In the previous lesson, we extended the transformation parameters to the entire complex number spectrum and introduced the Laplace transform from the Fourier transform. The Laplace transform has developed as a convenient method for solving differential equations, but this time we will discuss the "transfer function," which is the most important analytical technique in electrical circuits and control systems. Furthermore, we will explain how to obtain the frequency response function that represents the frequency characteristics of the system from this transfer function.
As discussed in "Fourier Transform and Convolution" (Examples of Sound Measurement - Part 3 "Various Measurements Using a Sound Level Meter") and "Transfer Function" (Frequency Analysis from the Basics (17)) in this series "Frequency Analysis from the Basics," when a time signal a(t) is input to a linear system of impulse response h(t), the output b(t) of the system can be expressed as a convolution integral.
.................................（1）
(Note: h(t)=0 (t<0) is used to apply the Laplace transform.)
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Figure 1: Input/Output Relationship and Laplace Transform of the System
As explained in "Transfer Function" (Frequency Analysis from the Basics (17)), if we apply the Laplace transform to both sides of equation (1), we obtain the following relationship due to the properties of the Laplace transform.
B(s)=H(s)A(s) .................................（2）
(Note: The initial value is set to 0)
Thus, the relationship between the input and output of the system is a convolution integral on the time axis (equation (1)), but in the Laplace transform (s-domain), the output is given by the product of the input and the system, which is a simple relationship. From equation (2)
.................................（3）
This H(s) is called the transfer function of the system (more precisely, a linear time-invariant system) in Figure 1. In other words, the transfer function is the Laplace transform of the system's impulse response, and is defined as the ratio of the Laplace transforms of the input and output time signals.
As a concrete example, in the one-degree-of-freedom damped vibration system shown in Figure 2 (left), we consider the input signal as an external force f(t) and the output signal as the response displacement x(t) (Figure 2, right), and use the Laplace transform to find its transfer function.
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Figure 2. Model of a 1-degree-of-freedom damped forced vibration system
As explained in "Fundamentals of Vibration Measurement - 2" (Frequency Analysis from the Basics (23)), the resultant force of the inertial force, viscous resistance, and restoring force present in this vibration system balances the external force, and therefore the equation of motion is:
This is the result.
If we find the Laplace transform of both sides of equation (4),
From this, the transfer function H(s) of the system is,
.................................（6）
This can be calculated as follows.
The standard form of a second-order lag element system (a system where the denominator is a quadratic expression in s, as in equation (6)) is
.................................（7）
Here, ωn: natural angular frequency, ζ: damping ratio (damping coefficient), K: gain constant.
Therefore,
When the transfer function of the vibration system in equation (6) is applied to the standard form, the parameters are:
Natural angular frequency
Damping ratio
Gain constant
.................................（8）
This can be calculated as follows.
The transfer function of a typical system can often be expressed as a rational function with real coefficients.
.................................（9）
(P(s) and Q(s) are polynomials in s with real coefficients, where the degree of P(s) is less than the degree of Q(s))
This can be written as follows. In equation (9), the solutions to the denominator root Q(s)=0 are called poles, and the solutions to the numerator root P(s)=0 are called zeros. For example,
.................................（10）
So, the zeros are at Z1, and the poles are p1 and p2. In transfer functions, the poles are particularly important.
As an example, let's find the impulse response of the second-order lag element system in Figure 2 when it oscillates (0 < ζ < 1).
As explained last time, the Laplace transform of an impulse is 1, so we set F(s)=1.
.................................（11）
Here, if we let s1 and s2 be the roots (i.e., poles) of the denominator = 0 in equation (11),
.................................（12）
𝜔𝑑: Attenuation Natural Angular Frequency
Therefore, we expand the right-hand side of equation (11) using partial fractions,
.................................（13）
In equation (13), we take the inverse Laplace transform of X(s)
Euler's formula
twist
.................................（14）
From this result (equation (14)), the impulse response of the transfer function H(s) is a time waveform that oscillates with decay when ζ < 1, as shown in Figure 3. From equations (12) and (14), it can be seen that the real part (negative value) of the pole determines the amount of decay, and the imaginary part of the pole determines the frequency of oscillation (damped natural angular frequency). Also, the two poles (equation (12)) of the transfer function H(s) (equation (7)) are complex conjugates of complex numbers, and as shown in Figure 4, they are located on a circle of radius ωn (natural angular frequency) and are in the left half (real part < 0) of the s-plane.
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Figure 3 Impulse response of a second-order lag element system
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Figure 4 shows the pole positions of H(s) on the s-plane.
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Figure 5 shows the polar position of the transfer function on the s-plane and the system's response.
X is the pole position, and all other poles on the real axis are complex conjugates (pairs of two).
Based on these factors, the time response of the poles and the system on the s-plane can be summarized as follows: (Figure 5)
- When the pole lies in the left half-plane of the s-plane (real part < 0), the response is attenuated and the system is stable.
- When the pole lies in the right half of the s-plane (real part > 0), the response diverges and the system is unstable.
- When the poles are complex numbers, the system oscillates at a specific frequency (damped natural angular frequency).
- As the pole approaches the imaginary axis (ζ -> 0), the damping decreases.
