Stability Analysis of Sensorless Speed Control for PMSM

IEEJ Journal of Industry ApplicationsVol.8 No.4 pp.736–744 DOI: 10.1541/ieejjia.8.736Translated from IEEJ Transactions on Industry Applications, Vol.138 No.11 pp.848–856

Paper(Translation ofIEEJ Trans. IA)

Stability Analysis of Sensorless Speed Control for PMSMConsidered Current Control System

Sari Maekawa∗ Member, Mariko Sugimoto∗ Non-member

Keiichi Ishida∗∗ Member, Masaya Nogi∗∗ Member

Masaki Kanamori∗∗ Member

For the control of permanent magnet synchronous motors, speed control, current control, sensorless control exist,and control gain design is required. In general, the control gain needs to be designed in consideration of stability andresponsibility, and various researches have been conducted on this.

In this paper, the influence of the control band of the three types of control on the stability is analyzed by combiningthe analysis of the pole placement of the closed-loop transfer function and stability judgment considering the axial errorwhich is the difference between the actual position and the estimated position, the relationship between each controlband that can secure the results is discussed.

Keywords: PMSM, sensorless control, ASR, ACR, stability

1. Introduction

Although a method for estimating rotor magnetic flux andinduced voltage in the medium to high speed range is used insensorless control of permanent magnet synchronous motor(PMSM), various factors cause the problem of instability insensorless control.

Instability is caused by factors due to changes in d-q-axesinductances used in rotor flux estimation occurring due tomagnetic saturation (1)–(3) and inter-axial interference (4) (5), andfactors due to the motor model having nonsinusoidal wave-form (6), such as the spatial harmonics observed in motorswith concentrated windings. Moreover, various studies arebeing conducted on topics such as the stability limit (7) asso-ciated with the effect of the difference between the d-q-axesinductances due to interior permanent magnet synchronousmotor (IPMSM) polarity appearing as 2θ component in sen-sorless control, and mathematical models (8) (19) (21) such as theextended electromotive force model.

Furthermore, for the control gain and natural angular fre-quency (frequency) set in each control as required in config-uring the control system based on speed command, althoughdesign methods taking into account stability have been pro-posed concerning methods of current control (9) (10), speed con-trol (11) (12), and different sensorless controls (13)–(17), with respectto quantitative verification studies taking into account the sta-bility of sensorless control of the control system as a wholeconsidering the relationship between speed control and minor

∗ TOSHIBA CORPORATION YOKOHAMA COMPLEX33, Shin-Isogo-cho, Isogo-ku, Yokohama, Kanagawa 235-0017,Japan

∗∗ TOSHIBA CARRIER CORPORATION FUJI FACTORY &ENGINEERING CENTER336, Tadehara, Fuji, Shizuoka 416-8521, Japan

loop which is the frequency of current control, only stud-ies such as on induction motor (18) (19) can be found, but to thebest of the authors’ knowledge no such study has yet beenreported on permanent magnet synchronous motor.

Accordingly, with respect to the effect of frequency on thethree controls of current control, speed control, and sensor-less control in a permanent magnet synchronous motor sen-sorless control system, this study reports on determining sta-bility using analysis of pole placement of the closed-looptransfer function, and discusses the relationship between thefrequency of each control and securing stability, by consid-ering in combination the stability judgement that takes intoaccount the upper limit of the axial error - the difference be-tween the actual position and the estimated position - whichis a problem encountered in cases such as step load.

Section 2 describes a motor model where the simultaneousdifferential equation obtained form the motor voltage typeand speed equation is linearized over a very short interval oftime from the point of stable equilibrium, and discusses thederivation of the closed-loop transfer function and the charac-teristic equation of the sensorless speed control system basedon the control configuration.

Section 3 explains how the poles are calculated from theestimated transfer function, and with respect to the differencein stability with that of a speed control system with sensors,compares the effect of frequency for each of the controls.

Section 4 examines the frequency of sensorless controlnecessary according to the theory of axial error generated instep load and its upper limit.

