Figure 1: The Essential Components Of An Induction Energy Meter. (file: energy1.gif)

A voltage coil of several thousand turns of fine copper wire is wound around the center leg of the E-core of the voltage magnet. A smaller number of about 10 to 20 turns of thick copper wire are wound around the two legs of the U-shaped current magnet. In most industrial applications, the load current lags the applied voltage. This is called a lagging power-factor load. The induction energy-meter is normally calibrated for this type of loading. The induced emf due to the voltage coil is 180 deg. out-of-phase w.r.t. the main voltage. Since the disc is predominantly resistive, therefore the eddy current is in phase with the induced emf. The load current is predominantly lagging behind the main voltage. The flux due to this load current enters the disc normally and acts on the eddy current generated by the voltage coil creating time varying but periodic forces. The phase relation of this flux w.r.t. to the eddy current is almost exactly the same as the power-factor relation between the main voltage and load current. For unity power-factor, the torque generated is maximum and of course for zero power-factor no real power is consumed and thus the rotor should stay stationary although it seldom does. This is because the eddy current in the rotating disc encounters a resistive path which consumes a small amount of real power which rightfully should be recorded.

In the investigation, we are only interested in per unit magnitudes of harmonic torques w.r.t. the fundamental torque, i.e., the torque generated by the 50 Hz voltage and current. Harmonic torques are generated by the electromagnetic interactions between the 50 Hz main voltage and integer multiples of 50 Hz load current. Fundamental to 5th harmonic torques are computed for three choices of power-factors at 1.0, 0.866, and 0.5.


2. Formula For Harmonic Forces

The formula is developed using sequence algebra [1,2]. Torque and force differ only by a constant radius term and are used interchangeably in this paper. Only circumsferential forces will generate useful torques. Radial forces can be ignored although these do cause wears by vibrations of the spindle in the jewelled bearings. The formula developed assumes that the applied voltage will always stay at 50 Hz as a pure sinusoidal waveform. On the other hand, the load current could deviate from pure sinusoidality by superposition of higher harmonic currents. The sites of torque generation are confined to two positions marked as x and xx along the circumsference where the current flux will interact with the eddy current (see figure 1). For a perfectly uniform airgap the force generated at these two positions will be perfectly identical. Since this state of affair seldom prevails, the unsymmetrical airgap causes unsymmetrical tangential forces which could affect the accurcay of the induction energy meter. We only investigate the ideal case of uniform airgap. The two forces will always be in the same direction for each harmonic current component although the directions could alternate between clockwise and anticlockwise directions. The formula developed to predict the harmonic forces contains two proportionality constants k1 and k2 since the geometrical factor and circuit parameters are unknown. Currentv represents eddy current induced by the voltage coil and Fluxi represents the flux generated by the mmf of the current coil. To derive the force equation, we do not take the sequence product (or cartesian product) of Currentv and Fluxi but the scalar product, i.e., products taken with terms of the same order. The force equation Force in equation (3) is derived from equations (1) and (2). Note that in equation (2), the frequency f has an addition integer multiplier m > 1 to take care of harmonic current.

................................./iv sin(2 pi f dt k)........iv sin(2 pi f dt k)\
.......Currentv := k1 |------------------- - ----------------- |
.................................|............x^t ...................xx^t...............| .....................(1).
.................................\........................................................./

............................../iphi sin(2 pi m*f dt k - theta).....iphi sin(2 pi m*f dt k - theta)..\
............Fluxi := k2 |----------------------------- - ------------------------------- |
..............................|...................x^t...........................................xx^t.....................| .........(2).
..............................\............................................................................................./

