Novel current sensor for PWM AC drives

J.T. Boys

Indexing terms: Inverters, Induction motors

Abstract: A novel current sensor which deter-mines the currents flowing in an inverter driveninduction motor from observations of the pulsecurrents existing in the DC busbar of the PWMinverter is described. The sensor can determine thereal part of the current, and, if the motor is sup-plying mechanical power, the current magnitude.It is low cost and easy to implement, and experi-mental results show good accuracy over a range ofoperating conditions.

List of principal symbols

a

i

hlavIktTVR,0

4>i

= modulation depth= harmonic amplitudes= line current phasor magnitude

IY, lB = u n e currents= instantaneous filter-and-hold current= average filter-and-hold current= filter-and-hold current magnitude= pulse height sampling fraction= pulse width= time between pulses

VY, VB— phase to neutral voltages= angle variable= motor current phase angle

Introduction

Pulse width modulated inverters are now accepted as areliable means for controlling the speed of three phaseAC induction motors in industrial and otherenvironments. These inverters conventionally use uncon-trolled rectification of the incoming 3-phase AC supply toproduce a DC supply (typically 500-650 V) which is theninverted with six power switching devices to producewaveforms suitable for driving the motor. The powerswitching devices are switched in such a way that boththe output magnitude and frequency of the motor voltageare varied to give whatever control option is required.With modern inverters, power transistors are convenient-ly chosen as the switching elements and microprocessorsare used to determine the switching instants [1-3].

A fundamental aim of all AC drives is to achieve therequired motor output without exceeding either theshort-term or the long-term motor and inverter ratings.These restraints inevitably mean that a number ofworking currents within the drive must be monitored. Inparticular, the switched currents in the inverter must bemeasured and checked for overload and shoot through

Paper 5723B (P1/P6), first received 15th June and in revised form 7thSeptember 1987The author is with the Department of Electrical and Electronic Engin-eering, University of Auckland, New Zealand

conditions, and the motor currents must be measured forcontrol and motor protection purposes. All of these cur-rents are difficult to measure in the presence of high di/dtand dv/dt switching transients. Further, for a 3-phasemotor at least two motor line currents must be measured(the third is often also measured to detect a fault-to-ground condition) and it is difficult to get current sensorswith equal gains over the wide range of frequencies, volt-ages and currents used in a practical inverter. Theproblem is exacerbated if the motor windings are not per-fectly balanced or if the current sensors have some DCoffset.

In this paper, a novel circuit configuration called a'filter-and-hold' circuit is used to determine the motorcurrent by measurements of the current in the DC busbarof a sinusoidal PWM inverter using switching waveformsgenerated by the triangulation method [1-3]. The tech-nique is applicable to transistorised inverters with negli-gible snubber charging currents; other circuits mayrequire some additional circuitry. Only one sensor is usedso that intrinsically all three phase currents are measuredwith the same gain and no DC offset can occur. The samesensor output can also be used to detect a shoot-throughcondition and, subject to some restraints, a fault-to-ground. The natural output from the circuit is the realpart of the motor current but under most conditionswhere the drive is operating as a motor producingmechanical power and not regenerating, the magnitude ofthe motor current can also be determined with goodaccuracy.

The analytical method presented in this paper isapplicable only to sinusoidal PWM inverters, but workto extend the technique to harmonic elimination and dis-tortion minimised PWM (described in [4-6]) is currentlyunder way. The intention is to find the motor current formotor control purposes and not the DC busbar currentfor subsequent analysis [7].

2 The filter-and-hold technique

We consider an application where an electrical signalconsists of a series of pulses and the information is con-veyed essentially by the average amplitude of thosepulses. The filter-and-hold technique proposed in thispaper enables such information to be extracted.

A simple pulse train suitable for use with the techniqueis shown in Fig. 1 and consists of pulses of variableheight, variable period and variable duty cycle. If theinformation required from this pulse train is the averagepulse amplitude a, simple lowpass filtering giving anaverage output axt/T may be an acceptable solution pro-vided that t/T is constant; in many circumstances,however, this is not the case.

