Power and energy measurements

The power measure with the volt-ammetric method cannot be applied to the circuits in alternated state, since the product U · I (where U and I are the effective voltage and current values) defines only the apparent power S, but not the active power.

We use therefore an instrument called WATTMETER.

It has four terminals, two voltmetric and two ammetric, since it has to evaluate in the same time both the voltage and the current; we can distinguish therefore, at its inside, a voltmetric circuit and an ammetric circuit.

We remember that the ammetric terminal have to be series connected to the load, while the voltmetric terminals have to be parallel connected.


Fig. 8.2.1

We distinguish:

voltmetric range Un
ammetric range In 

that represent the rated values of voltage and current that brings the instrument index to full scale; their product:

wattmetric range  

represents the maximum instrument power that can be measured, by considering its  cos j = 1.

The most used wattmeter is an electrodynamical instrument: it is composed of a fixed coil which links up with the ammetric terminals and of a mobile coil, to with an additional resistance Ra is added, which links up with the voltmetric terminals, schematically:

where:

A and B are the voltmetric terminals connected to the mobile coil;
C and D are the ammetric terminals connected to the fixed coil;
If  is the current flowing in the fixed coil;
Im  is the current flowing in the mobile coil.


Fig. 8.2.2

The active power measure is made at the balance between the motor torque that stresses the mobile coil and the resistant torque given by the spring torsion blocked to the fixed coil.

The wattmeter connection in a circuit can be upstream of downstream, and it is always accompanied by the connection of a voltmeter or of a ammeter to check the voltage and the current:

Fig. 8.2.3

Fig. 8.2.3a

where:

Um and Im are the voltage and current values given by the instruments;
RVW is the resistance of the voltmetric coil of the wattmeter;
RAW is the resistance of the ammetric coil of the wattmeter;
RA and RV are respectively the general resistances of the ammetric and voltmetric circuits.
RC is the load.

By referring to the "downstream" connection it is possible to calculate the quantities:

voltage at the load ends: U = Um
absorbed active power:

where the terms U2/R represent the self-consumption of the voltmetric circuit and of the voltmetric coil.
current absorbed by the coil and by the voltmetric circuit:

this current results in phase with Um; while the current Im of the ammeter is out of phase of an angle jm defined by:
absorbed apparent power:
load power factor:
absorbed reactive power:

Remember at this point that the idea of energy, defined as:

 is strictly bond to the idea of power: where:

is the time interval where the energy W flows.

Also the energy is produced or used and its measure is the contact element between the energy producer and its user..

The measure instrument of the electrical energy is the COUNTER; the measure unit is the Kilowatt-hour.

The counter operation is based upon the carrying out of a small electric motor developing a motor torque, proportional to the circuit power, to which a resistant torque proportional to the motor rotation speed has to be opposed.

The motor includes a wattmetric circuit, composed of a series ammeteric winding, composed of a series ammetric winding and a parallel voltmetric winding between the line wires.

On the axis around which the motor turns an aluminium disk is fixed that is able to flow between the polar expansions of a permanent magnet, representing the resistant torque proportional to the angular speed of the disk rotation.


Fig. 8.2.4

The counter axis is stressed on one side by the motor by a motor torque proportional to the circuit power and on one side by the disk that produces a resistant torque proportional to the angular speed.

Supposed the friction as null, the balance condition of the system is given by the fact that the axis reaches and keeps the speed for which motor torque and resistant torque equalize.
Therefore the electrical power of the circuit is proportional to the angular speed w of the disk.
In other terms, the energy measure in nothing but the count of the turn number made by the disk.