......................./ iv sin(2 pi f dt k)...........................iv sin(2 pi f dt k)..................\
.......................| *iphi sin(2 pi m*f dt k - theta)......*iphi sin(2 pi m*f dt k - theta) |
Force := k1k2 |----------------------------- - -----------------------------.......|
.......................|.....................x^t ........................................xx^t.........................| ..............(3).
.......................\................................................................................................./

where iv = eddy current induced by voltage coil.
iphi=load current in current coil.
pi = 3.14159265 rad.
m*f = integer multiples of fundamental frequence of 50 Hz. m =1,2.....
dt = time interval increment of 0.0001 second. 100 *dt = T = 0.01 second.
T = period = 0.01 second.
k1,k2 are proportionality factors related to the geometry and circuit parameters of the meter. iv and iphi are converted into magnetic fluxes which interact as eddy currents within the rotor disc. The relations are too complicated and are taken care of by two linear constants. This is admissible because of the presence of a large air-gap.
m = harmonic multiplier factor. m = 1 for 50 hz, m=2 for second harmonic ...
theta = phase lag angle in radians of load current w.r.t. main voltage.
x, xx = the two positions inside the rotor disc where electromagnetic interactions take place.
t = discrete time = k*dt. Normally you read time as follows: x^1/100 means time at position x = 1/100 second. Of course both x and xx must post the same time t since there is no space-time distortion in an induction energy meter.

Equation (3) has an unconventional feature in the denominator for x^t and xx^t. If we keep x and xx as order variables, then forces at x and xx will be expressed as a discrete function of time t. On the other hand, by putting x and xx equal to unity, all the forces will be summed algebraically to give the total forces acting circumferentially on the rotor. Since 100 samples have been taken within a period T, therefore the average torque is total torque divided by 100. This dual purpose served by equation (3) is a handy trick not encountered in conventional algebra.

In section 3, we will compute these forces (or torques) for fundamental and harmonic components.


3. Computations

The force (or torque) formula in equation (3) is computed using Maple V R 3 based on following program line. The average force or torque is computed by putting x=xx=1. The values are reported in per unit w.r.t. the fundamental torque at p.f.=1.

sort(eval(sum(iv*sin(2*pi*f*dt*k)/(x^(dt*k))*iphi*sin(2*pi*m*f*dt*k- theta)+iv*sin(2*pi*f*dt*k)/(xx^(dt*k))*iphi*sin(2*pi*m*f*dt*k- theta),k=0..100))); ......(4).

Case (1): Fundamental Frequency Operation

(i): f = 50, P.F. = 1, ie. theta = 0, iv = iphi = 1 per unit.

We give sample full computations only for the fundamental and 5th harmonic component. Table 1 summarises all computations carried out. Instantaneous torques are unidirectional and peak at T/2=1/200 second. All peaks and troughs are marked in bold fonts. Only the peak components are of interest so that there is no point to display all the 100 points within a period. In the denominator of each term, the x and xx indicate positions of force generation and the indices the discrete time of occurrence of these time-varying forces.

.0009866357835 .0009866357835 .003942649334 .003942649334 .008856374615 .008856374615
-------------- .......+ --------------..... + ------------- .....+ ------------- .....+ -------------.... + -------------
.........1/10000...............1/10000..............1/5000................1/5000.............3/10000..........3/10000
.......x.........................xx........................x........................xx........................x....................xx

+....................................................................................+.......

.9990133641 .9990133641........1. ..............1. ..... .9990133643 .9990133643 .9960573508
+ ----------- .+ ----------- ......+ ------ .....+ -------.. + ----------- ...+ ----------- ...+ -----------
...........49..................49............1/200........1/200............51...................51..................13
...........-----..............-----.........x...............xx...................-----...............-----................----
........10000.............10000............................................10000............10000..............2500
......x.....................xx..................................................x...................xx.....................x

+........................................................................................+............

..........................-16................................-16
.1288661550*10........ .1288661550*10
+ ----------------- ........+ -----------------
............1/100...........................1/100
...........x................................xx

The average force can be summed algebraically by putting x = xx = 1 and divide by 100. This gives:
Average force for fundamental current component = 1.0 per unit.

(ii): f = 50, P.F. = .8660254040 , ie. theta = 30 deg. = .5235987751c, iv = iphi = 1 per unit.

