Piezoelectric high-frequency multi-dimensional thrust vector test system quasi-static calibration device and method

Abstract

This invention provides a quasi-static calibration device and method for a piezoelectric high-frequency multidimensional thrust vector testing system. The calibration device includes a measurement unit, a leaf spring unit, an excitation unit, and a control unit. The measurement unit is connected to the excitation unit, the excitation unit is connected to the leaf spring unit, and the control unit is connected to both the measurement unit and the excitation unit. The measurement unit monitors force and displacement signals in real time. The leaf spring unit, as the core component for force transmission, provides a stable static force reference. The excitation unit applies high-frequency micro-vibrations to eliminate the influence of static friction. The control unit controls the excitation unit to output excitation loads with different pulse widths, amplitudes, and waveforms. This invention solves the technical problem of long on-site static calibration cycles leading to charge leakage and affecting the accurate acquisition of low-frequency force-to-electric conversion coefficients, effectively improving low-frequency testing accuracy and providing reliable data support for the thrust performance evaluation of next-generation attitude and orbit control engines.

Description

Technical Field

[0001] This invention relates to the field of calibration technology for piezoelectric thrust testing systems, and specifically to a quasi-static calibration device and method for a piezoelectric high-frequency multidimensional thrust vector test system. Background Technology

[0002] As the direct power source for attitude adjustment and maneuvering in missile weapon systems, the thrust performance of attitude control engines is a key parameter characterizing engine performance, directly affecting the maneuverability and target accuracy of missile weapon systems. Therefore, high-precision testing of the engine's thrust vector characteristics is necessary during ground trials. Due to the inherent characteristics of this type of engine's thrust output—rapid switching between high and low frequencies and wide-range output for attitude and orbit control—currently, multi-dimensional thrust vector testing often employs piezoelectric multi-dimensional force measurement components as the core component of the testing system, which offer good dynamic characteristics, fast response speed, high sensitivity, and wide frequency response. However, the inherent deficiency of piezoelectric measurement systems in low-frequency characteristics affects the reliability of low-frequency test data.

[0003] In traditional engine thrust vector testing, to obtain the force-to-electric conversion coefficient, most thrust vector engineering tests in practice rely on GJB 2365B-2021 "Test Method for Static Test Parameters of Solid Rocket Engines" for the calibration of the thrust test system. This type of calibration falls under the category of static calibration. However, the "charge leakage" characteristic of piezoelectric sensors limits the acquisition of high-precision force-to-electric conversion coefficients for this type of test system under traditional static calibration methods. Since engine thrust vector testing requires in-situ calibration in six directions, the entire calibration cycle is lengthy. For the dynamic signals measured by the piezoelectric test system, the shorter the measurement loading time, the less charge leakage. To ensure the accuracy of the measurement system calibration, the load loading time during calibration should be much shorter than the system's time constant. Empirically, when the ratio of the time constant to the loading time is satisfied, the charge leakage rate can be kept below 0.05%. However, during in-situ static calibration, the standard load loading time is long, and the time constant of the charge amplifier used in practice cannot reach the aforementioned level. This causes the charge between the two plates to leak rapidly through impedance, resulting in drift. Therefore, the results of static calibration cannot reflect the true variation pattern. The on-site quasi-static calibration loading time is short (generally in the millisecond range), which can meet the above-mentioned ratio between the time constant and the loading time, effectively control the charge leakage rate of the test system, and achieve accurate calibration and value transfer in the low-frequency band.

[0004] In the prior art, patent CN202310866939.5 discloses a quasi-zero stiffness vibration isolation device based on the negative stiffness generated by a rotatable leaf spring, which improves vibration isolation efficiency through negative stiffness components, but does not involve the calibration application of piezoelectric high-frequency multidimensional thrust vector testing systems. Patent CN118797926A discloses a tire motion verification method based on leaf spring suspension, mainly used for motion verification of automotive leaf spring suspension, but does not involve quasi-static force calibration technology. Although leaf spring structures have been widely used as force transmission mechanisms in high-precision test benches for the precise thrust testing of traditional solid rocket engines, they have not yet been combined with exciters for the on-site quasi-static calibration scenario of attitude and orbit control engine piezoelectric thrust vector testing systems, nor have matching on-site quasi-static calibration devices been designed. Therefore, there is an urgent need to develop an on-site quasi-static calibration device based on leaf spring structures and suitable for attitude and orbit control engine piezoelectric multidimensional thrust vector testing systems to compensate for the shortcomings of traditional static calibration methods. Summary of the Invention