Section 5 verifies the validity of the proposed method, withrespect to the stability constrained by the frequency of thethree control systems, by comparing the results of the cal-culations based on the proposed method with the results ofactual control tests.

c© 2019 The Institute of Electrical Engineers of Japan. 736

Stability Analysis of Sensorless Speed Control for PMSMConsidered Current Control System（Sari Maekawa et al.）

Fig. 1. Control configuration

2. Configuration of Motor and Control System

2.1 Motor Model The d-q-axes voltages of the targetmotor for control is given by equation (1):(

Vd

Vq

)=

(R + pLd −ωLq

ωLq R + pLq

) (Id

Iq

)+

(0ωφf

)· · · · · · · · · (1)

where, Id,q are d-q-axes currents, ω is electrical frequency,Vd,q are d-q-axes voltages, Ld,q are d-q-axes inductances, Ris winding (stator) resistance, and φ f is linked flux of perma-nent magnet.

Moreover, the PMSM output torque Tm is given by equa-tion (2), and the speed defined by load torque, mechanicalfriction, and moment of inertia can be expressed as in equa-tion (3):

Tm = Pφ f Iq + P(Ld − Lq

)IdIq · · · · · · · · · · · · · · · · · · (2)

Tm − Tl = Jdωm

dt+ Dωm · · · · · · · · · · · · · · · · · · · · · · · · (3)

where, ωm is mechanical frequency, P is pole pair number, Jis moment of inertia (inertia), D is friction coefficient, and Tl

is load torque.Here, by linearizing the minute changes in each state vari-

able from a stable equilibrium point (ωm0, Id0, Iq0) in equa-tions (1) and (3), state equation (4) can be derived.

ddt

⎛⎜⎜⎜⎜⎜⎜⎜⎝∂ωm

∂Id

∂Iq

⎞⎟⎟⎟⎟⎟⎟⎟⎠ =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

−DJ

PL1Iq0

J−

P(φ f + L1Id0

)J

−Iq0 − RLd

ωLq

Ld

−φ f

Lq−ωLd

Lq− R

Lq

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

⎛⎜⎜⎜⎜⎜⎜⎜⎝∂ωm

∂Id

∂Iq

⎞⎟⎟⎟⎟⎟⎟⎟⎠

+

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 01Ld

0

01Lq

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠(Vd

Vq

)=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝1J00

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ Tl · · · · · · · · · (4)

where, L1 = Ld − Lq.

2.2 Configuration of Control System Figure 1shows the control configuration used in the analysis. Thecontrol method used is sensorless vector control, and utilizesspeed control (auto speed regulator: ASR) based on controlof motor speed command by detecting the motor current anddirect current voltage. As shown in Fig. 1, sensorless con-trol utilizes a general phase locked loop (PLL) configurationwhereby, based on the voltages Vdc, Vqc, and currents Idc, Iqc

along the estimated axes dc, qc, axial error Δθ is calculated asthe difference between the actual position and the estimatedposition, and the estimated speed ωˆ and position θˆ are cal-culated using PI controllers.

Each of the controllers is a PI controller, and the auto cur-rent regulator (ACR) CACR(s), the auto speed regulator (ASR)CASR(s), and the PLL controller CPLL(s) are expressed byequations (5)∼(7):

CACR(s) =KpACRs + KiACR

s· · · · · · · · · · · · · · · · · · · · · (5)

CASR(s) =KpASRs + KiAS R

s· · · · · · · · · · · · · · · · · · · · · · (6)

Moreover, the position estimation part is configured to cal-culate the axial error Δθ from the estimated information onvoltage/current/speed and 1st order lag filter (low pass filter:LPF) given by equation (8):

CPLL(s) =KpPLLs + KiPLL

s· · · · · · · · · · · · · · · · · · · · · · (7)

GLPF(s) =ωLPF

s + ωLPF· · · · · · · · · · · · · · · · · · · · · · · · · · · · (8)

where, K pACR and KiACR are proportional and integral gainsin current control respectively, K pAS R and KiAS R are propor-tional and integral gains in speed control respectively, K pPLL

and KiPLL are proportional and integral gains in sensorlesscontrol respectively, and ωLPF is the filter cut off frequency.

