Download Chapter-6 Electrical power and its Components I

Chapter-6
Electrical Power & Its Components.
6.1 The Measurement of Power
In direct-current circuits power may be measured either by a watt-meter or by an
ammeter and voltmeter, the product of whose readings gives the power in the circuit. In
alternating-current circuits the power at any instant is given by :
p = e i…………………………………… (11.6.1)
Where p = instantaneous power
e = instantaneous voltage
i = instantaneous current
Thus, if both the current and voltage waves are sinusoidal, the current lagging in
phase by an angle Φ, then
e = Emax Sin wt
i = Imax Sin (wt-Φ)
The instantaneous power p is therefore given by
p=ei
= Emax Imax Sin wt Sin (wt-Φ)
Or, writing θ for wt
= Emax Imax Sin θ Sin (θ -Φ)
The mean power is :
P= 1
2Π
∫02 Π Emax Imax Sin θ Sin (θ -Φ) d θ……….(11.6.2)
Limits 0 to 2 Π
= Emax Imax
2Π
= Emax Imax
4Π
∫02 Π Cos Φ Cos (2θ -Φ) d θ
2
[ θ Cos Φ - Sin (2θ -Φ) ] 02 Π
2
6-1
= Emax Imax
Cos Φ
2
or
P= E I
Cos Φ ………………………………..(11.6.3)
Where E and I are r.m.s. values of voltage and current.
The fact that the power factor (Cos Φ) is involved in the expression for the power
means that a wattmeter must be used instead of merely an ammeter and voltmeter, since the
latter method takes no account of power factor.
6.2 Wattmeter Measurements in Single-Phase A.C. Circuits.
Following Figure (Fig-6.1) shows a wattmeter connected in such a circuit. The
“current coil” of the instrument caries the load current, while the “voltage coil” carries a
current proportional to, and in phase with, the voltage.
Fig-6.1
Wattmeter connections
6-2
The deflection of the wattmeter depends upon the currents in these two coils and upon
the power factor. Inductance in the voltage-coil should be avoided as far as possible, since it
causes the voltage-coil current to lag behind the applied voltage. A high non-inductive
resistance is connected in series with the voltage coil in order that the reactance of the coil
itself shall be small in comparison with the resistance of the whole voltage-coil circuit and
also, of course, to ensure that the current taken by the voltage-coil shall be small.
6.2.1 Wattmeter Errors.
(a)
VOLTAGE-COIL INDUCTANCE
rp = resistance of the voltage-coil
If
lp = inductance of the voltage-coil
R = resistance in series with voltage-coil
V = voltage applied to the voltage-coil circuit
Then the current through the voltage coil is
ip =
V
( rp + R )2 + ω2 lp2
The phase of this current relative to the voltage is lagging by a small angle such
that:
tan β =
ω lp
rp + R
By increasing the non-inductive resistance R, this angle is reduced, although the
current ip is reduced by such increase. The effect of an increase of frequency is to increase β
( since it increases ω), and to reduce the current ip, slightly, by its effect in increasing the
impedance of the voltage-coil circuit.
6-3
Thus, if α is the phase angle (lagging) of the load circuit, the wattmeter deflection is
proportional to
I ip cos (α – β)
I V cos (α – β)
i.e.
Zp
Where Zp is the impedance of the voltage-coil circuit.
Zp = rp + R
cos ( β )
Thus the deflection is proportional to
I
cos ( β ) cos (α – β)
V
(rp + R)
If the inductance of the voltage-coil circuit was zero, the deflection would be
proportional to
I
V
cos (α )
(rp + R)
And the wattmeter would read correctly at all frequencies and power factors.
The ratio of the true reading of the instrument to the actual reading is therefore,
I
V
cos (α )
cos (α )
(rp + R)
=
I
=
cos ( β ) cos (α – β)
cos ( β ) cos (α – β)
V
(rp + R)
Therefore, true reading =
cos (α )
X
cos ( β ) cos (α – β)
6-4
Actual reading. …….(11.6.4)
Correction Factor.
The “correction factor” by which the actual reading must be multiplied, in order to
obtain the true reading, is
cos (α )
.
cos ( β ) cos (α – β)
The wattmeter will read high on lagging power factors of the load, since the effect of
the inductance of the voltage-coil circuit is to bring the current in it more nearly into phase
with the load current than would be the case if this inductance were zero. If the power factor
of the load is very low, a serious error may be introduced by voltage-coil inductance unless
special precautions are taken to reduce its effect.
Fig-6.2 gives curves showing the variation in the value of the correction factors as the
power factor of the load varies, the phase angle β of the voltage-coil circuit being 1o in one
case and 0o in the other.
Fig-6.2 : Wattmeter Correction Factors
The error, in terms of the actual instrument deflection, is;
Actual reading – True reading
6-5
[ 1 -
cos (α )
] X
Actual reading.
cos ( β ) cos (α – β)
[ 1 -
cos (α )
] X
Actual reading.
cos (α – β)
if cos β is assumed equal to unity.
Therefore, Error = [ 1 -
cos (α )
] X
Actual reading.
cos (α ) + sin α sin β
=
sin α sin β
[
] X
Actual reading.
cos (α ) + sin α sin β
=
[
sin β
] X
Actual reading…..(11.6.5)
cot (α ) + sin β
(b)
VOLTAGE-COIL CAPACITANCE.
The voltage-coil circuit may have capacitance as well as inductance, this being
largely due to inter-turn capacitance in the high-value series resistor. The effect produced is
similar to that of inductance in this circuit, except that the voltage-coil current tends to lead
the applied voltage instead of to lag behind it. This causes the wattmeter to read low, on
lagging power factors of the load, by increasing the angle between the load and voltage-coil
currents.
The effect of frequency will be, of course, to vary the phase angle between V and ip,
the angle increasing with increase of frequency.
If the capacitive reactance of the voltage-coil circuit is equal to its inductive
reactance, there will be no error due to these effects since the two individual errors will
neutralize one another.
