Embed Size (px)
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 Presentation
Citation preview
1
AC POWER CALCULATIONAC POWER CALCULATIONInstantaneous, average and reactive powerInstantaneous, average and reactive power
Apparent Power and Power FactorApparent Power and Power Factor
Complex 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
v
i
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(2
IVt2cos)cos(
2
IV)cos(
2
IViv
mmiv
mmiv
mm p =
p = P + Pcos(2t) Qsin(2t)
Using trigonometry functions, it can be shown that:
)cos(2
IVP iv
mm = AVERAGE POWER (watt)
)sin(2
IVQ iv
mm = REACTIVE POWER (var)
Which can be written as
5
t2sin)sin(2
IVt2cos)cos(
2
IV)cos(
2
IViv
mmiv
mmiv
mm p =
6
t2sin)sin(2
IVt2cos)cos(
2
IV)cos(
2
IViv
mmiv
mmiv
mm 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(2
IVt2cos)cos(
2
IV)cos(
2
IViv
mmiv
mmiv
mm 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(2
IVt2cos)cos(
2
IV)cos(
2
IViv
mmiv
mmiv
mm p =
0)( iv Voltage and current are in phase,
t2cos0cos2
IV0cos
2
IV mmmm p = )t2cos1(2
IV 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 = 2
IV mm
Q = reactive power = 0
10
Power for an inductorPower for an inductor
t2sin)sin(2
IVt2cos)cos(
2
IV)cos(
2
IViv
mmiv
mmiv
mm p =
oiv 90)( Voltage leads current by 90o,
P = average power = 0
t2sin)90sin(2
IV Omm p = t2sin2
IV mm p =
Q = reactive power = 2
IV 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(2
IVt2cos)cos(
2
IV)cos(
2
IViv
mmiv
mmiv
mm p =
oiv 90)( Voltage lags current by 90o,
P = average power = 0
t2sin)90sin(2
IV Omm p = t2sin2
IV mm p =
Q = reactive power = 2
IV 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(2
IVP iv
mm
2
I
2
V 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(S
PpfFACTORPOWER 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(S
PpfFACTORPOWER iv
15
Apparent Power and Power FactorApparent Power and Power Factor
)cos(S
PpfFACTORPOWER 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 VimI 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 IV2
1S
)(IV2
1ivmm
)(IV ivrmsrms
)sin(IjV)cos(IV ivrmsrmsivrmsrms
jQP
)-(S) iv cosIV(P rmsrmsRe )-(S) iv sinIV(Q rmsrmsIm
18
Complex PowerComplex Power
jQP S
The 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
2
rmsIZS
)jXR(2
rmsIS
)XjR(22
rmsrms IIS
PP QQ
20
Complex PowerComplex Power
rmsrms *IVS
Other useful forms of complex powerOther useful forms of complex power
Zrms
rms
VI We know that
*
Z
rms
rms
VVS
Z
2
rmsVS
For a pure resistive element, R
2
rmsVP
For a pure reactive element, X
2
rmsVQ
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
L
2R
2
1P I P max when 0
R
P
L
0X
P
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 0R
P
L
0X
P
L
and
)jXR()jXR( LLThTh
Th
VI
L
2R
2
1P I
2
R
)XX()RR(L
2ThL
2LTh
2
ThV
P
XL = XTh , RL= RTh
![Instantaneous Power Control for Suppressing the Second ...eprints.whiterose.ac.uk/107210/1/Instantaneous Power Control for... · For example, the notch filter was employed in [9-11]](https://img.pdfslide.us/doc/110x75/5f965d1957cc2f7df86aed03/instantaneous-power-control-for-suppressing-the-second-power-control-for.jpg)
![Power Quality Solutions · Q2 = S2 — P2 Q Q 2 C Q 1 S 2 S 1 Apparent Power [kVA] S2 = P2 + Q2 Active Power [kW] P2 = S2 — Q2 sin With power factor correction the apparent power](https://img.pdfslide.us/doc/110x75/5f3199cd411b7c7d2516b80c/power-quality-solutions-q2-s2-a-p2-q-q-2-c-q-1-s-2-s-1-apparent-power-kva.jpg)
