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Electric Circuits II Power Measurement

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Dr. Firas Obeidat

Dr. Firas Obeidat – Philadelphia University

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Table of contents

1 • Single-Phase Power Measurement

2 • Three-Phase Power Measurement


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Single-Phase Power Measurement The wattmeter is the instrument used for measuring the average

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.

The varmeter is the instrument used for measuring the reactive power.

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).


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Single-Phase Power Measurement

if 𝑣 𝑡 = 𝑉𝑚cos(𝜔𝑡 + 𝜃𝑣) 𝑖 𝑡 = 𝐼𝑚cos(𝜔𝑡 + 𝜃𝑖)

then

and

𝐕𝐫𝐦𝐬 = 𝑉𝑟𝑚𝑠∠𝜃𝑣 =𝑉𝑚

2∠𝜃𝑣 𝐈𝐫𝐦𝐬 = 𝐼𝑟𝑚𝑠∠𝜃𝑖 =

𝐼𝑚

2∠𝜃𝑖

and

The wattmeter measures the average power given by

𝑃 = 𝐕𝐫𝐦𝐬 𝐈𝐫𝐦𝐬 cos 𝜃𝑣 − 𝜃𝑖 = 𝑉𝑟𝑚𝑠𝐼𝑟𝑚𝑠cos 𝜃𝑣 − 𝜃𝑖

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


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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: 1.437 kW


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Three-Phase Power Measurement

A single wattmeter can also measure the average power in a three-phase system

that is balanced, so that P1=P2=P3; 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 P1,P2 and P3 correspond to the readings of wattmeters W1,W2 and W3

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 delta-

connected 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.


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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 delta-

connected, 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 ZY=ZY∠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 30o. Thus,

the total phase difference between the phase current Ia and line voltage Vab is

θ+30o and the average power read by wattmeter is


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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


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The tangent of the power factor is given by

The total apparent power can be obtained as

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.


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Three-Phase Power Measurement Example: Three wattmeters W1,W2 and W3 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.

(a)

(b) The total power absorbed is

or


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(a)

Example: The two-wattmeter method produces wattmeter readings P1=1560 W and

P1=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.

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


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(d)

It is a lagging pf because QT is positive or P1>P2

The phase impedance is Zp=Zp∠θ, θ is the same as pf angle, so θ=14.33o

For a delta-connected load VL=Vp=220 V

(c) The power angle is


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(a)

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 W1 and W2. Find PT and QT.

The impedance per phase is

The pf angle is 36.87o

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