.01484317820 .01484317820 .02791887385 .02791887385 .03917548315 .03917548315
- ------------ .... - ------------ ... - ------------ ... - ------------ .... - ------------ .... - ------------
........1/10000..........1/10000...........1/5000.............1/5000...........3/10000...........3/10000
.......x....................xx.....................x.....................xx...................x......................xx

+.................................................+..................

.06523372762 .06523372762 .06687763972 .06687763972 .06654871303 .06654871303
- ------------.... - ------------ .....- ------------..... - ------------..... - ------------..... - ------------
........7/10000...........7/10000.............1/1250............1/1250..........9/10000.........9/10000
......x.....................xx........................x.....................xx...................x....................xx

+........................................................................+....................

.9329030437 .9329030437 .9325741169 .9325741169 .9302736499 .9302736499
+ ----------- .. + ----------- .. + ----------- .. + ----------- .. + ----------- . + -----------
..........29...................29..................59....................59................3/500.............3/500
..........----.................----................-----...................-----............x...................xx
........5000................5000.............10000..............10000
......x.....................xx..................x.....................xx

+....................................................+.........................

........................... -8............................-8
.1794896629*10..... .1794896629*10
+ ----------------..... + ----------------
..............1/100.....................1/100
............x...........................xx

The average torque can be found by putting x = xx = 1 and divide by 100. iv=iphi=1 per unit. This gives:
Average force = 0.8660254050 per unit.

(iii): f = 50, P.F. = .5 , ie. theta = 60 deg. = 1.047197550c, iv = iphi = 1 per unit.

.02669577458 .02669577458 .05229955734 .05229955734 .07671030185 .07671030185
- ------------ .... - ------------ .... - ------------ .... - ------------ .... - ------------ .... - ------------
..........1/10000..........1/10000.............1/5000............1/5000..........3/10000..........3/10000
.........x...................xx........................x....................xx...................x......................xx

+...............................................................+..............

.2495614145 .2495614145 .2498903412 .2498903412 .2482464291 .2482464291
- ----------- ... - ----------- ... - ----------- ... - ----------- ... - ----------- ... - -----------
.....1/625..............1/625...................17.................17.................9/5000.........9/5000
....x...................xx...........................-----................-----............x..................xx
.....................................................10000.............10000
...................................................x....................xx

+.........................................................+..................

.7498903423 .7498903423 .7482464303 .7482464303 .7446361672 .7446361672
+ ----------- .. + ----------- .. + ----------- .. + ----------- .. + ----------- .. + -----------
............67.....................67....................17..................17...............69..................69
...........-----.................-----...................----.................----..............-----................-----
.........10000...............10000...............2500..............2500...........10000...........10000
........x....................xx........................x...................xx..................x................xx

+....................................................................+.............

.......................... -8..........................-8
.3108852144*10 .3108852144*10
+ ---------------- .. + ----------------
.................1/100...................1/100
...............x........................xx

The average force can be found by putting x = xx = 1 and divide by 100. iv=iphi=1 per unit. This gives: Average force = 0.5 unit.

==============================================================

Case (2): 5th Harmonic Current Operations

(i): f = 250, P.F. = 1, ie. theta = 0, iv = iphi = 1 per unit.

5th harmonic, m = 5. There are five peaks/troughs with the largest positive peak at T/2 = 1/200. The other four peaks are also marked in bold fonts and are symmetrically displaced in time about the largest peak. The signs sequence is + - + - +.

.004913725283 .004913725283 .01940333758 .01940333758 .04272428010 .04272428010
------------- ......+ ------------- .....+ ------------ ....+ ------------ ...+ ------------ ....+ ------------
..........1/10000..........1/10000............1/5000...............1/5000...........3/10000............3/10000
........x....................xx.......................x.......................xx....................x........................xx

+.............................................................+...............