6-6
(c)
EDDY CURRENTS ERROR.
Eddy currents induced in the solid metal parts of the instrument, by the alternating
magnetic field of the current coil, alter the magnitude and phase of the field, and so produce
an error. The phase of the induced eddy e.m.f.s. will be 90o behind the inducing flux – i.e.
rather more than 90o behind the main current in the current coil. The eddy current is
practically in phase with its e.m.f., and this current set up a magnetic field which, combined
with that of the current-coil produces a resultant magnetic field which is less than that of the
current-coil alone and which also lags behind the current-coil field by a small angle.
This eddy current error is not easily calculatable, and may be serious if care is not
taken to ensure that any solid metal parts (which should be avoided as far as possible) are
well removed from the current-coil. If the current–coil is designed for heavy currents, it
should consist of stranded conductors in order to minimize the eddy currents flowing in the
current-coil itself.
6.3
METHODS OF WATTMETER CONNECTIONS IN
THE CIRCUIT.
There are two obvious methods of connecting a wattmeter in circuit as shown
in Fig-6.3, in which C.C. and V.C. indicate current-coil and voltage-coil respectively. Neither
measures the power in the load directly, without correction, even neglecting the errors
discussed above.
Fig-6.3 Alternative Wattmeter Connections.
6-7
In the method of diagram (a), in which the voltage-coil is connected on the “supply”
side of the current-coil, the voltage applied to the voltage coil is higher than that of the load
on account of the voltage drop in the current coil. In diagram (b) the current-coil carries the
small current taken by the voltage-coil, in addition to the load current.
In the first case, the instrument measures the power, I2 Rc, lost in the current coil, and
in the second case the power lost in the voltage coil, as well as the power in the load.
If the load current is small, the voltage drop in the current coil is small, so that the
first method of connection introduces a very small error. On the other hand, if the load
current is large, the power lost in the voltage coil will be small compared with the power in
the load, and the second method of connection is better.
6.3.1
COMPENSATION FOR POWER LOSS IN VOLTAGE
COIL.
In some wattmeters, a compensating coil is used to eliminate error due to current-coil
carrying the voltage-coil current in addition to the load current, when the connections are as
shown in Fig-6.3 (b).
This compensating coil is as nearly as possible identical and coincident with the
current-coil, so that if it were connected in series with the latter, and a current passed through
the two coils-connected so that their magnetic effects were in opposition-the resultant
magnetic field would be zero.
Actually the compensating coil is connected in series with the voltage-coil, but in
such a way that the magnetic effect opposes that of current-coil and neutralizes the voltagecoil component of the current in the current-coil. Thus, if no-load current flows in the
instrument, the deflection should be zero, since the resultant current-coil field should be zero.
The connections of this method of compensation are shown in Fig: 6.4 below.
6-8
Fig: 6.4 Connections of compensating coil.
6.4
THE USE OF INSTRUMENT TRANSFORMERS
WITH WATTMETERS.
Current and voltage transformers may be used with wattmeters just as they are used
with ammeters and voltmeters. By using a number of current transformers, of different ratios,
for the supply of wattmeter current-coil, and a number of voltage transformers for its voltagecoil, the same wattmeter may be used to cover a very large range of power measurements.
The connections of the wattmeter when so used are shown in Fig: 6.5, in which an ammeter
and a voltmeter are connected in circuit, supplied from the same transformers as the
wattmeter.
Fig-6.5 : Use of Instrument Transformers with A Wattmeter
6-9
When such transformers are used, corrections must be applied to allow for their ratio
and phase angle errors. Fig: 6.6 gives the vector diagram (a) refers to load with the lagging
power factor, and the diagram (b) to the load with the leading power factor.
Fig: 6.6 Vector diagrams for the currents and voltages of the load, and
in the wattmeter coils.
It is assumed that both current and voltage transformers are used.
In the diagram:E = Voltage of the load.
I = Load current.
a = Phase angle between load current and voltage.
γ = Phase angle between the currents in the current and voltage-coils of the
wattmeter.
Is = Current in the wattmeter current-coil.
= Secondary current of the current transformer.
Es = Voltage applied to wattmeter voltage-coil.
ip = Current in wattmeter voltage-coil.
β = Angle by which ip lags Es on account of inductance of voltage-coil.
δ = Phase angle of voltage transformer.
θ = Phase angle of current transformer.
6-10
The vectors shown dotted are E and I reversed.
Obviously, when the load has a lagging phase angle a, we have
a=γ+θ+δ+β
or, more strictly,
a=γ+θ+δ+β
Since the phase angle of the voltage transformer may be either lagging or leading.
In the case of leading power factor of the load,
a=γ+δ-θ-β
Correction Factor.
The correction factors-neglecting, for the present, the ratios of the transformersbecome:
Cos a
Cos β Cos ( a - θ + δ - β )
in the case of lagging power factor of the load, and
Cos a
Cos β Cos ( a + δ + θ + β )
in the case of leading power factor of the load.
Writing K for the correction factor, we have for the general expression for the power
to be measured,
Power = K X Wattmeter reading X Actual ratio of current transformer X
Actual ratio of voltage transformer……………………(11.6.6)
The transformer ratios to be used in the expression are the actual ratios as distinct
from the nominal ratios. These ratios are not constant but depend upon the load conditions,
so the calibration curves of the transformers are necessary if accurate power measurement are
to be made. It should be noted, also, that in the above the power losses in the instruments are
not considered.
6-11
6.5
MEASUREMENT OF POWER WITHOUT USING A
WATTMETER.
It is possible to measure the power in a circuit without a wattmeter by using either
three voltmeters or three ammeters, in conjunction with non-inductive resistors.
6.5.1
THREE-VOLTMETER METHOD.
The connections are as shown in Fig: 6.7, in which V1, V2, V3 are the three
voltmeters and R is a non-inductive resistor which is connected in series with the
load.