Here, by specifying the current controller using non-interference method along with the ones described in Refs.(9) and (10), the respective control gains can be representedby equation (9) using the frequency of ACR ωACR.

737 IEEJ Journal IA, Vol.8, No.4, 2019


{KpACR = ωACRLq

KiACR = ωACRR· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (9)

Moreover, since the loop transfer function in a closed loopis a second order system, using the methods described in ref-erences such as (20) and (21), gains of speed control and sen-sorless control are expressed by equations (10) and (11) usingthe respective damping coefficients ξASR and ξPLL, and the re-spective frequencies of the systems ωASR and ωPLL.⎧⎪⎪⎨⎪⎪⎩

KpASR = 2ξASRωASRJ(φ f P2

)KiASR = ωASRJ

(φ f P2

) · · · · · · · · · · · · · · · · · · (10)

{KpPLL = 2ξPLLω

2PLL

KiPLL = ω2PLL

· · · · · · · · · · · · · · · · · · · · · · · · · · (11)

In the pole placement analysis to be described later, thestability is judged not using control gain but the frequenciesωACR, ωASR, and ωPLL obtained using equations (9)∼(11).

Next, using the controllers CACR(s), CASR(s), CPLL(s) andthe motor model, the closed-loop transfer function of the sen-sorless control system is derived. Figure 2 shows the blockdiagram of the sensorless speed control. Here, for the cur-rent control system, the current control gain is defined byequation (9), and since the pole and zero point cancel eachother when non-interference control operates ideally, the cor-responding closed-loop transfer function GACR(s) can be re-duced to a lower order through simplification using equation(12):

GACR(s) =ωACR

s + ωACR· · · · · · · · · · · · · · · · · · · · · · · · · · · (12)

Moreover, with respect to the speed control system, assum-ing that the friction coefficient D can be ignored being suffi-ciently lower compared to the moment of inertia, if the d axiscurrent control command Idc∗ is taken as zero in a simplifiedconfiguration, the transfer function G1(s) from speed controlcommand value ω∗ to actual speed ω can be expressed as inequation (13):

G1(s) = CASR (s) GACR (s) Pφ fPJs· · · · · · · · · · · · · · · (13)

Note that, the configuration of Id � 0, that is, consider-ing the case of applying d axis current control command in

Fig. 2. Block diagram of sensorless speed control

controllers such as MTPA, if as an example a configuration isassumed where Id∗ and Iq∗ are generated according to MTPAangle β such that Iq∗ = ASR output, and Id∗ = tan(β)Iq∗, thenthe motor output torque generated from the resulting currentflow, using linear approximation of the equilibrium points Id0

and Iq0 in equation (4), and current phase β0 at the point ofequilibrium, is given by equation (14):

ΔTm=ΔIqP{φ f +

(Ld−Lq

) (Id0−tan β0Iq0

)}· · · · · · · (14)

Here, the transfer function G1(s) from speed commandvalue ω∗ to actual speed ω is given by equation (15). Com-pared to equation (13), the configuration becomes complexsince the reluctance torque can now be taken into account,but for a certain

G1(s) = CASR (s) GACR (s)

×P{ϕ f + (Ld − Lq)

(Id0 − Iq0 tan β0

)} PJs· · · · · · (15)

degree of saliency ratio, it is presumed that there is no prob-lem in approximating by equation (13). For example, assum-ing pole pair number P = 3, φ f = 0.1 Wb, Ld = 5 mH, Lq

= 10 mH, equilibrium points as Id0 = −3 A and Iq0 = 5 A,and current phase β0 = 31 deg, although the torque coeffi-cient for converting current into torque under the conditionof Id = 0 is calculated as Pφ f = 0.3 in equation (13), for theaforementioned condition of Id � 0, in equation (15) the termcorresponding to saliency is incorporated causing an increaseto about 0.4. In the analysis hereafter, the verification is per-formed using the simplified equation (13).

