Power and Energy Measurement
ENE 240 Electrical and Electronic Measurement
Class 11, February 4, 2009
werapon.chi@kmutt.ac.th
1
Work, Energy and Power
• Work is an activity of force and movement in
the direction of force (Joules)
• Energy is the capacity for doing work (Joules)
• Power is the rate of using energy (Watt)
P = W / t , Work done in the time
= E / t , Energy transferred in the time
2
1
Electrical Power
• DC: Instantaneous power at time t
P(t) = I(t) V(t)
= I2(t) R
= V2(t) / R
• AC: Mean power per one cycle
Pavg = 1/T ∫t=0→T P(t) dt
= Irms2 R
= Vrms2 / R
3
Apparent and True Power
• True power is the power used by the load
in Watt unit (W),
P = I2R = V2/R
• Reactive power in Volt-Amps-Reactive unit
(VAR),
Q = I2X = V2/X
• Apparent power is the power that is
applied to the load in Volt-Amps unit (VA),
S = I2Z = V2/Z
where impedance Z = R + jX
4
2
Power Factor
cos φ = True Power / Apparent Power
• Pure resistance, cos 0° = 1 ⇒ perfect
• Pure inductance, cos 90° = 0
• Pure capacitance, cos –90° = 0
Im
Reactive Power
Apparent Power
φ
0
True Power
Re
5
Power Factor (Cont’d)
P = I V cosφ
•
•
•
•
•
•
Up to a kind of loads being powered
In phase or φ = 0 (resistor)
Lagging or current lags voltage, φ > 0 (inductor)
Leading or current leads voltage, φ < 0 (capacitor)
Lower power factor, higher losses
Need to improve power factor
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3
Power Factor (Cont’d)
I = sinω(t+∆t)
= sin(ωt+ω∆t)
= sin(θ+φ)
V = sinθ
I = sin(θ+φ)
t
∆t
7
Wattmeter
• To measure the true power
• Two fixed coil is used to
provide a magnetic field
that is proportional to a
current passing through it,
instead of using a
permanent magnet.
B
I
Scale
Pointer
Spring
Fixed Coil
I2
I1
Moving Coil
(Rotatable)
Meter Terminal
Electrodynamometer
Right Hand’s Rule
8
4
Wattmeter (Cont’d)
• Fixed coils, B ∝ I1
• Moving coil, Torgue ∝ B I2
θ = K I1 I2
= K I2
∝P
M
g
in
ov
Fixed
, I = I1 = I2
Fixed
9
Wattmeter (Cont’d)
• Full-scale deflection current 5-100 mA
• For larger FSD up to 20 A, it is shunted by
a low resistance.
• Electrodynamometer can be used for both
DC and AC (rms) measurement.
M
g
in
ov
Fixed
Fixed
Shunt
10
5
Single-Phase Wattmeter
• Based on the principles of the dynamometer
• The current passing through moving coil and
load is proportional to the potential difference
across the load.
il
Co
ge
• θ = K I1 I2
lta
o
V
= K I (V/R)
Load
= (K/R) (IV) I Current Coil
∝P
V
11
Single-Phase Wattmeter (Cont’d)
• Low-current high-voltage load
Fixed
Preading = Iload2(Rfixed+Rload)
Iload
Supply
Load
Moving
Pload = Iload2Rload
Wattmeter reads high due to the potential drop
across the fixed coil (if neglect the reactance).
• High-current low-voltage load
Fixed
Supply
Moving
+
Vload
–
Load
Preading = Vload2 / (Rmoving || Rload)
= Vload2 RmovingRload
(Rmoving+Rload)
Pload = Vload2 / Rload
Wattmeter reads high due to the current through
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the movable coil (if neglect the reactance).
6
Single-Phase Wattmeter (Cont’d)
• Compensated wattmeter
Fixed
Compensating
Supply
Moving
Load
To reduce the systematic error, the compensating
coil current is, however, in the opposite direction to
the load current and cancels out the proportion of
the magnetic flux due to the moving coil current.
13
Crossed-Coil Power Factor Meter
• Two moving coils are
crossed and connected in
series to an inductor and
resistor.
0.8
Lag
• For PF = 1, the coil with R
will be in phase with the
Supply Fixed
line current (moving),
Coil
while the coil with L will be
out of phase by 90° (no
moving).
• For PF = 0, coil L moves
and coil R does not move.
