In the previous two installments, we discussed "transfer functions." In this third installment, we will discuss specific methods for calculating transfer functions, how to visualize them, and provide practical examples from electrical and mechanical systems.
The purpose of determining the transfer function is to determine the transfer characteristics of the transfer system. Generally, the transfer function is defined as the ratio of the Laplace transforms of the input and output of the transfer system, but here we consider it as a function of the measurable frequency.
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Figure 1: Example of a transmission system
Now, consider the transfer system shown in Figure 1. When a sinusoidal function x(t) = a sin (2πft) with a constant frequency f is input, since it is a linear system, its steady-state response, the output, is also a sinusoidal function with the same frequency f, and is given by y(t) = b sin (2πft + θ) (Figure 2).
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Figure 2. Steady-state response of a linear system when a sine wave is input.
At this time, the gain and phase of the transfer function H(f) are:
Gain
(amplitude ratio)
Phase ∠H(f)=θ(phase difference)
This means that the transfer function H(f) is determined by two quantities (gay) that depend on the value of the frequency.
It has a phase difference (and a specific frequency response), which is specifically called the frequency response of the system.
Furthermore, the transfer function H(f) has two pieces of information for each frequency: gain and phase, so it is complex.
It can be expressed as a number.
In Figure 3, if we let H real be the real part of the complex function H(f) and H imag be the imaginary part, then:
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Figure 3 H(f) on the complex plane
H(f) = Hreal + j Himag
This means that Figure 3 shows the transfer function H(f) at a certain frequency on the complex plane as a vector.
This is what it represents.
Furthermore, the gain and phase are expressed using the real and imaginary parts;
Gain
phase
It can be written as follows:
Since the transfer function H(f) is a complex function with frequency as a variable, there are various ways to represent it graphically.
Here, we will explain using the transfer system of a resonant low-pass filter as an example.
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Figure 4: Example of a Bode plot graph
Figure 4 shows the gain characteristics and phase characteristics plotted against each frequency f, and the Bode plot and
It is called a logarithmic graph, and the horizontal axis (frequency axis) is usually displayed on a logarithmic scale. The vertical axis of the gain characteristic is...
The value obtained from equation (3) in dB is 20log|H(f)|, and the vertical axis of the phase characteristic is ±180 degrees (or ±200 degrees).
This is how it will be displayed. The features of this graph are: ① The resonant frequency and its magnitude are easy to understand.
② Although there are two graphs, the frequency axis is clear, and ③ the relationship between gain and phase is easy to understand (resonance).
For example, the phase will always lag by 180 degrees at each point.
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Figure 5: Example of a Nyquist plot graph
Similar to Figure 3, Figure 5 shows the transfer function as a vector on the complex plane while varying the frequency, with its vertices plotted. This is called a Nyquist plot (vector locus), where the horizontal axis represents the real part and the vertical axis represents the imaginary part. The characteristics of this graph are: ① it makes it easy to see the phase rotation near the resonance point, and ② it uses only one graph, but the frequency axis is not visible.
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Figure 6: Example of a Coquad diagram graph.
Figure 6 plots the real and imaginary parts for each frequency f, and is called a coquad diagram. The characteristics of this graph include: ① it is easy to find resonance points in the imaginary part, and ② gain and phase are not directly represented.
In addition, there is the Nichols diagram (gain-phase diagram), which is used in control systems and other applications, but we will omit it here.
How are the transfer functions H(f) determined in specific measuring instruments such as FFT analyzers and servo analyzers?
Recent frequency analyzers employ two methods: the FRA method and the FFT method.
The Frequency Response Analysis (FRA) method involves sweeping a single frequency while using an auto-ranging function to repeatedly measure (Fourier integrate) one point at a time, thereby determining the specified frequency.
We determine the frequency response in the frequency band (Figure 7).
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Figure 7 Measurement principle of the FRA method
The features of this method are:
- Since only one frequency point is measured per measurement, the auto-ranging function of the measuring instrument is used.
This enables measurements with a very high dynamic range. - Logarithmic resolution sweeps (log sine sweeps) are possible.
- It is possible to measure across a wide bandwidth and any frequency range.
- The measurement time is long.
is.
In contrast, the FFT (Fast Fourier Transform) method is linked to a pre-defined analysis frequency band.
The signal source is then applied to the object being measured, and the entire bandwidth is measured simultaneously using FFT technology (Figure 8).
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Figure 8 Measurement principle of the FFT method
The signal sources used in the FFT method include random noise, pseudo-random noise, and swept-sight noise.
Chirp sine (chirp sine), impulse, and all of these signals are the same across the analysis frequency.
It contains the frequency components of the same power.
The features of this method are:
- Since the frequency band of interest is determined simultaneously, the system's characteristics can be measured at high speed.
- Analysis results can only be obtained with linear resolution.
- The coherence function, which allows us to check the reliability of the transfer function, can also be calculated at the same time.
is.
Finally, I will introduce some specific measurement examples of transmission systems.
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Figure 9. Second-order lag element system (electrical) -
Figure 10 Second-order lag element system (machine)
Figure 9 shows an example of an LCR resonant circuit in an electrical system, and Figure 10 shows an example of a one-degree-of-freedom resonant system in a mechanical system; both are,
It can be expressed as a transfer function of a second-order lag system.
The standard form of the transfer function of a second-order lag system is obtained using the Laplace transform:
ω n (=2πf): Natural angular frequency, ζ: Damping ratio, K: Gain constant
It will be.
In Figure 9 (Electrical System), if L is an inductor, R is a resistor, and C is a capacitor, then:
Here,
Far away;
Thus, equation (8) is equal to equation (5).
Similarly, in Figure 10 (mechanical system), if m is the mass, u is the viscous damping coefficient, and k is the spring constant, then:
Here;
If we set this, equation (9) becomes equation (5);
This is equivalent to:
Let's actually use a servo analyzer to measure the circuit transfer characteristics (equation (8)) shown in Figure 9.
To make equation (8) a function of frequency, let s = jω (ω = 2πf);
The shape of the graph for G(jω) in equation (11) changes depending on the value of the damping ratio ζ.
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Figure 11 Bode plot with ζ varied and superimposed.
Figure 11 is a Bode plot overlaid with measurements taken by varying ζ in the transfer system shown in Figure 9.
In this example, L = approximately 800 (μH) and C = approximately 0.1 (μF), so the natural angular frequency ω n is 18.64 kHz
It can be confirmed that this is the case. Furthermore, this circuit is a low-pass filter with resonant characteristics.
This can be seen from the frequency response.
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Figure 12 Example of a measured Bode plot -
Figure 13: Example of a measured Nyquist diagram
Measurement of Bode plots (Figure 12) and Nyquist plots (Figure 13) obtained using the servo analyzer DS-0342.
This is a regular occurrence.
(Excerpt from the email newsletter issued on January 22, 2015)
