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By C. Rhodes, P.Eng., Ph.D.

This web page reviews basic electrical engineering concepts relating to three phase circuit energy metering.

A wye electricity service presents three AC phases with 120 degree (2 Pi / 3 radians) phase separations. The voltage reference point is the neutral, at the junction of the wye, which is normally close to ground potential. The individual phase voltages are measured with respect to the neutral. If the individual phases are denoted by x, y, z, the instantaneous phase voltages are Vx, Vy, Vz with respect to neutral and the instantaneous phase currents feeding a wye load are Ix, Iy, Iz. The instantaneous power Px fed to load phase x is:
Px = Vx Ix.
If the voltage source is the electricity grid, to a good approximation the voltage is sinusoidal. Hence:
Vx = Vxo sin(WT)
Vxo = peak voltage on phase x
W = 2 Pi F
Pi = 3.1415928
F = 60 Hz
T = time in seconds relative to an arbitrary initial time
sin = sine

For a 3 phase wye fed system the total instantaneous power P is defined as:
P = Px + Py + Pz
P = (Vx Ix) + (Vy Iy) + (Vz Iz)
Vy = instantaneous voltage on phase y with respect to neutral
Iy = instantaneous current through phase y
Vz = instantaneous voltage on phase z with respect to neutral
Iz = instantaneous current through phase z

Thus a precision three phase power meter for a wye connected load requires three multiplication elements.

In some applications the three phase voltages are sinusoidal, are well balanced and are phase separated by precisely 120 degrees.
If the voltages are balanced:
Vxo = Vyo = Vzo
and if the phases are exactly (2 Pi/3) radians (120 degrees) apart, the voltage on phase x can be expressed as:
Vx = Vxo sin(WT),
the voltage on phase y can be expressed as:
Vy = Vxo sin(WT + (2Pi / 3))
= Vxo sin(WT) cos(2Pi/3) + Vxo cos(WT)sin(2Pi / 3)
= Vxo sin(WT)(-0.5) + Vxo cos(WT)((3^0.5)/2)
and the voltage on phase z can be expressed as:
Vz = Vxo sin(WT + (4 Pi / 3))
= Vxo sin(WT) cos(4 Pi/3) + Vxo cos(WT) sin(4Pi/3)
= Vxo sin(WT)(-0.5) + Vxo cos(WT)(-(3^0.5)/2)

Summing these expressions gives for any instant in time:
Vx + Vy + Vz = 0
Vz = -Vx -Vy
Then substituting for Vz in the equation for instantaneous power P gives:
P = (Vx Ix) + (Vy Iy) + (Vz Iz)
= (Vx Ix) + (Vy Iy) + ((-Vx -Vy) Iz)
P = Vx(Ix - Iz) + Vy (Iy - Iz)

Thus, with balanced three phase sinusoidal voltages a three phase wye load can be energy metered using only two multiplication elements. This metering methodology was standard practice for commercial revenue metering for many years. However, in recent years increasing use of power inverters and line carrier signalling has caused significant voltage distortion. If there is significant voltage distortion the quantity:
(Vx + Vy + Vz)
is non-zero, which causes power measurement error. In that event three multiplication elements must be used to accurately measure energy flow.

Recall that the instantaneous power P is given by:
P = Vx Ix + Vy Iy + Vz Iz
For an isolated delta connected three phase load there is no neutral. At any instant in time the instantaneous phase currents Ia, Ib, Ic exactly conform to:
Iz = -Iy - Ix
Substituting this equation for Iz into the exact three phase formula for P gives:
P = (Vx Ix) + (Vy Iy) + (Vz Iz)
= (Vx Ix) + (Vy Iy) + (Vz (-Iy - Ix))
P = Ix (Vx - Vz) + Iy (Vy - Vz)
In this equation the required voltages are the voltage differences between phases.

Thus the power to an isolated delta connected three phase load can be precisely measured using only two multiplication elements because, if the z phase is used as a voltage reference, then:
Pz = 0.

Accurate measurement of energy flows requires two requires two multiplication elements for an isolated delta fed grid customer and requires three multiplication elements for a wye fed grid customer. However, two multiplication elements are sufficient for a wye fed customer if the voltages are sinusoidal, balanced and of precise phase such that:
(Vx + Vy + Vz) = 0.

This web page last updated March 16, 2009.

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