- When the poles are on the imaginary axis, the vibrations are sustained (they vibrate at the natural angular frequency without damping), and the frequency increases as the poles move away from the origin.
Next, we will explain the relationship between the transfer function and the frequency response function.
In a system with transfer function H(s) (Figure 1)
Let the input signal be a(t) = sinωt and the output signal be b(t).
.................................（15）
Assuming that all poles of H(s) are unipolar and can be expanded using partial fractions as follows
.................................（16）
As can be seen from equations (13) and (14), the first n terms on the right-hand side above represent a transient phenomenon.
In a steady state, only the last two terms remain, so if we let the time signal of the steady-state term be b s (t),
Using the method for finding residues, we find C1 and C2 and rearrange the expression,
.................................（18）
From the result of equation (19), the output signal b(t) with the transient response removed has an amplitude equal to that of the input sinusoidal signal a(t).
It is found that the amplitude is multiplied and the phase shifts by ∠H(jω). In other words, the frequency response (amplitude ratio and phase difference) when a steady sine wave is applied to a system with transfer function H(s) is equal to the absolute value and phase angle of H(jω) obtained by setting s=jω. When considered on the S-plane, the value of H(s) on the imaginary axis (S=jω) is the frequency characteristic of the system itself. This H(jω) becomes the frequency response function (FRF) in an FFT analyzer.
Figure 6 Relationship between the response of a stationary sinusoidal wave and the frequency response function.
By setting s = jω in the transfer function H(s) of equation (7), we can find the amplitude magnification explained in "Fundamentals of Vibration Measurement Part 2" (Frequency Analysis from the Basics (23)).
.................................（20）
Setting the gain constant K=1, the gain of the amplitude multiplier is
.................................（21）
The phase angle Φ of the amplitude multiplier is (with the lag being a negative angle).
.................................（22）
When equations (21) and (22) are converted into fluffs with the frequency of the input forcing external force as a variable, we obtain Figure 7. We can see that the amplitude is maximized near the system's natural angular frequency ωn, i.e., a resonance phenomenon occurs. Furthermore, we can see that the phase angle is rotated (lags) by 180 degrees (π) around ωn.
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Figure 7 Amplitude and phase in forced oscillation of a 1-degree-of-freedom damped system
Next, we will discuss the relationship between the poles and zeros of a system and the frequency response of that system.
Let's consider the transfer function in equation (10) again as an example. (We'll call it equation (23))
.................................（23）
z1:零点
p 1,p 2:pole
The frequency response of a system with the transfer function of equation (23) can be obtained by substituting s=jω into the equation.
.................................（24）
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Figure 8 shows the length and angle of a vector when the starting point is a pole or zero in the s-plane and the ending point P is a point moving along the imaginary axis.
Consider a point P moving along the imaginary axis (jω). If we set the lengths and angles of the vectors from the poles and zeros to P as shown in Figure 8, then the gain and phase of H(jω) are:
.................................（25）
This can be written as follows: In other words, by examining how these values change when point P on the imaginary axis is moved (the frequency is swept), we can predict the frequency characteristics of the system.
As a concrete example, let's consider the second-order lag system in equation (7).
.................................（27）
When ζ < 1
.................................（28）
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Figure 9: Example in a second-order lag element system
Substitute s=jω
.................................（29）
In Figure 9, the gain and phase are
.................................（30）
From Figure 9 and equations (30) and (31), we can see the following:
- The gain is maximum around the attenuation angular frequency (ωd).
- The phase is 0 degrees at the origin (DC) and decreases monotonically thereafter.
- The natural angular frequency is (ωn), and the phase is always -90 degrees (-π/2).
- At infinite frequency, the phase is -180 degrees (-π).
For reference, the Bode plot of a second-order lag system measured with a servo analyzer is shown in the figure below.
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Figure 10 Bode plot of a second-order lag element system (example)
In conclusion, here's a summary.
- The transfer function is defined as the ratio of the Laplace transforms of the system's input and output time signals, and is the Laplace transform of the system's impulse response.
- The transfer function of a typical system is often expressed as a rational function with real coefficients, and the roots of its denominator are called poles, while the roots of its numerator are called zeros.
- By analyzing the poles and zeros of a transfer function, we can understand the behavior and frequency characteristics of the system.
- The position of the poles of the transfer function on the s-plane allows us to evaluate the stability and damping characteristics of the system.
- By substituting s = jω into the transfer function of the system, we can determine the frequency characteristics of the system, or its frequency response function.
- By moving a point P on the imaginary axis (sweeping the frequency) and examining the relationship between the poles and zeros, we can estimate the frequency characteristics of the system.
【keyword】
Fourier transform, Laplace transform, differential equation, transfer function, frequency response function, convolution integral, s-domain, impulse response, inertial force, viscous resistance, restoring force, equation of motion, second-order lag system, natural angular frequency, damping ratio, gain constant, rational function, pole, zero, damped natural angular frequency, Euler's formula, s-plane, transient phenomena, transient response, amplitude magnification, resonance phenomena, servo analyzer, Bode plot
【reference】
Hiroshi Harashima and Yoichi Hori, "Laplace Transforms and Z-Transforms," Suuri Kogakusha (2004).
"Automatic Control" by Fumio Matsumura, Asakura Shoten (1984)
(Excerpt from the email newsletter issued on September 21, 2017)