.3538613616 .3538613616 .3444626830 .3444626830 .3210197613 .3210197613
+ ----------- ..+ ----------- ..+ ----------- ....+ -----------... + -----------... + -----------
...........13................13.............7/5000.............7/5000............3/2000..............3/2000
...........-----............-----..........x.....................xx.....................x.....................xx
........10000..........10000
.......x.................xx

+.........................................................+......................

.8090169937 .8168978399 .8168978399 .8030035764 .8030035764 .7669267636
- -----------... - -----------..... - ----------- ...- ----------- ....- -----------.... - -----------
..........3/1000...........31..................31...............2/625.............2/625................33
.......xx....................-----..................-----..........x..................xx.........................-----
.............................10000............10000.........................................................10000
...........................x...................xx...............................................................x

+...................................................+..............

..........1.........................1
+ --------------- + ----------------- +
............1/200..................1/200
.........x.......................xx

+...................................................+..............

.7669267589 .7669267589 .8030035735 .8030035735 .8168978392 .8168978392
- -----------.... - ----------- ....- -----------....- ----------- ....- -----------.... - -----------
.........67....................67..................17...................17...................69...................69
........-----................-----.................----...................----...............-----.................-----
.......10000.............10000............2500................2500.............10000.............10000
.....x...................xx....................x....................xx...................x......................xx

+..........................................................................+.....................

.2831677005 .3210197568 .3210197568 .3444626824 .3444626824 .3538613614
+ ----------- ..+ -----------... + ----------- ...+ ----------- ...+ -----------... + -----------
...........21...................17...............17...................43..................43..................87
...........----.................----..............----.................----.................----................-----
.........2500..............2000............2000...............5000...............5000.............10000
......xx...................x..................xx.....................x....................xx...................x

.3538613614 .3501072556 .3501072556 .3345674959 .3345674959 .3090169974
+ ----------- ..+ ----------- ...+ ----------- ...+ -----------... + ----------- ...+ -----------
.........87.....................11.................11..................89...............89.................9/1000
........----- ..................---- ...............----............... ----- ...........-----............x
.......10000...............1250.............1250.............10000.........10000
....xx......................x..................xx...................x................xx

+...........................................................................+....................

....................................................-16.......................-16
.004913726243 .6443307748*10 .6443307748*10
+ ------------- ...+ ----------------- ..+ -----------------
................99................1/100...............1/100
................-----............x....................xx
..............10000
...........xx

.......................................-9

Average force = -.134*10 per unit.

(ii): f = 250, P.F. = .8660254040 , ie. theta = 30 deg. = .5235987751c, iv = iphi = 1 per unit.

Comments: Only the forces at position xx are shown as those at x are identical.

.01125660930 .01305488306 .004925248566 .01310089025 .04048821876 .07621484669
- ------------.... - ------------..... - -------------..... + ------------ ....+ ------------ ....+ ------------
.......1/10000...........1/5000..............3/10000............1/2500............1/2000.............3/5000
.....xx....................xx.....................xx......................xx....................xx....................xx

+....................................................................................+....................

.3966036128 .4234468277 .4385211485 .4401038807 .4269180536 .3981969105
+ ----------- ...+ ----------- ...+ ----------- ...+ -----------... + ----------- ...+ -----------
.......13....................7/5000..........3/2000..........1/625.................17...............9/5000
..... -----...............xx..................xx.................xx.......................... ----- .........xx
....10000..................................................................................10000
..xx.......................................................................................xx

+........................................................................+.............................

.7721462330 .8258773331 .8595624094 .8715061803 .8606462142 .8266006453
- -----------.... - -----------.... - -----------.... - ----------- ....- -----------.... - -----------
...........31..............2/625................33...................17.............7/2000...........9/2500
...........----- .......xx.........................-----.................----........xx...................xx
........10000.................................10000...............5000
.....xx.......................................xx.....................xx

+........................................................................................+...........................