Consider the circuit shown in Fig. 2. Switch S is closedwhenever a pulse is present, otherwise it is open, thus thecircuit acts as a lowpass filter during time t (for the pulse

IEE PROCEEDINGS, Vol. 135, Pt. B, No. I, JANUARY 1988 27

train of Fig. 1), and simply holds that value for the timewhen no pulses are present. The output of the circuitsettles to the value at irrespective of t or T. When the

I I

'1 -

Fig. 1 Pulse trains suitable for use with filter-and-hold technique

output

input1+S7

Fig. 2 A filter-and-hold circuit

amplitude of the pulse train changes to a2 the circuitoutput also changes to a2 as required. During thesechanges, however, the output can change only whilepulses are actually present (only during t) so that theeffective filter time constant is magnified by the switchingaction in inverse proportion to the duty cycle of the pulsetrain. Pulse trains with low duty cycles have consequentlylarge time constants associated with them. The physicalRC time constant of the filter should be chosen for theparticular application: it should be large compared with tto give a stable output but if it is too large then the mag-nified effective time constant will cause the circuit to havea sluggish response.

3 Motor current measurements in practicalinverter systems

In the typical PWM inverter system shown schematicallyin Fig. 3, incoming power from the mains is rectified and

3-phase*

50 Hz «

input «

dioderectifier

filtercapacitor 3-phase

inverter

SCIM

Fig. 3 Simple PWM inverter system showing where DC busbar cur-rents can be measured

used to power a DC busbar. A 3-phase transistorised (forexample) inverter switches between the positive and nega-tive rails of this busbar to synthesise the required outputvoltages for driving the motor. In these switching pro-cesses discontinuous currents flow from the DC busbarthrough the inverter to the motor, and these segments ofthe motor current that exist in the DC busbar are of theform shown in Fig. 1, so that the filter-and-hold tech-nique is appropriate for examining them. Note that if ashoot-through fault condition should occur in one leg ofthe 3-phase inverter the fault current will be detected bythe sensor and some protection strategy can be imple-mented. Thus the one sensor can serve a dual role.

To apply the filter-and-hold technique in this way asensor in either side of the DC busbar is required to givean output in the form of a series of pulses. This sensor

should have a good frequency response to follow faith-fully the current pulses and should also be accurate downto DC. An exact analysis showing the effect of subjectingthe current pulses thus detected to a filter-and-holdcircuit is intractable. But under the assumptions that theswitching rate is much higher than the modulating sine-wave frequency, that the ripple currents are small andthat there are no snubber charging currents, a simplifiedanalysis is relatively straightforward. For this analysis weconsider true sinusoidal PWM produced by the triangu-lation process, preferentially with regular sampling [8]and with a modulation depth less than 1. As will beshown subsequently, the errors are negligible up tomodulation depths of 2 and a justification for keeping themodulation depth below 2 is given in Section 5.

The 3-phase PWM voltage waveforms produced bythe triangulation process and the associated currentpulses in the DC busbar are then as shown in Fig. 4 for a

modulatingtrianglewaveform

l I ioutput PWM

voltage

waveforms

l I

I I i

AA

DC busbarcurrent pulses

Fig. 4 Relationship between PWM voltages and DC busbar currents

positive going transition of the generating triangle wave-form. The triangle wave is here assumed to have a peakamplitude of ± 1 while the three phase sinewaves vary inboth amplitude and frequency.

For the switching sequence shown, current flows in theDC busbar only when all the output PWM voltages areneither high nor low. Thus during times t% and t2 currentflows in the DC busbar. During tx the blue phase is at adifferent voltage from those of the other two phases, sothat the DC current must be a sample of the blue phasecurrent IB. Similarly during t2 the busbar current is asample of IY • On the negative transition of the trianglewaveform exactly the same sampling process takes place.For these pulses the filter-and-hold circuit of Fig. 2 isused to take the average of the busbar current while itexists and simply hold that average while the busbarcurrent is at zero: all switches high or all switches low.

