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ADE7761A
∞
i ( t ) = I O + 2 × ∑ I h × sin( h ω t + β h )
h ≠ 0
where:
i(t) is the instantaneous current.
I O is the dc component.
I h is the rms value of current harmonic h.
β h is the phase angle of the current harmonic.
Using Equation 1 and Equation 2, the active power P can be
expressed in terms of its fundamental active power (P 1 ) and
harmonic active power (P H ).
P = P 1 + P H
where:
(2)
The HPF in Channel 1 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 ADE7761A 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
V 1 × I 1
2
V 1 × I 0
V 0 × I 1
P 1 = V 1 × I 1 cos(Φ 1 )
0 ?
2 ω
FREQUENCY (RAD/S)
and
Φ 1 = α 1 ? β 1
(3)
Figure 24. Effect of Channel Offsets on the Active Power Calculation
∑ V h h I × cos( Φ h )
P H =
∞
h = 2
×
0.30
0.25
Φ h = α h ? β h
(4)
0.20
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.
0.15
0.10
0.05
0
–0.05
Note that the input bandwidth of the analog inputs is 7 kHz
–0.10
0
100
200
300
400 500 600 700
800
900
1000
with an internal oscillator frequency of 450 kHz.
FREQUENCY (Hz)
Figure 25. Phase Error Between Channels (0 Hz to 1 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.
V ( t ) × I ( t ) =
0.30
0.25
0.20
( V 0 + V 1 × cos( ω t )) × ( I 0 + I 1 × cos( ω t )) =
(5)
0.15
V 0 × I 1 +
V 1 × I 1
2
+ V 0 × I 1 × cos( ω t ) + V 1 × I 0 × cos( ω t )
0.10
0.05
As can be seen in Equation 5 and Figure 24, 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
0
–0.05
generate the active power information, the offsets contribute a
constant error to the active power calculation. This problem is
easily avoided in the ADE7761A 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(ω t )
are removed by the LPF and the digital-to-frequency conversion
(see the Digital-to-Frequency Conversion section).
Rev. 0 | Page 15 of 24
–0.10
40 45 50 55 60 65
FREQUENCY (Hz)
Figure 26. Phase Error Between Channels (40 Hz to 70 Hz)
70
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