How Does Phase Resistance Change Torque And Speed Bldc

i. Introduction

The brushless DC motor is quickly becoming more popular than the brushed DC motor. Information technology needs less maintenance and has a longer life span since it contains no wearable brushes, and has flat speed-torque characteristics, higher efficiency, and meliorate thermal treatment every bit windings on the stator allow constructive heat dissipation and better dynamic response with lower rotor inertia. The brushless DC motor is more expensive, however, and has a more complex controller blueprint [1]. It also has safety problems, demagnetisation problems, and a high risk of inverter failure [ii]. The differences between brushed and brushless DC motors are that the latter has its windings built on the stator and permanent magnets on the rotor and commutation is done electronically instead of via brushes. There are two kinds of brushless motors; namely, brushless AC motors and brushless DC motors. The divergence between the brushless AC motor (also known every bit the permanent magnet synchronous motor) and the brushless DC motor depend on the current waveform supplied; the brushless AC motor is fed by sinusoidal current whereas the brushless DC motor is fed past rectangular electric current [3].

Enquiry has been conducted to measure out torque ripple in a brushless DC motor [4, 5]. Papers that draw the adding of electromagnetic torque produced by a brushless DC motor using the data of stage current solitary are, notwithstanding uncommon. Motor torque can be measured directly by a torque sensor, which is expensive and sometimes may be too bulky to be implemented in certain applications. Therefore, electromagnetic torque estimations with measurable parameters such as back EMF, rotor speed and phase current are highly desirable. The electromagnetic torque of brushless DC motor tin exist estimated by measuring phase currents [half dozen]. It was claimed past [6] that at to the lowest degree 2 current sensors were required to estimate electromagnetic torque. Also, DC bus current can also be used to guess the output torque as shown by [seven]. Instantaneous torque was calculated by multiplying the torque constant with DC link current with the aid of lineto- line back EMF shape function in [eight, nine]. Reference [x] derived an analytical equation from classical synchronous motor theory which correlated electromagnetic torque with motor speed. The equation required information such as back EMF abiding, phase resistance and inductance, motor speed and DC source voltage to depict electromagnetic torque. Reference [11] estimated instantaneous electromagnetic torque with back EMF and rotor speed whereby back EMF was estimated with a sliding-mode observer and rotor speed was estimated with a simplified extended Kalman filter.

This paper describes how electromagnetic torque produced past a three-stage brushless DC motor can be calculated through single-phase electric current sensing. An equation has been derived to calculate electromagnetic torque from the theory of the brushless DC motor. In conjunction, an algorithm has been implemented in an Arduino microcontroller in order to deport out the electromagnetic torque measurement in real-time. Motor dynamometer testing has been conducted to verify the capability of the proposed equation and the measurement algorithm.

It is interesting to note that the abc (stage variable) reference frame of a permanent magnet synchronous motor is usually transformed to the dq0 (space vector) reference frame to simplify the analysis of a three-stage excursion fed past sinusoidal moving ridge (hence sinusoidal dorsum EMF). dq0 transformation is not appropriate and has not been implemented in this paper because the brushless DC motor is fed past rectangular wave and has a trapezoidal back EMF.

ii. Mathematics of the Brushless DC Motor

A schematic diagram of a brushless DC motor drive (wye-connected) is shown in Fig. 1. The equations shown in the latter function of this newspaper are based on this system. Wye-connection is chosen in this report because it is applied in nigh applications thanks to its simpler commutation logic, which is less prone to bridge-shorting [12].

Fig. 1.Schematic of brushless DC motor bulldoze.

A few assumptions are fabricated in club to reduce the mathematical complexity of the brushless DC motor drive. They are: (one) symmetrical three-phase winding, (2) no magnetic saturation, (three) no hysteresis and eddy electric current losses, (four) compatible air-gap, (v) mutual inductance is ignored, and (6) armature reaction is ignored.

Mathematical model of armature winding is

where

Va, Vb, Vc - terminal voltages of phase a, b, and c [5] ia, ib, ic - stator electric current of phase a, b, and c [A] ea,eb,ec - back EMF of phase a, b, and c[5]

L - per phase armature self-inductance [H] R - per stage armature resistance [Ω]

The back EMF are displaced by 120 electrical degrees from one phase to another, and they can be expressed every bit

where

ωm - mechanical rotor speed [rad.s−1] Ke - back emf constant [V/rad.s−1] f(θe) - trapezoidal function θe - electrical angle of rotor [rad]

Subtracting (two) from (1) and (3) from (2) yield

For a wye-connected three-phase winding, the full phase electric current is naught according to Kirchoff'due south Current Police force, thus

Substituting ic from (9) to (viii) gives

The electromagnetic torque produced by a brushless DC motor can be expressed equally

or

where

Te - electromagnetic torque [Nm] Kt - torque constant [Nm/A]

Under steady-country performance, electromagnetic torque volition exist counter-balanced by load torque, inertia torque and friction torque. Therefore,

where

TL - load torque [Nm] J - inertia of the rotor and coupled shaft [kgm2] β - friction factor [Nm.due south.rad−1]

Eqs. (7, 10), and (13) can be transformed into Laplace domain with the Differentiation Theorem, which can then be expressed in the form of state-space equations giving a multiple-input-multiple-output (MIMO) correlation which is crucial in computer simulation. State space equations of the brushless DC motor are

3. Adding of Electromagnetic Torque

Theoretically, the dorsum EMF of each phase of a brushless DC motor has a trapezoidal waveform and the current has a pulse-like waveform which alternates its direction every one-half moving ridge, every bit shown in Fig. 2.

