Electric Circuits II Power Measurement. Dr. Firas Obeidat

Transcription

1 Electric Circuits II Power Measurement Dr. Firas Obeidat 1

2 Table of contents 1 Single-Phase Power Measurement 2 Three-Phase Power Measurement 2

3 Single-Phase Power Measurement The wattmeter is the instrument used for measuring the average power. The varmeter is the instrument used for measuring the reactive power. The wattmeter consists essentially of two coils: the current coil and the voltage coil. A current coil with very low impedance (ideally zero) is connected in series with the load and responds to the load current. The voltage coil with very high impedance (ideally infinite) is connected in parallel with the load and responds to the load voltage. The current coil acts like a short circuit because of its low impedance; the voltage coil behaves like an open circuit because of its high impedance. When the two coils are energized, the mechanical inertia of the moving system produces a deflection angle that is proportional to the average value of the product v(t)i(t). 3

4 Single-Phase Power Measurement if v t = V m cos (ωt + θ v ) and i t = I m cos (ωt + θ i ) then V rms = V rms θ v = V m 2 θ v and I rms = I rms θ i = I m 2 θ i The wattmeter measures the average power given by P = V rms I rms cos θ v θ i = V rms I rms cos θ v θ i Example: Find the wattmeter reading of the circuit The wattmeter reads the average power absorbed by the impedance (8-j6) because the current coil is in series with the impedance while the voltage coil is in parallel with it. The current through the circuit is 4

5 Single-Phase Power Measurement The voltage across the impedance (8-j6) is The complex power is The wattmeter reads Q: For the circuit, find the wattmeter reading. Answer: kw 5

6 A single wattmeter can also measure the average power in a three-phase system that is balanced, so that P 1 =P 2 =P 3 ; the total power is three times the reading of that one wattmeter. Two or three single-phase wattmeters are necessary to measure power if the system is unbalanced. The three wattmeter method of power measurement will work regardless of whether the load is balanced or unbalanced, wye or delta-connected. The total average power is the algebraic sum of the three wattmeter readings Where P 1,P 2 and P 3 correspond to the readings of wattmeters W 1,W 2 and W 3 respectively. The common or reference point o is selected arbitrarily. If the load is wye-connected, point o can be connected to the neutral point n. For a deltaconnected load, point o can be connected to any point. If point o is connected to point b, for example, the voltage coil in wattmeter reads zero and indicating that wattmeter is not necessary. Thus, two wattmeters are sufficient to measure the total power. 6

7 The two-wattmeter method is the most commonly used method for three-phase power measurement. The two wattmeters must be properly connected to any two phases. The current coil of each wattmeter measures the line current, while the respective voltage coil is connected between the line and the third line and measures the line voltage. The algebraic sum of the two wattmeter readings equals the total average power absorbed by the load, regardless of whether it is wye- or deltaconnected, balanced or unbalanced. The total real power is equal to the algebraic sum of the two wattmeter readings, For a balanced three-phase system wye-connected load Assume the source is in the abc sequence and the load impedance Z Y =Z Y θ. Due to the load impedance, each voltage coil leads its current coil by θ, so that the power factor is cosθ. each line voltage leads the corresponding phase voltage by 30 o. Thus, the total phase difference between the phase current I a and line voltage V ab is θ+30 o and the average power read by wattmeter is 7

8 Use the trigonometric identities The sum of the wattmeter readings gives the total average power The difference of the wattmeter readings is proportional to the total reactive power, or 8

9 The total apparent power can be obtained as The tangent of the power factor is given by The equations derived for balanced wye-connected load are equally valid for a balanced delta-connected load. The two-wattmeter method cannot be used for power measurement in a three-phase four-wire system unless the current through the neutral line is zero. The three-wattmeter method are used to measure the real power in a three-phase four-wire system. 9

10 (a) Example: Three wattmeters W 1,W 2 and W 3 are connected, respectively, to phases a, b, and c to measure the total power absorbed by the unbalanced wye-connected load. The three phase load is connected to balanced voltages of 100 V and the acb sequence. (a) Predict the wattmeter readings. (b) Find the total power absorbed. (b) The total power absorbed is or 10

11 Example: The two-wattmeter method produces wattmeter readings P 1 =1560 W and P 1 =2100 W when connected to a delta-connected load. If the line voltage is 220 V, calculate: (a) the per-phase average power, (b) the perphase reactive power, (c) the power factor, and (d) the phase impedance. (a) The total real or average power is The per-phase average power is then (b) The total reactive power is The per-phase reactive power is 11

12 (c) The power angle is It is a lagging pf because Q T is positive or P 1 >P 2 (d) The phase impedance is Z p =Z p θ, θ is the same as pf angle, so θ=14.33 o For a delta-connected load V L =V p =220 V 12

13 Example: The three-phase balanced load shown in the circuit has impedance per phase of ZY=8+j6 Ω. If the load is connected to 208-V lines, predict the readings of the wattmeters W 1 and W 2. Find P T and Q T. (a) The impedance per phase is The pf angle is o 13

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