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Measurement of Power and Wattmeters

1. Introduction

Power may be defined as the rate at which energy is transformed or made available. The

power in a circuit at any instant is equal to the product of the current in the circuit and the

voltage across its terminals at that instant. In a d.c. circuit, if the current and voltage are

constant, P=VI so that it is necessary only to determine the current and voltage and to take

their product inorder to obtain the value of power in the circuit.

In almost all cases the power in a d.c. circuit is best measured by separately measuring

quantities, V and I and by computing power by the formula P=VI rather than measuring the

power directly with a wattmeter. If voltage and current are measured simultaneously,

allowance for power required for operation of voltmeter and ammeter must be made. Of

course the power loss in the ammeter or in the voltmeter is often very small compared to the

load power and may be safely neglected.

In the case of a.c. circuits the instantaneous power varies continuously as the current

and voltage go through a cycle of values. If the voltage and current are both sinusoidal the

average power over a cycle is given y expression cosVIP watts, where V and I are r.m.s.

values of voltage and current and is the phase angle by which current lags behind or leads

the voltage.

The fact that the power factor ( cos ) 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.

2. Wattmeter

A wattmeter is essentially an inherent combination of an ammeter and a voltmeter and,

therefore, consists of two coils known as current coil and pressure coil. The operating torque

is produced due to interaction of fluxes on account of currents in current and pressure coils.

The current coil is inserted in series with the line carrying current to be measured and the

pressure coil in series with a high non-inductive resistance R is connected across the load or

supply terminals, as shown in Fig. 1.

2

FIGURE 8-1 Wattmeter connection

3. Wattmeter Errors

(i) Error due to Inductance of Pressure Coil : Inductance of pressure coil may cause an

error in the reading of the wattmeter.

Let rp, Lp and R be the resistance of pressure coil, inductance of pressure coil and

resistance connected in series respectively.

Current through pressure coil,

222

pp

p

LRr

VI

...................................................................................(1)

lagging behind the supply voltage V by a small angle

Rr

L

p

p

1tan ................................................................................................(2)

where V is the supply voltage, and is the angular velocity of supply.

If is the phase angle (lagging) between load current I and supply voltage V as seen in

Fig. 2, then phase angle between the currents of current coil and pressure coil will be

then

deflection of wattmeter coscp II ..........................................................(3)

or deflection cosIZ

V

p

since p

pZ

VI ............................................................(4)

where Zp is the impedance of pressure coil circuit and Ic=I

or deflection

coscosIrR

V

p

since cos

p

p

rRZ

......................................(5)

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If the inductance Lp of pressure coil circuit were zero, then phase angle between supply

voltage V and pressure coil current Ip i.e. would be zero and wattmeter deflection would be

proportional to cosIrR

V

p at all frequencies and power factors,

coscos

cos

coscos

cos

p

p

rR

VI

rR

VI

instrumentthebyindicatedPower

PowerTrueHence...............(6)

or

wattmeterofadingpowerTrue Recoscos

cos

.........................................(7)

Hence true reading may be obtained by multiplying the actual reading of the wattmeter

by

coscos

cos where

coscos

cos is known as correction factor.

sincot

sin

sinsincos

cos1

cos

cos1

coscoscoscos

cos1

coscoscos

readingActual

readingActual

readingActual

rR

VI

rR

VI

rR

VIError

p

pp

.................................................(8)

FIGURE 2 Phasor diagram of V, Ip, and I

From the above expression it is obvious that the instrument gives high reading on

lagging power factor and low reading on leading power factor.

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(ii) Error due to Pressure Coil Capacitance : The pressure coil circuit may have

capacitance also in addition to inductance. Due to capacitance of pressure coil circuit, the

pressure coil current will tend to lead the supply voltage. In case of no inductance in pressure

coil circuit error will be introduced due to capacitance in pressure coil circuit whose value can

be determined in the same way as has been done in case of inductance and will be equal to

actual reading multiplied by

cotsin

sin

Generally both inductance and capacitance are present in pressure coil circuit and

therefore cancel the effect of each other.

(iii) Error due to Eddy Currents : The alternating magnetic field of current coil induces

eddy current in the solid metal parts nearby the current coil. These eddy currents set up their

own magnetic field and thus alter the magnitude and phase of the magnetic field causing

deflection. Thus the error is introduced in the instrument readings.

Since the error due to eddy currents cannot be detemined and may be seious if no care is

taken to minimise it, therefore, solid metal parts are removed as far away from the current coil

as possible.

