Improved Sensorless Control of Interior Permanent Magnet Sensorless Motors Using an Active Damping Control Strategy

Abstract

This paper proposes the active damping control strategy for position sensorless operation of an interior permanent magnet (IPM) motor. The proposed method is applied to both the current controller and the position estimator to control damping characteristics of the IPM drive system. By actively increasing the damping characteristics of the system with the proposed method, the current control and the position estimation loops become immune to parameter variation of the stator resistance which may degrade the accuracy of the position estimator. To analyze the accuracy of the position estimator with and without the proposed method, a small-signal analysis is carried out for low speed operation where the effect of the parameter variation is relatively large due to a low signal-to-noise ratio (SNR). Additionally, an open-loop voltage to angular velocity transfer function including the electrical and the mechanical parameters is investigated. Since no hardware modifications are necessary, the proposed method can be easily implemented just in software routines. Both the simulations and the experimental validations in which the proposed active damping control strategy is incorporated with the existing extended electromotive force (EMF)- based sensorless algorithm are provided to support the effectiveness of the proposed method.

1. Introduction

Position sensors are necessary for high performance vector control of AC motors. However, popular position sensors such as optical encoders and resolvers are usually bulky, mechanically vulnerable, and may increase the implementation cost of the entire motor drive system. If the accuracy of the position sensor is unreliable, it may cause severe instability problems in the control loop. To overcome these drawbacks of position sensors, numerous control techniques where no position sensors are employed have been proposed [1]. The main purpose of sensorless control algorithms is to estimate the position and the speed of the motor without using a position or a speed sensor. In general, the realization of sensorless control can be classified into two categories: signal injection [2,3,4,5,6,7] and model-based methods [8,9,10,11,12]. In the signal injection methods, a high frequency voltage or current signal is directly utilized to estimate electrical position of the rotor by detecting inductance variation depending on rotor position and inductor saturation. These methods are generally suitable for low speed or zero speed operation due to additional losses and acoustic noise induced by injecting the voltage or current signal. The computational burden of implementing signal injection and filtering algorithms is an extra drawback.

The model-based methods mostly use electrical or electro-mechanical models of AC motors to estimate control parameters. Mostly, they extract the estimated states from back electromotive force (EMF) information. It means these methods cannot be directly applied under low speed or zero speed operation where back EMF information is rarely obtained. Many modern control schemes using a model reference adaptive system (MRAS), an adaptive observer, nonlinear control, and so on are proposed for sensorless AC motor drives [8,9,10,11,12]. In some cases, hybrid methods incorporating signal injection and model based methods are utilized to cover full operation range of the AC motor [13,14,15,16,17]. In many applications such as fans, compressors and blowers that normally have medium or high speed regions, the model-based methods are attractive [18,19,20].

Among many model-based sensorless methods, so-called extended EMF (EEMF)-based methods have been popularly adapted for IPM and synchronous reluctance motors where magnetic saliency exists [21,22,23,24]. However, the problem with these methods is that sensorless operation can fail due to parameter mismatches or variations, especially in low speed operation or mode transition [25]. In [26], online stator flux and resistance estimators are proposed to improve sensorless operation at low speed. A flux estimation algorithm that uses the phase current derivative without signal injection in low speed is presented in [27]. In [28], an online parameter estimation technique is employed to minimize the effects of parameter variations. In [29], the current measurement error and the inverter irregularities are analyzed, then a stator resistance estimator is proposed. The parameter identification and inverter error compensation algorithms based on the extended EMF method are presented in [30]. In [31,32], non-ideal characteristics of inverters such as output voltage phase delay and on-state resistance of the power devices are compensated to achieve performance improvement of the sensorless drive. However, the drawbacks of the aforementioned methods are that they are somewhat complicated to implement, and moreover, the performance of sensorless control is highly dependent on the accuracy of the position estimators and the compensation methods.