Fig: 6.7 Three-voltmeter method of measuring single phase power.
From the vector diagram of Fig: 6.7 (b) we have,
V12 = V22 V32 + 2 V2 V3 cos Φ
Neglecting the currents taken by voltmeters V2 and V3, the current in R is the same as
the load current I. Thus V2 = I R and
V12 = V22 V32 + 2 I R V3 cos Φ
Now, I V3 cos Φ is the power in the load, so that
Power in load = I V3 cos Φ
= V12- V22 -V32 ……….(11.6.7)
2R
The power factor is given by;
cos Φ = V12- V22 -V32 ………………………… (11.6.8)
2 V2 V3
The assumptions are made that the current in the resistor R is the same as the load
current, and this resistor is entirely non-inductive.
6-12
6.5.2
THREE-AMMETER METHOD.
This method is somewhat similar to the above. The necessary connections are shown in
Fig: 6.8.
Fig: 6.8 Three-Ammeter method of measuring single phase power
The current measured by ammeter A1 is the vector sum of the load current and that
taken by the non-inductive resistor R (this latter being in phase with the voltage E). From the
vector diagram,
I12 = I22 + I32 + 2 I2 I3 cos Φ
But
I2 = E / R
Therefore, I12 = I22 + I32 + 2 E/R. I3 cos Φ
Hence, the power E I3 cos Φ is given by
E I3 cos Φ = ( I12 – I22 - I32 ) R
2
cos Φ = ( I12 – I22 - I32 )
2 I2 I3
6-13
6.6
MEASUREMENT OF THREE-PHASE POWER.
6.6.1
THREE-WATTMETER METHOD.
The connections for this method are shown in Fig: 6.9, in which the load is starconnected. W1, W2, and W3 are three wattmeters connected as shown.
Fig-6.9 : Three Wattmeter-Method of Measuring Three-Phase Power
The arrows denote the direction of current and voltage which are conventionally
considered positive. If the symbols representing currents and voltages denote instantaneous
values, then
Instantaneous power in the load = e1 i1 + e2 i2 + e3 i3
Let v be the potential difference between the star point of the load and the star point
O of the wattmeter voltage coils. Then we have
e1 / + v = e 1
e2 / + v = e 2
e3 / + v = e 3
6-14
Therefore, Total instantaneous power, by substitution, is
(e1/ + v) i1 + (e2/ + v) i2 + (e3/ + v) i3
= e1/ i1 + e2/ i2 + e3/ i3 + v (i1 + i2 + i3)
= e1/ i1 + e2/ i2 + e3/ i3
Since i1 + i2 + i3 = 0 in any three-phase 3-wire system, weather balanced or not.
Now, e1/ i1 + e2/ i2 + e3/ i3 is the total instantaneous power measured by the three
wattmeters, and thus the sum of the readings of the wattmeters will give the mean value of
the total power.
6.6.2
TWO-WATTMETER METHOD
This is the commonest method of measuring three-phase power. It is practically
useful when the load is unbalanced. The connections for the measurement of power in the
case of a star connected three-phase load are shown in Fig : 6.10.
Fig : 6.10
Two Wattmeter-Method of Measuring Three-Phase Power
6-15
The current coils of the wattmeters are connected in lines(1) and (3), and their voltage
coils between lines(1) and (2) and (3) and (2) respectively.
Fig : 6.11 gives the vector diagram for the load circuit, assuming a balanced load- i.e.
the load currents and power factors are the same for all three phases. E10, E20, E30 are the
vectors representing the phase voltages, and are supposed to be equal, while I1, I2 and I3 are
vectors representing the line currents.
Fig :6.11
Vector Diagram, Two-Wattmeter Method
The voltages applied to voltage-coil circuits of the wattmeters are E12 and E22, which
are the vector sums of the phase voltages as shown.
Then, total instantaneous power in the load
= e1 i1 + e2 i2 + e3 i3
Where e1,
e2, e3 are the instantaneous phase-voltages and i1, i2, i3 are the
instantaneous line currents.
6-16
Since i1 + i2 + i3 = 0,
i2 = - i1 - i3
Therefore, total instantaneous power = e1 i1 + e2 (- i1 - i3) + e3 i3
= i1 (e1 - e2 ) + i3 (e3 - e2 )
Now, i1 (e1 - e2 ) is the instantaneous power deflecting wattmeter W1 and i3 (e3 -
e2 ) is that deflecting wattmeter W2. These wattmeters measures I1 E12 cos α and I3 E32 cos β
respectively, where α and β are the phase angles between I1 and E12 and between I3 and E32.
The sum of the wattmeter readings thus gives the mean value of the total power in the load.
Now α = 30 o + Φ
β = 30 o - Φ
E12 = E32 = √3 E
Where E is the phase voltage.
Therefore the sum of the wattmeter readings is
W = √3 I E cos(30 o + Φ) + √3 I E cos(30 o - Φ)
If I1 = I2 = I3 = I
W = √3 I E [cos(30 o + Φ) + cos(30 o - Φ)]
= √3 I E [ 2 cos 30 o cos Φ ]
= 3 I E cos Φ
Which is, of course, the total power in the load.
It should be noted that, if one of the voltages(such as E12) is more than 90o out of
phase with the current associated with this voltage in the wattmeter, the voltage-coil
connections must be reversed in order that the instrument may give a forward reading.
Under these circumstances the wattmeter reading must be reckoned as negative, and
the algebraic sum of the readings of the two instruments gives the mean value of the total
power.
Another important point is that, if the power factor of the load is 0.5 – so that I1 lags
60o behind E10 ( cos 60o being 0.5) – than the phase angle between E12 and I1 is 90o and
wattmeter W1 should read zero.
6-17
Power Factor : If W1 and W2 are the two wattmeters readings, than W1 + W2
gives the total power (as seen above).