(iv) Errors due to Power Loss in Pressure Coil or Current Coil : There are two methods

of connecting wattmeters in the circuit for measurement of power, as shown in Fig. 3.

FIGURE 3 Wattmeter connections

The first method is shown in Fig. 3(a). In this case wattmeter reading is given by

crIWreadingWattmeter 2 ..........................................................................(9)

The second method is shown in Fig. 3(b). In this case wattmeter reading is given by

prR

VWreadingWattmeter

2

.....................................................................(10)

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4. Measurement of Power in Single Phase A.C. Circuits

(a) Measurement of power without watt-meters : Usually watt-meters are used or

measuring the power in inductive a.c. circuits but in case if watt-meter is not available or

conditions are such that measurement of power by watt-meter may be incorrect, then use of 3

volt-meters of of 3 ammeters can be made for this purpose as explained below.

(i) 3-voltmeter method : The inductive circuit, Z in which the power is required to be

measured is connected in series with a non-inductive resistance R. Three voltmeters V1, V2,

and V3 are connected across the inductive circuit Z, non-inductive resistance R and across the

whole combination respectively, as shown in Fig. 4(a).

FIGURE 4(a) Measurement of power by 3-voltmeter method

Let the power and p.f. of the inductive circuit be P and cos respectively. Let the

voltage drop measured by voltmeter across inductive circuit, pure resistance and across the

whole circuit be V1, V2, and V3 volts respectively.

Now V1 will lead the current I by an angle and V2 will be equal to IR and in phase

with current I.

From vector diagram shown in Fig. 4(b)

cos2 21

2

2

2

1

2

3 VVVVV ..................................................................................(11)

Since IRV 2

cos2 1

2

2

2

1

2

3 IRVVVV

or PRVVV 22

2

2

1

2

3 ..............................................................................................(12)

Since power in the inductive circuit, cos1IVP

or R

VVVP

2

2

2

2

1

2

3 ..................................................................................................(13)

21

2

2

2

1

2

3

2cos

VV

VVV ............................................................................................(14)

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Disadvantages : (i) Even small errors in measurement of voltages may cause serious

errors in the value of power determined by this method. (ii) Supply voltage higher than

normal voltage is required because an additional resistance R is connected in series with the

load Z (inductive circuit).

FIGURE 4(b) Vector diagram of Fig. 4(a)

(ii) 3-ammeter method : The disadvantages of measurement of power by 3 voltmeters

are overcome in this method. The other advantage of this method is that the value of power

determined is independent of supply frequency and wave forms.

FIGURE 5(a) Measurement of power by 3-ammter method

In this method across the inductive circuit Z in which the power is to be determined, a

non-inductive resistance R is connected, as shown in Fig. 5(a).

Three ammeters A1, A2, and A3 are connected in the circuit to measure currents flowing

through the inductive circuit Z in which the power is to be determined, non-inductive

resistance R and whole circuit respectively, as shown in Fig. 5(a).

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Let the power and power factor of the inductive circuit be P and cos respectively and

currents flowing through inductive circuit, non-inductive resistance and whole circuit be I1, I2,

and I3 respectively.

Now current I1 will lag behind the supply voltage V by an angle current I2 will be

equal to R

V and will be in phase with supply voltage V.

From vector diagram shown i Fig. 5(b)

cos2 21

2

2

2

1

2

3 IIIII .......................................................................................(15)

or 21

2

2

2

1

2

3

2cos

II

III ................................................................................................(16)

also since R

VI 2

cos2 1

2

2

2

1

2

3R

VIIII .....................................................................................(17)

or R

PIII 22

2

2

1

2

3 ..................................................................................................(18)

Since power in inductive circuit, cos1VIP

2

2

2

1

2

32

IIIR

P ...............................................................................................(19)

FIGURE 5(b) Vector diagram of Fig. 5(a)

(b) Measurement of Power With Wattmeters : This method has already been described.

(c) Measurement of Power in Conjuction With Instrument Transformers : When the

currents and voltages of the circuits to be measured are high then use of instrument

transformers is made with wattmeters just as with ammeters and voltmeters. The connection

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diagram of wattmeter, when used in conjuction instrument transformers, for measurement of

power is shown in Fig. 6.

FIGURE 6. Measurement of power with wattmeter in conjunction with instrument

transformers in single phase A.C. circuits

When the wattmeter is used in conjuction with instrument transformer, then correction

should be applied for ratio and phase angle errors of instrument transformers.