In this paper, an active damping control strategy is applied to improve the sensorless operation of the IPM motor. It is analyzed that low damping component in the entire control loop causes the failure of the sensorless operation under low speed or the mode transition between the open-loop acceleration and the sensorless mode. The proposed method artificially increases the damping component of the drive system so that the effects of parameter variations on the stator resistance are reduced. Therefore, stable sensorless operation in the low speed region where the signal-to-noise (SNR) is extremely low can be achieved, even with parameter mismatches between the physical system and the plant model. The small-signal analysis is taken to show the accuracy of the position estimator considering a practical parameter mismatch with and without the proposed method. From this, it is theoretically verified that the proposed method actively changes the damping component, and reduces the position estimation error. The open-loop voltage to angular velocity model is also derived to show the performance of the active damping control algorithm. Both the simulation and the experimental results for the IPM drive system show good agreement with the analyses, and the effectiveness of the proposed method are verified.

2. Extended EMF Based Sensorless Control of IPM Motor

In an IPM motor, the extended EMF E_ex is defined as [21,22]:

$$E_{ex}={\omega }_r\left[\left(L_d-L_q\right)i_d+{\lambda }_f\right]-\left(L_d-L_q\right)\left(s{i}_q\right)$$

(1)

where s, ω_r, L_d, L_q, λ_f, i_d and i_q are the Laplace variable, the electrical angular velocity, the d-axis inductance, the q-axis inductance, the magnet flux linkage, and the d- and q-axes currents, respectively. By using Equation (1), the electrical model of IPM motors in the synchronous rotating frame is written as follows:

$$\left[\begin{array}{l}{v}_d\\ v_q\end{array}\right]=\left[\begin{array}{cc}\begin{array}{l}{R}_s+s{L}_d\\ {\omega }_r{L}_q\end{array}& \begin{array}{l}-{\omega }_r{L}_q\\ R_s+s{L}_d\end{array}\end{array}\right]\left[\begin{array}{l}{i}_d\\ i_q\end{array}\right]+\left[\begin{array}{l} 0\\ E_{ex}\end{array}\right]$$

(2)

where R_s, v_d, and v_q represents the stator resistance, the d-, and the q-axes voltages. It should be noted that L_q appears in both of d- and q-axes cross-coupling terms, and L_d is used in non-cross-coupling terms unlike the typical electrical model of IPM motors. Let us define γ-δ coordinate as the estimated axes obtained by using a proper sensorless algorithm. Then, the electrical model of IPM motors oriented to γ-δ coordinate is written as:

$$\left[\begin{array}{l}{v}_{\gamma }\\ v_{\delta }\end{array}\right]=\left[\begin{array}{cc}\begin{array}{l}{R}_s+s{L}_d\\ {\omega }_r{L}_q\end{array}& \begin{array}{l}-{\omega }_r{L}_q\\ R_s+s{L}_d\end{array}\end{array}\right]\left[\begin{array}{l}{i}_{\gamma }\\ i_{\delta }\end{array}\right]+\left[\begin{array}{l}{e}_{\gamma }\\ e_{\delta }\end{array}\right]$$

(3)

where i_γ and i_δ represent the estimated γ- and δ-axis currents in γ-δ coordinate, and e_γ and e_δ are the estimated EMF components in γ-δ coordinate. Here, e_γ and e_δ are defined as [22]:

$$\left[\begin{array}{l}{e}_{\gamma }\\ e_{\delta }\end{array}\right]=E_{ex}\left[\begin{array}{r}-\sin \left({\theta }_r-{\widehat{\theta }}_r\right)\\ \cos \left({\theta }_r-{\widehat{\theta }}_r\right)\end{array}\right]+\left({\widehat{\omega }}_r-{\omega }_r\right)L_d\left[\begin{array}{r}-i_{\delta }\\ i_{\gamma }\end{array}\right]$$

(4)

where θ_r, ${\widehat{\theta }}_r$, and ${\widehat{\omega }}_r$ are the actual electrical position, the estimated electrical position, and the estimated electrical angular velocity of the rotor.

