ac power calculation instantaneous, average and reactive power apparent power and power factor
DESCRIPTION
AC POWER CALCULATION Instantaneous, average and reactive power Apparent Power and Power Factor Complex Power. SEE 1023 Circuit Theory. Dr. Nik Rumzi Nik Idris. i(t). Passive, linear network. Instantaneous, Average and Reactive Power. + v(t) . - PowerPoint PPT PresentationTRANSCRIPT
1
AC POWER CALCULATIONAC POWER CALCULATIONInstantaneous, average and reactive powerInstantaneous, average and reactive power
Apparent Power and Power FactorApparent Power and Power FactorComplex PowerComplex Power
Dr. Nik Rumzi Nik IdrisDr. Nik Rumzi Nik Idris
SEE 1023 Circuit TheorySEE 1023 Circuit Theory
2
Instantaneous, Average and Reactive PowerInstantaneous, Average and Reactive Power
+v(t)
i(t)
Passive, linear network
Instantaneous power absorbed by the network is, p =v(t).i(t)
Let v(t) = Vm cos (t + v) and i(t) = Imcos(t + i)
Which can be written as
v(t) = Vm cos (t + v i) and i(t) = Imcos(t)
3
v(t) = Vm cos (t + v i) and i(t) = Imcos(t)
p = Vm cos(t + v – i ) . Im cos(t)
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
-2
-1
0
1
2
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08-0.5
0
0.5
1
1.5
2
vi
Instantaneous Power (p)
Example when v i = 45o
positivepositive p p = power transferred from source to network
negativenegative p p = power transferred from network to source
45o
4
v(t) = Vm cos (t + v i) and i(t) = Imcos(t)
p = Vm cos(t + v – i ) . Im cos(t)
t2sin)sin(2IVt2cos)cos(
2IV)cos(
2IV
ivmm
ivmm
ivmm p =
p = P + Pcos(2t) Qsin(2t)
Using trigonometry functions, it can be shown that:
)cos(2IVP ivmm = AVERAGE POWER (watt)
)sin(2IVQ ivmm = REACTIVE POWER (var)
Which can be written as
5
t2sin)sin(2IVt2cos)cos(
2IV)cos(
2IV
ivmm
ivmm
ivmm p =
6
t2sin)sin(2IVt2cos)cos(
2IV)cos(
2IV
ivmm
ivmm
ivmm p =
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08-1
0
1
2
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.080
0.5
1
1.5
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08-1
-0.5
0
0.5
1
Example for v-i = 45o
7
t2sin)sin(2IVt2cos)cos(
2IV)cos(
2IV
ivmm
ivmm
ivmm p =
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08-1
0
1
2
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.080
0.5
1
1.5
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08-1
-0.5
0
0.5
1
P = average power
Q = reactive power
p = P + P cos(2t) Q sin(2t)
8
P = AVERAGE POWER
Q = REACTIVE POWER
p = P + P cos(2t) Q sin(2t)
• Useful power – also known as ACTIVE POWER
• Converted to other useful form of energy – heat, light, sound, etc
• Power charged by TNB
• Power that is being transferred back and forth between load and source
• Associated with L or C – energy storage element – no losses
• Is not charged by TNB
• Inductive load: Q positive, Capacitive load: Q negative
9
Power for a resistorPower for a resistor
t2sin)sin(2IVt2cos)cos(
2IV)cos(
2IV
ivmm
ivmm
ivmm p =
0)( iv Voltage and current are in phase,
t2cos0cos2IV0cos
2IV mmmm p = )t2cos1(
2IV mm p =
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08-2
-1
0
1
2
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08-0.5
0
0.5
1
1.5
2
2.5
P = average power = 2IV mm
Q = reactive power = 0
10
Power for an inductorPower for an inductor
t2sin)sin(2IVt2cos)cos(
2IV)cos(
2IV
ivmm
ivmm
ivmm p =
oiv 90)( Voltage leads current by 90o,
P = average power = 0
t2sin)90sin(2IV Omm p = t2sin
2IV mm p =
Q = reactive power = 2IV mm
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08-2
-1
0
1
2
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08-2
-1
0
1
2
v i
11
Power for a capacitorPower for a capacitor
t2sin)sin(2IVt2cos)cos(
2IV)cos(
2IV
ivmm
ivmm
ivmm p =
oiv 90)( Voltage lags current by 90o,
P = average power = 0
t2sin)90sin(2IV Omm p = t2sin
2IV mm p =
