21 Measurement of Power and Wattmeters

Vinay Gupta

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    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 DC 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 in order to obtain the value of the power in the circuit. Alternatively, if the circuit resistance is known, power may be computed from one of the equivalent formula, P = V2/R = I2R.

 

In almost all cases, the power in a DC circuit is best measured by separately measuring quantities, V and I and by computing 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. Ofcourse 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 AC circuits, the instantaneous power varies continuously as the current and voltage go through a cycle of values. The cyclic variation of power has period so short that it can be followed only by special instruments such as oscillographs. However, we are not usually interested in the instantaneous power (except where transient phenomena are being studied), but in its time average. If the voltage and current are both sinusoidal, the average power over a cycle is given by the expression P = VIcos ɸ watts where V and I are the rms values of voltage and current and ɸ is the phase angle by which current lags behind or leads the voltage.

 

Power in D.C. Circuits

 

The power taken by a load from a d.c. supply is given by the product of readings of an ammeter and a voltmeter when connected in the circuit as shown in Fig. 1.

 

Power P=V X lwatt

 

It should be taken in account that both ammeter and voltmeter require power for their operation. One can use any of the following two connections as shown in Figs. 1 (a) and (b)

 

 

Power measurement in DC circuit

 

In Fig. 1 (a), the ammeter is connected between load and voltmeter. Therefore, the voltmeter not only indicates the voltage VL across the load, but in addition voltage drop Vs across the ammeter. If Ra is the resistance of the ammeter, voltage drop

 

Va=I Ra

 

Power consumed by load= VL I=(V- Va) l= Vl-Val= VI-12Ra

 

=power indicated by instruments – power loss in ammeter.

 

In Fig. 1(b), voltmeter is connected between load and ammeter. Therefore, ammeter not only indicates the current through the load but in addition current in the voltmeter also.

 

Current through the voltmeter Iv= V/ Rv where Rv =resistance of voltmeter.

 

Power consumed by load = VIL = V (l – lv ) = V ( I – V/Rv ) = VI – V2/Rv

 

=power indicated by instruments – power loss in Voltmeter.

 

Thus, in both the cases, the power indicated by the instruments is equal to the power consumed by the load plus the power consumed by the instrument nearest to the load terminals.

 

In order to obtain the true power, corrections must be made for power loss in instruments. Under normal conditions the value of power loss in instruments is quite less as compared with the load power and, therefore, the error introduced on this account is very low. However, when the output of the source to be measured is limited, the ammeter and voltmeter may load the circuit too much thereby causing errors.

 

For a permanently wired installation, when power measurements are required, it would be advantageous to install a wattmeter in place of voltmeter and ammeter. Wattmeter gives direct indication of power and hence multiplying of two readings as is the case when voltmeter and ammeter is saved. This increases the accuracy of the measurements.

 

Power in A.C. circuits

 

The instantaneous power varies continuously as the current and voltage go through a cycle, in the case of alternating currents. However, only the average value over a cycle is more useful as the average power multiplied by time measures the transfer of energy over a time interval in steady state conditions.

 

The instantaneous power is given by p=vi

 

where p, v, i are the instantaneous values of power, voltage and current respectively

 

Thus, if both current and voltage waves are sinusoidal, the current lagging in phase by an angle ɸ, then:

 

v= Vm sin ωt ;      i=lm sin (ωt – ɸ).

 

Therefore, the instantaneous power is given by:

 

p = vi = Vm lm sin ωt sin (ωt – ɸ)  taking Ө = ωt , we get

 

p  = Vm lm sin Ө sin (Ө – ɸ) = Vm lm [cos ɸ – cos (2Ө – ɸ)] /2 Therefore, average power over a cycle is given by:

 

 

 

Thus, P = VI cos ɸ

 

where V and I are rms values of voltage and current, cos ɸ gives the power factor of the load.

 

The presence of cos ɸ in the above expression for power explains that a wattmeter should be utilized for the measurement of power in a.c. circuits. This is because of the fact that the use of ammeter and a voltmeter do not take account of the power factor.

 

Eletrodynamometer Wattmeters

 

It consists of two coils which are connected in different circuits for measurement of power. The fixed coils or ”field coils” are connected in series with the load and so carry the current in the circuit. The fixed coils, therefore, form the “current coil” of-the watt meter. The moving coil is connected across the voltage and therefore, carries a current proportional to the voltage. A high non-inductive resistance is connected in series with the moving coil to limit the current to a small value. Since the moving coil carries a current proportional to the voltage, it is called the “pressure coil” or “voltage coil” of the wattmeter.