Figure 1 shows the relationship between d-q and γ-δ coordinates. In sensorless operation, the second term in right-hand-side (RHS) in Equation (4) can be neglected by assuming L_d and the error between ${\widehat{\omega }}_r$ and ${\omega }_r$ are small enough to be ignored after they are multiplied. By doing so, Equation (4) is simplified as Equation (5).

$$\left[\begin{array}{l}{e}_{\gamma }\\ e_{\delta }\end{array}\right]\approx E_{ex}\left[\begin{array}{r}-\sin \left({\theta }_r-{\widehat{\theta }}_r\right)\\ \cos \left({\theta }_r-{\widehat{\theta }}_r\right)\end{array}\right]$$

(5)

From Equation (5), the angle error θ_e between d-q and γ-δ coordinates is obtained as below:

$${\theta }_e={\theta }_r-{\widehat{\theta }}_r={\tan }^{-1}\left(-\frac{e_{\gamma }}{e_{\delta }}\right)$$

(6)

To calculate θ_e from Equation (6), Equation (3) is rearranged as follows:

$$\begin{array}{l}{e}_{\gamma }=v_{\gamma }+{\omega }_r{L}_q{i}_{\delta }-\left(R_s+s{L}_d\right)i_{\gamma }\\ e_{\delta }=v_{\delta }-{\omega }_r{L}_q{i}_{\gamma }-\left(R_s+s{L}_d\right)i_{\delta }\end{array}$$

(7)

where v_γ and v_δ represent γ- and δ-axes voltages while i_γ and i_δ represent γ- and δ-axes currents. By substituting Equation (7) into Equation (6), θ_e can be calculated. In practice, a low pass filter is multiplied on the equations in (7) as:

$$\left[\begin{array}{l}{e}_{\gamma LPF}\\ e_{\delta LPF}\end{array}\right]=\frac{{\omega }_{ci}}{s+{\omega }_{ci}}\left[\begin{array}{l}{e}_{\gamma }\\ e_{\delta }\end{array}\right]$$

(8)

where ω_ci is the radian expression of the cutoff frequency f_ci for the first order low pass filter. The low pass filter which was reported as a part of the observer in [22] helps to avoid direct implementation of the differential terms which may be vulnerable for noise components in feedback paths. Then, e_γLPF and e_δLPF are rewritten as follows:

$$\begin{array}{l}{e}_{\gamma LPF}=\frac{{\omega }_{ci}}{s+{\omega }_{ci}}\left(v_{\gamma }+{\omega }_r{L}_q{i}_{\delta }-R_s{i}_{\gamma }\right)-\frac{{\omega }_{ci}s}{s+{\omega }_{ci}}{L}_d{i}_{\gamma }\\ e_{\delta LPF}=\frac{{\omega }_{ci}}{s+{\omega }_{ci}}\left(v_{\delta }-{\omega }_r{L}_q{i}_{\gamma }-R_s{i}_{\delta }\right)-\frac{{\omega }_{ci}s}{s+{\omega }_{ci}}{L}_d{i}_{\delta }\end{array}$$

(9)

In fact, by introducing the low pass filter, the first term ω_ci/(s + ω_ci) in RHS of Equation (9) is analyzed as the first order Pseudo integrator with the cut-off angular frequency ω_ci while the second term sω_ci/(s + ω_ci) operates as the first order Pseudo differentiator which may be the same form with a high pass filter. Compared to Equation (7), Equation (9) gives much more stable operation in steady state with slightly less, but acceptable, dynamic performance in back EMF estimation.

Figure 2 shows the structures of the position and speed estimator. The fundamental idea shown in Figure 2a is to estimate ${\widehat{\theta }}_r$ by tracking θ_e to be zero. In practice, θ_e is directly obtained from Equation 6 as can be seen in Figure 2b, because ω_r or θ_r is not directly available. The estimator G_est(s) in Figure 2 can be implemented with a simple proportional-integrator (PI) compensator. The detailed design procedure of G_est(s) is explained in [22]. The electrical angle estimation algorithm is widely used in many applications due to its simple implementation and good estimation performance. However, there are mainly two problems with the algorithm. One is that the estimation performance is not very good with inaccurate system parameters. Another is that low speed operation may be hard or even impossible due to the difficulty of obtaining the extended EMF components in such a low signal-to-noise ratio (SNR) condition.