In addition, the closed-loop transfer function GASRˆ(s) ofestimated speed ωˆ corresponding to speed command valueω∗ including the current control, speed control, and sensor-less control systems is given by equation (16).

GASRˆ(s) =ωˆω∗=

G1 (s) G3(s)1 +G1 (s) G3(s)

· · · · · · · · · · · · · · (16)

where,

G2(s) =1s

GLPF (s) CPLL(s) · · · · · · · · · · · · · · · · · · · · · (17)

G3(s) =ωˆω=

G2 (s)1 +G2 (s)

· · · · · · · · · · · · · · · · · · · · · · (18)

and the characteristic equation for the closed-loop transferfunction of the sensorless speed control system given byequation (16) becomes equation (19), and has six poles. Notethat in case of control with sensors there are only three poles.

DAsRˆ(s) = s6 + a5s5 + a4s4 + a3s3 + a2s2 + a1s + a0

· · · · · · · · · · · · · · · · · · · (19)

where,

a5 = ωACR + ωLPF · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (20)

a4 = ωLPF (2ωPLL + ωACR) · · · · · · · · · · · · · · · · · · · · (21)

a3 = ωLPF

(ω2

PLL + 2ξPLLωACRωPLL

)· · · · · · · · · · · (22)

a2 = ωLPFωACRωPLL

(ωPLL + 4ξASRξPLLωASRP4ϕ2

f

)· · · · · · · · · · · · · · · · · · · (23)

a1=2ωLPFωACRωASRωPLLP4ϕ2f (ξASRωPLL

+4ξPLLωASR) · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (24)

a0 = ωLPFωACRω2ASRω

2PLLP4ϕ2

f · · · · · · · · · · · · · · · · · (25)



(a) Sensorless (b) With sensor

Fig. 3. Root locus when the frequency of ACR isdecreased

(a) Sensorless (b) With sensor

Fig. 4. Root locus when the frequency of ASR isincreased

3. Stability Judgement of Sensorless Speed Con-trol by Pole Placement

Figure 3 shows the root locus when the frequency of ACRis decreased. The graph in (a) corresponds to sensorless con-trol, while that in (b) corresponds to control with sensors.The 4 Hz speed control frequency (frequency of ASR), the64 Hz sensorless control frequency (frequency of PLL), andthe 100 Hz cut-off frequency of low pass filter GLPF(s) for ax-ial error computation are held constant, and the frequency ofPLL and the LPF for axial error computation are not used inverification in (b). In all the cases, although the two poles ofA and B approach the right-half of the complex plane withdecrease in frequency of ACR, in sensorless control, theyreach the right-half plane faster than in control with sensors.This shows that in case of sensorless control, to secure stabil-ity it is necessary to set the frequency of ACR ωACR to a valuehigher than what is necessary in control with sensors. Notethat the three poles, pole E at a location quite distant on thenegative side of real axis, and poles C and D on either side ofthe imaginary axis, are generated only when operating undersensorless control.

Figures 4(a) and (b) show the root locus when the fre-quency of ASR is increased. The 64 Hz frequency of ACR,the 64 Hz frequency of PLL, and the 100 Hz cut-off frequencyof LPF GLPF(s) for axial error computation are held constant.Compared to current control, although the differences be-tween sensorless control and control with sensors are small,in sensorless control, the poles A and B shift to the right-halfplane resulting in instability. It is presumed that, although thepoles E and F along the imaginary axis have the characteristicof moving in the opposite direction of poles A and B, sinceeven with frequency of ACR at about 1 Hz, they do not reachthe right-half plane, their effect on stability can be consideredsmall.