W1 - W2 = √3 I E [cos(30 o + Φ) - cos(30 o - Φ)]
= √3 I E [ -2 sin 30 o sin Φ ]
= - √3 I E sin Φ
Therefore,
W1 - W2
W1 + W2
Or
tan Φ =
=
- √3 I E sin Φ = - tan Φ
√3 I E cos Φ
√3
√3 (W2 - W1)
W1 + W2
From which Φ and the power factor cos Φ of the load may be found.
6.6.3
ONE-WATTMETER METHOD
This method can be used only when the load is balanced. The connections for a starconnected system are shown in Fig-6.12.
Fig-6.12 One-Wattmeter Method For A Balanced Three-Phase Load.
6-18
The current coil of the wattmeter is connected in one of the lines, and one end
of the voltage coil is connected to the same line, the other end being connected
alternately to first one and then the other of the remaining two lines by means of the
switch S.
Fig-6.13 Vector Diagram, One-Wattmeter Method
The vector diagram for the method of measurement is given in Fig-6.13, in which
E01, E02, and E03 represent the three phase voltages, and I1, I2, and I3 the three line currents. In
a balanced system these three voltages are each equal to E, and the three currents are each
equal to I. The phase angles are also all equal to Φ. The vector E12 is the vector difference
between E01 and E02 and is the voltage applied to the voltage coil when the switch S is on
contact “a”. Similarly, E13 is the vector difference of E01 and E03, and is applied to the
wattmeter voltage coil when switch S is on contact “b”.
Then
E12 = E13 = √3 E
Wattmeter reading when switch S is on contact “a”
= √3 E I cos (30 o + Φ)
6-19
(30 o + Φ) being the phase angle between E12 and I1.
Wattmeter reading when switch S is on contact “b”
= √3 E I cos (30 o - Φ)
Thus, the sum of these readings is 3 E I cos Φ, as shown in the analysis of
the two-wattmeter method, and this is the total power in the circuit.
In the same way the angle Φ is given by
tan Φ =
√3 (W2 - W1)
W1 + W2
And the power factor is cos ( Φ ) or cos [ tan -1 √3 (W2 - W1) ]
W1 + W2
6.7
TYPES OF WATTMETRS
Four types of wattmeter will be considered:(a) Dynamometer.
(b) Induction.
(c) Electrostatic.
(d) Electronic / Digital.
(a)
Dynamometer Type Wattmeter.
These instruments are similar in design and principle to the dynamometer ammeter.
The fixed coil carry the current in the circuit, while the moving coil acts as wattmeter voltage
coil and carries a current proportional to the voltage of the circuit across which it is
connected. A high non-inductive resistance is connected in series with the voltage coil.
Dynamometer wattmeters may be divided into two classes:(i)
Suspended-coil, torsion instruments.
(ii)
Pivoted-coil, direct indicating instruments.
6-20
(i)
Torsion Wattmeters.
These instruments are used largely as standard wattmeters.
The moving, or voltage, coil is suspended from a torsion head by a metallic
suspension which serves as a lead to the coil. This coil is situated entirely inside the current,
or fixed, coils and the winding is such that the system is astatic. Errors due to external
magnetic fields are thus avoided. The torsion head carries a scale, and when in use, the
moving coil is brought back to the zero position by turning this head, the number of divisions
turned through, when multiplied by a constant for the instrument, giving the power.
Eddy current errors are eliminated as far as possible by winding the current coils of
stranded wire and by using no metal parts within the region of the magnetic field of the
instrument.
In the Drysdale single-phase astatic wattmeter, manufactured by Messrs. H. Tinsley,
the moving coil, which is stitched to a flat strip of mica, is divided into two equal portions
wound so that the current (proportional to the applied voltage) circulates in a clockwise
direction in one-half and in an anti-clockwise direction in the other. This coil is suspended by
a silk fiber together with a spiral spring which gives the required torsion.
The fixed coil also is in two halves, which are wound so as to have opposite
directions of current circulation in them. The cable used for the fixed coil consists of ten
strands, insulated from one another. Thus, in effect, there are ten current coils which run
together and are thus as nearly as possible coincident in space.
These ten coils, or cores, are brought out to a commutator so that a number of current
ranges of the instrument may be obtained by grouping them all in parallel, all in series, or in
a series-parallel combination.
The current is led into the moving coil by two fine phosphor-bronze ligments, the
spring, which is of “Iso-elastic” wire and is and is annealed, merely serving as a torsion
control. The spring has a number of turns, and by carefully adjusting its length the constant
of the instrument can be made an exact figure. The moving system carries a knife-edge
pointer moving over a short scale at the front of the instrument, so that the zero position of
the moving coil can be easily determined.
Damping is provided by the mica vane which carries the voltage coil. Drysdale states
that, in order to reduce error due to the voltage-coil inductance to a negligible amount, the
resistance of the voltage-coil circuit should be at least 3,000 ohms per millihenry of
inductance.
6-21
(ii)
Direct-Indicating Dynamometer Wattmeters.
Like the standard or torsion wattmeters described above, the instruments have a
moving voltage-coil which is almost entirely embraced by the fixed current-coils. The
moving coil is carried on a pivoted spindle and the movement is spring-controlled. The
moving system carries a pointer and a damping vane, the latter moving in a sector-shaped
box. The current-coils are usually stranded or laminated, especially when heavy currents are
to be carried. Currents up to about 200 amp can be dealt with in direct indicating wattmeters
of suitable design. For currents above this, a low range wattmeter is usually employed, in
conjunction with current transformers. Similarly, voltages up to 600 volts are applied to the
wattmeter voltage-coil directly, but for higher voltages the voltage-coil circuit is designed for
110 volts, and a voltage transformer used to step down the voltage.
Metal parts should be removed from the magnetic field of the instrument as far as
possible, although care must be taken to ensure that this elimination of metal does not result
in any relative movement of the working coils due to warping of the materials substituted for
the metal.
Figs :
6.14, 6.15, 6.16, on the next pages,
dynamometer wattmeters.