3. Sensorless Control with Active Damping Control Strategy

The active damping technique used in this paper has been studied in several references [33,34,35,36] where the same concept may be called by different names. The basic concept of the active damping control is to artificially adjust the electrical damping of the motor drive system within the possible physical range by adding current feedback paths with a proper proportional gain. If the proportional gain is well selected, the stability and the dynamic performance of the control system can be improved [33,34,35,36]. In this paper, the active damping technique is combined with the back EMF estimation which is a part of the sensorless algorithm to achieve stable operation and superior dynamics of the IPM motor drive system.

3.1. Back EMF Estimation with Active Damping Control

The proposed back EMF estimation scheme using the active damping resistance R_dp is shown in Figure 3 where $v_{\gamma \_ff}={\widehat{\omega }}_r{L}_q{i}_{\delta }$ and $v_{\delta \_ff}=-{\widehat{\omega }}_r{L}_q{i}_{\gamma }$. Here, R_dp is added only to the estimated current feedback paths.

The estimated back EMF components after passing the low pass filter can be written as follows.

$$\begin{array}{l}{e}_{\gamma LPF}=\frac{{\omega }_{ci}}{s+{\omega }_{ci}}\left(v_{\gamma }+v_{\gamma \_ff}-\left(R_s+R_{dp}\right)i_{\gamma }\right)-\frac{{\omega }_{ci}s}{s+{\omega }_{ci}}{L}_d{i}_{\gamma }\\ e_{\delta LPF}=\frac{{\omega }_{ci}}{s+{\omega }_{ci}}\left(v_{\delta }+v_{\delta \_ff}-\left(R_s+R_{dp}\right)i_{\delta }\right)-\frac{{\omega }_{ci}s}{s+{\omega }_{ci}}{L}_d{i}_{\delta }\end{array}$$

(10)

Note that Equation (10) has the same form as Equation (9) except R_dp is added to R_s. The equations in (10) imply that the tendency of the back EMF and angle estimation characteristics can be adjusted by R_dp. In order to see how R_dp affects the estimation performance, the small signal analysis for θ_e will be discussed hereafter. For convenience of the analysis, the first order Pseudo integrator and differentiator are neglected. Then, i_δ can be rearranged as in Equation (11):

$$i_{\delta }=\frac{v_{\delta }-{\omega }_r{L}_q{i}_{\gamma }-e_{\delta }}{L_{d}s+(R_s+R_{dp})}$$

(11)

By substituting Equation (11) into e_γ, we have:

$$e_{\gamma }=v_{\gamma }+{\omega }_r{L}_q\left(\frac{v_{\delta }-{\omega }_r{L}_q{i}_{\gamma }-e_{\delta }}{L_{d}s+(R_s+R_{dp})}\right)-\left((R_s+R_{dp})+s{L}_d\right)i_{\gamma }$$

(12)

If the operational speed and the inertia of the IPM motor are low and high enough, it is possible to assume that the differentiation of the mechanical speed is zero. After that, e_γ can be simplified as in Equation (13) by considering the perturbation:

$${\tilde{e}}_{\gamma }=-\frac{{\omega }_r{L}_q{\tilde{e}}_{\delta }}{L_{d}s+(R_s+R_{dp})}$$

(13)

Equation (13) can be rearranged as below:

$$-\frac{{\tilde{e}}_{\gamma }}{{\tilde{e}}_{\delta }}=\frac{{\omega }_r{L}_q}{L_{d}s+(R_s+R_{dp})}$$

(14)

By substituting Equation (14) into Equation (6), the small signal model of the electrical angle error ${\tilde{\theta }}_e$ is obtained as follows:

$${\tilde{\theta }}_e={\tan }^{-1}\left(-\frac{{\tilde{e}}_{\gamma }}{{\tilde{e}}_{\delta }}\right)={\tan }^{-1}\left(\frac{{\omega }_r{L}_q}{L_{d}s+(R_s+R_{dp})}\right)$$

(15)

If ${\tilde{\theta }}_e$ changes within a very limited range, e.g., –0.2 rad < ${\tilde{\theta }}_e$ < 0.2 rad, (15) can be simplified even more as below:

$${\tilde{\theta }}_e={\tan }^{-1}\left(\frac{{\omega }_r{L}_q}{L_{d}s+(R_s+R_{dp})}\right)\approx \frac{{\omega }_r{L}_q}{L_{d}s+(R_s+R_{dp})}$$