[0005] Based on this, the present invention provides a quasi-static calibration device and method for a piezoelectric high-frequency multidimensional thrust vector test system, which solves the technical problem that the long on-site static calibration cycle of traditional methods leads to charge leakage and affects the accurate acquisition of the force-to-electric conversion coefficient in the low-frequency range. It realizes on-site quasi-static calibration of the piezoelectric multidimensional thrust vector test system for attitude and orbit control engines in the 0-10Hz frequency range, effectively improves the test accuracy in the low-frequency range, and provides reliable data support for the thrust performance evaluation of the next generation of attitude and orbit control engines.

[0006] To address the aforementioned technical problems, one objective of this invention is to provide a quasi-static calibration device for a piezoelectric high-frequency multidimensional thrust vector testing system, comprising: The measuring unit is used to monitor force and displacement signals in real time; A leaf spring unit, which is used for force transmission; The excitation unit is used to apply high-frequency micro-vibration to eliminate static friction; The control unit is used to control the excitation unit to output excitation loads with different pulse widths, amplitudes, and waveforms. The measuring unit is connected to the excitation unit, the excitation unit is connected to the leaf spring unit, and the control unit is connected to both the measuring unit and the excitation unit.

[0007] Furthermore, the measuring unit includes a piezoelectric combined force measuring component, a dynamic force sensor, a charge amplifier, and a data acquisition unit; the dynamic force sensor is connected to the piezoelectric combined force measuring component; the piezoelectric combined force measuring component is connected to the charge amplifier via a cable; and the charge amplifier is connected to the data acquisition unit.

[0008] Furthermore, the leaf spring unit includes a leaf spring suspension bracket and a leaf spring, with one end of the leaf spring connected to the leaf spring suspension bracket; the excitation unit includes an exciter, an exciter mounting base, and a flexible rod, with the exciter mounted on the exciter mounting base, and the other end of the leaf spring connected to the exciter mounting base to ensure that the exciter axis is perpendicular to the leaf spring surface. The flexible rod is mounted on the exciter, and the pitch angle of the flexible rod is adjusted by adjusting knobs at both ends of the exciter to make the excitation direction consistent with the force direction of the leaf spring. The dynamic force sensor is threadedly connected to the flexible rod at the front end of the exciter.

[0009] Furthermore, the control unit includes a power amplifier, a dynamic controller, and a host computer. The power amplifier is connected to the excitation unit via a cable, the dynamic controller is connected to the power amplifier via a cable, and the host computer is connected to the dynamic controller via a cable.

[0010] Furthermore, the piezoelectric combined force measuring component has a natural frequency of 2.9kHz for the X and Y axes, a natural frequency of 3kHz for the Z axis, and a maximum measurement range of 30kN across the three axes; the dynamic force sensor has a range of ±400N, an accuracy of 0.03%FS, and a sensitivity of 10pC / N; the data acquisition unit uses 24-bit resolution, a sampling frequency of 1000Hz, and 8 channels for synchronous acquisition.

[0011] Furthermore, the leaf spring is made of 50CrVA spring steel with an elastic modulus of 206GPa, a yield strength of 1200MPa, a leaf spring width of 100mm, a thickness of 5mm, an effective length of 400mm, a stiffness coefficient of 10059N / m, a natural frequency of 24.76Hz, and a linearity error of ≤0.02%FS.

[0012] Furthermore, the exciter is a low-frequency electromagnetic exciter with a peak excitation force of 500N, a frequency range of 0.1Hz to 50Hz, a weight of 15kg, a first-order resonant frequency >6500Hz, a force constant of 10N / A, an inductance of 0.1mH, and a back electromotive force constant of 10 V / (m / s).

[0013] Furthermore, the power amplifier has a frequency range of DC-20kHz, a rated power of 500VA, a signal-to-noise ratio of >90dB, and a dynamic controller capable of PID closed-loop control of sinusoidal, random, and transient impulse signals. The input channel has a dynamic range of >110dB, an impedance of 100kΩ, a synchronous sampling rate of 256kHz, and an output channel frequency range of 50kHz with a resolution of 24bit.

