In the vibration and noise analysis of motors , we often see umbrella-shaped orders as shown in Figure 1. These orders are significantly different from conventional orders: 1) the starting point is not zero (0 RPM, 0 Hz); 2) they scatter in an umbrella shape on both sides of the center frequency, rather than the traditional one-sided scattering. Therefore, this article will discuss this special umbrella-shaped order.
1. What is off-zero order?
We know that the physical meaning of order is the number of events (such as vibration and noise) occurring per revolution of a rotating structure, while frequency represents the number of events occurring per second. The rotational frequency of a rotating structure represents the number of revolutions per second. Therefore, the frequency of an event corresponding to a given order is the order multiplied by the rotational frequency. Thus, the frequency of an event corresponding to order 1 is 1 times the rotational frequency, and the frequency of an event corresponding to order k is k times the rotational frequency. Assuming the rotational speed of the reference shaft is R rpm, the relationship between order and rotational speed is as follows:
When the rotational speed R=0, the rotational frequency is also 0Hz. As can be seen from the above formula, for any order, the starting frequency of the order is 0Hz. That is to say, regardless of the order, it starts at 0Hz (zero point) corresponding to 0 rotational speed.
However, in NVH analysis of motors, umbrella-shaped orders are frequently observed around 5000Hz, 10000Hz, and 15000Hz, as shown in Figure 1. These orders do not originate from zero, but rather from a non-zero frequency, scattering out in umbrella-shaped lines on both sides of this non-zero frequency, with the starting point deviating from zero. Therefore, these umbrella-shaped orders are called off-zero orders.
These off-zero orders, frequently observed in motor NVH analysis, have the following characteristics:
1) The starting point is not zero, but certain specific frequencies, because the signal is modulated;
2) These specific frequencies are carrier frequencies, and the umbrella-shaped orders on both sides are modulation wave frequencies and their harmonics;
3) The carrier frequency is the frequency corresponding to the original pulse, and the modulation wave is the frequency of the desired sine wave.
Therefore, the umbrella-shaped order appearing in the motor is actually the result of a modulation process using the fundamental frequency corresponding to the original pulse square wave as the carrier signal and the desired sine wave as the modulation wave. So, why do these orders deviating from zero appear in the motor?
2. Why do umbrella-shaped sequences occur?
In the process of converting DC to AC, pulse width modulation (PWM) technology is commonly used. It uses a sequence of rectangular pulses (PWM waves) with equal amplitude and width proportional to the function value of the modulating wave (such as a sine wave or square wave) to represent the modulating wave. Analog signals are replaced by digital signals, and DC is converted to AC by controlling the on/off state of the inverter's switching transistors. PWM technology can be used in both voltage-source and current-source inverters. Many readily available PWM strategies are available, such as sinusoidal PWM, balanced PWM, optimized PWM, triangular PWM, random PWM, equal-area PWM, hysteresis PWM, and space vector PWM. Current-controlled hysteresis PWM and space vector PWM are widely used in driving induction motors in electric vehicles. Voltage-source controlled equal-area PWM strategies are specifically used for driving battery-powered induction motors in electric vehicles. Since most electric vehicles currently manufactured are battery-powered, voltage-source controlled equal-area PWM strategies are primarily used. Previously, in the article "What is PWM", we also introduced the equal area pulse width modulation technology.
In equal-area pulse width modulation (PWM), the excitation signal supplied to the stator windings of the motor is a square wave pulse signal with a fixed fundamental frequency. By adjusting the pulse width of the square wave, the voltage amplitude is adjusted, thereby converting direct current (DC) into alternating current (AC). PWM uses a sine wave with the same frequency as the desired waveform as the modulating wave, and a square wave with a frequency much higher than the modulating wave as the carrier wave. Thus, the carrier signal is modulated by the modulating wave. When performing waterfall plot analysis on the signal, an umbrella-shaped order pattern appears in the colormap. The starting frequency of the umbrella-shaped order (orders deviating from zero) is the carrier frequency, and it scatters outwards on both sides of the carrier frequency as the rotational speed increases. These switching frequencies and their harmonics gradually move away from the carrier frequency as the rotational speed increases, thus forming the umbrella-shaped order lines.
These orders of deviation from zero are the switching frequencies of the motor controller's PWM (Pulse Width Modulation) used to control the motor. These switching frequencies, generated by the pulse width modulation signal, are used to convert DC voltage into AC voltage to drive the motor.
To obtain a sinusoidal AC voltage, pulse width modulation (PWM) requires switching the circuit on and off in a specific sequence to generate a square pulse wave (PWM wave) with the same amplitude as the sinusoidal wave, as shown in Figure 5. The frequency of this specific sequence of switching on and off is the so-called switching frequency, which is also the frequency of the sinusoidal wave, and the corresponding umbrella-shaped order frequency in the colormap. This frequency forms an order as the rotational speed increases.