.7767624813 .8660253996 .9331197577 .9762174481 .9941975812 .9866797944
+ ----------- ...+ -----------... + ----------- ...+ ----------- ...+ ----------- ...+ -----------
...........49.................1/200................51..................13..................53..................27
...........----- ..........xx.........................-----...............----................-----................----
........10000...................................10000..............2500.............10000.............5000
.....xx.........................................xx...................xx..................xx..................xx

+................................................................................+........................

.4687936974 .5649656472 .6427623241 .7006292657 .7376732504 .7536756785
- -----------... - ----------- .....- -----------... - ----------- .....- ----------- ....- -----------
.............67................17.................69................7/1000...............71...............9/1250
............-----................----.............-----............xx.........................-----.........xx
.........10000............2500.............10000....................................10000
......xx.................xx.................xx............................................xx

+.......................................................+........................

.1175013829 .1731800351 .2163022394 .2463234016 .2632488060 .2676165675
+ ----------- ...+ -----------... + -----------... + -----------... + -----------... + -----------
....................17.................43..................87................11.................89............9/1000
....................----...............----................-----..............----...............-----........xx
..................2000.............5000............10000...........1250............10000
...............xx.................xx................xx..................xx.................xx

+...............................................+........................

..................................................................................................-8
.07892587580 .04666245300 .01976743359 .1794896689*10
+ ------------ ...+ ------------... + ------------... + ---------------- ...+
..........97.....................49.....................99......................1/100
.........-----..................----....................-----..................xx
.......10000................5000................10000
....xx......................xx....................xx

......................................-9
Average force = -.10*10 per unit.

(iii): f = 250, P.F. = .5 , ie. theta = 60 deg. = 1.047197550c, iv = iphi = 1 per unit.

Comments: Only the forces at position x are shown as those at xx are identical.

.02441074451 .04201505832 .05125506085 .05097761856 .04048821878 .01958668081
- ------------ ...- ------------..... - ------------..... - ------------..... - ------------..... - ------------
.....1/10000.............1/5000..............3/10000............1/2500..............1/2000...........3/5000
....x........................x......................x........................x......................x.......................x

+..........................................................................+................

.3330762464 .3889687371 .4385211480 .4791145765 .5083437917 .5241176936
+ ----------- ...+ -----------... + -----------... + ----------- ...+ ----------- ...+ -----------
........13.................7/5000...........3/2000............1/625................17..............9/5000
........-----............x....................x......................x.........................-----............x
......10000..................................................................................10000
.....x.........................................................................................x

.5247500394 .5090369607 .4763182419 .4265199016 .3601762572 .2784303936
+ ----------- ...+ ----------- ...+ ----------- ...+ ----------- ...+ ----------- ...+ -----------
..........19.................1/500...............21...................11...................23..............3/1250
..........-----.............x......................-----.................----...................-----..........x
.......10000..................................10000..............5000...............10000
......x.........................................x......................x....................x

+...................................................+..................

.9164968748 .9094586395 .8783878710 .8236391061 .7462883628 .6481086423
- ----------- ...- -----------..... - -----------.... - -----------.... - -----------.... - -----------
..........37..............19...................39................1/250.................41..................21
.........-----.............----................-----.................x.....................-----................----
.......10000..........5000.............10000.....................................10000.............5000
.....x..................x...................x.............................................x.....................x

+.................................................+........................

.9540336746 .9769061778 .9745793027 .9474172974 .8965113921 .8236391087
+ -----------... + -----------... + ----------- ...+ ----------- ...+ -----------... + -----------
............11..............7/1250..............57...................29.................59.................3/500
............----...........x.........................-----................----...............-----..............x
..........2000...................................10000..............5000............10000
........x..........................................x.....................x..................x

+...................................................+................

.6290960721 .6659459779 .6830127016 .6807970863 .6604055626 .6234947494
- -----------... - -----------.... - -----------.... - -----------.... - -----------.... - -----------
.........73....................37.................3/400.............19.....................77...............39
........-----...................---- ............x.......................----..................-----..............----
......10000...............5000...................................2500.................10000..........5000
....x.......................x.........................................x.......................x..................x

+..............................................+................

