ADE7761B

Rev. 0 | Page 15 of 24

)

sin(

2

)

(

0

h

h

h

O

t

h

I

I

t

i

β

+

ω

×

×

+

=

∑

∞

≠

(2)

where:

i(t)

is the instantaneous current.

is the dc component.

is the rms value of Current Harmonic h.

Using Equation 1 and Equation 2, the Active Power P can be

P

where:

(3)

and

)

cos(

2

∑

∞

=

Φ

×

×

=

h

h

h

h

H

I

V

P

(4)

h

h

h

β

−

α

=

Φ

As can be seen in Equation 4, a harmonic active power component

is generated for every harmonic provided that the harmonic is

present in both the voltage and current waveforms. The power

factor calculation was previously shown to be accurate in the

case of a pure sinusoid; therefore, the harmonic active power

must also correctly account for the power factor because it is

made up of a series of pure sinusoids.

Note that the input bandwidth of the analog inputs is 7 kHz

with an internal oscillator frequency of 450 kHz.

HPF and Offset Effects

Equation 5 shows the effect of offsets on the active power

calculation. Figure 24 shows the effect of offsets on the active

power calculation in the frequency domain.

)

cos(

)

cos(

2

))

cos(

(

))

cos(

(

)

(

)

(

t

I

V

t

I

V

I

V

I

V

t

I

I

t

V

V

t

I

t

V

0

1

1

0

1

1

1

0

1

0

1

0

ω

×

×

+

ω

×

×

+

×

+

×

=

ω

×

+

×

ω

×

+

=

×

(5)

As shown in Equation 5 and Figure 24, an offset on Channel V1

and Channel V2 contributes a dc component after multiplication.

Because this dc component is extracted by the LPF and used to

generate the active power information, the offsets contribute

a constant error to the active power calculation. This problem is

easily avoided in the ADE7761B with the HPF in Channel V1. By

removing the offset from at least one channel, no error component

can be generated at dc by the multiplication. Error terms at cos(ωt)

are removed by the LPF and the digital-to-frequency conversion

(see the Digital-to-Frequency Conversion section).

The HPF in Channel V1 has an associated phase response that

is compensated for on-chip. Figure 25 and Figure 26 show the

phase error between channels with the compensation network

activated. The ADE7761B is phase compensated up to 1 kHz as

shown, which ensures a correct active harmonic power calculation

even at low power factors.

DC COMPONENT (INCLUDING ERROR TERM)

IS EXTRACTED BY THE LPF FOR ACTIVE

POWER CALCULATION

2ω

FREQUENCY (Rad/s)

0ω

2

Figure 24. Effect of Channel Offsets on the Active Power Calculation

FREQUENCY (Hz)

0

100

–0.05

–0.10

0

0.05

0.10

0.15

0.20

0.25

0.30

200

300

400

500

600

700

800

900 1000

Figure 25. Phase Error Between Channels (0 Hz to 1 kHz)

FREQUENCY (Hz)

40

–0.05

–0.10

0.05

0

0.10

0.15

0.20

0.25

0.30

45

50

55

60

65

70

Figure 26. Phase Error Between Channels (40 Hz to 70 Hz)