28 IEE PROCEEDINGS, Vol. 135, Pt. B, No. 1, JANUARY 1988

In the particular case shown in Fig. 4, the DC busbarcurrent pulse is made up from samples of two of themotor phase currents, the blue and the yellow phases,corresponding to the instantaneous phase voltages whichat that time are the most positive or the most negative.As the modulation process proceeds in time, differentpairs of the motor currents are sampled, at different timescorresponding to each n/3 phase change in the modulat-ing waveforms. For a balanced machine under steady-state conditions essentially the same pattern of pulsestherefore occurs at six times the modulating frequency,thereby simplifying the analysis.

Thus for the condition shown, the instantaneous'filter-and-hold' current output I0 is given using the nota-tion of Fig. 4 by the weighted average of the currentpulses as follows:

t , • (-.

+ h)(1)

and the average filter-and-hold current Iav is then givenby

3 f+7t/6

iav = - /71 J-n/6

(2)

where the integral has been evaluated over a limitedrange as it repeats each n/3 (of the modulatingwaveforms).

With the assumptions given the triangle wave is at ahigh frequency so that

t2az{VY-VR)

where VR, RY and VB refer to the instantaneous voltagesbeing used to produce the PWM waveform. For regular-ly sampled PWM these proportionalities are exact evenfor modest to low switching frequencies. If VR, VY, VB aresinewaves of peak value a and the current waveforms areassumed to be balanced with phase angle <$>, then

a sin (0 + n/6) • lY + a sin (0 + 5n/6) • (-IB)

Now

IY - i sin (0 + n/3 - (f>)

and

- 7 B = is in(0 + 2 7 r / 3 - 0

so that, on simplifying,11/6 cos n/6 cos

I - n / 6 COS 0dd

In 3 cos <f> = 0.9085i cos 0

(5)

(6)

Thus the filter-and-hold current in the DC busbar isdirectly proportional to the real component of the motorcurrent.

3.1 Effect of ripple currentIn all practical inverter systems motor currents are notperfectly sinusoidal so that sampled currents in the DCbusbar are not segments of sinewaves. Provided,however, that the switching rate is reasonably high, thedeviations from a perfect sinewave are small and occur

essentially equally above and below the ideal sinewave.Furthermore, with these high switching rates both thesinewave and the actual current waveform can beapproximated by straight line segments for each switch-ing state. The integrated charges J idt given by the areaunder these straight line approximations for the real DCbusbar current and the ideal sinusoidal sample are essen-tially equal, so that the technique maintains reasonableaccuracy. In practice, therefore, the sensor responds tothe essentially sinusoidal component of the motor currentand ignores the distortion or ripple components.

As the frequency and output voltage from the inverterchange during the normal operation of the drive, the DCbusbar current pulse duty cycle varies so that the effectivetime constant of the circuit varies. In general, however,for an AC induction motor, low duty cycles correspondto low voltage amplitudes at low frequencies while highduty cycles correspond to high amplitudes and high fre-quencies. Thus the circuit automatically has more filter-ing action (a large effective time constant) at lowfrequencies and less filtering action at high frequencies.This characteristic is usually beneficial.

3.2 Curren t magnitude measuremen tsThe actual current pulses used by the filter-and-holdcircuit are determined or produced by the PWM switch-ing sequence itself, as shown in Fig. 4. For most motorloading conditions, however, some samples of currentcorrespond to the peaks of the current waveform and toform a magnitude estimate largely independent of themotor current displacement factor (the apparent powerfactor of the fundamental component of the motorcurrent) these current segments must be preferentially se-lected at the expense of lower currents. If i corresponds tothe maximum or peak of the assumed sinusoidal funda-mental current then by eliminating all currents belowi/y/(2) we have for example:

IY = i sin (0 + n/3 - <f>) -n/6 < 6 < + n/6

-IB = i sin (0 + 2n/3 - -n/6 < 6 < + n/6

In calculating the output of the filter-and-hold circuit theactual integration limits must be chosen (with the 1/^/2restraint) so that the argument of the sine functions isalways between n/4 and 3n/4. Thus for IY the lower limitis ( — n/Yl + (p) while for — IB there is no restraint provid-ed that <j) ^ n/4.