Q = reactive power = 2IV mm
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08-2
-1
0
1
2
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08-1
-0.5
0
0.5
1
1.5
v i
12
Apparent Power and Power FactorApparent Power and Power Factor
Consider v(t) = Vm cos (t + v) and i(t) = Imcos(t + i)
We have seen, )cos(2IVP ivmm
2I
2V mm
rmsrms IV = Is known as the APPARENT POWERAPPARENT POWER
rmsrms IVS VA
13
Apparent Power and Power FactorApparent Power and Power Factor
rmsrms IVS
We can now write,
)cos(SP iv
The term )cos( iv is known as the POWER FACTORPOWER FACTOR
)cos(SPpfFACTORPOWER iv
For inductive load, (v i) is positive current lags voltage lagging pflagging pf
For capacitive load, (v i) is negative current leads voltage leading pfleading pf
14
Apparent Power and Power FactorApparent Power and Power Factor
)cos(SPpfFACTORPOWER iv
15
Apparent Power and Power FactorApparent Power and Power Factor
)cos(SPpfFACTORPOWER iv
Irms = 5- 40o
Vrms = 25010o
Load+Source
+
VL
Active power absorbed by the load is 250(5) cos (50o)= 1250(0.6428) = 803.5 watt
Power factor of the load = cos (10-(-40)) = cos (50o) = 0.6428
Apparent power, S = 1250 VA
Reactive power absorbed by load is 250(5) sin (50o)= 1250(0.6428) = 957.56 var
(lagging)
16
Complex PowerComplex Power
Defined as:
2*IVS
(VA)
Where, vmV V imI I imI I*and
If we let vrmsvm V2
VrmsV irmsi
m I2
IrmsIand
rmsrms *IVS (VA)
17
Complex PowerComplex Power
2*IVS
(VA)
Where, imvm IV21
S
)(IV21
ivmm
)(IV ivrmsrms
)sin(IjV)cos(IV ivrmsrmsivrmsrms
jQP
)-(S) iv cosIV(P rmsrmsRe )-(S) iv sinIV(Q rmsrmsIm
18
Complex PowerComplex Power
jQP SThe complex power contains all information about the load
Irms = 5- 40o
Vrms = 25010o
Load+Source
+
VL
We have seen before:
Active power, P = 803.5 watt
Apparent power, S = 1250 VA
Reactive power, Q = 957.56 var
803.5 watt
957.56 varS S = (803.5 + j957.56) VA
S = 1250 50o VA
|S| = S = Apparent power
S = 25010o (5-40o) VA
= 1250 VA
With complex power,
50o
19
Complex PowerComplex Power
rmsrms *IVS
Other useful forms of complex powerOther useful forms of complex power
rmsrms ZIV We know that
rmsrms *IZIS
2rmsIZS
)jXR(2 rmsIS
)XjR( 22rmsrms IIS
PP QQ
20
Complex PowerComplex Power
rmsrms *IVS
Other useful forms of complex powerOther useful forms of complex power
Zrms
rmsVI We know that
*
Z
rms
rmsVVS
Z
2rmsVS
For a pure resistive element, R
2rmsVP
For a pure reactive element, X
2rmsVQ
21
Conservation of AC PowerConservation of AC Power
Complex, real, and reactive powers of the sources equal the respective sums of the complex, real and reactive powers of the individual loads
22
Conservation of AC PowerConservation of AC Power
Complex, real, and reactive powers of the sources equal the respective sums of the complex, real and reactive powers of the individual loads
Ss = Ps +jQs = (P1 + P2 + P3) + j (Q1 + Q2 + Q3)
But
23
Maximum Average Power TransferMaximum Average Power Transfer
Max power transfer in DC circuit can be applied to AC circuit analysis
ZL
+
V
IZTh
VTh +
AC linear circuit
What is the value of ZL so that maximum averageaverage power is transferred to it?
24
Maximum Average Power TransferMaximum Average Power Transfer
ZL
+
V
IZTh
VTh +
What is the value of ZL so that maximum averageaverage power is transferred to it?
25
Maximum Average Power TransferMaximum Average Power Transfer
ZL
+
V
IZTh
VTh +
What is the value of ZL so that maximum averageaverage power is transferred to it?
ZTh= RTh + jXTh
ZL= RL + jXL
L2R
21P I P max when 0
RP
L
0
XP
L
and
26
Maximum Average Power TransferMaximum Average Power Transfer
ZL
+
V
IZTh
VTh +
What is the value of ZL so that maximum averageaverage power is transferred to it?
P max when 0RP
L
0
XP
L
and
)jXR()jXR( LLThTh
Th
VI
L2R
21P I
2R
)XX()RR(L
2ThL
2LTh
2
ThVP
XL = XTh , RL= RTh