Figure 5 shows the root locus when the frequency of PLL isdecreased. The 64 Hz frequency of ACR, the 4 Hz frequency

Fig. 5. Root locus when the frequency of PLL isdecreased

of ASR, and the 100 Hz cut-off frequency for LPF GLPF(s)for axial error computation are held constant. It is revealedthat the poles A and B that were tracked in Fig. 3 and Fig. 4,move along the direction of the arrows towards the right-halfplane as the frequency of PLL decreases, thereby causing in-stability. However, even when the frequency of PLL is setexcessively high, they reach the right-half plane in a similarmanner, causing instability. This is due to the effect of cut-off frequency used in LPF for axial error computation, anddemonstrates that the upper limit of the sensorless controlfrequency for stability is constrained due to LPF.

4. Limit of Frequency of Sensorless Control dueto Axial Error

In the previous section the stability of the sensorless con-trol system was analyzed using pole placement. However, insensorless control, analysis is necessary by additionally con-sidering the aspect of axial error limit. This becomes moreprominent not in the steady state but in cases such as stepload where the load torque changes rapidly causing speedchange. Reference (22) proposed a control method where theaxial error during acceleration is reduced using a feedbackconfiguration. In this section, the axial error limit is analyzedusing a conventional PLL method.

First, the characteristics of applying step load during sen-sorless control is explained by using the case shown inFig. 6(a), where a constant current command is applied with-out controlling the speed. Before applying step load, the mo-tor output torque Tm and the load torque Tl are in balance,with the difference between the two, the disturbance torqueΔT, being zero, and the speed ω is constant. When a stepload is applied at time t0, the disturbance torque ΔT generatesΔTmax in the negative direction, and although the actual mo-tor speed ω and estimated speed ωˆ gradually decrease undera constant acceleration, there is a lag in ωˆ corresponding toω due to the lag in response of the sensorless control, whichresults in generation of axial error Δθ.

Here, corresponding to the block diagram of sensorlessspeed control in Fig. 2, the transfer function G4(s) fromtorque disturbance ΔT to the axial error Δθ can be calculatedusing equation (26):

G4(s) =PJs

1s

(ωLPF

s + ωLPF

)

1 +1s

(ωLPF

s + ωLPF

) (KpPLLs + KiPLL

s

)

· · · · · · · · · · · · · · · · · · · (26)



(a) Constant torque mode (b) ASR mode

Fig. 6. Transient waveform under step load

In addition, arranging the expressions of the proportionaland integral gains of sensorless control using ξPLL and ωPLL

in equation (11), the axial error Δθ is given by equation (27):

Δθ(s) =

ΔT PωLPF

Js3 + s2ωLPF + 2ξPLLωPLLωLPF s + ωPLLωLPF

· · · · · · · · · · · · · · · · · · · (27)

Here, corresponding to Fig. 6(a), the converged Δθmax

value for axial error Δθ at the time of applying step load, canbe derived using the expected disturbance torque ΔTmax, polepair number P, inertia J, and frequency of sensorless controlωPLL in equation (28).

lims→0Δθ(s) =

ΔTmaxP

Jω2PLL

· · · · · · · · · · · · · · · · · · · · · · · · · · · (28)

On the other hand, in case of sensorless speed control ana-lyzed in this study, the response characteristics are as shownin Fig. 6(b). Although decrease in speed occurs correspond-ing to the step load, due to speed control, in an effort to holdthe speed constant the motor torque increases, whereby theaxial error Δθ converges to zero. When the the step load risetime is of the order of a few ms and is extremely fast com-pared to the response of the speed control, the maximum axialerror generated at this time can be approximated by the sen-sorless loop response only similar to that in Fig. 6(a), and canbe expressed as in equation (28).

Next, let us consider the allowable value of axial error Δθto stabilize sensorless control under step load. When an axialerror Δθ is generated, the actual current flow Iq along the qaxis is given by equation (29) using the axial currents Idc andIqc along the estimated axes dc and qc.

