ADE7760

Rev. 0 | Page 14 of 24

Power Factor Considerations

The method used to extract the active power information from

the instantaneous power signal (by low-pass filtering) is still

valid even when the voltage and current signals are not in

phase. Figure 18 displays the unity power factor condition and

a displacement power factor (DPF = 0.5), that is, current signal

lagging the voltage by 60°. If one assumes the voltage and

current waveforms are sinusoidal, the active power component

of the instantaneous power signal (dc term) is given by

(V × I/2) × cos(60°). This is the correct active power calculation.

INSTANTANEOUS

POWER SIGNAL

INSTANTANEOUS

ACTIVE POWER SIGNAL

INSTANTANEOUS

POWER SIGNAL

INSTANTANEOUS

ACTIVE POWER SIGNAL

60°

CURRENT

CURRENT

VOLTAGE

0V

0V

VOLTAGE

V× I

2

V× I

2

× cos(60°)

Figure 18. Active Power Calculation over PF

Nonsinusoidal Voltage and Current

The active power calculation method also holds true for

nonsinusoidal current and voltage waveforms. All voltage and

current waveforms in practical applications have some har-

monic content. Using the Fourier transform, instantaneous

voltage and current waveforms can be expressed in terms of

their harmonic content:

)

sin(

2

)

(

0

h

h

h

o

t

h

V

V

t

V

α

+

ω

×

×

+

=

∑

∞

≠

(1)

where:

v(t)

is the instantaneous voltage.

is the rms value of voltage harmonic h.

is the phase angle of the voltage harmonic.

)

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.

is the phase angle of the current harmonic.

Using Equations 1 and 2, the active power P can be expressed in

H

1

P

P

P

+

=

where:

1

1

1

1

1

1

1

I

V

P

β

−

α

=

Φ

Φ

×

=

)

cos(

(3)

and

h

h

h

h

h

h

h

H

I

V

P

β

−

α

=

Φ

Φ

×

×

=

∑

∞

=

)

cos(

2

(4)

As can be seen from Equation 4, a harmonic active power

component is generated for every harmonic, provided that

harmonic is present in both the voltage and current waveforms.

The power factor calculation has previously been shown to be

accurate in the case of a pure sinusoid; therefore, the harmonic

active power must also correctly account for 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 the internal oscillator frequency of 450 kHz.

HPF and Offset Effects

Equation 5 shows the effect of offset on the active power

calculation. Figure 19 shows the effect of offsets on the active

power calculation in the frequency domain.

)

cos(

)

cos(

2

))

cos(

(

))

cos(

(

)

(

)

(

0

0

1

1

0

t

I

V

t

I

V

I

V

I

V

t

I

I

t

V

V

t

I

t

V

1

1

1

0

1

1

0

ω

×

×

+

ω

×

×

+

×

+

×

=

ω

×

+

×

ω

×

+

=

×

(5)

As can be seen from Equation 5 and Figure 19, an offset on

Channel 1 and Channel 2 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 calcula-

tion. This problem is easily avoided in the ADE7760 with the

HPF in Channel 1. 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).