stability Criterion. So This criterion focuses on the phase margin and gain margin of the system’s open-loop response as it analyzes the frequency response of a system.

The Bode Stability Criterion are listed below:

1. For a Stable System: The phase margin must be greater than the gain margin, or both margins must be positive.
2. So For Marginal Stable System: The gain margin should be equal to the phase margin, or the margins should be zero.
3. For Unstable System: The phase margin should be less than the gain margin if any of them are negative.

Advantages of a Bode Plot

The advantages of the Bode plot include the following:

1. So The asymptotic approximation, which offers a quick way to plot the logarithmic magnitude curve, serves as the foundation for this method.
2. The transfer function’s multiplication of different magnitudes can be thought of as an addition. In contrast, using a logarithmic scale, division can be thought of as a subtraction.
3. So Only with the aid of this plot, without the aid of any calculations, can we directly comment on the stability of the system.
4. When it comes to gain margin and phase margin, Bode plots offer comparatively stable results.
5. They cover a wide frequency range (low to high-frequency).