Measurement of Power in a Three phase Circuit | Basic Electrical Technology - Electrical Engineering (EE) PDF Download

Two-wattmeter Method of Power Measurement in a Threephase Circuit 

Fig. 20.1 Connection diagram for two-wattmeter method of power measurement  in a three-phase balanced system with  star-connected load  \

The connection diagram for the measurement of power in a three-phase circuit using two wattmeters, is given in Fig. 20.1. This is irrespective of the circuit connection – star or delta. The circuit may be taken as unbalanced one, balanced type being only a special case. Please note the connection of the two wattmeters. The current coils of the wattmeters, 1 & 2, are in series with the two phases, R & B , with the pressure or voltage coils being connected across R − Y and B − Y respectively. Y is the third phase, in which no current coil is connected.  

If star-connected circuit is taken as an example, the total instantaneous power consumed in the circuit is,

Each of the terms in the above expression is the instantaneous power consumed for the phases. From the connection diagram, the current in, and the voltage across the respective (current, and pressure or voltage) coils in the wattmeter,W₁  are   and   So, the instantaneous power measured by the wattmeter,W₁is,

Similarly, the instantaneous power measured by the wattmeter, W₂ is,  

The sum of the two readings as given above is, 

Since, 

Substituting the above expression for in the earlier one,

If this expression is compared with the earlier expression for the total instantaneous power consumed in the circuit, they are found to be the same. So, it can be concluded that the sum of the two wattmeter readings is the total power consumed in the three-phase circuit, assumed here as a star-connected one. This may also be easily proved for deltaconnected circuit. As no other condition is imposed, the circuit can be taken as an unbalanced one, the balanced type being only a special case, as stated earlier. 

Phasor diagram for a three-phase balanced star-connected circuit

Fig. 20.2 Phasor diagram for two-wattmeter method of power measurement   in a three-phase system with balanced star-connected load  

The phasor diagram using the two-wattmeter method, for a three-phase balanced starconnected circuit is shown in Fig. 20.2. Please refer to the phasor diagrams shown in the figures 18.4 &18.6b. As given in lesson No. 18, the phase currents lags the respective phase voltages by φ = φ _p , the angle of the load impedance per phase. The angle, φ  is taken as positive for inductive load. Also the neutral point on the load (N ) is same as the neutral point on the source (N' ), if it is assumed to be connected in star. The voltage at that point is zero (0).

The reading of the first wattmeter is,

The reading of the second wattmeter is, 

The line voltage,  leads the respective phase voltage,  by  , and the phase voltage, leads the phase current,   by φ . So, the phase difference between& is   Similarly, the phase difference between &  in the second case,   &   in the second case, can be found and also checked from the phasor diagram.

The sum of the two wattmeter readings is, 

So, ( W₁ + W₂ ) is equal to the total power consumed by the balanced load. This method is also valid for balanced delta-connected load, and can be easily obtained. The phasor diagram for this case is shown in the example No. 20.2. 

Determination of power factor for the balanced load 

The difference of the two wattmeter readings is, 

 If the two sides is multiplied by √3 , we get  

From the two expressions, we get, 

The power factor, cosφ of the balanced load can be obtained as given here, using two wattmeter readings.

The two relations, cosφ and sin φ can also be found as, 

  and 

Comments on Two Wattmeter Readings 

When the balanced load is only resistive (φ = 0° ), i.e. power factor ( cosφ = 1.0 ), the readings of the two wattmeters   are equal and positive.  Before taking the case of purely reactive (inductive/capacitive) load, let us take first lagging power factor as ( cosφ = 0.5 ), i.e. φ = +60° . Under this condition,   

 \

It may be noted that the magnitudes of the phase or line voltage and also phase current are assumed to be constant, which means that the magnitude of the load impedance (inductive) is constant, but the angle, φ varies as stated. 