(16)

Figure 4 compares the frequency responses of ${\tilde{\theta }}_e$ at different R_dp values where L_d = 2.51 mH, L_q = 6.94 mH, R_s = 0.09 Ω, and ω_r = 18.85 rad/s. As can be seen in the figure, the magnitude of ${\tilde{\theta }}_e$ is getting lower as R_dp increasing. If all parameters and variables are constant in (16) except R_dp, the gain of the transfer function is reduced as R_dp increases. This implies that even if the back EMF estimation is not very stable, the electrical angle error is smaller with larger R_dp. In other words, the angle estimation is better with R_dp in transients and some conditions where the back EMF estimation is oscillatory.

The trajectories of the position error variation depending on the ratio of the BEMF variation are plotted in Figure 5. As illustrated in the figure, the variation of the position error becomes smaller as R_dp is increased. This trend also matches the analysis taken previously.

3.2. Current Control Strategy with Active Damping Control

If R_dp is used for the back EMF estimation, it is better to use R_dp even in the current controller to reduce the model mismatch between the actual and the estimated control system as shown in Figure 6.

Figure 7 shows the current controller implementation with R_dp. Note that the current controller consists of a PI controller as a feedback controller and a decoupling terms v_{dq_ff} as follows:

$$\left[\begin{array}{l}{v}_{d\_ff}\\ v_{q\_ff}\end{array}\right]=\left[\begin{array}{cc}\begin{array}{l}0\\ {\omega }_r\end{array}& \begin{array}{l}-{\omega }_r\\ 0\end{array}\end{array}\right]\left[\begin{array}{cc}\begin{array}{l}{L}_d\\ 0\end{array}& \begin{array}{l}0\\ L_q\end{array}\end{array}\right]\left[\begin{array}{l}{i}_d\\ i_q\end{array}\right]$$

(17)

The gain of the PI controllers is selected as:

$$\begin{array}{l}{K}_{pd}={\omega }_c{L}_d\hspace{1em}\hspace{1em}\hspace{1em}{K}_{pq}={\omega }_c{L}_q\\ K_{id}=K_{iq}={\omega }_c\left(R_s+R_{dp}\right)\end{array}$$

(18)

where K_pd, K_pq, K_id, K_iq, and ω_c represent d-, q-axes proportional gains, d-, q-axes integral gain, and desired bandwidth of the current controller, respectively. Note that R_dp is included to the integral gain. By selecting the control gain this way and using the feed-forward terms, the overall characteristics of the entire control system can be emulated as a simple low pass filter whose cutoff frequency is the same to the bandwidth of the control system, as expressed in Equation (19).

$$\frac{i_{dq}(s)}{i_{dq}^{*}(s)}\simeq \frac{{\omega }_c}{s+{\omega }_c}$$

(19)

This is so-called the technical optimization of current controllers [37]. Equation (19) implies that the current control bandwidth is identical to ω_c, and the phase margin is 90°.

To examine the effect of R_dp in the current control loop, a simple voltage to mechanical speed model will be derived hereafter. Let us consider the torque equation of the IPM motor as below:

$$T_e=\frac{3}{2}\frac{P}{2}\left({\lambda }_f{i}_q+\left(L_d-L_q\right)i_d{i}_q\right)$$

(20)

where P is the number of poles. In order to simplify the analysis, i_d is assumed to be zero, then we have:

$$T_e=\frac{3}{2}\frac{P}{2}{\lambda }_f{i}_q=K_T{i}_q , \text{where} K_T\equiv \frac{3}{4}{K}_E \text{and} K_E\equiv P{\lambda }_f$$

(21)

The block diagram of q-axis voltage-to-mechanical speed is shown in Figure 8, whose parameters are denoted in

Table 1. Motor parameters.