 

Construction

 

Fixed Coils: The fixed coils carry the current of the circuit. They are divided into two halves. The reason for using fixed coils as current coils is that they can be made more massive and can be easily constructed to carry considerable current since they present no problem of getting the current in or out. The fixed coils are wound with heavy wire. This wire is stranded or laminated especially when carrying heavy current in order to avoid eddy current losses in conductors. They are first varnished and baked to exclude moisture and clamped into place, thus forming a rigid solid.

 

 

Fig. 2 Dynamo-Wattmeter

 

Moving Coil: The moving coil is mounted on a pivoted spindle and is entirely embraced by the fixed current coils. Spring control is used for the movement. Fig. 2 shows an electrodynamometer type wattmeter. The use of moving coil as pressure coil is a natural consequence of design requirements. Since the current of the moving coil is carried by the instrument springs it is limited to values which can be carried by springs without appreciable heating. A series resistor is used in the voltage circuit, and the current limited to a small value, usually between 10 to 50 mA.

 

Control: The instrument is controlled by using spring control.

 

Damping: Air friction damping is used. The moving system carries a light aluminum vane which moves in a sector shaped box.

 

Scales and Pointers: They are equipped with mirror type scales and knife edge pointers to remove reading errors due to parallax.

 

Theory

 

The instantaneous torque of an electrodynamometer instruments is given by:

 

 

 

where i1 and i2 are instantaneous values of currents in two coils. Let V and I be the r. m.s. values of voltage and current being measured.

Fig. 3 Circuit of electrodynamometer- wattmeter

 

Instantaneous value of voltage across the pressure coil circuit:

 

 

If the pressure coil circuit has a very high resistance, it can be treated as purely resistive. Therefore, current ip, in the pressure coil is in phase with the voltage and its instantaneous value is given by:

 

 

 

is the rms value of current in pressure coil circuit and  is the resistance of pressure coil circuit.

 

If the current in the current coil lags the voltage in phase by an angle ɸ instantaneous value of current through current coil is :

 

 

Controlling torque exerted by springs, T = KӨ

 

Where K = spring constant and Ө = final steady deflection

 

Since the moving system of the instrument cannot follow the rapid variations in torque (the torque has a double frequency, component), it will take up a position at which the average deflection torque is equal to the restoring torque of the springs.

 

Thus, at balance position

 

 

 

 

 

 

 

 

 

Where P = power measured = VI cos ɸ and K1 = 1/RpK

 

Wattmeter Errors

 

Pressure Coil Inductance

 

We have, rp = resistance of pressure coil,

 

L = inductance of pressure coil,

 

R =resistance in series with pressure coil

 

Rp =total resistance of pressure coil circuit = rp + R

 

V = voltage applied to pressure coil circuit

 

I=current in the ·current coil circuit

 

Ip = current in the pressure· coil circuit,

 

Zp = impedance of pressure coil circuit

 

 

 

It was implied in the discussions of the idealized wattmeter that the current in the pressure coil is in phase with the applied voltage. If the pressure coils of the wattmeter have an inductance, the current in it will lag the voltage by an angle β given by:

 

 

 

 

Fig. 5 Wattmeter phasor diagram for lagging p.f.

 

It will be seen from Fig. 4, that for lagging power factor, the angle between current in the current coil circuit and the current in the pressure coil circuit is less than ɸ, by which the load current lags the applied voltage. The angle between the pressure coil current and the current in the current coil is:

 

ɸ’ = ɸ – β

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Correction Factor:

 

It is defined as a factor by which the actual wattmeter reading is multiplied to get the true power.

 

 

 

 

 

 

 

 

 

 

 

Fig. 6 Wattmeter phasor diagram for leading p.f.

 

It is clear from the phasor diagram ( Fig. 5) that on lagging loads the wattmeter will read high, as the effect of the inductance of the pressure coil circuit is to bring the pressure Coil current more nearly into phase with the load current than would be the case if this inductance were zero. Very serious errors may be introduced by pressure coil inductance at low power factors unless special precautions are taken.

 

The wattmeter will read low when the load power factor is leading as in that case the effect of pressure coil inductance is to increase the phase angle between load current and pressure coil current (See Fig. 6).

 

 

 

Questionnaire

  1. Explain the power measurement in DC circuit.
  2. Explain the power measurement in AC circuit.
  3. Describe the construction of Eletrodynamometer Wattmeters.
  4. Explain the working of Eletrodynamometer Wattmeters in detail.
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    References:

  1. Electronic Measurements and Instrumentation by Bernard M. Oliver and John M. Cage.
  2. Measurement and Instrumentation Principles by Alan S. Morris.
  3. Instrumentation and Measurement in Electrical Engineering by Roman Malaric.
  4. Measurement and Instrumentation Systems by William Bolton.
  5. Engineering Measurements and Instrumentation by Leslie Frank Adams.
  6. Electrical Measurements and Instrumentation by U. A. Bakshi.
  7. Introduction to Measurements and Instrumentation by Arun K Ghosh.