Fig. 2.Waveforms of back EMFs and currents of the three phases as a function of electrical angle (degree).

The waveforms showed in Fig. 2 result from the switching sequence shown in Tabular array 1. The switching sequence shows the motor is rotating in the clockwise direction. Referring to Fig. 1, S1 to S6 are power switches (e.g. MOSFET, IGBT, etc.) and D1 to D6 are the associated diodes.

Tabular array 1.Switching sequence (clockwise direction)

Assuming the windings of the three phases are symmetrical, the magnitudes of dorsum EMFs and currents should be equal for the three phases. From Eq. (11) and Fig. 2 it tin can be seen that the electromagnetic torque adult by a brushless DC motor at any instant is

where

ep - phase back EMF [V] ip - not-zero stage current [A]

Too, Eq. (16) can too be written equally

The accented sign has to exist implemented in Eq. (17) because multiplication of stage back EMF and phase current in Eq. (sixteen) gives simply a positive value. In other words, stage back EMF and phase electric current are e'er in phase for the motoring way. Assuming the absolute in Eq. (17) is particularly important in the real-fourth dimension application, as lack of the absolute operation will give negative torque value because of the alternating nature of stage current as can exist seen in Fig. two.

Eq. (17) is theoretically correct, merely it is not directly implementable in the real-time awarding because a consummate single phase electric current waveform is equanimous of 240 electrical degrees of non-zero ampere and 120 electrical degrees of zero ampere. These zeroes' current moment will give an incorrect issue, as shown in Fig. 3.

Fig. three.Per-unit electromagnetic torque derived from Eq. (17).

In order to get rid of the error introduced by Eq. (17) and to obtain a torque value as in the actual condition, some mail-processing is required. The suggested postal service-processing is expressed as

in which α is an average value of electromagnetic torque over a number of sampling information. The abiding 0.75 is derived from the fact that the alternating square pulse (phase current) is equanimous of 2/3 high pulse and 1/3 low pulse; only the average magnitude of high pulse is of interest and it has to be one-half of the magnitude of the high pulse, so the abiding 0.75 is obtained by dividing 0.v by 2/iii. The derivation of the constant 0.75 is illustrated in Fig. 4.

Fig. 4.Per-unit torque, average torque (α), and 0.75 of average torque (0.75α) of the brushless DC motor.

The thought backside Eq. (18) is to find an average value produced by Eq. (17) and capsize the low side of the calculated electromagnetic torque to the high side nearly that boilerplate level. Without violating the theory of the brushless DC motor, Eq. (eighteen) gives a precise estimation of the electromagnetic torque produced past a brushless DC motor using information on the single-phase current lone.

Eq. (eighteen) is theoretically verified by comparing the result generated past Eq. (18) and the result generated by Eq. (eleven). The results are shown in Figs. five(a) and (b).

Fig. 5.(a) Adding of torque with Eq. (18); (b) Calculation of torque with Eq. (11).

Figs. 5(a) and (b) bear witness identical results and hence Eq. (18) is able to summate electromagnetic torque correctly past theory.

four. Experimental Verification of Torque Estimation Equation

Section 3 theoretically shows how Eq. (18) correctly calculates the electromagnetic torque produced past a brushless DC motor. In club to verify the validity of Eq. (xviii), laboratory experiments need to exist conducted.

A brushless DC motor of one.2 kW is tested with a motor dynamometer to validate Eq. (18). Measurement shows that the respective brushless DC motor has a back EMF abiding and hence a torque constant of 0.07 Nm/A (V/rad.s−one). Measurement of motor parameters similar phase resistance, inductance, and rotor inertia is not needed because these parameters are irrelevant in the evaluation of electromagnetic torque. The motor dynamometer test set-up is shown in Fig. vi. For ameliorate interpretation, it is expressed equally a schematic diagram in Fig. 7.

Fig. vi.Experimental set-up.

Fig. 7.Schematic diagram of the experimental setup.

Eq. (18) is implemented with an Arduino micro-controller (a low price open up source microcontroller) in an algorithmic mode to estimate the instantaneous electro-magnetic torque. The algorithm used for instantaneous electromagnetic torque estimation is shown in Fig. 8.

