Page 1 Module 5 Three-phase AC Circuits Version 2 EE IIT, Kharagpur Page 2 Module 5 Three-phase AC Circuits Version 2 EE IIT, Kharagpur Lesson 20 Measurement of Power in a Three-phase Circuit Version 2 EE IIT, Kharagpur Page 3 Module 5 Three-phase AC Circuits Version 2 EE IIT, Kharagpur Lesson 20 Measurement of Power in a Three-phase Circuit Version 2 EE IIT, Kharagpur In the previous lesson, the phase and line currents for balanced delta-connected load fed from a three-phase supply, along with the expression for total power, are presented. In this lesson, the measurement of total power in a three-phase circuit, both balanced and unbalanced, is discussed. The connection diagram for two-wattmeter method, along with the relevant phasor diagram for balanced load, is described. Keywords: power measurement, two-wattmeter method, balanced and unbalanced loads, star- and delta-connections. After going through this lesson, the students will be able to answer the following questions: 1. How to connect the two-wattmeter to measure the total power in a three-phase circuit – both balanced and unbalanced? 2. Also how to find the power factor for the case of the above balanced load, from the reading of the two-wattmeter, for the two types of connections – star and delta? Two-wattmeter Method of Power Measurement in a Three- phase Circuit • • • • • R c L c R a L a L b R b I RN 1 I YN 1 I BN 1 W 2 Y B R W 1 V RY • 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 Version 2 EE IIT, Kharagpur Page 4 Module 5 Three-phase AC Circuits Version 2 EE IIT, Kharagpur Lesson 20 Measurement of Power in a Three-phase Circuit Version 2 EE IIT, Kharagpur In the previous lesson, the phase and line currents for balanced delta-connected load fed from a three-phase supply, along with the expression for total power, are presented. In this lesson, the measurement of total power in a three-phase circuit, both balanced and unbalanced, is discussed. The connection diagram for two-wattmeter method, along with the relevant phasor diagram for balanced load, is described. Keywords: power measurement, two-wattmeter method, balanced and unbalanced loads, star- and delta-connections. After going through this lesson, the students will be able to answer the following questions: 1. How to connect the two-wattmeter to measure the total power in a three-phase circuit – both balanced and unbalanced? 2. Also how to find the power factor for the case of the above balanced load, from the reading of the two-wattmeter, for the two types of connections – star and delta? Two-wattmeter Method of Power Measurement in a Three- phase Circuit • • • • • R c L c R a L a L b R b I RN 1 I YN 1 I BN 1 W 2 Y B R W 1 V RY • 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 Version 2 EE IIT, Kharagpur coils being connected across Y R - and Y B - 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, N B N B N Y N Y N R N R v i v i v i W ' ' ' ' ' ' · + · + · = 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, are and . So, the instantaneous power measured by the wattmeter, is, 1 W N R i ' N Y N R RY v v v ' ' - = 1 W () N Y N R N R RY N R v v i v i W ' ' ' ' - · = · = 1 Similarly, the instantaneous power measured by the wattmeter, is, 2 W () N Y N B N B BY N B v v i v i W ' ' ' ' - · = · = 2 The sum of the two readings as given above is, () ()( N B N R N Y N B N B N R N R N Y N B N B N Y N R N R i i v v i v i v v i v v i W W ' ' ' ' ' ' ' ' ' ' ' ' ' + ·) - · + · = - · + - · = + 2 1 Since, or, 0 = + + ' ' ' N B N Y N R i i i ( ) N B N R N Y i i i ' ' ' + = Substituting the above expression for in the earlier one, N Y i ' N Y N Y N B N B N R N R v i v i v i W W ' ' ' ' ' ' · + · + · = + 2 1 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 delta- connected 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. Version 2 EE IIT, Kharagpur Page 5 Module 5 Three-phase AC Circuits Version 2 EE IIT, Kharagpur Lesson 20 Measurement of Power in a Three-phase Circuit Version 2 EE IIT, Kharagpur In the previous lesson, the phase and line currents for balanced delta-connected load fed from a three-phase supply, along with the expression for total power, are presented. In this lesson, the measurement of total power in a three-phase circuit, both balanced and unbalanced, is discussed. The connection diagram for two-wattmeter method, along with the relevant phasor diagram for balanced load, is described. Keywords: power measurement, two-wattmeter method, balanced and unbalanced loads, star- and delta-connections. After going through this lesson, the students will be able to answer the following questions: 1. How to connect the two-wattmeter to measure the total power in a three-phase circuit – both balanced and unbalanced? 2. Also how to find the power factor for the case of the above balanced load, from the reading of the two-wattmeter, for the two types of connections – star and delta? Two-wattmeter Method of Power Measurement in a Three- phase Circuit • • • • • R c L c R a L a L b R b I RN 1 I YN 1 I BN 1 W 2 Y B R W 1 V RY • 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 Version 2 EE IIT, Kharagpur coils being connected across Y R - and Y B - 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, N B N B N Y N Y N R N R v i v i v i W ' ' ' ' ' ' · + · + · = 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, are and . So, the instantaneous power measured by the wattmeter, is, 1 W N R i ' N Y N R RY v v v ' ' - = 1 W () N Y N R N R RY N R v v i v i W ' ' ' ' - · = · = 1 Similarly, the instantaneous power measured by the wattmeter, is, 2 W () N Y N B N B BY N B v v i v i W ' ' ' ' - · = · = 2 The sum of the two readings as given above is, () ()( N B N R N Y N B N B N R N R N Y N B N B N Y N R N R i i v v i v i v v i v v i W W ' ' ' ' ' ' ' ' ' ' ' ' ' + ·) - · + · = - · + - · = + 2 1 Since, or, 0 = + + ' ' ' N B N Y N R i i i ( ) N B N R N Y i i i ' ' ' + = Substituting the above expression for in the earlier one, N Y i ' N Y N Y N B N B N R N R v i v i v i W W ' ' ' ' ' ' · + · + · = + 2 1 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 delta- connected 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. Version 2 EE IIT, Kharagpur Phasor diagram for a three-phase balanced star-connected circuit V BY V RY V YN I YN V BN V RN I RN F F F F-30° I BN 30°+ F 30° 30° Fig. 20.2 Phasor diagram for two-wattmeter method of power measurement in a three-phase system with balanced star-connected load V YB 30° The phasor diagram using the two-wattmeter method, for a three-phase balanced star- connected 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 p f f = , the angle of the load impedance per phase. The angle, f is taken as positive for inductive load. Also the neutral point on the load ( ) is same as the neutral point on the source ( ), if it is assumed to be connected in star. The voltage at that point is zero (0). N ' N The reading of the first wattmeter is, () ( ) () ? ? + ° · · · = + ° · · = · · = 30 cos 3 30 cos , cos 1 p p RN RY RN RY RN RY I V I V I V I V W The reading of the second wattmeter is, () ( ) () ? ? - ° · · · = - ° · · = · · = 30 cos 3 30 cos , cos 2 p p BN BY BN BY BN BY I V I V I V I V W The line voltage, leads the respective phase voltage, by , and the phase voltage, leads the phase current, by RY V RN V ° 30 RN V RN I f . So, the phase difference between & is RY V RN I ( ) f + ° 30 . Similarly, the phase difference between & in the second case, can be found and also checked from the phasor diagram. BY V BN I Version 2 EE IIT, KharagpurRead More

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