Transcript 1.0 - Power 101

Solid State Electricity Metrology
1.0 - Power 101
What is Electrical Energy ?
Kinetic Energy of the generator or Chemical Energy of the cell is converted
into Electrical Energy which flows through the circuit and is converted back
into Light, Heat, and kinetic Energy.
Energy flows from source to load
1.0 - Power 101
What is Electrical Energy ?
1.0 - Power 101
Electrical Energy in a dc circuit
p(t) = v(t)  i(t)
Total Energy (E) is equal to integral of p(t) over time
t
E =
 p(t)dt
0
= v  i  t = joules = watts - sec
1.0 - Power 101
Electrical Energy in a dc circuit
Total Energy Vs Time
The slope of this graph is the product of v and i.
E increases over time.
The slope represents the rate of Energy flow.
1.0 - Power 101
What is Electrical Power?
Electrical Power is the rate of flow of Electrical Energy in a circuit.
The unit of power is the watt and is defined as joules per second.
Therefore we can see that power is a rate expression
P =
dE
 v.i
dt
Power is equal to the rate of flow of Electrical Energy
1.0 - Power 101
Energy - Power Relationship
E =
 P dt
dE
P =
dt
1.0 - Power 101
What is Electrical Power?
Power can also be defined as the time average of the instantaneous power p(t).
t
P =
1
t
 p(t)dt
= v  i = joules / sec = watts
0
This is equivalent to
P =
1
t
P =
 P dt 
dE
dt
1
t

dE
dt 
dt
1
t
 dE 
E
 v.i
t
1.0 - Power 101
Electrical Energy in an ac circuit
v(t)  2.V sin(t)
i(t)  2.I sin(t)
where:
V = rms voltage
I = rms current
p(t) = v(t)  i(t)
p(t)  V.I - V.I cos2t
1.0 - Power 101
Total Electrical Energy in an ac circuit
t
E =
 p(t)dt
= joules = watts - sec
0
E  V.I.t -
V.I
sin 2t
2
1.0 - Power 101
Electrical Energy in an ac circuit
• Energy increases with time
• The rate of energy flow is not constant as in the dc case
• For a pure resistive load the rate of energy flow is > = 0
1.0 - Power 101
Electrical Power in an ac circuit
kT
P =
1
kT
 p(t)dt
0
= joules = watts - sec
time average over an integral
number of line cycles (kT)
where:
k = integer
T = period of v(t)
P  V.I
P is often referred to as Active or Real Power
in ac systems
NOTE: The Real Power (P) calculated over an integral number of line cycles
is equal to the steady state component (dc) of the instantaneous power signal p(t)
p(t)  V.I - V.I cos2t
1.0 - Power 101
Electrical Energy in an ac circuit
Energy flow in an ac circuit over an integral number of line cycles:
E  V.I.t -
V.I
sin 2t
2
averages to zero
E  V.I.t = P.t
Energy = Power x time
1.0 - Power 101
Watts, VARS and VA
Unfortunately practical loads are not purely resistive…………...
E.g., the windings of an electrical motor contain a significant amount of
Inductive Reactance as well as resistance
1.0 - Power 101
v(t)  2.V sin(t)
i(t)  2.I sin(t - )
p(t) = v(t)  i(t)
p(t) = v(t)  ia(t) +iq(t)
p(t)  2.V sin(t)   2.I cossin(t)+ 2.I sincos(t)


p(t)  V.I cos(1- cos2t)- V.I sinsin2t
Pp(t)
Pq(t)
1.0 - Power 101
P = VI cos …active power…watts
Q = VI sin …reactive power…VARS
S = VI …apparent power …VA
p(t)  P(1- cos2t)- Qsin2t
1.0 - Power 101
Why do we need to measure VARS & VA ?
Cost to distribute Energy is higher when the load is reactive……….
If V is constant and the load is reactive (current leads or lags V),
then more current (I) must be transmitted in order to deliver a fixed amount
of Electrical Energy (Watts).
More current means more losses and higher transmission costs!
2.0 - AD775x Theory of Operation
See page 1 of AN-545
FUNCTIONAL BLOCK DIAGRAM
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2.1 - Theory of Operation
See page 1 of AN-545
• Analog Signals on Channel 1 and Channel 2 are digitized by the two ADCs
• These signals (Current and Voltage) are multiplied in a digital multiplier
• Multiplication generates an INSTANTANEOUS POWER SIGNAL
• The REAL POWER is extracted from the INSTANTANEOUS POWER
by an LPF
• REAL POWER is converted to a FREQUENCY by the DIGITAL-TOFREQUENCY Converters
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2.1 - Theory of Operation
See page 2 of AN-545
Non unity Power Factor
Active Energy Calculation
p(t)  Vcos(t)Icos(t  60o)
p(t)=
V  Icos(60o ) + cos(2 t - 60o )




2
REAL POWER is still equal to
the dc component of the
INSTANTANEOUS POWER
when PF <> 1
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2.1 - Theory of Operation
See page 2 of AN-545
Nonsinusoidal / Harmonic
Active Energy Calculation

v(t)  V0  2.  Vh.sin(ht  h)
h0

i(t)  I0  2. Ih.sin(ht  h)
h0
P  P1  PH
Where:
P1  V1  I1cos1
1  1  1
PH 

 Vh  Ihcosh
h1
h  h  h
Total Harmonic Power is equal to the
sum of the Real Power for each
harmonic component !
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2.1 - Theory of Operation
See page 3 of AN-545
Digital-to-Frequency Conversion
FOUT - High frequency output
for meter calibration
F1, F2 - Low frequency outputs
for direct drive of counter
FOUT has frequency ripple due to
2 instantaneous power signal.
The ripple is removed by
averaging the output frequency!
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2.1 - Theory of Operation
AD7755 typical watt-hour implementation