1: Bode Plots for Frequency Analysis
In the 1930s, Hendrik Wade Bode created an intuitive gain/phase approach
that had circuit stability as its sole goal. This became what we now know as
the Bode plot, an intuitive graphical representation of the gain, phase, and
feedback system of a circuit or amplifier at frequency.
Given its usefulness and importance, let’s take some time to apply a Bode
stability analysis technique to observe the magnitude of the feedback factor
of an open-loop circuit and amplifier, in decibels (dB) and phase response (in
degrees ). This blog will discuss these concepts and suggest how to avoid
designing an unstable circuit when your primary goal is frequency stability.
To practice this technique, you can download a printable version of a Bode
plot from the resources included in the online DigiKey Innovation Handbook.
Simple pole Bode plot.
With the single-pole circuit configuration, signals from the DC voltage input
(DC VIN) can go directly to the voltage output (VOUT), while at higher input
frequencies, the VOUT is equal to zero decibels. (dB). Creating the Bode plot
is simple. The units on the y-axis constitute the logarithmic frequency, and
the linear x-axis constitutes the gain in decibels or the phase in degrees.
There are a lot of formulas that can be applied when designing a Bode plot,
but let’s get to the point with the quick solution.
The simplicity of the Bode plot is that drawing the graphs only requires a
straight edge tool and knowing a few rules (Figure 1).
The two diagrams in Figure 1 represent the frequency versus gain and phase
of a pair of single-pole capacitors/resistors. The x-axis frequency ranges in
the top and bottom diagrams are from 1 hertz (Hz) to 10 megahertz (MHz).
The range of the y-axis of the diagram above is from 0 decibels (dB) to 100
dB, where the value of the 1 Hz signal is equal to 100 dB. This value is
consistent with a gain factor of 100,000 x VIN. The blue curve is the gain
response with a single pole at fP or at 100 Hz, where R is equal to 159
kilohms (kΩ) and C is equal to 10 nanofarads (nF).
As the frequency exceeds the pole frequency (fP), the blue curve falls at a
rate of -20 dB/decade or -6 dB/octave. This attenuation rate is the first
golden rule of the Bode plot to remember: Each pole in the circuit decays at
a rate of -20 dB/decade, starting with the frequency of the pole. Therefore, if
two poles attenuate the VOUT in the same frequency range, the atenuarían
rate is -40 dB/decade.
The bottom diagram in Figure 1 represents the phase of this single pole
circuit. At 1 Hz, the R/C network phase is 0 degrees (°). At a decade before
fP or, in this case, 10 Hz, the single pole phase begins to fall at a rate of -
45°/decade toward its target of -90°.
Several rules apply to the phase response of the pole. The second golden
rule of the Bode plot for the pole circuit is that the phase must be equal to -
45° in the fP. The third and fourth rules of the Bode plot describe the point of
attenuation and end of the phase.
2: Nyquist Plots for Frequency Analysis
It is a method of measuring impedance by applying an AC signal. The AC
signal used to measure a battery’s impedance typically has a fixed
frequency of 1 kHz. There is also a method to measure impedance using
multiple frequencies instead of a single frequency. It’s called electrochemical
impedance spectroscopy or EIS. This method does not require disassembling
the battery, so it provides a safe, non-destructive way to observe behavior
within a battery.
The internal resistance of the battery in general can be broadly divided into
the following three categories.
1. Resistance to electrolytes
2. Reaction resistance
3. Diffusion resistance
As shown in Fig. 2, the physical phenomena that domínate impedance differ
depending on the frequency band. For example, the impedance at high
frequencies (around 1 kHz) is mainly due to the migration of lithium ions in
the electrolyte. Lithium ion diffusion occurs within the electrode at low
frequencies (less than 1 Hz) and lithium ion transfer reactions at
intermediate frequencies (1 to several hundred Hz).
In other words, a detailed analysis of the Nyquist diagram allows us to
evaluate various phenomena in different parts of the battery.
Typical relationship between the Nyquist diagram and the internal resistance
of a battery
Fig. 2: Typical relationship between the Nyquist diagram and the internal
resistance of a battery
A typical analysis method is known as equivalent circuit analysis, which uses
an equivalent circuit model as shown in Fig. 3. In the circuit model, each
phenomenon within the battery is modeled by different equivalent circuit
elements. The values of the elements calculated by the analysis can be
considered to indicate the characteristics of the physical phenomena
represented by the elements.
3: Nichols Plots for Frequency Analysis.
There are two tuning rules proposed by engineers Ziegler and Nichols in
1942 to determine the parameters of the PID controller based on the
transient response of the system.
Therefore, this type of tuning is especially useful, mainly when THE
MATHEMATICAL MODEL OF THE PLANT IS NOT KNOWN. Obviously it can also
be applied if the plant model is known, however, we can agree that there
are more interesting strategies when the model is known.
The two Ziegler-Nichols methods aim to obtain a maximum overshoot of
25% when faced with a step type entry.
We will cover the Ziegler Nichols rules in detail below.
It is believed that at that time Ziegler, who was from the sales department,
needed a tuning procedure for the PIDs to increase sales of the controller,
which is how he began working with Nichols in the research department,
performing hundreds of tests on systems. . with the most varied dynamics.
Ziegler and Nichols – Method 1 (Open Loop)
This Ziegler Nichols method is carried out with the open loop system, where
the controller is placed in manual mode to be able to generate a step type
variation in the output of the PID controller itself.