Here, if Δθ exceeds the range −π2 ∼ π/2, polarity reversaloccurs causing generation of negative torque in a state thatneeds positive torque in the speed control system, resultingin increasing the disturbance torque ΔT. That means, in equa-tion (27), with increase in disturbance torque ΔT the axialerror Δθ increases further. Accordingly, the allowable upperlimit of Δθ becomes π/2. And finally, substituting Δθ < π/2in equation (28), the minimum frequency of sensorless con-trol ωPLLmin needed for stable sensorless control is given byequation (30).

Table 1. Verification conditions

5. Comparison of Frequency Stability Verifica-tion and Transient Response Among the ThreeControls

Corresponding to the stability analysis using pole place-ment described in section 3, and the method of stability ver-ification corresponding to load explained in section 4, thissection discusses the confirmation of transient response op-eration under actual sensorless control, and the verificationof validity of the proposed method.(

Id

Iq

)=

(cosΔθ − sinΔθsinΔθ cosΔθ

) (Idc

Iqc

)· · · · · · · · · · · · · · · · (29)

ωPLLmin ≥√

2ΔTmaxPπJ

· · · · · · · · · · · · · · · · · · · · · · · · (30)

5.1 Verification Conditions Table 1 shows the var-ious conditions used in the verification such as the motor,control specification, operating speed, and step load. Basedon the proposed sensorless speed control stability judgementmethod, 6 poles can be obtained from the characteristic equa-tion (19) used in pole analysis, and instability occurs if one ofthese is located on the right-half plane. Accordingly, amongthe 6 poles, the real part value of the one with the highestreal part is confirmed, and if it is positive then it is consid-ered unstable, but if it is negative then it is considered sta-ble. Additionally, considering the upper limit of axial errorΔθ from equation (30), and incorporating the stable range offrequency of sensorless control computed under verificationconditions shown in Table 1, the stable range of frequenciesfor the three controls enabling stable sensorless speed controlare also shown.

Next, the operating characteristics under step load obtainedby actually operating a motor under sensorless control usingthe proposed method described above are compared.

Figure 7(a) shows the response waveforms of speed com-mand value ω∗, estimated speed value ω ,̂ actual speed ω,



(a) The waveform when the control system is stable.f ACR = 256 Hz, f ASR = 4 Hz, f PLL = 32 Hz

(b) The waveform when the control system is unstable.f ACR = 256 Hz, f ASR = 4 Hz, f PLL = 4 Hz

Fig. 7. Transient response waveform

axial error Δθ, the currents Id and Iq along d and q axes, and Uphase current Iu under the condition of constant speed com-mand value and application of step load at time 2.0 s. Thefrequency of ACR is 256 Hz, the frequency of ACS is 4 Hz,and the frequency of PLL is 32 Hz. When the step load isapplied, although the estimated speed value temporarily de-viates form the speed command value, the operation of thesystem is stable. In contrast, Fig. 7(b) shows the results whenthe frequency of ACR is 256 Hz, the frequency of ASR is4 Hz, and the frequency of PLL is 4 Hz, and with applica-tion of the step load the operation becomes unstable and theoutput diverges. Under such diverging conditions, since cor-responding to the speed command value ω∗, the differenceωerr with the estimated speed value ωˆ becomes extremelylarge, in this study, verification was performed using the max-imum value of ωerr, the difference between ω∗ and ωˆ usedas an indicator of stability in the transient response, duringthe simulation period (the time span of 0∼3.0 s as shown inFig. 7). And finally, regarding the actual threshold value for

Table 2. Specification of simulator and verificationconditions

judgement of stability/instability, assuming that the opera-tion will be stopped in the event of divergence under sucha speed command value, when ωerr exceeded ω∗, the systemwas considered to have become unstable.5.2 Simulator Specification In verification of opera-

tion, considering the number of verifications, the results fromthe simulator modelling the motor and inverter were used inthe study. The simulator was implemented using indepen-dently developed numerical computation.

