Pixabay

Overview

Introduction

As with any Alternating Current (AC) System, power in a three phase system depends on the magnitudes of the currents and voltages and on the angles (θ) between them – i.e. the power factor.   

Balanced System

Considering a balanced system, the equations for power, regardless of whether Star or Delta connection, are:

Apparent or Complex Power (measured in Volt Amps = VA) =

$S=3\times V_{\text{phase}}\times I_{\text{phase}}=\sqrt{3}\times V_{\text{line}}\times I_{\text{line}}$ 

Active Power (measured in Watts = W) =

$P=3\times V_{\text{phase}}\times I_{\text{phase}}\times \text{cos}\theta =\sqrt{3}\times V_{\text{line}}\times I_{\text{line}}\times \text{cos}\theta$ 

Reactive Power (measured in Volts Amps reactive = VAr) =

$3\times V_{\text{phase}}\times I_{\text{phase}}\times \text{sin}\theta =\sqrt{3}\times V_{\text{line}}\times I_{\text{line}}\times \text{sin}\theta$ 

Since we are usually only interested in Active Power, we can measure this using one or more instruments called Wattmeters.   You may have come across these when studying single phase. They have two coils:

1. A very low impedance current coil is connected in series with the load, to measure the current flowing through it. Ideally this coil looks like a short circuit, so develops no voltage drop across it.
2. A very high impedance current coil is connected in parallel with the load, to measure the voltage across it. Ideally this coil looks like an open circuit so that no current flows through it.

© L.Gray UHI

When both coils are energised, the interaction between the voltage and current causes a reading which is proportional to the magnitudes of the two quantities and the angle between them, thus the Active Power can be calculated and displayed.   Some more advanced digital wattmeters can also measure Reactive and Apparent power, and Power Factor.

Using a Single Wattmeter to Measure Three Phase Power

A single wattmeter can be used to measure three phase power, but only in a balanced system (and these rarely occur in practice).  The method is simply to measure the power consumed by a single phase of the load, and then calculate the total three phase power as three times this measurement.  The connections for both Delta and Star loads are shown in Figure 1.

Figure 1: The single Wattmeter measurement method for Balanced Loads

© L.Gray UHI

Using Three Wattmeters to Measure Three Phase Power

If the load is unbalanced, as is most likely to be the case, then a three wattmeter method can be used as seen in Figure 2.  The application here is obvious – the power in each phase is measured and then the three measurements are added together to give total power.

Figure 2: The Three Wattmeter method for both Delta and Star connected loads

© L.Gray UHI

Disadvantages of both the 1 wattmeter and the three wattmeter methods are:

1. For the Delta load, it might not be possible to access a single phase. For example, if the load is a motor, it may not be possible to break the phase connections to insert the current coil/coils.
2. For the Star load, the neutral connection might not be accessible.

A further disadvantage of the three wattmeter method is that is uses a lot of wattmeters and a lot of connections.

Using Two Wattmeters to Measure Three Phase Power

A better way of measuring three phase power, is the two wattmeter method.  This doesn’t require a neutral point for Star connected loads, nor does it require access to the phases for Delta connected loads.  The two wattmeter method may be used in the case of a three-phase three-wire system where the load is connected either in Star or Delta.  It can be used for either balanced or unbalanced loads.  The current coils of the wattmeters are connected in any two lines of the supply and the voltage coils are connected between the corresponding lines and the third line.  Example circuit diagrams are shown in Figure 3, although other configurations are possible.

Figure 3: Two Wattmeter method connections for both Delta and Star connected loads

© L.Gray UHI

Using this method, the three phase power has to be calculated in different ways for different types of load.

Self-Assessment Questions

Now do the questions in the Self-Assessment Questions folder for Week 11.

Unbalanced Loads

Considering the Star connections in Figure 3 (Delta works the same way, but the Wattmeter is in Lines 1 and 2 so the references would be different).  Remember any two phases with respect to the third, can be used for either Delta or Star connected loads.

Wattmeter 1 (W1) reads V_L23 x I_L3

Wattmeter 2 (W2) reads V_L12 x I_L1

So

W1 + W2 = (V_L23 x I_L3) + (V_L12 x I_L1) = (V_P3 – V_P2)I_P3 + (V_P1 – V_P2) I_P1 = V_P3 I_P3 + V_P1 I_P1 – V_P2(I_P3 + I_P1)

But because the currents must sum to zero at the neutral point (Kirchoff’s Current Law):

I_P3 + I_P2 + I_P1 = 0, so I_P3 + I_P1 = -I_P2

Substituting this gives W1 + W2 = V_P3 I_P3 + V_P1 I_P1 + V_P2I_P2 = Total three phase power

Balanced Loads

With balanced loads, how the total power is calculated depends on the load Power Factor (PF).  Again, using the Star connection in Figure 3: 

Wattmeter 1 (W1) reads V_L23 x I_L3 cos (30° + ∅) = V_L I_L cos (30° + ∅)

Wattmeter 2 (W2) reads V_L12 x I_L1 cos (30° - ∅) = V_L I_L cos (30° - ∅)

where ∅ is the angle between the line voltage and the line current

W1 + W2  = V_L I_L cos (30° + ∅) + V_L I_L cos (30° - ∅) = V_LI_L [cos (30° + ∅) + cos (30° - ∅)]

= V_L I_L [2 cos(30°).cos∅]          (from trigonometric formula)

$=V_L{I}_L\left[\frac{2\sqrt{3}}{2}\text{cos}\varnothing \right]$                         (cos(30°) = $\frac{\sqrt{3}}{2}$ from trigonometry) 

$=\sqrt{3}{V}_L{I}_L\text{cos}\varnothing$                          which is well known as the formula for total 3 phase Active power

Reactive Power and Power Factor

The two wattmeter method allows calculation of Reactive Power (Q = VAr) from the readings:

W1 – W2 = V_L I_L [cos (30˚ + ∅ ) – cos (30˚ - ∅) ] = V_L I_L [2sin(30˚)sin∅] = V_LI_L [2 x 0.5 sin Ø]

= V_L I_L sin∅

It is also possible to calculate the load power factor from the readings:

$\frac{W_1-W_2}{W_1+W_2}$$=\frac{V_L{I}_L\text{sin}\varnothing }{\sqrt{3}{V}_L{I}_L\text{cos}\varnothing }$$=\frac{\text{tan}\varnothing }{\sqrt{3}}$ 

From this equation, Φ and thus cos(Φ) = PF, can easily be calculated.

Self-Assessment Questions

Now do the questions in the Self-Assessment Questions folder for Week 12.

Conclusion

You have reached the end of this Unit.  In this final section you studied power measurement in three-phase systems, using the 1 wattmeter, three wattmeter and two wattmeter methods.  You learnt how to calculate the total power, from these three methods in balanced and unbalanced loads.   You now have the pre-requisite knowledge required to study a more advanced course in electrical power.