[0014] Based on the same inventive concept, this invention also provides a quasi-static calibration method for a piezoelectric high-frequency multidimensional thrust vector test system, comprising the following steps: S1: Set the sampling frequency of the data acquisition unit and perform pre-calibration data acquisition; S2: In the host computer dynamic control software interface, set half-sine excitation loads with pulse widths of 1ms, 2ms, 3ms, 4ms, 5ms, 6ms, 7ms, 8ms, 9ms, 10ms, 11ms, and 12ms respectively. Within the range of 10N to 100N, uniformly select 10 calibration points, with each loading interval of 10N. Repeat the test 6 times at each calibration point, and collect and record the average standard thrust value corresponding to each test. Response voltage of the calibrated system ; S3: Based on the data curve obtained from the S2 calibration test, curve fitting is performed according to the mathematical model of the voltage response formula (1) of the piezoelectric force measurement system under half-sinusoidal excitation load to identify the system voltage sensitivity. and the discharge time constant of the test system ; ; In the formula: -t The output voltage of the piezoelectric force measurement system at any given time, in mV, characterizes the dynamic voltage response of the test system under pulse excitation; - Voltage sensitivity of a piezoelectric force measurement system, measured in mV / N, characterizes the ability of the test system to convert force signals into voltage signals; - Pulse load amplitude, in N; - The discharge time constant of the test system, in seconds, characterizes the rate of charge leakage and directly affects the low-frequency characteristics of the test system; - The pulse width of the load excitation, in milliseconds, characterizes the duration of the load force signal. S4: Introduce a dimensionless parameter into formula (1) in S3: This parameter, as a key factor determining output distortion, allows us to obtain the voltage sensitivity of piezoelectric testing systems with different pulse widths based on this fitting model. With discharge time constant As shown in formula (2): ; In the formula: -No. i Voltage sensitivity of the force measurement system under secondary excitation, in mV / N; -No. i The amplitude of the pulse load during the second excitation, in N; -No.i The discharge time constant of the secondary excitation test system, in seconds; S5: Based on the different pulse widths and amplitudes identified in step S4, the voltage sensitivity of the piezoelectric testing system under excitation. With discharge time constant According to equation (3), the low-frequency compensation function of the piezoelectric force measurement system is obtained: ; In the formula: -No. i Voltage sensitivity of the force measurement system under secondary excitation, in mV / N; - indicates the imaginary unit; - The angular frequency of the input signal, in rad / s.

[0015] The above-described one or more technical solutions of the present invention have at least one or more of the following technical effects: (1) High calibration accuracy: The static friction effect is eliminated by the vibrator, the static calibration accuracy is ≤0.05%FS, and the dynamic error is ≤0.01%FS; (2) Calibration range: Applicable to force sensor calibration from 1N to 100N, with a frequency coverage of 0 to 10Hz quasi-static frequency band; (3) Good system stability: The natural frequency of the leaf spring system is 24.76Hz, which avoids the calibration frequency band of 0-10Hz and avoids resonance interference; (4) Simple structure and easy to operate: It adopts a modular design, which makes installation and maintenance convenient. It can realize automated calibration and data processing, and improve work efficiency. Attached Figure Description

[0016] Figure 1 Schematic diagram of the quasi-static calibration device for a piezoelectric high-frequency multidimensional thrust vector test system; Figure 2 Schematic diagram of the quasi-static calibration device for a piezoelectric high-frequency multidimensional thrust vector test system; Figure 3 Connection structure diagram of leaf spring unit and excitation unit; Figure 4 Quasi-static calibration process for piezoelectric high-frequency multidimensional thrust vector test system; Among them, 1. leaf spring suspension bracket, 2. leaf spring, 3. vibrator, 4. flexible rod, and 5. dynamic force sensor. Detailed Implementation

[0017] This invention provides a quasi-static calibration device and method for a piezoelectric high-frequency multidimensional thrust vector test system, solving the technical problem that the long on-site static calibration cycle of traditional methods leads to charge leakage, affecting the accurate acquisition of the force-to-electric conversion coefficient in the low-frequency range. It enables on-site quasi-static calibration of the piezoelectric multidimensional thrust vector test system for attitude and orbit control engines in the 0-10Hz frequency range, effectively improving the testing accuracy in the low-frequency range and providing reliable data support for the thrust performance evaluation of the next-generation attitude and orbit control engine.