0.9 1 0.9
0.8
Lead
C
L
ro
ss
ed
Load
R
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7
Three-Phase Power Distribution
The three-phase system uses less conductor material
to transmit electric power that the others.
Y-Type,
120°
VAB = VAN – VBN
|VAB| = |VAN| cos30° + |VBN| cos30°
, |VAN| = |VBN|
= √3 |VAN|
A
VAB
N
30°
30°
–VBN
C
120°
B
VAN
12
0°
VBN
15
Three-Phase Loads
3-Phase
Generator
3-Phase Transmission Line
IPhase
ILine
VPhase
VLine
Neutral
ILine
3-Phase
Loads
Star-Type,
VLine = √3 VPhase
ILine = IPhase
Balanced loads,
P1 = P2 = P3 = P = IL VP cosφ
Ptotal = P×3 = 3 IL VP cosφ
= 3 IL (VL / √3 ) cosφ
= √3 IL VL cosφ
Similarly, Q = √3 IL VL sinφ
IPhase
Delta-Type,
VL = VP
IL = √3 IP
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8
Three-Phase Wattmeter
• For three-phase system with balanced
load, just using single-phase wattmeter,
P×3
Current Coil
Voltage Coil
Reading
P = √3 /3 ILVLcosφ
17
Unbalanced-Load 3-Phase Wattmeter
P1
P2
Using 3 Meters
P3
By using 3 wattmeters,
Ptotal = P1 + P2 + P3
18
9
Balanced-Load 3-Phase Wattmeter
Brondel’s theory,
P = I1V1 + I2V2 + I3V3
P1
Kirchoff’s law,
V1 = V3 + V1′
V2 = V3 + V2′
I1
V1′
V2
Common
S
P2
Therefore, 0
P = (I1+I2+I3)V3
+ I1V1′ + I2V2′
= P1 + P2
V1
I3
I2
V3
V2′
19
Balanced-Load 3-Phase Wattmeter (Cont’d)
P1
By using 2 wattmeters, R
I
R
P1 = IR VRB cosφ1
S
B
VRB = VRS – VBS
P2
Y
Using 2 Meters
Balanced, |VRS| = |VBS|
VBS
φ1 = 30° – φ
120°
–VBS
φ1
φ lagging
VRS
60°
IR
30°
VRB
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10
Balanced-Load 3-Phase Wattmeter (Cont’d)
P1 = ILVLcos(30° – φ)
= ILVL ( √3 /2 cosφ + 1/2 sinφ )
Similarly,
VBS
P2 = ILVL ( √3 /2 cosφ – 1/2 sinφ )
120°
φ
φ2 = 30° + φ
30°
IY
VYS
–VBS
VYB
Note: cos(θ φ) = cosθ cosφ ± sinθ sinφ
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±
Balanced-Load 3-Phase Wattmeter (Cont’d)
Total Power,
Ptotal = P1 + P2
= √3 ILVL cosφ
The same as using 1 meter for balanced
load system
and P1 – P2 = ILVL sinφ
22
11
Balanced-Load 3-Phase Wattmeter (Cont’d)
Reactive Power,
Q = √3 ILVL sinφ = √3 (P1 – P2)
(P1–P2)/(P1+P2) = ILVLsinφ / √3 ILVLcosφ
= tanφ / √3
tanφ = √3 (P1 – P2) / (P1 + P2)
23
Balanced-Load 3-Phase Wattmeter (Cont’d)
sin2φ + cos2φ = 1
tan2φ + 1 = 1/cos2φ
3(P1–P2)2/(P1+P2)2 + 1 = 1/cos2φ
Power factor,
cosφ = √ 1 / [ 1 + 3(P1–P2)2/(P1+P2)2 ]
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12
Watt-Hour-Meter
To measure the electrical energy supplied to industrial
and domestic consumers in unit (kW-hr)
Generated Torque by Eddy Current = kg I V cosφ
Magnetic Breaking Torque = kb N
where N is number of revolution per unit time
(gearing to a mechanical counter to count watt-hours)
kb N = kg I V cosφ
N = (kg/kb) IVcosφ
25
Watt-Hour-Meter (Cont’d)
Gear Wheels for Dial Rotation
N
Voltage Coil
S
S
Magnetic break
Eddy Currrent
N
Current Coil
Rotating Aluminium Disk
Therefore, number of revolution refers to the
energy used by loads.
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