1 Introduction
PWM drive systems have advantages such as economy and energy saving, and are widely used in permanent magnet synchronous motor systems. However, PWM drive systems also have some disadvantages. One of them is that the dv/dt value will be very large at the moment when the inverter's switching devices are turned on and off, which will have a great impact on the motor. When the inverter and motor require long cables to transmit voltage signals, there may be a mismatch between the cable characteristic parameters and the motor characteristic parameters, resulting in overvoltage at the motor end, high-frequency damped oscillation, and even burnout of the motor [1-2]. Therefore, the research on the analysis and suppression of overvoltage at the end of high-speed permanent magnet synchronous motors driven by long cables is of great significance.
This paper first analyzes the voltage reflection phenomenon in PWM drive systems, and theoretically introduces the influence of pulse rise time, cable characteristics, etc. on motor terminal voltage [3-5]. Then, it discusses two common measures to suppress motor terminal voltage: RC filter and LRC filter. Relevant models were built using Matlab/Simulink software, and their effectiveness was verified through simulation.
2. Voltage reflection phenomenon
When using long cables, the PWM pulses on the transmission line are similar to traveling waves on the same line. The PWM pulse consists of an incident wave and a reflected wave, which are transmitted from the inverter to the motor and from the motor to the inverter, respectively. It's like a mirror reflecting the incident wave V+ to produce a reflected wave V-, where V- equals V+ multiplied by the corresponding voltage reflection coefficient. The expression for the load-side reflection coefficient N2 is...
(1)
(2)
Where ZC is the characteristic impedance of the cable; ZL is the load impedance; L0 is the inductance per unit length of the cable; and C0 is the capacitance per unit length of the cable.
The initial voltage reflection coefficient N1 is...
(3)
Where ZS is the resistance at the starting end, and under normal circumstances ZS≈0, then N1≈-1. Since ZL>>ZC, it can be seen from equation (1) that N2≈1, which makes the voltage at the motor end about twice the original voltage. And from the above analysis, it can be seen that this voltage reflection phenomenon is related to the rise time of the pulse and the length of the cable [6-7].
3. Relationship between motor terminal overvoltage, rise time, and cable characteristics
3.1 Theoretical Analysis
When the time tt required for the output pulse to be transmitted from the inverter to the permanent magnet synchronous motor is less than the rise time tr of the PWM voltage output by the inverter, the amplitude of the motor terminal voltage is [value missing].
(4)
Where N2 is the voltage reflection coefficient and UDC is the DC bus voltage.
When tt≥tr, its amplitude is
(5)
From equation (5), we know that when tt≥tr, the rise time is independent of the reflected voltage. When tt≥tr, the reflected voltage is reflected at the end. For a typical low-impedance network, the starting end N1≈-1, which will make the amplitude of the reflected wave transmitted to the motor negative. As a result, after the PWM pulse is transmitted 3 times on the cable, the value of the motor terminal voltage will decrease.
3.2 Simulation Analysis of the Influence of PWM Rise Time
The simulation parameters are set as follows: PWM pulse voltage amplitude is 500V, pulse frequency is 500Hz. Inverter impedance is Zs=0.5Ω/km, resistance is R=0.02Ω/km, inductance is L0=1×e-3H/km, capacitance is C0=13×e-9F/km, and cable length is l=1km. The simulation block diagram is shown in Figure 1.
Observe the changes in the terminal voltage waveform as the rise time changes. The simulated waveform is shown in Figure 2.
(a) Voltage waveform with a rise time of 0.01 ms
(b) Voltage waveform with a rise time of 0.05 ms
(c) Voltage waveform with a rise time of 0.1 ms
Figure 2. Voltage waveforms with different rise times
As shown in the simulation diagrams of Figure 2(a), (b), and (c), when the cable length is constant, the longer the pulse rise time, the smaller the overvoltage amplitude at the motor end and the shorter the oscillation period.
3.3 Simulation Analysis of the Influence of Cable Length
With a fixed rise time of 0.05 ms, and cable lengths of 1 km, 2 km, and 4 km, the changes in the terminal voltage waveform were observed. The simulation results are shown in Figure 3. From the simulation graphs in Figures 3(a), (b), and (c), it can be seen that, with a constant pulse rise time, as the cable length increases, the amplitude of the motor terminal voltage increases, and the oscillation period becomes longer.