Table 2 shows the specification of the simulator and theverification conditions. Although it is possible to incorpo-rate the nonlinear features of the motor and the inverter inthis simulator, the first set of verification is undertaken underideal conditions. This verification confirms the efficacy ofthe stability range analysis taking into account the proposedmethod of pole analysis and upper limit of axial error. Next,in the second set of verification, the nonlinear features of themotor and inverter such as the PWM harmonics and deadtimeare incorporated into the simulation, and the usefulness of theproposed method considering an actual machine is verified.5.3 Verification Results under Ideal ConditionsFigure 8(a)∼(c) show the results of root locus analysis when

the frequencies of ACR, ASR, and PLL are changed. Whenthe maximum value is negative it is stable region, whereaswhen the maximum value is positive it is unstable region.Note that the plot scale was set such that only positive andnegative are shown. As a result, it was revealed that to makethe frequency of ASR high, it is necessary to increase thefrequency of PLL. However, its upper limit is constrainedby the frequency of ACR. Moreover, the upper limit of fre-quency of PLL is also constrained by the cut off frequencyωLPF of the LPF of axial error. Since the cut off frequencyωLPF is 100 Hz in this study, in the region with frequency ofPLL higher than 100 Hz, the real part of the pole becomespositive, thereby causing instability. Moreover, the minimumfrequency of PLL ωPLL (f PLL in the verification conditions)needed to secure stability, calculated using equation (28) withthe verification conditions of this study including unexpecteddisturbance torque Δtmax, pole pair number P, and inertia J,is 11.4 Hz (71.4 rad/s), and in Figs. 8(a)∼(c) the region be-low the dotted line shows the unstable region. That means,



(a) Root locus, f ACR = 16 Hz (d) Ideal Transient analysis, f ACR = 16 Hz (g) Transient analysis f ACR = 16 Hz(Simulating real condition,)

(b) Root locus, f ACR = 64 Hz (e) Ideal Transient analysis, f ACR = 64 Hz (h) Transient analysis f ACR = 64 Hz(Simulating real condition,)

(c) Root locus, f ACR = 256 Hz (f) Ideal Transient analysis, f ACR = 256 Hz (i) Transient analysis f ACR = 256 Hz(Simulating real condition,)

Fig. 8. The comparison of root locus analysis and transient analysis when frequencies of ACR, ASR, PLL are changed

in the analysis of placement of poles, the region of stabilityis constrained even furthe. Corresponding to the aforemen-tioned results of stability analysis, plots of ωerr obtained intransient response tests under ideal conditions (ideal transientanalysis) are shown in Figs. 8(d)∼(f). The transient analysisresults showed that, with increase in frequency of ACR, thetrend of change in range of frequency of ASR and frequencyof PLL to enable stable operation, closely agreed with theresults of pole (placement) analysis. For example, as shownFig. 8 it is observed that, in the case of frequency of PLL of64 Hz, the frequency of ASR needed for stable operation un-der frequency of ACR of 64 Hz is lower than 4 Hz, and inthe corresponding transient analysis results also, for a fre-quency lower than 8 Hz the speed change corresponding toinput command is smaller, enabling stable operation withoutdiverging. In the pole placement analysis with frequency ofACR of 256 Hz, the stable region is similarly below 4 Hz,while in the transient analysis, the stable region is below16 Hz, which is slightly larger, but the trend is similar.

Next, the results of analysis of lower limit of frequency ofPLL are compared. Substituting the verification conditionsfrom Table 1 in equation (30), the calculated frequency ofPLL for stable operation is above 11.4 Hz (71.4 rad/s), andsince the transient analysis results as shown in Figs. 8(d)∼(f)also indicate that the region above 16 Hz is stable, togetherthese results confirm the usefulness of the present method.

In contrast, the upper limit of frequency PLL for stableoperation constrained by the low pass filter of axial error,obtained from pole placement analysis is about two timessmaller than that obtained from transient analysis. One likelyreason for this is that, as shown in Fig. 5, the movements ofthe poles A and B near the real axis are very large, and even aslight difference in frequency of PLL can cause reaching theright-half plane making the operation unstable. In both thepole placement analysis and the transient analysis the resolu-tion of frequency of PLL was set as multiple of 2, and it isconsidered likely that nearly similar results could be obtainedby setting a higher resolution.