6-22
shows the construction of some
Fig : 6.14
Movement of Reflecting Dynamometer Wattmeter.
6-23
Fig : 6.15 Internal Arrangement of Portable Dynamometer Wattmeter.
6-24
Fig : 6.16 Single Element Indicating wattmeter.
In Fig : 6.14 the details of the suspended movement of a reflecting dynamometer
wattmeter {Elliot Brothers (London), Ltd} are shown. This instrument is intended for the
measurement of power when the power consumption of the instrument itself must be small,
e.g. when measuring the power taken by a low-power fluorescent lamp.
6-25
In this particular measurement it is an advantage that the wattmeter has no iron in its
magnetic circuit since the lamp voltage has a high harmonic content with an appreciable a.c.
component. The wattmeter current-coil takes 0.25 VA and the voltage winding is about 750
Ω per volt on 50 c/s.
Fig : 6.15 shows the internal arrangements of the Everett Edgcumbe wattmeter, with
the moving system suspended, by a robust ligment, from a torsion head which carries a
pointer moving over an evenly divided scale. The instrument is designed for measurement at
very low power factors, e.g. iron loss and dielectric loss measurements. The power loss in the
current element is very small, e.g. 05 microwatts when carrying a current corresponding to
full-scale deflection at unity power factor, the instrument being scaled to 2 watts at 12,000
volts.
In the deflection type of dynamometer instrument the relative position of current and
voltage coils change with the deflection, whereas in the torsion type the relative positions are
the same for all loads, since the moving coil is returned to the zero position in all cases.
Cambridge Reflecting Wattmeter.
Precision dynamometer wattmeters manufactured by the Cambridge
Instruments Co. employ nickel-iron cores arranged as shown in Fig : 6.17 (on next page).
6-26
Fig : 6.17 Construction of Nickel-Iron Core Dynamometer.
To obtain high electromagnetic efficiency together with good screening and a
uniform redial field in which the moving coil can rotate. Because of the high permeability of
the material used, the reluctance of the magnetic circuit is almost entirely in the air gap and
hysteresis effects negligible. By using thin laminations, eddy-current losses, which would
cause phase-angle errors, are so reduced that they are easily compensated, over a wide range
of frequencies, by connecting a small capacitor across the moving coil and part of the
voltage-circuit resistor.
6-27
In a reflecting laboratory-type instrument suitable for the calibration of precisiongrade meters on d.c. or a.c., a carefully-aged quartz fiber is used as a control spring to ensure
consistency of calibration and low temperature coefficient. The current winding is arranged
to provide four current ranges of 0.5, 1.0, 2.5, and 5.0 amp, and these, together with nine
voltage ranges between 50 and 500 volts, are selected by switches mounted in a separate
control box. A further switch increases the sensitivity of the system by a factor of 5, when
full-scale deflection will be obtained at 0.2 power factor or with voltage of 1/5 of the marked
value.
It is important that wattmeter of this type shall give the same indication on d.c. or a.c.,
and the best determinations indicate that the difference does not exceed 0.01 per cent.
The scale is 2.5 m long, and is in three parallel sections with three separate lamps and
optical system which are brought into use in turn. The makers claim that the accuracy of
calibration is 0.05 per cent over the upper three-fourths of the scale length.
(b)
Induction Wattmeters.
Induction wattmeters, the principle of which is same as that of induction ammeters
and voltmeters, can only be used in alternating-current circuits : dynamometer wattmeters
can be used in either a.c. or d.c. circuits.
Induction instruments are useful only when the frequency and supply voltage are
approximately constant.
Fig : 6.18 Induction Wattmeters.
6-28
Construction.
These instruments have two laminated electromagnets; one is excited by the load
current ( or a definite fraction of it), and the other by a current proportional to the voltage of
the circuit in which the power is to be measured.
A thin aluminium disc is mounted so that it is cut by the flux from both of these
magnets, and the deflecting torque is produced by the interactions between these fluxes and
the eddy currents which they induce in the disc. One or more copper rings are fitted on one
limb of the “shunt” magnet- i.e. the magnet excited by the voltage coil and its current- in
order to cause the resultant flux in the magnet to lag in phase by exactly 90
o
behind the
applied voltage.
Fig : 6.18 shows two common forms of magnets with their windings, the magnets
being placed, in each case, one above and one below the moving disc of the instrument. The
positions and shapes of the magnets are, such that the flux from both “shunt” and “series”
magnets cuts the moving disc.
In the form of instrument shown in Fig: 6.18 (a), the two voltage coils, connected
in series, are wound so that they both send flux through the centre limb. The series magnet in
the instrument carries two small current coils in series, these being wound so that they both
magnetize the core, upon which they are wound, in the same direction. The positions of the
copper shading bands are adjustable in order that the correct phase displacement between the
shunt and series magnet fluxes may be obtained.
In the instrument shown in Fig: 6.18 (b) there is only one voltage coil and one current
coil. A copper shading band, whose position is adjustable, surrounds the two projecting pole
pieces of the shunt magnet.
Both types are spring-controlled and have the advantage of a long and uniform scale
(up to 300o).
Currents up to about 100 amp can be dealt with directly in such instruments. For
currents above this, current-transformers are used in conjunction with the wattmeter. Unlike
the dynamometer wattmeter, the voltage-coil circuit of the induction instrument is made as
inductive as possible, in order that the flux of the shunt magnet may lag by nearly 90o behind
the applied voltage.
6-29
Fig : 6.19 gives a simplified vector diagram for the wattmeter. The flux Φsh of the
shunt magnet is assumed to lag by exactly 90o behind the applied voltage. As previously
stated, this is actually brought about by adjustment of the shading bands, since the angle of
lag would be somewhat less than 90o unless such bands were used. It is assumed that flux
Φsc of the series magnet is proportional to and in phase with the line current, and that the
hysteresis and saturation effects in the iron core are negligible. Owing to the larger air gap in
the core these assumptions are justifiable.