 As the lagging power factor decreases from 1.0 to 0.5, with φ increasing from  to   the reading of the first wattmeter w₁ decreases from a certain positive value to zero (0). But the reading of the second wattmeter W₂,  increases from a certain positive value to positive maximum, as the lagging power factor is decreased from 1.0 to 0.866 (= cos 30°) φ increasing from 0° to + 30° . As the lagging power factor decreases from 0.866 to 0.5, with φ increasing from + 30° to + 60° , the reading of the second wattmeter,  decreases from positive maximum to a certain positive value. It may be noted that, in all these cases,   , with both the readings being positive. If the lagging power factor is 0.0 (φ = +90° ), the circuit being purely inductive, the readings of the two wattmeters   are equal and opposite, i.e.,  is negative and  is positive. The total power consumed is zero, being the sum of the two wattmeter readings, as the circuit is purely inductive. This means that, as the lagging power decreases from 0.5 to 0.0, with φ increasing from   to , the reading of the first wattmeter,W₁  decreases from zero (0) to a certain negative value, while the reading of the second wattmeter W₂,  decreases from a certain positive value to lower positive one. It may be noted that   which means that the total power consumed, i.e., ( W₁ + W₂ ) is positive, with only  being negative. The variation of two wattmeter readings as stated earlier, with change in power factor (or phase angle) is now summarized in Table 20.1. The power factor [pf] (=cosθ) is taken as lagging, the phase current lagging the phase voltage by the angle, φ (taken as positive (+ve)), as shown for balanced star-connected load in Fig. 20.2. The circuit is shown in Fig. 20.1. All these are also valid for balanced delta-connected load. 

Sl. No.	Power factor [pf] (Phase angle)	Wattmeter readings (W)		Remarks
		W1	W2
1.	pf = unity [1.0] (φ = 0° )	+ve	+ve	W 1 = W2
2.	0.5 < pf < 1.0 ( 60° >φ > 0° )	+ve	+ve	W 1 > W2
3.	pf = 0.5 (φ = 60° )	+ve	zero (0.0)	Total power = W1
4.	0.0 < pf < 0.5 ( 90° >φ > 6 0° )	+ve	-ve
5.	pf  = zero [0.0] (φ = 90° )	+ve	-ve

Table 20.1 Variation of two wattmeter readings with change in power factor of the load current  

It may be noted that, if the power factor is leading (φ = negative (-ve)), the circuit being capacitive, the readings of the two wattmeters change with the readings interchanging, i.e.,W₁ taking the value of W₂ and vice versa. All the points as stated earlier, remain valid, with the comments as given earlier. The first one (#1) in Table 20.1 is a special case, neither lagging, nor leading, with pf = 1.0. But in second one (#2), both readings remain +ve, with W₁ < W₂ Same is the case in fourth one (#4), where W₁ is –ve and W₂  is +ve, with    total power being positive (+ve). For third case (#3), W₁ = 0.0 and W₂ is +ve, with total power= W₂ For last (fifth) case (#5), W₁   is –ve and  W₂ is +ve, with    total power being zero (0.0).    

Power measurement using one wattmeter only for a balanced load 

Fig. 20.4 Connection diagram for power measurement using only one wattmeter in a three-phase system with balanced star-connected load  

Fig. 20.5 Connection diagram for power measurement using only one wattmeter in a three-phase system with balanced delta-connected load

The circuit diagram for measuring power for a balanced three-phase load is shown in Fig. 20.3. The only assumption made is that, either the neutral point on the load or source side is available. The wattmeter measures the power consumed for one phase only, and the reading is  . The total power is three times the above reading, as the circuit is balanced. So, the load must be star-connected and of course balanced one, with the load neutral point being available. The load may also be delta-connected balanced one, if the neutral pinpoint on the source side is available. Otherwise for measuring total power for delta-connected balanced load using one wattmeter only, the connection diagram is given in Fig. 20.4. The wattmeter as stated earlier, measures power for one phase only, with the total power consumed may be obtained by multiplying it by three.  

Example 20.1  

Calculate the readings of the two wattmeters ( W₁& W₂) connected to measure the total power for a balanced star-connected load shown in Fig. 20.6a, fed from a three phase, 400 V balanced supply with phase sequence as R-Y-B. The load impedance per phase is   Also find the line and phase currents, power factor, total power, total reactive VA and total VA. 

Fig. 20.6 (a) Circuit diagram for a three-phase system with balanced starconnected load (Example 20.1)  (b) Phasor di agram