1 A Novel Third Harmonic Injection Method for closed Loop control of PMSM Motors Bharath Kumar Suda James David .. Eaton Corporation Abstract - In closed loop control using Field Oriented II. Third Harmonic Injection METHODOLOGY. control (FOC), three phase voltage outputs of FOC. controller are not in a manner that the resultant stator A. Field Oriented control voltage vector and rotor magnetic axis are 90 electrical For simplicity, we consider 3-phase, 2 pole star- apart. Calculation of the Third Harmonic Value based on rotor angle measurement can lead to phase difference connected PMSM motor with isolated neutral point. between fundamental and Third Harmonic voltages, and Consider the dq reference frame as shown in the Figure 1. can lead to degraded performance. This paper presents a Now, in any given position of the rotor, we can resolve Novel methodology to dynamically calculate Third Harmonic the magnetic fields from A, B and C phases into dq axes value using the three phase voltage outputs from the FOC direction if we know the rotor position.
2 Controller. Simulation and experimental results are also presented for a three phase Permanent Magnet Synchronous Motor (PMSM) in closed loop current control with FOC and with FOC plus Third Harmonic Injection to confirm the methodology. I. INTRODUCTION. Field Oriented control (FOC)   and Third Harmonic Injection - are well known methods for controlling Permanent Magnet Synchronous Motors (PMSM) and Induction Motors. In the case of PMSM. motors, if field weakening is not desired, FOC provides optimal performance by orthogonally aligning rotor magnetic field and torque producing stator magnetic field. For 3-phase motors, Third Harmonic Injection (THI). along with FOC enables an increase of inverter DC bus utilization by thereby allowing better motor speed-torque performance. One of the simplest methods of inducing a Third Harmonic is by calculating instantaneous average of the minimum and maximum of Figure 1.
3 Dq Axes Convention 3-phase voltage outputs from FOC control , subtracting this value from the phase voltages, and multiplying the Using the above convention and IA+IB+IC = 0, FOC. results by . This technique gives a Third equations can be written as Harmonic waveform in a triangular profile. 3 3. = + + 3 ..( 1 ). In this paper, we discuss on analytical basis for Third 2 2. Harmonic Injection justifying the magnitude of the Third 3 3. Harmonic for best DC bus utilization. Additionally, we = 3 ..( 2 ). 2 2. will present a different strategy for obtaining Where, instantaneous values of sinusoidal Third Harmonic using instantaneous 3-phase voltage outputs from FOC control . IA = Phase A Current, Calculating the sinusoidal Third Harmonic value from IB = Phase B Current, rotor angle measurement directly in an open-loop fashion IC = Phase C Current, and adding it to the three-phase voltage outputs of FOC Iq = Quadrature Axis Current, leads to degraded motor performance as the phase angle of fundamental voltage will not match with the calculated Id = Direct Axis Current, Third Harmonic .
4 Determining instantaneous magnitude = Angle between Phase A magnetic axis and q Axis. and phase of fundamental voltage waveforms is These are standard FOC equations of Clark inevitable for correct implementation of the strategy. transformation with ABC to transformation, and then Park transformation from to dq transformation, as found in literature. Eq. (1) can be represented in matrix form as follows 3 From the above analysis, it is clear that the amplitude 0. 2 . = ..( 3 ) of the Third Harmonic should be 1/6th of the amplitude of 3 3 fundamental for best modulation factor and thereby for 2. the best DC bus utilization. Substituting x = 1/6 and =. In order to calculate the phase voltages from the FOC 60 in Eq. (6) gives , which is the peak of resultant controller output quadrature axis and direct axis voltages, waveform with Third Harmonic . This gives us a we use the following equation modulation factor of 1 , , , giving 2 more DC bus utilization.
5 0 . 3 Now, for a given instantaneous Vq and Vd output = 1 1 ..( 4 ). combination of FOC control , we need to find the 3 3. instantaneous Third Harmonic value to be induced. For and, that we need to find the phase and the magnitude of the fundamental phase waveform. For a balanced 3-Phase = ( + ) ..( 5 ). system, phase voltages can be expressed as Where, VA = Phase A Voltage, ( ). VB = Phase B Voltage, = ( 120) ..( 9 ). VC = Phase C Voltage, ( 240). Vq = Quadrature Axis Voltage, Where, Vd = Direct Axis Voltage. V = Instantaneous Magnitude of Fundamental, = Instantaneous Phase of the Fundamental. B. Third Harmonic Injection - Sinusoidal If we know instantaneous V and , we can easily calculate the instantaneous Third Harmonic value . For a well-controlled system, under steady operating sin(3 ). Using Eq. (4), (5) and (9), and trigonometric 6. conditions, phase voltages extracted from Eq.