The filter-and-hold integral for the current magnitudenow becomes

I = —

*

n/6 cos (n/6) cos 6i dO

-it/12 +<t> cos 6f-K/12 + * "I

- /B ^J —Ji/6 J

(7)

where the first expression is exactly as before (eqn. 5) butover a limited integration range, and the second expres-sion obtains over the remainder of the range.

Evaluating these integrals yields

/ =2n

[J3 cos ((j> - 7t/12)1cos <p In T :—— ——

— (sin7T

s m (8)

For displacement factors of greater than 0.7 this expres-sion varies by less than 1 % (/ ~ 0.925i) so that the

IEE PROCEEDINGS, Vol. 135, Pt. B, No. 1, JANUARY 1988 29

observed value is reasonably independent of the displace-ment factor of the load.

3.3 Effect of overmodulationAt higher modulation frequencies the PWM waveformsmust be allowed to saturate progressively to get thehigher voltages that are needed to fully flux the motor.The true sinusoidal nature of the PWM disappears andthe DC busbar current becomes continuous. Surprisingly,however, the filter and hold current expressions maintaingood accuracy up to modulation depths of 2. Forexample, at a modulation depth of 2 the switchingpattern gives a continuous DC busbar current alternatelyselecting between two phase currents. For this or anyovermodulated condition any instantaneous sinewavemagnitudes greater than 1 may be considered to beclamped at 1 so that the filter-and-hold output currentmay be easily expressed. At the modulation depth of twowe have

3[/y(0.5 + sin 0) + (-/B)(0.5 - sin 0)] dd

- 7 T / 6

The integrations yield

Iav = - i cosn - = 0.913/ cos 0

(9)

(10)

expressionwhich is only a 0.5% difference from thederived previously for true sinusoidal PWM.

The same procedure can also be used for determiningthe magnitude of the motor current under overmodulatedconditions. The integrations show no degradation inaccuracy as the displacement factor of the motor currentvaries from 1 to 0.7 at modulation depths up to 2. Atpoorer displacement factors the accuracy of the techniquedeteriorates but the analysis adds little value as theexpressions become cumbersome.

4 Practical current measurement method for aPWM AC drive

With a PWM inverter driving an AC induction motorthe filter-and-hold techniques described above provide aneasy and convenient method for measuring the motorcurrents. To determine the real part of the motor currentthe circuit of Fig. 2 may be used directly (with an appro-priate current sensor) by simply opening switch S when-ever all three PWM switches are in the same state, eitherall high or all low. The real part of the motor current isuseful for torque and speed control purposes [9, 10].

For protection purposes and some other control stra-tegies, however, measurement of the motor current mag-nitude may be more appropriate and the methodoutlined here involves selecting the pieces of the DCbusbar current which are to be included in the filter-and-hold averaging process. In cases where the displacementfactor is unknown (the usual condition) logical selectionis difficult, but an alternative strategy shown in Fig. 5enables the measurement to be made by using an essen-tially recursive analogue circuit.

Here switch S is held closed whenever the input pulseheight is greater than some proportion k of the circuit'soutput. For essentially sinusoidal current samples a selec-tion angle ±6 relative to the crest of the sinewave isequivalent to k = 6 cot (0) so that, for 6 = n/4 as before,then k = 0.785.

In the presence of distorted current pulses the ripplecurrents cause switch S to have an imperfect switchingaction so that some current segments are selected when

alternativeoutput

output

inputpulses

(l-k)R

kR

Fig. 5 Pulse selecting filter-and-hold circuit for selecting pulses or por-tions of pulses of magnitude greater than k times the average output

they should be excluded while others are excluded whenthey should be included. In practice it has been foundthat these errors are small and of little consequence sothat the motor current measured by this method is sur-prisingly accurate. The actual ratio k is not very criticaland a value of 2/3 has been found to give excellentresults. High values for k (approaching 1) should beavoided in a practical inverter as in that case a suddenreduction in the motor current may leave the circuit in alatched-up state where the incoming pulses are too smallto trigger switch S into conduction.

