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208 VAC Two-Wire Applications

Overview

Some applications require metering two wires of a three-phase 208 VAC circuit:

Generally, for this application, a 3D-240 (such as WNC-3D-240-MB) model would be used, but a 3Y-208 (such as WNC-3Y-208-MB) could also be used if neutral is available. Theoretically, only a single current transformer (CT) should be needed, because there are only two active wires, but the current WattNode® meters (WNB and WNC series) generally require two CTs, because WattNode meters uses neutral or ground as the measurement reference point.

Issues

To save money and simplify the installation, can I use a single CT instead of two CTs?

V_{AVG} = \frac{V_1 + V_2}{2} Average line voltage
V_{ib}(%) = 100 \cdot \frac{V_1 - V_2}{V_{AVG}}

V_{ib}(%) = 100 \cdot \frac{2 \cdot (V_1 - V_2)}{V_1 + V_2}

Voltage imbalance (percent)
E_r(%) = 100 \cdot \frac{V_1 - V_2}{V_1 + V_2} Power and energy reading error (percent)

Detailed Analysis

The following analysis shows the power of a two-wire 208 VAC load (or generator) and the theoretical metering results using one vs. two CTs.

\mathbf{V_a} = v \angle 0^\circ
\mathbf{V_b} = v \angle -\!120^\circ
\mathbf{V_{ab}} = \left ( v \sqrt{3} \right ) \angle 30^\circ
\mathbf{I_{ab}} = i \angle \left ( \theta + 30^\circ \right )
P_F = \cos(\theta)
\theta = \arccos(P_F)
\mathbf{I_a} = \mathbf{I_{ab}}
\mathbf{I_b} = -\mathbf{I_{ab}} = i \angle \left ( \theta + 30^\circ + 180^\circ \right ) = i \angle \left ( \theta + 210^\circ \right )
P_A = |V_a||I_a| \cos ( \theta_{Va} - \theta_{Ia} ) = vi \cos ( -\theta - 30^\circ )
P_B = |V_b||I_b| \cos ( \theta_{Vb} - \theta_{Ib} ) = vi \cos ( -\theta + 30^\circ )
P_{AB} = v i \sqrt{3} \cos( \theta_{Vab} - \theta_{Iab} ) = v i \sqrt{3} \cos(-\theta)

Example

The following table shows some example data assuming a constant 120 VAC line-to-neutral (208 VAC line-to-line), and a constant 10 amp current. The table shows the effect of varying power factor on the measured phase A and phase B power. In all cases, the sum of the phase A and B power matches the direct P_{AB} value.

v i P_F \theta P_A
(W)
P_B
(W)
P_A+P_B
(W)
P_{AB}
(W)
Q_A
(VAR)
Q_B
(VAR)
S_A
(VA)
S_B
(VA)
120 10 1.00 1039.2 1039.2 2078.5 2078.5 -600.0 600.0 1200.0 1200.0
120 10 0.71 45° 310.6 1159.1 1469.7 1469.7 -1159.1 -310.6 1200.0 1200.0
120 10 0.50 60° 0.0 1039.2 1039.2 1039.2 -1200.0 -600.0 1200.0 1200.0
120 10 0.00 90° -600.0 600.0 0.0 0.0 -1039.2 -1039.2 1200.0 1200.0

Correction

It is possible to largely compensate for the errors of monitoring a two-wire line-to-line (delta) load with only one CT and one input channel of a WattNode meter using the following equations. These require models that report reactive power.

The following assumes your load (or generating source) is between phases A and B and that you are just monitoring phase A with the WattNode.

P_{\!A} Real power measured on phase A
Q_A Reactive power measured on phase A
S_{\!A} = \sqrt{P_{\!A}^2 + Q_A^2} Apparent power measured on phase A
Some WattNode models provide this value directly.
P_{Fa} = \frac{P_{\!A}}{S_{\!A}} Power factor measured for phase A
Most WattNode models provide this value directly.
\theta_{\!A} = \arccos(P_{Fa}) Voltage to current phase angle measured on phase A
\theta_{\!A}' = \theta_{\!A} - 30^\circ Corrected voltage to current phase angle
P_{\!A}' = \sqrt{3} \, S_{\!A} \cos(\theta_{\!A}') Corrected real power for line-to-line load
P_{Fa}' = \frac{P_{\!A}'}{S_{\!A}} Corrected power factor

The following table shows the correction applied to the data above, and demonstrates that the theoretical P_{Adj} is correct.

v i P_F \theta P_A
(W)
P_{AB}
(W)
Q_A
(VAR)
S_A
(VA)
P_{Fa} \theta_A \theta_A' P_{Adj}
(W)
120 10 1.00 1039.2 2078.5 -600.0 1200.0 0.87 30° 2078.5
120 10 0.71 45° 310.6 1469.7 -1159.1 1200.0 0.26 75° 45° 1469.7
120 10 0.50 60° 0.0 1039.2 -1200.0 1200.0 0.00 90° 60° 1039.2
120 10 0.00 90° -600.0 0.0 -1039.2 1200.0 -0.50 120° 90° 0.0

See Also

 

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