Loop and Nodal Analysis and Op Amps
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Transcript Loop and Nodal Analysis and Op Amps
Circuits II
EE221
Unit 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:
ˆ
V
V
m
v
ˆ
I
I
m
i
Then the complex power is expressed as:
*
ˆ
ˆ
V
I
V
I
m
m
ˆ
S
V
I
V
I
v
i
rms
v
rms
i
rms
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:
ˆ
S
V
I
V
I
cos
j
V
I
sin
rms
rms
v
i
rms
rms
v
i
rms
rms
v
i
ˆ
S
P
jQ
Note that previously described power quantities can be
obtained from complex power
ˆ
Apparent
Power
S
S
V
I
units
VAs
rms
rms
ˆ
Real
(Average)
Power
Re
S
P
V
I
cos
units
Wat
rms
rms
v
i
ˆ
Reactive
Power
Im
S
Q
V
I
sin
units
VARs
rms
rms
v
i
P
Power
Factor
cos
v
i
S
Complex Power with Impedance
Load impedance can be expressed as:
ˆ
V
V
rms
rms
ˆ
Z
R
jX
v
i
ˆ
I
I
rms
rms
The above relationship can be used to express power in
terms of load impedance and either current or voltage
magnitude.
2
ˆ
ˆ I
ˆ* Z
ˆI
ˆ I* Z
ˆI
ˆ
S
V
rms
rms
rms
rms
rms
2
ˆ
V
V
rms
*
rms
ˆ
ˆ
ˆ
ˆ
S
V
Irms
V
*
rms
rms
Z
ˆ
Z
ˆ
*
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:
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
i1
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 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).
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:
S
cos(
)(tan(
)
tan(
))
1
1
1
2
C
2
V
rms
1
cos
(
x
)for
lagg
1
2
where
cos
(
x
)and
1
1
2
1
cos
(
x
)
for
lead
2
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:
2
V
rms
L
S
cos(
)(tan(
)
tan(
))
1
1
2
1
cos
(
x
)for
lagg
where
cos
(
x
)and
1
1
1
1
2
2
1
2
cos
(
x
)
for
lead
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.
(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.