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 ?

Electrical Energy in a dc circuit

1.0 - Power 101

E = p(t)dt = v i t = joules = watts - sec0

t

Total Energy (E) is equal to integral of p(t) over time

p(t) = v(t) i(t)

Electrical Energy in a dc circuit

1.0 - Power 101

The slope of this graph is the product of v and i.E increases over time.The slope represents the rate of Energy flow.

Total Energy Vs Time

What is Electrical Power?

1.0 - Power 101

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

Power is equal to the rate of flow of Electrical Energy

P = dE

dtv.i

Energy - Power Relationship

1.0 - Power 101

P = dE

dtE = P dt

What is Electrical Power?

1.0 - Power 101

Power can also be defined as the time average of the instantaneous power p(t).

P = p(t)dt = v i = joules / sec = watts1t

0

t

P = dE

dt

P = P dtdE

dtdt dE

E

tv.i1

t1t

1t

This is equivalent to

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 voltageI = rms current p(t) V.I - V.I cos2 t

p(t) = v(t) i(t)

1.0 - Power 101

Total Electrical Energy in an ac circuit

E = p(t)dt = joules = watts - sec0

t

E V.I.t - V.I

2 sin 2 t

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

P = p(t)dt = joules = watts - sec1kT

0

kT

where: k = integerT = period of v(t)

P V.I

time average over an integral number of line cycles (kT)

NOTE: The Real Power (P) calculated over an integral number of line cyclesis equal to the steady state component (dc) of the instantaneous power signal p(t)

p(t) V.I - V.I cos2 t

P is often referred to as Active or Real Powerin ac systems

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

2 sin 2 t

Energy = Power x time

E V.I.t = P.t

averages to zero

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) i (t)+ i (t)a q

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)

P = VI cos …active power…wattsQ = VI sin …reactive power…VARSS = VI …apparent power …VA

1.0 - Power 101

p(t) (1- cos2 t) - sin2 tP Q

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 amountof Electrical Energy (Watts).

More current means more losses and higher transmission costs!

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2.0 - AD775x Theory of Operation

FUNCTIONAL BLOCK DIAGRAM

See page 1 of AN-545

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2.1 - Theory of Operation

• 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-TO- FREQUENCY Converters

See page 1 of AN-545

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2.1 - Theory of Operation

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 POWERwhen PF <> 1

See page 2 of AN-545

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2.1 - Theory of Operation

Nonsinusoidal / Harmonic

Active Energy Calculation

v(t) V 2. V .sin(h t )0 h h

h 0

i(t) I 2. I .sin(h t )0 h h

h 0

P P P1 H

Where:

P V I cos1 1 1 1

1 1 1

P V I cosH h

h h h

h h

h 1

Total Harmonic Power is equal to the sum of the Real Power for each harmonic component !

See page 2 of AN-545

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2.1 - Theory of Operation

Digital-to-Frequency Conversion

FOUT - High frequency output for meter calibrationF1, 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!

See page 3 of AN-545

AD7755 typical watt-hour implementation

2.1 - Theory of Operation