Fig. 8.Algorithm implemented in the Arduino microcontroller for torque estimation.

The symbol i shown in Fig. viii indicates the current number of iterations. The symbol northward indicates the total number of iterations which also represents the number of sampling information used in the calculation of α. The average value (α) is expressed as:

The value of n should exist carefully selected so that it is large plenty to smooth out the measurements and small enough to be utilised for real-time application.

The loop determination block (Loop?) is added for the sake of completeness of the algorithm. It indicates the termination criteria of a measurement. For example, the algorithm tin be designed in such a way that it volition learn 10 readings and and so exit; then the loop conclusion cake will exist "No" upon 10 cycles of looping. It has no significant in the determination of electromagnetic torque.

With the algorithm shown in Fig. 8, the torque is estimated at a motor speed of 3150 RPM with a load of 0.30kW and is shown in Fig. 9. In the same figure, the actual torque generated by the brushless DC motor is superimposed. Since the brushless DC motor has 89% efficiency at 3150rpm, the electrical power produced by the motor is 0.33kW. The term efficiency has taken the load torque imposed by the dynamometer, motor friction and inertia into account as described by Eq. (xiii).

Fig. nine.The electromagnetic torque produced and estimated (Eq. 18) at 3150 rpm with 0.30kW loading.

The brushless DC motor has been tested under various speeds and loads with the motor dynamometer; results obtained for a motor speed of 1960 rpm with a load of 0.14kW and speed of 1550rpm with a load of 0.094kW are shown in Figs. x and 11 respectively.

Fig. ten.The electromagnetic torque produced and estimated at 1960 rpm with 0.14kW loading.

Fig. 11.The electromagnetic torque produced and estimated at 1550 rpm with 0.094kW loading.

five. Discussion of Experimental Results

There are several types of dynamometer; for case, engine dynamometer, motor dynamometer and chassis dynamometer. The motor dynamometer is commonly used to measure the force, torque, or power of an electric motor, and it is used in this inquiry. It is observed from Figs. 9, 10 and xi that the electromagnetic torque evaluated with Eq. (18) is comparable to the actual torque developed to counterbalance load torque imposed by the dynamometer. Therefore, Eq. (eighteen) can approximate the electromagnetic torque accurately with single-phase current measurement alone, which is price-effective and computationally efficient.

Theoretically, electromagnetic torque developed should be of a constant value. Measurement showed, notwithstanding, that there was torque fluctuation nearly a mean value, which can be seen from Fig. 9, 10 and 11. This fluctuation is known as ripples in the context of the brushless DC motor. Torque ripple is unavoidable because the phase flux linkage and current in stator windings are not sinusoidal [four]. The greatest torque fluctuation is institute to exist 0.025Nm in Fig. 9, 0.033Nm in Fig. 10, and 0.032Nm in Fig. 11, which correspond to 2.5%, 3.9% and 4.5% of measurement deviation respectively. The average of the estimated value showed that the bodily and the calculated electromagnetic torque were identical in each test.

Intendance should be taken while implementing Eq. (18) because it has the following limitations

i. Applicative to three-stage brushless DC motor only. 2. Identical dorsum EMF spatial distribution with 120 electrical degrees offset among the three phases. 3. Half-wave symmetry and cipher boilerplate back EMF and excitation current. iv. Constant and identical phase resistance and inductance with negligible mutual inductance among motor phases. 5. Excitation currents are in phase with their respective phase back EMF.

Limitation 1 can be extended to the multi-stage brushless DC motor by using the technique discussed in department three as long every bit condition 4 and condition 5 are fulfilled. In that case, condition 2 and status iii must be adjusted accordingly.

6. Conclusions

There are several ways in which electromagnetic torque can exist evaluated, and the simplest solution is through torque sensor measurement. A torque sensor is very expensive, still, and sometimes it may be as well bulky for sure applications. Therefore, torque interpretation with measurable parameters such as back EMF, currents and rotor speed is highly desirable. This paper shows how the electromagnetic torque of a brushless DC motor can exist evaluated with unmarried-phase current measurement alone and with proper mathematical manipulation of the theory of the brushless DC motor. A flow chart has been proposed to implement the electromagnetic torque estimation equation in an algorithmic way. This algorithm can exist used as it is or it can be plugged into whatever algorithm for estimation of brushless DC motor electromagnetic torque. The measurement technique suggested in this newspaper is cost effective because only 1 current sensor is needed, information technology is measurement-friendly with minimum intrusion in the system of interest compared with the torque sensor, and it is computationally constructive since no complex mathematics are involved. Furthermore, only the motor torque constant is required to evaluate the electromagnetic torque; no other parameters such every bit winding resistances, inductances, etc. are needed. The equation and the algorithm suggested accept been verified experimentally and prove a high caste of accuracy in evaluating electromagnetic torque in real-time.

Source: http://koreascience.or.kr/article/JAKO201414938221834.page

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