Transcript ee221_4a

Circuits II
EE221
Unit 4
Instructor: Kevin D. Donohue
Complex Power, Power Conservation, Power
Factor Correction, and Applications
Complex Power


Complex power represents its real and reactive
components.
Let the sinusoidal voltage and current in a load be
denoted by:
Vˆ  Vmv

Iˆ  I mi
Then the complex power is expressed as:
ˆIˆ*  V
V
 I

Sˆ 
  m  v  m    i   Vrms v I rms   i   Vrms I rms  v   i 
2  2
 2

Complex Power

The real and imaginary terms of complex power
represent the real (P) and reactive (Q) components of
the power:
Sˆ  Vrms I rms  v   i   Vrms I rms  cos v   i   j Vrms I rms sin  v   i 
Sˆ  P  jQ

Note that many previously described power quantities
can be obtained from complex power
Apparent Power  Sˆ  S  Vrms I rms  units  VAs
Real (Average) Power  Re Sˆ   P  Vrms I rms  cos v   i  units  Watts
Reactive Power  ImSˆ   Q  Vrms I rms sin  v   i  units  VARs
Power Factor 
P
 cos v   i 
S
Complex Power with Impedance

Load impedance can be expressed as:
Vˆrms  Vrms 
ˆ
 v  i   R  jX
Z
 
ˆI
 I rms 
rms

The above relationship can be used to express power in
terms of the impedance and either current or voltage
magnitudes.
2
*
*
Sˆ  Vˆrms Iˆrms
 ZˆIˆrms I rms
 Zˆ Iˆrms
*
Vˆrms
 Vrms 
*
ˆ
ˆ
ˆ
ˆ
 
S  Vrms I rms  Vrms 
*
ˆ
ˆ
Z
Z


2
Power Triangle
The real and reactive terms of a load (R, X) can be represented by a
triangle from the vector addition. This triangle will be similar to the
triangle formed by the real (P) and reactive (Q) components and complex
power:
Power Triangle
The power triangle provides a
graphic representation of
leading and lagging properties
of the load:
Q  0  Resistive Load
Q  0  Capacitive Load (leading)
Q  0  Inductive Load (lagging)
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:
0
N

Sˆi
i 1
Note that the above is only true for the real and reactive
components. This is not true for apparent power.
Power Factor Correction

For a fixed generator voltage and average power in a load, 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).
IˆS
Ẑ Line
VˆS
Ẑ Load
Ẑ C
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:
S1 cos(1 )(tan( 1 )  tan( 2 ))
C
2
Vrms
1


cos
( x2 ) for lagging
1
where 1  cos ( x1 ) and  2  
1

cos
( x2 ) for leading

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
leading 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:
2
Vrms
L
S1 cos(1 )(tan(  2 )  tan(1 ))
1


cos
( x2 ) for lagging
1
where 1   cos ( x1 ) and  2  
1

cos
( x2 ) for leading

Power Meters
Power meters must
simultaneously measure
the voltage (in parallel)
and the current (in series)
associated with load of
interest.
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.
The cost/rate of the kWh may very 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).
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 may consume excessive power.
A fixed overhead amount is charged simply to maintain the power
delivery system, even if you use no power.