Besides the switching frequency, PWM also has a fundamental switching frequency (base frequency), which is the carrier frequency. The PWM base frequency is typically 2500Hz, 5000Hz, 10000Hz, or higher. Therefore, these switching frequencies radiate outwards from the base frequency in an umbrella-like pattern. So how is this carrier frequency generated?
For PWM, the original frequency of the pulse remains unchanged; that is, the width between adjacent rising edges of the pulse (changing this width is called pulse frequency modulation) remains constant, as shown by the adjacent blue dashed lines in the diagram. PWM modulates the time of the middle green dashed line, which is the pulse width. The pulse width is controlled by adjusting the on-time of the switch to achieve a PWM wave (signal 2) with the same amplitude as the desired sine wave. The original square wave (signal 1) is modulated with pulse width modulation according to the desired sine wave period to obtain signal 2. Signal 1 and signal 2 have the same original frequency, only different pulse widths. This frequency is the fundamental frequency of the PWM, which is also the carrier frequency. Even though signal 1 is pulse-width modulated to obtain signal 2, because the time between adjacent rising edges of these two signals does not change, this frequency remains the same. This is why the carrier frequency remains constant during pulse width modulation.
Based on the carrier frequency, there is also a modulation frequency, which is the frequency of the desired sine wave. Therefore, the umbrella-shaped order in the motor is the result of modulation between the high-frequency carrier signal and the low-frequency modulation signal. We know that when a carrier signal is modulated by a modulation signal, sidebands are formed on both sides of the carrier signal, and these sidebands are symmetrically distributed on both sides of the carrier frequency. If both the carrier signal and the modulation signal in amplitude modulation are sine waves, only one pair of sidebands will be generated, while frequency modulation will generate an infinite number of pairs of sidebands. In pulse width modulation, the carrier signal is a square wave, and the modulation signal is an irregular sine wave whose frequency varies with the rotational speed. Therefore, multiple pairs of sidebands will inevitably be formed on both sides of the carrier signal. For more information on modulation, please refer to the article "Modulation Effect of Gears".
Because the carrier signal is modulated by the modulating wave, sidebands appear on both sides of the carrier signal. Therefore, the frequencies corresponding to these sidebands are the carrier signal frequency ± the modulating wave frequency and its harmonics. Thus, due to the presence of modulation, the umbrella-shaped order in the motor no longer starts from zero, but from the carrier frequency.
The original PWM wave has equal on and off times. To generate the desired sine wave (bottom curve) for the motor, the relative on and off times of the PWM wave need to be varied according to a specific pattern. We hope to obtain a smooth, ideal sine wave, but in reality, it is very difficult to achieve a smooth, ideal sine wave; in fact, there is a hysteresis band.
If the modulated sine wave is an ideal sine wave, like Asinωt, then describing this sine wave only requires one term (a single spectral line) from the Fourier series. Therefore, as the rotational speed increases, there should only be one pair of orders on either side of the carrier frequency, but in reality, multiple pairs of harmonics exist. This is because the modulated sine wave is not ideal and has hysteresis bands.
The sine wave generated by pulse width modulation is used to control the motor and change its speed. Variations in the off-time generate harmonics around the fundamental frequency, as shown in Figure 3. As the controlled motor speed increases, these harmonics will move further away from the fundamental frequency.
3. Methods to improve switching noise
The fundamental frequency of the pulsed square wave also acts on the magnetic field generated in the windings. Therefore, the vibration and noise behavior of the motor will also be affected by this fixed high-frequency signal (the fundamental frequency of the pulsed square wave). The motor rotates at different speeds, which leads to the classic order modulation phenomenon, common in the NVH (noise, vibration, and harshness) of traditional automotive engines. From the above analysis, we know this is the result of signal modulation; the carrier wave is the fundamental frequency of the square wave signal, and the modulating signal is a modulated wave that varies with the rotational speed. The carrier signal in the PWM modulation process, and these modulation orders, will also generate significant vibration and noise problems in the motor. How can these switching noise problems caused by PWM be improved? There are two different methods to change these switching frequencies:
Base frequency – This can be changed. For example, the base frequency can be increased from 2500Hz to 15000Hz, which can reduce the sound that is audible to the human ear, but will also affect the efficiency of the motor.
Switching strategy – Change the switching strategy, such as changing from a discrete method to a random method.
We know that structural radiated noise is directly related to vibration velocity. The higher the frequency, the lower the vibration velocity, resulting in lower radiated noise and thus less noise entering the ear. On the other hand, the fundamental frequency cannot be increased indefinitely; the physical limitations of the converter must also be considered.
A switching strategy can be modified to replace a discrete PWM mode, which produces discrete umbrella-shaped orders that make switching noise more prominent. If a randomized PWM switching strategy is used instead of the discrete one, the discrete order noise becomes broadband noise, as shown in Figure 11. The left side shows the discrete switching scheme, and the right side shows the randomized switching scheme. The change in switching strategy reduces the amplitude of the switching frequency, and the pure tone component is significantly reduced.
When the switching strategy is changed, note that the motor frequency remains unchanged, as shown in the lower part of the two figures in Figure 11.