Parameters	Values
Number of poles (P)	6
Magnet flux linkage (${\lambda }_f$)	0.235 V/(rad/s)
d-axis inductance (Ld)	2.51 mH
q-axis inductance (Lq)	6.94 mH
Stator resistance (Rs)	0.09 $\Omega$
Rotor inertia (Jm)	0.003334 kg·m2
Friction constant (Bm)	0.425 × 10−3·m2/s
Rated speed	1600 rev/min
Rated torque	65 Nm

. In order to see the effect of R_dp only, the feedback controllers are excluded in this analysis. From Figure 8, the transfer function of v_q- to -ω_rm is given as Equation (22):

$$G_{v\omega }(s)=\frac{{\omega }_{rm}}{v_q}=\frac{K_T}{\left(s{L}_q+\left(R_s+R_{dp}\right)\right)\left(J_{m}s+B_m\right)+K_T{K}_E}$$

(22)

where J_m and B_m are the inertia moment and the friction coefficients of the mechanical system. Figure 9 compares the frequency response of Equation (22) with different R_dp values. The highest resonance occurs with no R_dp. The resonance means that ω_rm is oscillatory when v_q which is the output of the feedback controller changes quickly in cases such as a step load variation. If R_dp is increased, the resonance is getting damped. When R_dp is selected to be 10R_s, the resonance is almost nothing so that the frequency response of the transfer function is similar to a first order low pass filter. In this case, ω_rm may not fluctuate that much even in the step load variation. Accordingly, it is supposed that the use of R_dp definitely helps to stabilize the speed control loop of the motor drive system.

4. Simulation Study

Simulations have been carried out to examine the performance of the proposed method on the sensorless control. The parameters in the simulations are the same with Table 1. The speed reference profile and the operation mode for the simulations are shown in Figure 10. To start the IPM motor, the well-known open-loop synchronous acceleration method [19,20] is applied. The EEMF based sensorless algorithm is also utilized to estimate the position and the speed of the IPM motor. The mode transition from the open-loop synchronous acceleration to the sensorless control occurs at 60 RPM.

Figure 11 shows the simulation results where the parameter mismatches in the current controller and the position and the speed estimator are intentionally inserted. At the start, 10% of the rated torque is applied. After 0.5 s, the operation mode is switched from the open-loop synchronous acceleration to the sensorless mode. After the speed of the motor is increased to 500 RPM, 30% of the rated torque is applied at t = 1.25 s. In each figure, the red dashed line, the green dotted line, and the blue solid line represent the cases when R_dp equals to zero, 3$\widehat{R}$_s, and 5$\widehat{R}$_s.

In each case in Figure 11, the responses without R_dp show more fluctuation than the cases with R_dp, where the response is much more damped at the mode transition from the open-loop synchronous acceleration to the sensorless operation. This is especially apparent in Figure 11c where the system failed to switch the operation mode to the sensorless control. These results correspond to the analysis in the previous sections, and show that the mode transition is made more stable by using R_dp. This implies that R_dp helps the stabilization of the IPM motor drive system undergoing the mode transition in the low speed region. Once the motor speed is over 500 RPM, the responses of a 30% load step condition are similar in all of the cases. From this result, it can be inferred that the utilized R_dp does not affect the normal sensorless operation where the motor speed is high enough to overcome low SNR on the estimated EMF components. However, it should be noted that the feedback current multiplied by R_dp is not a real-time current, but a sampled current, which has a zero-order-hold (ZOH) effect. Hence, the feedback current may not be the same with an ideal case, because the high frequency components are delayed in the feedback. The limitation of the output voltage is also another issue. If the feedback current multiplied by R_dp is too high, the output voltage reference may be over the physical limitation of the output voltage. Therefore, too high value of R_dp may cause other instability issues in the system due to the ZOH effect and the output voltage limitation.

5. Experiments

Experiments have been performed to show the effectiveness of the proposed method. Figure 12 shows the experimental configuration.

The dc-link voltage is selected as 300V and it is fed by a dc power supply. The same motor parameters described in Table 1 are utilized for the experiments. The specifications of control parameters are shown in

Table 2. Control parameters.

Parameters	Values
Switching frequency	10 kHz
Sampling frequency	10 kHz
Current control bandwidth	300 Hz
Position estimator bandwidth	100 Hz
Crossover frequency of the Pseudo integrator and differentiator	100 Hz

.