(a) Voltage waveform over a 1km cable
(b) Voltage waveform over a 2km cable
(c) Voltage waveform with a rise time of 4km
Figure 3. Voltage waveforms with different rise times
3.4 Simulation Analysis of the Influence of Cable Capacitance Characteristics
With a fixed rise time and cable length, and keeping the cable inductance L and resistance R constant, the capacitance C=10nF/km and C=30nF/km are compared. The simulation results are shown in Figure 4.
(a) Voltage waveform with a capacitance of 10nF/km
(b) Voltage waveform with a capacitance of 30nF/km
Figure 4 shows different voltage waveforms of the capacitor.
As shown in the simulation diagrams of Figure 4(a) and (b), when the capacitance per unit length of the cable increases, the oscillation frequency of the motor terminal voltage decreases, the oscillation duration remains unchanged, and the voltage peak value also decreases.
3.5 Simulation Analysis of the Influence of Cable Inductance Characteristics
With the rise time and cable length fixed, and the capacitance C and resistance R of the cable kept constant, the inductance L=1mH/km and L=5mH/km are compared. The simulation diagram is shown in Figure 5.
(a) Voltage waveform with an inductance of 1mH/km
(b) Voltage waveform with an inductance of 5mH/km
Figure 5 shows the voltage waveforms of different inductors.
As shown in the simulation diagrams in Figures 5(a) and (b), when the inductance per unit length of the cable increases, the oscillation frequency of the motor terminal voltage decreases, the oscillation duration becomes longer, and the voltage peak value increases.
4. Measures to suppress overvoltage at motor terminals
To reduce the phenomenon of overvoltage at the motor terminals, this paper discusses measures for suppressing overvoltage using RC filters and LRC filters.
4.1RC Filter
The topology of a first-order RC filter is shown in Figure 7. The cables in the RC filter topology can be represented as a collection of capacitors and inductors, as shown in Figure 7.
The voltage uf across the RC filter is
(6)
The voltage and current at both ends of the first-order RC motor terminal filter exist.
(7)
To determine the values of the filter's resistance and capacitance, two principles must be met. First, the first reflected voltage pulse generated at the motor terminal should be zero; this condition is satisfied when Rf=Zc. Second, regarding the design of the filter's capacitor parameters, to ensure that the overvoltage generated at the motor terminal does not exceed a certain limit, the voltage reflected from the second reflected wave at the motor terminal should be less than 20% of the input voltage.
To satisfy the second principle above, we have:
(12)
Theoretical analysis of incident and reflected voltages shows that the time it takes for the incident wave to travel from one end of the transmission line to the other is τ. Therefore, the voltage pulse propagation time on the transmission line from the start of charging the filter capacitor to the appearance of the second voltage pulse reflection at the motor end is 2τ. Thus, we can obtain:
(13)
The obtained value of Cf is:
(14)
In the formula, l is the cable length and C0 is the cable capacitance.
4.2 LRC Filter
For ease of analysis, the LRC filter is equivalent to a single-phase circuit, as shown in Figure 8.
Its transfer function is:
As can be seen from the above formula, the values of the corresponding filter inductor and filter capacitor can be calculated from the cutoff frequency.
Figure 9 shows the simulated waveforms of the terminal voltages without a filter and with RC and LRC filters respectively.
From the simulation analysis of Figures 9(a), (b), and (c), the following conclusions can be drawn: (1) After filtering by the RC filter at the motor end, a better voltage waveform can be obtained at the motor end, but the rise time of the voltage pulse is still very short; while the LRC filter at the inverter output end can reduce the ratio of this voltage pulse, thereby reducing the amplitude of the common-mode current. (2) Since the switching frequency of the inverter is constrained by the requirements of the load and the inverter switching tube, when the inverter is working at the highest switching frequency, the loss of the RLC filter at the inverter end is smaller than that of the RC filter at the motor end. (3) Although the RC filtering technology at the motor end is simple to design, it is very difficult to install in special cases such as deep well oil pumps.
(a) No filter applied
(b) Add an RC filter
(c) Add an RLC filter
Figure 9 Voltage waveforms of different filters