[0018] The following describes the embodiments of the present invention and... Figure 1-4 The technical solutions in the embodiments of the present invention are clearly and completely described herein. Obviously, the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments obtained. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art are within the scope of protection of the present invention.

[0019] Example 1 like Figure 1-3 As shown, a quasi-static calibration device for a piezoelectric high-frequency multidimensional thrust vector testing system includes: a measuring unit, a leaf spring unit, an excitation unit, and a control unit. The measuring unit is connected to the excitation unit, the excitation unit is connected to the leaf spring unit, and the control unit is connected to both the measuring unit and the excitation unit. The measuring unit is used to monitor force and displacement signals in real time. The leaf spring unit, as the core component for force transmission, provides a stable static force reference. The excitation unit applies high-frequency micro-vibrations to eliminate the influence of static friction. The control unit controls the excitation unit to output excitation loads with different pulse widths, amplitudes, and waveforms.

[0020] The measurement unit includes a piezoelectric combined force measuring component, a dynamic force sensor, a charge amplifier, and a data acquisition unit. The dynamic force sensor is connected to the piezoelectric combined force measuring component; the piezoelectric combined force measuring component is connected to the charge amplifier via a cable; and the charge amplifier is connected to the data acquisition unit. The piezoelectric combined force measuring component has a natural frequency of 2.9 kHz on the X and Y axes, a natural frequency of 3 kHz on the Z axis, and a maximum measurement range of 30 kN across the three axes. The dynamic force sensor has a range of ±400 N, an accuracy of 0.03%FS, and a sensitivity of 10 pC / N. The data acquisition unit uses 24-bit resolution, a sampling frequency of 1000 Hz, and synchronous acquisition across 8 channels.

[0021] The leaf spring unit includes a leaf spring suspension bracket and a leaf spring, with one end of the leaf spring connected to the suspension bracket. The excitation unit includes a vibrator, a vibrator mounting base, and a flexible rod. The vibrator is mounted on the vibrator mounting base, and the other end of the leaf spring is connected to the vibrator mounting base, ensuring that the vibrator axis is perpendicular to the leaf spring surface. The flexible rod is mounted on the vibrator. The pitch angle of the flexible rod is adjusted by adjusting knobs at both ends of the vibrator to make the excitation direction consistent with the force direction of the leaf spring. The coaxiality error of the flexible rod is ≤0.1mm. The dynamic force sensor is threadedly connected to the flexible rod at the front end of the vibrator. The leaf spring is made of 50CrVA spring steel with an elastic modulus of 206GPa, a yield strength of 1200MPa, a width of 100mm, a thickness of 5mm, an effective length of 400mm, a stiffness coefficient of 10059N / m, a natural frequency of 24.76Hz, and a linearity error ≤0.02%FS. The exciter is a low-frequency electromagnetic exciter with a peak excitation force of 500N, a frequency range of 0.1Hz to 50Hz, a weight of 15kg, a first-order resonant frequency >6500Hz, a force constant of 10N / A, an inductance of 0.1mH, and a back electromotive force constant of 10 V / (m / s).

[0022] The control unit includes a power amplifier, a dynamic controller, and a host computer. The power amplifier is connected to the excitation unit via a cable, the dynamic controller is connected to the power amplifier via a cable, and the host computer is connected to the dynamic controller via a cable. The power amplifier has a frequency range of DC-20kHz, a rated power of 500VA, and a signal-to-noise ratio >90dB. The dynamic controller can realize PID closed-loop control of sinusoidal, random, and transient impact signals. The input channel has a dynamic range >110dB, an impedance of 100kΩ, a synchronous sampling rate of 256kHz, and an output channel frequency range of 50kHz with a resolution of 24bit.