The proposed algorithm is implemented with a custom made digital control board in which a 32-bit digital signal processor is employed. The control board includes a 4-channel digital-to-analog converter (DAC) to monitor the internal variables in the software routines. A 13-bit external encoder is connected to the IPM motor to compare the real and the estimated position of the rotor. A Magtrol dynamometer is utilized to apply loads and to measure the motor torque. Although the testing motor is an IPM motor, the d-axis current reference is selected as zero to simplify confirming the validation of the proposed algorithm where a maximum torque per ampere is not necessary. However, the operational principle should not be different even in flux weakening region. In the experiments, the magnet flux linkage, the stator resistance, the d- and the q-axes inductances in the controller is selected as 0.9λ_f, 0.75R_s, 1.15L_d, and 1.15L_q to artificially emulate the parameter mismatch condition. The active damping component R_dp is selected as 5$\widehat{R}$_s. In Figure 13, the sensorless operation with 10% of the rated torque is shown. Figure 13a shows the mode transition from the open-loop synchronous acceleration to the sensorless operation. The speed at the transition point is 120 RPM. As can be seen in the figure, the operation mode is smoothly switched. Figure 13b shows the steady state operation of the sensorless control, which is very stable. In both Figure 13a,b the oscilloscope channels 3 and channel 4 represent the actual electrical position θ_r measured by the 13-bit encoder and the estimated position $\widehat{\theta }$_r. Channel 1 and channel 2 indicate the estimated EMFs e_γ and e_δ. The values of e_γ and e_δ are evaluated as 0V and 8.86V. These numbers agree with the results obtained from (3) when no errors occur. Hence, the position and the EMFs estimation are smooth, even with R_dp.

Figure 14 shows the test results of a deceleration where the speed reference is changed from 500 RPM to 60 RPM in ramp. Here, 30% of the rated torque is applied as the load. In both cases, the estimated EMF value and the speed control performance are within the normal range at the beginning. However, as seen in Figure 14a, the position and the EMF estimation fail and the speed is not regulated as the operating speed decreases without Rdp. This is due to the low SNR on the iterated values in the controller, the parameter mismatches, and the low system damping characteristic. Meanwhile, in Figure 14b, the stable sensorless operation is achieved with Rdp after slight transients at the beginning of 60 RPM operation under the same test condition.

Figure 15 compares the trajectories of the position error θ_e which depends on the estimated EMF ratio e_γ/e_δ. The operating speed is 90 RPM and no load torque is applied. As analyzed in the previous section, the dispersion of the error trajectory with R_dp is narrower than the case without R_dp. It means the performance of the position estimation is accurate with the active damping resistance.

Figure 16 compares the position and the EMF estimation performance without and with R_dp. The same operating condition as in Figure 15 is assumed. As shown in the figure, the peak to peak fluctuation of the estimated EMF and the position are evaluated as about 20V and 0.35 rad before applying R_dp. However, the magnitudes are decreased to 3V and 0.18 rad with R_dp. The ripple components in the estimated EMF and the position are also reduced.

Figure 17 illustrates the test results on the torque and the speed response. The speed reference is 60 RPM and the applied load torque is 5% of the rated torque. In Figure 17a, the magnitude of the torque ripple is reduced from 3Nm to 1Nm after applying R_dp. The range of the speed fluctuation is also limited from 40 RPM to 10 RPM with R_dp.

Figure 18 compares the step load test performance. 50% of the rated torque is applied using the Magtrol dynamometer. As can be seen in the comparison, if R_dp is not utilized, the speed regulation fails when the load is applied. With R_dp, although the minimum speed reaches 10 RPM, the speed regulation is recovered in steady state. In the experimental results, by using the proposed method, it is confirmed that the EMF and the position estimation performance and the stability of the system are improved even with the parameter mismatches.

6. Conclusions

This paper proposed an active damping control strategy for the sensorless control of an IPM motor. The proposed method reduces the effects of the parameter mismatches, and increases the damping characteristics of the overall control system. Accordingly, the stability of the control system is improved. The small signal analysis for the position error and the frequency response of the open-loop voltage-to-speed transfer function are performed to evaluate the stability of the system using the proposed method. The proposed algorithm is adapted to the EEMF based sensorless method. Both the simulations and the experiments agree that the EMF and the position estimation performance are improved, and the torque and the speed ripple are reduced.