4.1 Practical measurementsFrom the analysis presented, the real part of the motorcurrent is easy to determine with relatively high con-fidence as the approximations are small, but the expres-sions for the average current are not as rigorous.Experimentally, however, the current magnitude can bemeasured by alternative means with high accuracy whileour resources do not allow any correlation at all for mea-surements of the real part. Accordingly, the results pre-sented here are for current magnitude measurements. Theaccuracy for these measurements is shown to be reason-able so that it may be inferred that the real part measure-ment accuracy will be high.

The operation of the motor current sensor using thecircuit of Fig. 5 is illustrated in Fig. 6. The lower wave-form shows the observed DC busbar current over an

8.0 ms

Fig. 6 Observed waveforms obtained with circuit of Fig. 5: both wave-forms are DC coupled with the same zero (as shown)a Alternative output waveform (with two times higher gain, for clarity)b Input pulses from the DC bus current sensor

8 ms period for an inverter operating at an output fre-quency of 38 Hz, with a switching frequency of 798 Hz.The upper waveform shows the alternative output over

30 IEE PROCEEDINGS, Vol. 135, Pt. B, No. 1, JANUARY 1988

the same period with a k value of 0.67 (the alternativeoutput is used here to show more clearly the current seg-ments that are being selected, but the average values foreither output are essentially identical). The circuit prefer-entially selects larger pulses (or portions of pulses) at theexpense of lower pulses. Note too that all the waveformsare essentially linear segments, so that the approx-imations involved in the analysis are reasonably valid.The actual output waveform is the same as the alterna-tive output waveform over the constant hold portionsand is filtered by the RC time constant at other times(during the essentially sawtooth segments). For thesemeasurements the RC filter time constant was chosen tobe 2 ms which is large compared with the pulse durationbut short compared with the machine time constants.

The accuracy of the technique is shown in Fig. 7 for arange of measurements at 25 and 50 Hz with switching

An ideal PWM modulator provides an output voltagewhich is essentially the same as the reference voltage usedto generate the PWM waveforms, so that for modulationdepths a > 1 the ideal output would be a truncated sine-wave, closely resembling a trapezoidal waveform, with apeak amplitude of 1. In this case (a > 1), simple Fourieranalysis of the ideal output gives spectral componentsgiven by

4 \a sin ((n — l)a)n \(n- l)(n + 1)

1

n(ncos (na) (11)

where

a = sin

10 r

3 0.8

0.6

0.4

0.2

00.04 0.2 0.4 0.6torque . p.u.

0.800 0.04 0.2 0.4 0.6

torque, p.u.

a b

Fig. 7 Comparison of true and indicated motor currents as motor load varies from 0 to full loada 25 Hz

0.8 1.0

frequencies of 798 and 900 Hz, respectively. Since 'powerfactor' has little meaning, these measurements were madeon an induction motor with a changing mechanical load.The particular machine used was a 7.5 kW 4 pole induc-tion motor with a no-load current of 0.4 times the ratedcurrent and a no-load power factor of ~0.1 at 50 Hz.For the measurements presented here the modulationdepths were 0.58 at 25 Hz and 1.40 at 50 Hz to give thecorrect rated flux in the machine at full load. As shownthe technique gives value within 2% FS of the true valuefrom full load to 40% of full load. The true current wasdetermined by using a current probe with a spectrumanalyser to determine the magnitude of the fundamentalcomponent and is essentially the useful working currentin the motor and not the effective RMS current. Themethod has been found to be accurate over the full speedrange (from slip frequency to rated frequency) providedthat the iron is not saturated. It is best when the motor isreasonably loaded.