Let the load voltage , current and load power factor be V, I and cos respectively and

let

Current in current coil of wattmeter = current in secondary C.T. = Is

Voltage across pressure coil of wattmeter = voltage across secondary of P.T. = Vs

Current in pressure coil of wattmeter = ip lagging behind Vs by a small angle due to

inductance of pressure coil

Phase angle of P.T. =

Phase angle of C.T. =

FIGURE 7 Vector diagram for inductive load

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FIGURE 8 Vector diagram for capacitive load

From vector diagram in Fig. 7 and 8.

Phase angle between the currents in current and potential coils of wattmeter

for inductive loads.................................................................(20a)

and for capacitive loads................................................................(20b)

Since phase angle of PT may be lagging or leading. Neglecting transformation ratio

errors of instrument transformers. Correction factor,

coscos

cosK for inductive loads................................................(21a)

and

coscos

cosK for capacitive loads..............................................(21b)

True power=KxActual ratio of CTxActual ratio of PTxWattmeter reading.

5. Measurement of Power in 3 Phase Circuits

(a) Three-wattmeter Method : as seen in FIGURE 9

Total power of load circuit,

321 WWWP ...............................................................................................(22)

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FIGURE 9(a) Three wattmeter methof of measuring three phase power

FIGURE 9(b) Three-wattmeter method of measuring three-phase power

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FIGURE 9(c) Three-wattmeter method of measuring three-phase power

(b) One-wattmeter method : As seen in Fig. 10.

FIGURE 10 One wattmeter method of measuring power in 3-phase 3 wire balanced load

circuits

For balanced load condition,

WP 3 ..................................................................................................................(23)

(c) Two-wattmeter method : As seen in Fig. 11.

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FIGURE 11 Two-wattmeter method of measuring 3-phase 3-wire power

332211 ivivivp ............................................................................................(24)

(i) Star-Connected System

0321 iii or 213 iii ...........................................................................(25)

Substituting Eq. (25) in Eq. (24) we get

322311

2132211

vvivvi

iivivivp

..................................................................................(26)

So, the total average power,

21 WWP ..........................................................................................................(27)

(ii) Delta-Connected System

0321 vvv or 321 vvv .......................................................................(28)

122313

3322132

332211

iiviiv

ivivivv

ivivivp

...............................................................................(29)

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So, the total average power,

21 WWP ......................................................................................................(30)

6. Determination of Power Factor From Wattmeter Readings

If load is balanced, then p.f. of the load can also be determined from the wattmeter

readings.

The vector diagram for a balanced star-connected inductive load is shown in Fig. 12(a).

Let V1, V2, and V3 be the r.m.s. values of phase voltages and I1, I2, and I3 be the r.m.s. values

of phase currents.

FIGURE 12(a) Vector diagram for balanced star connected inductive load

Since load is balanced therefore

(i) Phase voltages will be equal (say, equal to Vp).

(ii) Phase currents will be equal (say, equal to Ip or IL).

(iii) Phase angles between respective phase voltages and phase currents will be equal to, say

The current in current coil of wattmeter W1=I1=IL lagging behind V1 by . The p.d. across

pressure coil of wattmeter Lp VVVW 3131 lagging behind V1 by 300.

Therefore phase angle between voltage across potential coil and current through current

coil of wattmeter W1 is 030 and reading of wattmeter

0

1 30cosLL IVW ...........................................................................................(31)

The current in current coil of wattmeter W2=I2=IL lagging behind V2 by .

The p.d. across pressure coil of wattmeter Lp VVVW 3232 leading V2 by 300.

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Therefore phase angle between p.d. across potential coil and current through current coil of

wattmeter W2 is 030 .

Hence reading of wattmeter

0

2 30cosLL IVW .......................................................................................(32)

The sum of two wattmeter readings

loadofpowerTrueIV

IV

IVIVWW

LL

LL

LLLL

cos3

cos30cos2

30cos30cos

0

00

21

..............................................(33)

and

sin

sin30sin2

30cos30cos

0

00

21

LL

LL

LLLL

IV

IV

IVIVWW

..............................................(34)

Dividing Eq. (34) by Eq. (33) we get

3

tan

21

21

WW

WW.................................................................................................(35)

or 321

211

WW

WWTan ...................................................................................(36)

and p.f. of the load circuit,

21

211 3tancoscos

WW

WW

.............................................................................(37)

FIGURE 11(b) The watt-ratio curve