[0023] Example 2 like Figure 4 As shown, a quasi-static calibration method for a piezoelectric high-frequency multidimensional thrust vector test system includes the following steps: S1: Set the sampling frequency of the data acquisition unit and perform pre-calibration data acquisition; S2: In the host computer dynamic control software interface, set half-sine excitation loads with pulse widths of 1ms, 2ms, 3ms, 4ms, 5ms, 6ms, 7ms, 8ms, 9ms, 10ms, 11ms, and 12ms respectively. Within the range of 10N to 100N, uniformly select 10 calibration points, with each loading interval of 10N. Repeat the test 6 times at each calibration point, and collect and record the average standard thrust value corresponding to each test. Response voltage of the calibrated system ; S3: Based on the data curve obtained from the S2 calibration test, curve fitting is performed according to the mathematical model of the voltage response formula (1) of the piezoelectric force measurement system under half-sinusoidal excitation load to identify the system voltage sensitivity. and the discharge time constant of the test system ; ; In the formula: - t The output voltage of the piezoelectric force measurement system at any given time, in mV, characterizes the dynamic voltage response of the test system under pulse excitation; - Voltage sensitivity of a piezoelectric force measurement system, measured in mV / N, characterizes the ability of the test system to convert force signals into voltage signals; - Pulse load amplitude, in N; - The discharge time constant of the test system, in seconds, characterizes the rate of charge leakage and directly affects the low-frequency characteristics of the test system; - The pulse width of the load excitation, in milliseconds, characterizes the duration of the load force signal. S4: Introduce a dimensionless parameter into formula (1) in S3: This parameter, as a key factor determining output distortion, allows us to obtain the voltage sensitivity of piezoelectric testing systems with different pulse widths based on this fitting model. With discharge time constant As shown in formula (2): ; In the formula: -No. i Voltage sensitivity of the force measurement system under secondary excitation, in mV / N; -No. i The amplitude of the pulse load during the second excitation, in N; -No. i The discharge time constant of the secondary excitation test system, in seconds; S5: Based on the different pulse widths and amplitudes identified in step S4, the voltage sensitivity of the piezoelectric testing system under excitation. With discharge time constant According to equation (3), the low-frequency compensation function of the piezoelectric force measurement system is obtained: ; In the formula: -No. i Voltage sensitivity of the force measurement system under secondary excitation, in mV / N; - indicates the imaginary unit; - The angular frequency of the input signal, in rad / s.

Claims

1. A quasi-static calibration device for a piezoelectric high-frequency multidimensional thrust vector test system, characterized in that, include: The measuring unit is used to monitor force and displacement signals in real time; A leaf spring unit, which is used for force transmission; The excitation unit is used to apply high-frequency micro-vibration to eliminate static friction; The control unit is used to control the excitation unit to output excitation loads with different pulse widths, amplitudes, and waveforms. The measuring unit is connected to the excitation unit, the excitation unit is connected to the leaf spring unit, and the control unit is connected to both the measuring unit and the excitation unit.

2. The quasi-static calibration device for the piezoelectric high-frequency multidimensional thrust vector test system according to claim 1, characterized in that: The measurement unit includes a piezoelectric combined force measuring component, a dynamic force sensor, a charge amplifier, and a data acquisition unit; the dynamic force sensor is connected to the piezoelectric combined force measuring component; the piezoelectric combined force measuring component is connected to the charge amplifier via a cable; and the charge amplifier is connected to the data acquisition unit.

3. The quasi-static calibration device for the piezoelectric high-frequency multidimensional thrust vector test system according to claim 2, characterized in that: The leaf spring unit includes a leaf spring suspension bracket and a leaf spring, with one end of the leaf spring connected to the leaf spring suspension bracket. The excitation unit includes an exciter, an exciter mounting base, and a flexible rod. The exciter is mounted on the exciter mounting base, and the other end of the leaf spring is connected to the exciter mounting base to ensure that the exciter axis is perpendicular to the leaf spring surface. The flexible rod is mounted on the exciter, and the pitch angle of the flexible rod is adjusted by adjusting knobs at both ends of the exciter to make the excitation direction consistent with the force direction of the leaf spring. The dynamic force sensor is threadedly connected to the flexible rod at the front end of the exciter.

4. The quasi-static calibration device for the piezoelectric high-frequency multidimensional thrust vector test system according to claim 1, characterized in that: The control unit includes a power amplifier, a dynamic controller, and a host computer. The power amplifier is connected to the excitation unit via a cable, the dynamic controller is connected to the power amplifier via a cable, and the host computer is connected to the dynamic controller via a cable.

