Circuits IIEE221Unit 11
Instructor: Kevin D. Donohue
Complex Power, Power Conservation, Power Factor Correction, and Applications
Complex Power
Complex power has real and reactive components. Denote the sinusoidal voltage and current in a load by:
Then the complex power is expressed as:
imvm IIVV ˆ ˆ
ivrmsrmsirmsvrmsim
vm IVIV
IVIVS
222
ˆˆˆ
*
Complex Power The real and imaginary terms of complex power
represent the real (P) and reactive (Q) components of the power:
Note that previously described power quantities can be obtained from complex power
jQPS
IVjIVIVS ivrmsrmsivrmsrmsivrmsrms
ˆ
sincosˆ
iv
ivrmsrms
ivrmsrms
rmsrms
S
P
IVQS
IVPS
IVSS
cosFactorPower
VARs units sinˆImPower Reactive
Watts units cosˆRePower (Average) Real
VAs units ˆPowerApparent
Complex Power with Impedance
Load impedance can be expressed as:
The above relationship can be used to express power in terms of load impedance and either current or voltage magnitude.
jXRI
V
I
VZ iv
rms
rms
rms
rms
ˆ
ˆˆ
*
2*
*
2**
ˆ
ˆ
ˆˆˆˆˆ
ˆˆˆˆˆˆˆ
Z
V
Z
VVIVS
IZIIZIVS
rmsrmsrmsrmsrms
rmsrmsrmsrmsrms
Power Triangle
The real and reactive terms of a load (R, X) can be represented by a triangle modeling the vector addition. The legs of the triangle are the real (P) and reactive (Q) components and complex power:
Phase of Impedance
Power Triangle
The power triangle provides a graphic representation of leading and lagging properties of the load:
(lagging) Load Inductive0Q
(leading) Load Capacitive0Q
Load Resistive0
Q
Conservation of Power
In a given circuit the complex power absorb (denoted by positive values) equals the complex power delivered (denoted by negative values).
For a circuit with N elements the sum of all power is zero:
Note that the above is only true for the real and reactive components. This is not true for apparent power.
N
iiS
1
ˆ0
Power Factor Correction
For a fixed generator voltage and load average power, the output current should be minimized to limit losses over the power line. This is done by adding reactive components to the power systems to bring the PF to 1 (or close to it).
SV
SILineZ
LoadZ CZ
Power Factor Correction
For an inductive load (PF lagging) a purely capacitive load can be added to the line to bring the power factor closer to 1. Show that for a load with PF = x1 lagging and apparent power S1 = Irms Vrms that a new power factor of PF = x2 is achieved by placing a capacitor in parallel with the load (shunt) such that:
leadingfor )(cos
laggingfor )(cos and )(cos where
))tan())(tan(cos(
21
21
211
1
22111
x
xx
V
SC
rms
Power Factor Correction
For a capacitive load (PF leading) a purely inductive load can be added to the line to bring the power factor closer to 1. Show that for a load with PF = x1 leading and apparent power S1 = Irms Vrms that a new power factor of PF = x2 is achieved by placing a shunt inductor across the load such that:
leadingfor )(cos
laggingfor )(cos and )(cos where
))tan())(tan(cos(
21
21
211
1
1211
2
x
xx
S
VL rms
Power Meters
Power meters must simultaneously measure the voltage (in parallel) and the current (in series) associated with load ofinterest.
The meter deflection is proportional the average power.
Electricity Consumption Cost
The kilowatt-hours (kWh) to a customer is measured with a kWh meter corresponding to the average power consumed over a period of time.
(Energy Charge) The cost/rate of the kWh may vary depending on when the power is used (high vs. low demand), and how much total power has been consumed (cost may go down after so many kWh used).
(Demand Charge) A fixed overhead amount is charged simply to maintain the power delivery system, even if you use no power.
A penalty may also be imposed for having a pf below a set figure (i.e. 0.9) since it requires larger currents and the unmetered losses in the line to the customer.