6 (4) & (5) identities, we can prove that instantaneous magnitude of are sinusoidal. fundamental waveform is Now, for injecting a Sinusoidal Third Harmonic signal, ( 2 + 2 ). we have to consider a key point, which is the optimum = ..( 10 ). amplitude of Third Harmonic wave. The best DC bus utilization can be obtained using injected amplitude With the knowledge of instantaneous magnitude, we which brings down the resultant waveform's peak can find instantaneous Sin( ) from Eq. (9) as, amplitude the most . To find this amplitude let us . ( ) = ..( 11 ). consider a simple fundamental sine wave along with a . Third Harmonic of unknown amplitude, x as shown below. From Eq. (10) and (11), instantaneous Third Harmonic ( ) = ( ) + (3 ) ..( 6 ) value can be calculated as . Finding the extremum of the above equation using = (3 ) = (3 ( ) 4 3 ( )) ..( 12 ). 6 6. ( ) = 0 gives, Using the above instantaneous Sinusoidal Third 9 1.
7 Harmonic value, the net phase voltage with THI can be = 1 ..( 7 ) expressed as, 12 . = + Substituting the above in Eq. (6) gives = + (3 +1) 3 +1 3/2. ( ) = ( ) = (3 + 1) (12 ) 4 ..( 8 ) = + ..( 13 ). 12 . Finding the extremum of the above function using . ( ) = 0, gives two solutions for x, -1/3 and 1/6. C. Triangular Third Harmonic Injection . At x = 1/6, For triangular Third Harmonic Injection , 2 instantaneous value of Third Harmonic is obtained as ( ) = , follows 2. [ ( , , )+ ( , , )]. which shows this is a minimum. The other solution at x = = ..( 14 ). 2. -1/3 is infeasible. Using x = 1/6 and Eq. (8), we get =. 60 deg. At = 60 deg, Using the above instantaneous triangular Third 2 Harmonic value, phase voltages with THI are calculated ( ) = 2, as, 2. which shows this is the maximum of the function F( ). = = 1. SinTHI AB. SinTHI BC. = ..( 15 ) SinTHI CA. Line Voltages(pu).
8 TriangTHI AB. TriangTHI BC. TriangTHI CA. 0. III. SIMULATION RESULTS AND ANALYSIS. To understand the difference between Sinusoidal THI and Triangular THI, a simulation model has been made. A snapshot of the model is shown in Fig. 2. The model -1. consists of algorithms for both Sinusoidal and Triangular 0 1. Time (sec). THI with open-loop test vector inputs for FOC controller outputs Vq & Vd and . varies at a rate of 60 rpm. Vq Figure 5. Line Voltages and Vd are in per-unit, and Vd is considered zero for PMSM motor. Simulation results are shown in Figures 3 From Figure 3, we can clearly see the difference to 5. between Sinusoidal and Triangular Third harmonics . A. phase shift of 180 deg between the two is due to the fact that Sinusoidal TH is added to phase voltages and Triangular TH is subtracted from the phase voltages. Despite the difference between the two Third Harmonic waves, the maximum value of phase voltage with a modulation factor of is same, and is evident from Fig.
9 4. From Figure 5, we can also see that line voltages are exactly the same in both the cases. VI. EXPERIMENTAL VERIFICATION. The test setup consists of a PMSM motor connected to a fully controllable loading DC motor. A torque sensor and an encoder are mounted between the unit under test Figure 2. Simulation Models and the loading motor. PMSM motor is controlled through Freescale's PowerPC based motor control processor tailored for motor control . Loading motor can be controlled precisely in speed control mode or current control mode. The loading torque on the test motor can be Third Harmonic Voltage (pu). controlled by controlling the current through the loading motor. The torque supplied by the test motor can be 0. measured through the torque sensor. The picture of the test setup is shown below. Sinusoidal Triangular 0 1. Time (sec). Figure 3. Simulated Third Harmonic Voltages SinTHI A.
10 SinTHI B. SinTHI C. Phase Voltages (pu). TriangTHI A. TriangTHI B. TriangTHI C. 0. Figure 6. Experimental Setup For comparing Sinusoidal and Triangular THIs, PMSM motor is run in speed control mode and the 0 1. Loading motor is run in current control mode. Fig. 7. Time (sec) shows experimental data collected at 500 rpm and around Figure 4. Phase Voltages with Third harmonics Nm load torque. As can be seen clearly, the phase voltage profiles are matching with the simulation results. 90 Simulation and experimental results for the comparison SinTHI A. 80 SinTHI B of conventional Triangular THI and the proposed Phase Voltages Duty Cycle(perc). 70. SinTHI C. TriangTHI A. Sinusoidal THI have also been presented. The results 60. TriangTHI B. TriangTHI C. show that the motor performance with the proposed Sinusoidal THI technique is at par with the conventional 50. Triangular THI technique, thereby increasing the DC.