Fig : 6.19 Vector Diagram for Induction Wattmeters.
(c)
Electrostatic Wattmeters.
This form of wattmeter cannot be regarded as a commercial instrument like
the dynamometer and induction forms already discussed. It is however, a very useful
instrument for the measurement of small amounts of power, especially when the
voltage is high and the power factor low. Its use for the measurement of dielectric
power loss was considered especially. It is also useful for the calibration, in the
laboratory, of commercial forms of wattmeter and watt-hour meter.
6-30
The electrostatic wattmeter consists of quadrant electrometer used in conjunction
with a non-inductive resistor, the essential connections to the load circuit being shown in Fig
: 6.20.
Fig : 6.20 Connections of Electrostatic Wattmeter
Let the instantaneous load current be i and v, v1 and v2 be the instantaneous voltages
of the needle and of the two pairs of quadrants, as shown in the figure.
The instantaneous torque is given by
T directly proportional to
(v - v1)2 - (v – v2)2
Now (v - v1) is the instantaneous value e of the supply voltage.
Also (v – v2) - (v2 – v1) = e - i R, since the potential difference
v2 – v1 = i R .
Hence the instantaneous torque ( T ) is given by
T directly proportional to e2 – ( e - i R )2
6-31
T directly proportional to e2 –e2 + 2 e i R - i2 R 2
T directly proportional to 2 e i R - i2 R 2
or
T directly proportional to 2 R [ e i - i2 R ]
2
T is directly proportional to the instantaneous power supplied minus half the power
lost in the non-inductive resistor.
If e = instantaneous value of the voltage on the load side of the instrument,
T is directly proportional to 2 R [ e i + i2 R ]
2
T is directly proportional to the instantaneous power in the load plus half the power
lost in the non-inductive resistor.
Use of the electrostatic wattmeter for Calibration of Commercial Wattmeters or
Watt-hour Meters.
Fig : 6.21 shows the connections for the calibration of a commercial
instrument by means of the electrostatic wattmeter. The load is, in this case, fictitious.
Fig : 6.21 Use of electrostatic wattmeter for
Wattmeter testing.
6-32
The voltage for the voltage-coil of the wattmeter under test is supplied from one
phase of the secondary of a phase-shifting transformer, while the current-coil of the
instrument is fed from one phase of the supply through a step-down transformer. The phaseshifting transformer requires a three-phase supply.
Any required phase displacement between current and the voltage in the wattmeter
circuit can be obtained by turning the rotor of the phase-shifting transformer, by means of a
worm and hand-wheel. In this way the apparent power factor of the load is altered.
The needle of the electrostatic instrument is supplied from a tapping point on a
potential divider across the voltage-coil of the wattmeter. The two pairs of quadrants are
connected to the two ends of a non-inductive resistor R, through which the load current
flows. The mid-point of this resistor is connected to one end of the potential divider, as
shown.
Let
Voltage A to C
= n
Voltage B to C
Then, if e is the instantaneous value of the load voltage, the instantaneous value of the
voltage B to C is e / n , and hence
v - v2 = e - 1 i R
n
2
where i is the instantaneous value of the load current, and
v – v1 = e + 1 i R
n
2
Thus, the instantaneous torque is given by
T is directly proportional to ( v – v1 )2 - (v - v2)2
T is directly proportional to { e + 1 i R }2 - { e - 1 i R}2
n
2
n
2
T is directly proportional to
{ e + i R
n
+ e - i R} {e + i R
2
n
T is directly proportional
2
n
2
- e + i R}
n
2
{2e}{iR}
n
T is directly proportional e i
T is directly proportional
true instantaneous power in the load.
6-33
The mean value of the torque is
Tm =
2R
e i dt
n t/
2 R ( mean power in the load)
n
t/ being the periodic time of the current and the voltage waves.
If
θp is the steady deflection of the electrostatic instrument, which is spring-
controlled, we have
k θp = Tm,
where k is the constant.
Mean power = n k/ θp
2 R
Where k/ is the constant of the electrostatic instrument, and must be determined
experimentally by the use of standard resistors and electrostatic voltmeter.
(d)
Electronic / Digital Wattmeters.
The electronic or digital wattmeter does not employ any moving parts in it. Instead it
employs micro-processor, and a digital display. Micro-processor performs the measurement
of the power and also controls the output. Out put is directly sent to digital out put either
L.E.D (Light Emitting Diode) or L.C.D (Liquid Crystal Diode). Six current leads are brought
out (Two for each phase), and four voltage leads are brought out (one for each phase voltage,
and one for neutral).
6-34
6.8
POLYPHASE WATTMETERS.
In order that that the power in a poly-phase circuit may be measured without the use
of more than one wattmeter, poly-phase wattmeters have been developed. These consist of
two separate wattmeter movement mounted together in one case with the two moving coils (
assuming the instrument to be of the dynamometer type) mounted on the same spindle, so
that the total deflecting torque acting on the moving system will be the sum of the torques
produced by the two component wattmeter working systems. The readings of the instrument
thus give the total power in the circuit directly, the addition being carried out by the
instrument itself.
Thus, for example, in measuring three-phase power, the connections to the poly-phase
wattmeter are the same as those for the measurement of power by the two wattmeter method,
using two single-phase instruments. The only difference is that the two single-phase
instruments are combined in the poly-phase wattmeter and operate a single moving system
whose deflection gives the total power directly.
An important point in connection with such double-element wattmeter is that there
shall be no mutual interference between the two elements. Thus, the fixed coils of one of the
elements must not produce a torque by interacting with the field of the moving coil of the
other element. In order to eliminate any such action and ensure that the total torque shall be
merely the sum of the torques produced within the two component element them selves, a
laminated iron shield is placed between the two elements to provide magnetic screening.
Fig : 6.22
Poly-phase Wattmeter Connections.
Compensation for mutual interference can be made by a method due to
Weston, the connections for which are shown in Fig : 6.22.