5.4 Verification Results Assuming Real ConditionsLastly, the results of changing the motor and inverter model

under ideal conditions to a more realistic model are comparedbe referring to Figs. 8(g)∼(i). Compared to the stable rangeshown in Figs. 8(d)∼(f), a slight decrease can be observed.Although it may be considered that the range of stable fre-quency has narrowed due to inclusion of nonlinear featuressince the proposed method assumes an ideal linear modelof motor and inverter, the trends observed are nearly simi-lar confirming that even when considering real conditions theproposed method is useful. One reason for this change instable range is that, since significant difference is observedin case of f ACR = 256 Hz, it may be due to the increase indisturbance in current control caused by deadtime error andmotor harmonics, and the authors would like to report on thisin detail in a subsequent paper.

6. Conclusion

The effect of frequency on stability in permanent magnetsynchronous motor sensorless speed control system was stud-ied for three types of control, namely, current control, speedcontrol, and sensorless control. The characteristics of theproposed method of stability analysis are described as fol-lows.(1) The range of the frequency of the three controls thatenable stable sensorless control can be quantitatively deter-mined by analyzing the poles of the closed-loop transferfunction from the speed command value to the estimatedspeed.(2) The stable range of frequency considering transient statessuch as under step load, can be calculated by combining poleplacement analysis with stability judgement based on upperlimit of axial error.

In the aforementioned investigation, drive verification wasundertaken through actual sensorless control operation, andin addition to ideal conditions, under actual conditions as-suming real motors, it was confirmed that similar resultscould be obtained. The authors would like to study the ef-fect of the nonlinear features on stability in the future.

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Sari Maekawa (Member) received a Master’s degree in electrical en-gineering from Hosei University. In April of the sameyear, he went to work for Toshiba. In April 2014,he received a Doctorate Degree from the GraduateSchool of Science and Technology at Meiji Univer-sity. In 2019, he joined the Faculty of Science andTechnology of Seikei University, as an associate pro-fessor. He is primarily engaged in the research anddevelopment of power electronics, EMI, and motordrives. In 2013, Dr. Maekawa received an IEEJ Best

Paper Award (Award A) and an IEEJ Industry Applications Society BestPresentation Award. He also received the Ichimura Industry Award from theNew Technology Development Foundation in 2014. His Doctorate degree isin engineering. Dr. Maekawa is a member of IEEE.



Mariko Sugimoto (Non-member) received the B.E., M.E. degrees inelectrical engineering from Tokyo Institute of Tech-nology, Japan, in 2004, and 2006, respectively. She iswith Toshiba Corporation, Japan, and has been work-ing on the development of permanent magnet syn-chronous motor and magnetic application products.

Keiichi Ishida (Member) received the B.E., M.E. degrees in electri-cal engineering from Nagaoka University of Tech-nology, Japan, in 2002, and 2004, respectively. Heis with Toshiba Carrier Corporation, Japan, and hasbeen working on the development of the inverter forair conditioners. Mr. Ishida received the Institute ofElectrical Engineers of Japan (IEEJ) Excellent Pre-sentation Award in 2007. He also received the Elec-trical Science and Engineering Promotion Awards in2016.

Masaya Nogi (Member) received the associate degree in electrical en-gineering from National Institute of Technology, Nu-mazu College, Japan, in 2002. He is with ToshibaCarrier Corporation, Japan, and has been working onthe development of the inverter for air conditioner.

Masaki Kanamori (Member) received the B.E., M.E. degrees in elec-trical engineering from Shizuoka University, Japan,in 2007, and 2009, respectively. He is with ToshibaCarrier Corporation, Japan, and has been working onthe development of the inverter for air conditioner.


Documents

Stability Analysis of Sensorless Speed Control for PMSM