5 Discussion and conclusions

The filter-and-hold techniques outlined in this paper pro-vides a low-cost current-measuring system for 3-phasesinusoidal PWM inverters subject to the restraint thatthe modulation depth is less than two. At higher modula-tion depths more than one phase is in continuous con-duction and the technique loses accuracy.

while for a ^ 1 the ideal output is a single component ofequal magnitude to the reference sinewave. The impor-tant harmonics are the 1st, 5th, 7th, 11th, 13th, ..., andthese are shown plotted against the modulation depth ain Fig. 8. As shown, at a modulation depth of 2 the total

modulation depth

Fig. 8 Ideal harmonic amplitudes as a function of modulation depth forsinusoidal PWM

IEE PROCEEDINGS, Vol. 135, Pt. B, No. 1, JANUARY 1988 31

harmonic distortion is approximately 4.5% while the fun-damental amplitude is 95.7% of the maximum possibleamplitude. As a increases to 4 the fundamental (n = 1)component increases by 3%, but the total harmonic dis-tortion increases to 15%. Similarly if a is allowed toincrease to infinity the fundamental output increases byanother 1 % at the expense of a total harmonic distortionof greater than 30% (theoretically J(n2/9 - 1). Thusabove a modulation depth of 2 the distortion increases ata rate approximately 6 times faster than the (useful) fun-damental component, while for modulation depths of < 2the distortion is essentially negligible. In our opinionthere is therefore good cause for maintaining a modula-tion depth of less than 2 and we have implemented thisrestraint in a practical inverter to allow the current mea-surement system to be used.

In our application the scalar magnitude of the motorcurrent is used to control the motor and the measure-ment technique described here works extremely well. Ourcontroller uses a novel variable flux approach where withlight loads the motor flux is reduced thereby keeping thedisplacement factor high. Under this strategy the currentsensor gives better than 2% accuracy from DC to ratedfrequency for torque values from 0.1 p.u. to 1.1 p.u.Without this strategy the errors at low currents may needto be compensated for. Such compensation is easy with amicroprocessor-controlled inverter using a table-look-uptechnique, but is likely to be more difficult with otherinverters. In a general case the current-magnitudemethods cannot be used for an overhauling load, but inpractice this may not be a problem as often motor

control strategies must then be modified to control theDC busbar voltage so that the actual motor current islargely irrelevant. The technique for producing the realpart of the motor current is of course applicable under allconditions.

6 References

1 GRANT, D.A., HOULDSWORTH, J.A., and LOWER, K.N.: 'Anew high quality PWM AC drive', IEEE Trans., 1983, IA-19, pp.211-216

2 BOSE, B.K., and HUNT, S.A.: 'A high performance pulsewidthmodulator for an inverter-fed drive system using a microcomputer',ibid., pp. 235-243

3 VARNOVITSKY, M.: 'A microcomputer based control signal gen-erator for a three-phase switching power inverter', ibid., pp. 228-235

4 POLLMANN, A.: 'A digital pulsewidth modulator employingadvanced modulation techniques', ibid., pp. 409-414

5 DE BUCK, F., GISTELINCK, P., and DE BACKER, D.: 'Loss-optimal PWM waveforms for variable-speed induction-motordrives', IEE Proc. B, 1983,130, (5), pp. 310-320

6 TAKAHASHI, I., and MOCAIKAWA, H.: 'A new control of PWMinverter waveform for minimum loss operation of an inductionmotor drive', IEEE Trans., 1985, IA-21, pp. 580-587

7 EVANS, P.D., and HILL-COTTINGHAM, R.J.: 'DC link currentin PWM inverters', IEE Proc. B, 1986,133, (4), pp. 217-224

8 BOWES, S.R., and BIRD, B.M.: 'Novel approach to the analysisand synthesis of modulation processes in power converters', IEEProc. B, 1975,122, (5), pp. 507-513

9 POTTEBAUM, J.R.: 'Optimal characteristics of a variable-frequency centrifugal pump motor drive', IEEE Trans., 1984, IA-20,pp. 23-31

10 ABBONDANTI, A.: 'Method of control in motors driven by vari-able frequency, variable voltage supplies'. IEEE/IAS Intl. Sem.,Power Conv., Conf., 1977, pp. 177-184

32 IEE PROCEEDINGS, Vol. 135, Pt. B, No. I, JANUARY 1988