6-35
Instead of connecting the two voltage coils - each of resistance R (including their
series resistances)-directly to the line which does not contain a current coil, they are each
connected to one end of a resistor R/ whose other end is connected to the line as shown. By
suitably adjusting R/ the currents in the voltage-coil circuits can be altered to compensate for
the mutual interference between the wattmeter elements.
Fig:6.23 illustrates the internal construction of a commercial type of double-element
poly-phase wattmeter. A nickel-iron plate shield the two elements from one another. To
reduce the weight of the moving system, aluminium is used for many of the parts. The total
weight of the double-element movement is about 5.25 grammes and torque / weight ratio is
0.09 (average for 100o deflection).
Fig : 6.23 Double-Element Wattmeter.
6-36
Fig : 6.24, (on next page) shows the construction of the Drysdale standard
poly-phase wattmeter manufactured by H.Tinsely & Co (U.K).
Fig : 6.24 Drysdale Standard Poly-Phase Wattmeter
These wattmeters have ranges of 0.5, 1.0, 2.5, 5.0 amperes ( the range can be
extended to 2,400 A by the use of class AL current transformers). Voltages up to 1,000 volts
can be used by employing voltage resistance boxes.
6.9
SUMMATION METERING.
It is often necessary to measure the total power in a number of separate circuits, and it
is convenient to have this total indicated upon a single instrument. This is simple matter if the
circuits are interconnected so that one portion of the system carries the whole load. If,
however, there is no part of the system carrying the aggregate load, more complicated
methods of measurement must be employed.
6-37
A four-element wattmeter, which consists essentially of two poly-phase wattmeters
like those described in the preceding section, with all four moving systems ( two in each
poly-phase wattmeter) mounted on common spindle, can be used for the measurement of the
total power in two three-phase systems, but for more two systems current-transformers are
used in conjunction with a double-element “Summation” wattmeter.
6.25 Connections For Summation Method Of Power
Measurement.
The principle of such summation measurements is illustrated in 6.25, in which the
total power supplied to three circuits A, B, and C is measured by a two-element summation
wattmeter, in conjunction with six current-transformers, all of whose ratios are the same.
Actually voltage transformers may also be necessary if the line voltage is too great to be
applied directly to the voltage coils of the wattmeter, but these have been omitted for
simplicity.
It will be observed that the principle of measurement is essentially that of the twowattmeter method of measurement of three-phase power.
Current transformers a, b, and c together supply current coil F1, while transformers a/,
b/, and c/ supply current coil F2. Neglecting the small phase angles of the current
transformers, the currents in their secondary circuits are each 180o out of phase with their
primary currents, and hence the vector sum of the former will be equal to vector sum of the
primary currents divided by the ratio of the transformer ( the same in all cases), the ratio
errors of the transformers are also being neglected. It can be seen, therefore, that the current-
6-38
- supplied to current coil F1 is proportional to the vector sum of the currents supplied to the
three circuits A, B and C by line(3), and the current supplied to current coil F2 is proportional
to the vector sum of the currents supplied by line(1).
The voltage coils P1 and P2 are connected between lines (3) and (2) and between (1)
and (2) respectively.
The reading of the summation wattmeter is thus equal to the total power supplied to
the three circuits A, B, and C, divided by the ratio of the current transformers, the current
transformer errors being neglected.
6.10
COMPONENTS OF ELECTRIC POWER.
The electric power has three components as represented by a power
triangle ABC.
Active Power
Reactive Power
Apparent Power
Power Triangle ABC
6-39
Active Power
The base “AB” of the power triangle represents the Active Power. According to
pathagourus theorem; ( KVA2 = KW2 + KVAr2 )
Or KVA = √ (KW2 + KVAr2 ) It is designated by a capital letter “P”.
From this expression the Active Power Component (KW) can de easily
derived as follows :
KVA = √( KW2 + KVAr2 )
Taking square on both sides, we have
( KVA2 = KW2 + KVAr2 )
From above expression KW2 = KVA2 - KVAr2
KW = √ ( KVA2 - KVAr2 )
Or
The units are watts. Bigger units are Kilo Watts (KW), Megga Watts
( MW) and Giga Watts (GW). Its mathematical representation is as given below:
P = V . I . Cos Φ
This component is actually watt full components i.e. do some useful work and
actually is the power needed to drive the electrical appliances. Usually the nominal ratings of
consumer appliances is given in the component.
Reactive Power
The perpendicular “AC” of the power triangle represents the Reactive Power.
According to pathagourus theorem; ( KVA2 = KW2 + KVAr2 )
Or KVA = √ (KW2 + KVAr2 ), It is designated by a capital letter “Q”.
From this expression the Reactive Power Component (KVAr) can de easily derived as
follows :
KVA = √( KW2 + KVAr2 )
Taking square on both sides, we have
( KVA2 = KW2 + KVAr2 )
6-40
From above expression KVAr2 = KVA2 - KW2
KVAr = √ ( KVA2 - KW2 )
Or
The units being Volt-Ampere reactive (VAr). Bigger units are Kilo Volt-Ampere
reactive (KVAr), Megga Volt-Ampere reactive( MVAr) .
Its mathematical representation is as given below:
P = V . I . Sin Φ
This component is actually waste full components i.e. can not do any useful work,
and actually is the power just wondering “to end fro” in the system. It causes low power
factor and voltage drop in the system.
Reactive power also hampers the KVA or MVA rating of the electrical system
(Generator, transformer, transmission line etc.). It is being generated due to use of inductive
and capacitive load causing lagging or leading power factor. Lightly loaded transmission
lines act as charged capacitor due to closer vicinity of earth . This feature causes line
charging current usually with leading power factor. Heavy loaded lines usually have lagging
power factor due to inductive type load on the system.
Apparent Power
Apparent Power and Apparent Energy.