5. The quasi-static calibration device for the piezoelectric high-frequency multidimensional thrust vector test system according to claim 2, characterized in that: The piezoelectric combined force measuring component has a natural frequency of 2.9kHz on the X and Y axes, a natural frequency of 3kHz on the Z axis, and a maximum measurement range of 30kN across the three axes; the dynamic force sensor has a range of ±400N, an accuracy of 0.03%FS, and a sensitivity of 10pC / N; the data acquisition unit uses 24-bit resolution, a sampling frequency of 1000Hz, and 8 channels for synchronous acquisition.

6. The quasi-static calibration device for the piezoelectric high-frequency multidimensional thrust vector test system according to claim 3, characterized in that: The leaf spring is made of 50CrVA spring steel with an elastic modulus of 206GPa, a yield strength of 1200MPa, a leaf spring width of 100mm, a thickness of 5mm, an effective length of 400mm, a stiffness coefficient of 10059N / m, a natural frequency of 24.76Hz, and a linearity error of ≤0.02%FS.

7. The quasi-static calibration device for the piezoelectric high-frequency multidimensional thrust vector test system according to claim 3, characterized in that: The exciter is a low-frequency electromagnetic exciter with a peak excitation force of 500N, a frequency range of 0.1Hz to 50Hz, a weight of 15kg, a first-order resonant frequency >6500Hz, a force constant of 10N / A, an inductance of 0.1mH, and a back electromotive force constant of 10 V / (m / s).

8. The quasi-static calibration device for the piezoelectric high-frequency multidimensional thrust vector test system according to claim 4, characterized in that: The power amplifier has a frequency range of DC-20kHz, a rated power of 500VA, and a signal-to-noise ratio of >90dB. The dynamic controller can realize PID closed-loop control of sinusoidal, random, and transient impulse signals. The input channel has a dynamic range of >110dB, an impedance of 100kΩ, a synchronous sampling rate of 256kHz, and an output channel frequency range of 50kHz with a resolution of 24bit.

9. A quasi-static calibration method for a piezoelectric high-frequency multidimensional thrust vector test system using the calibration device described in any one of claims 1-8, characterized in that, Includes the following steps: S1: Set the sampling frequency of the data acquisition unit and perform pre-calibration data acquisition; S2: In the host computer dynamic control software interface, set half-sine excitation loads with pulse widths of 1ms, 2ms, 3ms, 4ms, 5ms, 6ms, 7ms, 8ms, 9ms, 10ms, 11ms, and 12ms respectively. Within the range of 10N to 100N, uniformly select 10 calibration points, with each loading interval of 10N. Repeat the test 6 times at each calibration point, and collect and record the average standard thrust value corresponding to each test. Response voltage of the calibrated system ; S3: Based on the data curve obtained from the S2 calibration test, curve fitting is performed according to the mathematical model of the voltage response formula (1) of the piezoelectric force measurement system under half-sinusoidal excitation load to identify the system voltage sensitivity. and the discharge time constant of the test system ; ； In the formula: -t The output voltage of the piezoelectric force measurement system at any given time, in mV, characterizes the dynamic voltage response of the test system under pulse excitation; - Voltage sensitivity of a piezoelectric force measurement system, measured in mV / N, characterizes the ability of the test system to convert force signals into voltage signals; - Pulse load amplitude, in N; - The discharge time constant of the test system, in seconds, characterizes the rate of charge leakage and directly affects the low-frequency characteristics of the test system; - The pulse width of the load excitation, in milliseconds (ms), characterizes the duration of the load force signal. S4: Introduce a dimensionless parameter into formula (1) in S3: This parameter, as a key factor determining output distortion, allows us to obtain the voltage sensitivity of piezoelectric testing systems with different pulse widths based on this fitting model. With discharge time constant As shown in formula (2): ； In the formula: -No. i Voltage sensitivity of the force measurement system under secondary excitation, in mV / N; -No. i The amplitude of the pulse load during the next excitation, in N; -No. i The discharge time constant of the secondary excitation test system, in seconds; S5: Based on the different pulse widths and amplitudes identified in step S4, the voltage sensitivity of the piezoelectric testing system under excitation. With discharge time constant According to equation (3), the low-frequency compensation function of the piezoelectric force measurement system is obtained: ； In the formula: -No. i Voltage sensitivity of the force measurement system under secondary excitation, in mV / N; - indicates the imaginary unit; - The angular frequency of the input signal, in rad / s.

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