.1755747909 .1848126026 .1829503164 .1711813510 .1511040921 .1246466482
+ ----------- ...+ -----------... + ----------- ...+ ----------- ...+ ----------- ...+ -----------
..........91...................23.................93....................47..................19..................6/625
.........-----.................----...............-----..................----.................----...............x
.......10000..............2500............10000...............5000..............2000
......x......................x...................x.......................x...................x

..........................................................................................................-8
.09397934420 .06141839961 .02932447286 .3108852170*10
+ ------------ ...+ ------------ ...+ ------------... + ---------------- ...+
...........97......................49...................99.......................1/100
...........-----...................----.................-----...................x
.........10000................5000..............10000
.......x........................x....................x

.....................................-9
Average force = -.84*10 per unit.



5. Table Of Average Torques From Fundamental to 5th Harmonic Currents
Only the average forces are listed in Table 1 below. Forces are expressed in an per unit since we are only interested in relative magnitudes referred to the fundamental force at pf=1 as the base. Note that forces and torques are interchangeable terms in this paper since these differ by a constant of radius only.

==================================================

(i) Fundament frequency current
pf = 1,
Average Force = 1 per unit
pf = 0.866,
Average Force = 0.866 per unit
pf =0.5,
Average Force = 0.5 per unit
--------------------------------------------------------------------------------

(ii) Second Harmonic Current
pf = 1,
Average Force = .22*10^(-9) per unit
pf = 0.866,
Average Force = .2122589527 per unit
pf = 0.5,
Average Force = .3676432900 per unit
----------------------------------------------------------------------------

(iii) Third Harmonic Current
pf = 1,
Average Force = -.65*10^(-11) per unit
pf = 0.866,
Average Force = -.5*10^(-11) per unit
pf = 0.5,
Average Force = -.1*10^(-10) per unit
--------------------------------------------------------------------------------

(iv) Fourth Harmonic CurrentM
pf = 1,
Average Force = -.49*10^(-10) per unit
pf = 0.866,
Average Force = .04249371981 per unit
pf = 0.5,
Average Force = .07360128181 per unit
--------------------------------------------------------------------------

(v) Fifth Harmonic Current
pf=1,
Average Force = -0.134*10^(-9) per unit
pf=0.866,
Average Force = -0.1*10^(-9) per unit
pf = 0.5,
Average Force = 0.84*10^(-9) per unit
==================================================

From Table 1, it can be seen that both the 2nd and 4th harmonic currents contribute surplus average forces (or torques) to the induction energy meter which increase with increasing lagging power-factor. Considering that the fundamental force (or torque) is falling with increasing power-factors, these contributions are significant and cannot be neglected. On the other hand, third and fifth harmonic current contribute negligible forces (or torques) to the induction energy meter. In other words, the induction enery meter will register real power due to the 2nd and 4th harmonics but not the 3rd and 5th harmonics.


4. Conclusions

The computations show that even harmonics contribute significant amount of torques to the induction energy meter. It would be difficult to calibrate an energy meter to be accurate to both the fundamental and the 2nd harmonic real power. However, one can neglect the effect of odd harmonics on the energy meter. Another finding is that the number of peaks/trough per fundamental period is equal to the maximum harmonic order in the load current. The reason why odd harmonic current contributes negligible average torque is because the forward and backward torques per period almost exactly cancel. The computing of average torque by summation of torques at discrete time is justifiable because the period of 0.01 second is very short and because of the inertia of the rotor disc which acts as an integrator.

5. Reference:

1. Huen Y.K.: A matrix map for primes and nonprimes, Int. J. Math.Educ.Sci.Technol., 1994, Vol.25, No.6, pp 913 - 920.

2. Huen Y.K.: Visual algebra and its applications, Int. J. Math. Educ. Sci. Technol., 1997, Vol.28, No.3, 333-344.

======================== END OF PAPER ======================