As previously discussed, according to pathagourus theorem;
( KVA)2
Apparent Power2
=
=
( KW)2
+ ( KVAr2 )
Active Power2
+ Reactive Power2
Or KVA = √ (KW2 + KVAr2 )
The units being Volt-Ampere (VA). Bigger units are Kilo Volt-Ampere (KVA),
Mega Volt-Ampere ( MVA) and Giga Volt-Ampere( GVA). Its mathematical representation
is as given below:
Apparent Power S = V . I .
6-41
The capacity or rating of the equipment installed by utility ( Generator, Transformer
and Transmission line) are designated in this component of the power
As mentioned earlier, Reactive power also hampers the KVA or MVA rating
(Apparent Power) of the electrical system ( Generator, transformer, transmission line etc.).
6.11
EASUREMENT OF REACTIVE POWER.
If E and I are the r.m.s. values of the voltage and current in a single-phase circuit
and Φ is the phase angle between them, then the actual, or active. Power in the circuit is, of
course, E I cos Φ.
The active power is equal to the voltage multiplied by the component of the current
which is in phase with it (i.e. I cos Φ). The component of the current which is 90o out of
phase with the voltage is I sin Φ and the product E I sin Φ is the “wattles” or “reactive”
component of the volt-amperes, or the reactive power.
The measurement of this reactive component is, however, useful, since the phase
angle Φ of the circuit can be obtained from the ratio Reactive Power / Active Power, which
equals E I sin Φ / E I cos Φ , or tan Φ.
It should first be observed that sin Φ = cos (90o – Φ), and therefore a wattmeter may
be used for the measurement if the current coil carries the load current I and the voltage
applied to the voltage coil is such that its phase displacement from the actual voltage of the
circuit is 90o . Under these circumstances the wattmeter will read I E cos (90o – Φ) or I E sin
Φ.
It may be more convenient in single-phase measurements to compensate the
wattmeter so that the field of its voltage coil lags 90o behind the phase it would have if the
wattmeter voltage were used for power measurement.
In the case of a balanced three-phase load when the power is being measured by one
wattmeter, it is a simple matter to use this wattmeter for the measurement of the reactive
power by connecting its current and voltage coils as shown in Fig : 6.26 (on next page).
6-42
The current coil is connected in one line and the voltage coil is connected across the
two other lines.
Fig : 6.26 Measurement of Reactive Power
Referring to the vector diagram of Fig: 6.26 (b), the vector OI2 represents the current
through the wattmeter current coil. The voltage applied to the voltage coil is the vector
difference of OE1 and OE3, i.e. the vector OE. Now, the angle between OE2 and OE3
reversed is 60o, and since the three phase voltages E1, E2 and E3 are equal, the angle between
OE3 reversed and vector OE is 30o. Hence the angle between OE2 and OE is 90o, and the
angle between OI2 and OE is (90o + Φ). This means that the wattmeter reads
OE X OI2 cos (90o + Φ) = √3 E I cos (90o + Φ)
= √3 E I sin Φ
= - Wr
Where E is the phase voltage and I the line current of the system.
The total reactive power of the circuit is
3 E I sin Φ = - √3 Wr
6-43
And hence, if W is the measured value of the total active power, the phase angle Φ of
the load is
Tan-1 ( - √3 Wr )
W
------------------------------------------------------------------------------
BIBLOGEAPHY AND REFERENCES
(1)
Commercial A.C.Measurements, G.W.Stubbings.
(2)
Electrical Measurements, F.A.Laws.
(3)
Alternating Current Electrical Engineering, W.T.Maccall.
(4)
Theory and Practice of Alternating Currents, A.T.Dower.
(5)
Electrical Measuring Instruments, C.V.Drysdale and A.C.Jolley.
(6)
Theory and Calculation of Electric Currents, J.L.La Cour and O.S.Bragstad.
(7)
British Standard Specifications, No.89 (1954).
(8)
Alternating Currents, A.Russell.
(9)
“Recent Developments in Electricity Meters, J.L Carr, Jour, I.E.E.,Vo.LXVII,
p.859.
(10)
“The
Metering
Arrangements
for
the
‘Grid’
Transmission
system,
C.W.Marshall, Jour. I.E.E., Vol. LXVIII, p.1497.
(11)
“ A New Method of Measuring the Total Power in a Balanced Three-Phase
Circuits, employing only One Wattmeter,” H.Monteagle Barlow, Jour.I.E.E.,
Vol. LXVI., p.1001.
(12)
“A New Concentric Standard Dynamometer Wattmeter for Heavy Currents.
A.E.Moore, Jour.I.E.E., Vol. LV, p.380.
(13)
P.G.Agnew, Bull. Bur.Sta., Vol. VIII.
(14)
“A Valve Wattmeter,”E. Mallet, Jour.I.E.E., Vol. LXXIII, p.295.
(15)
“The Development of a Sensitive Precision Wattmeter for the Measurement of
very Small Powers,” N.H.Searby, Jour. I.E.E., Vol. LXXVI, p.205.
6-44
BIBLOGEAPHY AND REFERENCES (Continued)
(16)
“Performance Limits in Electrical Instruments,” A. H. M. Arnold, Proc.I.E.E.,
Vol. 98, pt. II, p. 701.
(17)
“The Design and Application of a Portable Electrostatic wattmeter,” F. R.
Axworthy and J. K. Choudhury, Trans. Soc. Inst. Tech., Vol. IV, No. 2.
(18)
The Use of an Electrostatic Wattmeter for Magnetic-loss Measurements,” J. K.
Choudhury and G. Glynne, Proc. I.E.E., Vol.98.
(19)
“Some Applications of the Phase-sensitive Rectifier Bridge to Electrical
Measurements,” J. De Gruchy, Tran. Soc. Inst. Tech., Vol. II.
(20)
“Audio-frequency Power Measurement by Dynamometer Wattmeters,” A. H.
M. Arnold, Proc. I.E.E., Vol 102B.
-----------------------------------------------------------
6-45