# Measurement of power of three-phase system

At symmetrical three-phase load it is enough to determine the power _{of the USAn Federation} consumed in one fayoza as the measured power of three-phase loading P = _{ZRF}. The simplest conditions for such measurement are available when loading is connected by the star to the available origin. In these cases the current circuit of the wattmeter joins consistently from one of loading phases, and the chain the napryazheyoniya of the wattmeter joins on tension of that phase which current passes through the wattmeter.

If the origin is unavailable or loading is connected on the delta network, the artificial origin is applied.

So the origin of the star formed of branch circuit resistance _{vattmetrarn} is called_{. W } and two another equal to it additional _{soprotivleniy:rv} _{irc}.

At pravilyyony connection with the artificial origin the branch circuit of the wattmeter is under the phase tension and through ваттметр there passes the phase current. The wattmeter measures the phase power _{of the USAn Federation }in such conditions_{, }and the power of three-phase loading is defined by means of multiplication of the indication of the wattmeter again. Usually the manufacturer supplies the wattmeter with the artificial origin for measurement in three-phase systems.

Measurements of power in three-phase three-wire systems at asymmetrical loading vypolnyayotsya in most cases on the way of two wattmeters. The peculiar osoyobennost of this way is that circumstance that not only at asymmetrical, but even at the symmetric nagruzyoka of the indication of two wattmeters are in most cases not equal, and indications of one of wattmeters can become negative. Power of three-phase system in this case is necessary определять as the algebraic sum of indications of two vattyometr.

Justice of such way is proved on the basis of the equations of the instantaneous power expressed through instantaneous values of tension and currents. Instantaneous power of any fayoza is equal to the work of instantaneous values of phase napryazheyoniya and current, and the instantaneous power of three-phase system is equal to the sum of instantaneous phase powers. For example, at the soyedineyoniya the star:

- Р =
_{uAiA}_{+ }_{uBiB}_{+ }_{uCiC}_{}.

But according to the first law of Kirchhoff at the wye without zero wire:

_{iA}_{ + }_{iB + }_{iC}and , therefore;_{iC }= - (_{iA + }_{iB}) =0.

Having substituted this value in the power equation, we will receive:

- p = (
_{uA }-_{uC})_{iA}+ (_{uv-}_{ис})_{iB}.

The difference of phase tension is equal to the corresponding lineyyony tension:

_{uA-uC}_{}_{ }=_{uAC},_{uB }-_{ис }_{=uBC}_{,}on the basis of what;- p = +
_{+ +}+_{+ }+_{uBCiB}_{}_{}.

Therefore, the power of three-phase system can be expressed by the sum of two works, and these two works can be measured by two wattmeters included according to the scheme of the method.

There is no need especially to prove justice of the way of two wattmeters for delta connection as at opredeyolenny values of linear stresses and currents power does not depend on the way of connection of loading.

Let's note the peculiar feature of ways of two vattmetyor: the system of linear stresses in the normal posledovayotelnost is designated _{by IAV}, _{ивс}, _{ISA}, and enters the equation of this spoyosob _{IAS} tension. Such shift of indexes oboyoznachat that in relation to the first wattmeter it is necessary to change tension phase on 180 °. For this purpose it is enough to connect "beginning" (the clip to the sign of the asterisk) branch circuits of the first wattmeter with the wire And, and "end" of this chain (the clip, at kotoyory rated voltage is specified) with the Page wire.

**Distribution of power of three-phase system between pokazayoniya of two wattmeters depends, mainly, on size and the sign of phase shift.** Let's trace this dependence in the simplest case at the symmetrical load. If instead of the instant moshchyonost in the equation (101) to substitute the active (average) moshchyonost of three-phase system, then it is necessary to replace instantaneous values of tension and currents acting and to enter into uravneyony cosines of phase shifts between the corresponding napryazheyoniya and currents. Thus, the equation of power will take the following form:

- р =
_{r1 }+_{r2}=_{uaciacosf1}_{}_{}+_{ivsivcosf2}_{}_{}.

At the symmetrical load in size line currents:

_{iA =iB}_{}=_{Il}.

are equal among themselves as well as linear stresses:

_{uAC}=_{ивс}=_{silt}.

On the vectogram of the three-phase siyostema on which the vector _{of uAC} is constructed equal in size and opposite on _{the napravleniyuis} is constructed

On the basis of this chart the phase angle between _{vekyotoramiuac} _{iia} both the phase angle _{f2mezhdu} _{векторамиивс} and _{iB }will be respectively _{f1 }= ф – ^{30o} and _{f2 }= ф + 30 ^{lakes}. Therefore, the indications of two wattmeters making the power of three-phase system will be expressed as follows:

- р =
_{r1 }+_{r2}=_{ililcos}_{}(ф –^{30o}) +_{ililcos}_{}(ф +^{30o}).

This expression shows that at the symmetrical load of the indication of wattmeters are equal only at ф = 0. If ф> ^{60o}, then the arrow of the second wattmeter deviates for zero scale, and chtoyoba to count the indication of the second wattmeter in such conditions, it is necessary to switch (i.e. to trade places in the scheme) device branch circuit clips. Often for the current phase izyomeneniye on 180 ° the special switch is built in napryayozheniye chains in the body of the wattmeter. Indications of the second wattmeter after switching should be considered negative, and, chtoyoba to determine the power of the three-phase usyotanovka, it is necessary to subtract these indications from indications of the first wattmeter.

For measurement of power in three-phase four-wire systems to protozoa the way of three wattmeters is. Each of line conductors joins the current circuit of one of wattmeters, and the branch circuit of each of wattmeters vklyuyochatsya between the corresponding line conductor and the zero wire of system.

At such connection each of wattmeters measures the moshchyonost of one phase of system. Therefore, the active power of all three-phase system will be equal to the simple sum of indications of three wattmeters:

- р =
_{r1 }+_{r2+}_{r3}.

In plants on distribution boards wattmeters of the three-phase current are widely applied. They represent two (for three-wire system) or three (for four-wire system) measuring the mechanism, connected by the general axis and such way influencing the general arrow. These measuring mechanisms turn on in the trekhfazyony chain according to the way of two wattmeters or the way of three wattmeters.

- To add the comment

Vitya

If F =30 DEGREES and wattmeters are included in two consecutive links by the triangle then what there will be 0 and 60 degrees?