### At the end of this week, the students are expected to:

– Derive an expression for power in a three-phase circuit (balanced and unbalanced)

– Explain the two wattmeter and single wattmeter methods of measuring three-phase power

### 2.8 DERIVATION FOR POWER IN A 3-PHASE CIRCUIT (BALANCED AND UNBALANCED)

### 2.8.1 Power in a 3-Phase circuit (balanced-delta connection)

Consider the delta connection of a three phase circuit (fig 2.9). The total active power in a delta connected circuit is the sum of the three phase powers. Hence,

Power per phase = VphIphcos-

Total power = 3 x VphIph cos –

However, Vph = VL and Iph = IL/√3. In terms of line values, the above expression for power becomes

P = 3 x VL x IL x cos- = √3 VLILcos- √3

Similarly, total reactive power is given by

Q = √3 VLILsin- (2.10) and total apparent power of the three phase is

S = √3 VLIL (2.11)

### 2.8.2 Power in a three phase circuit (unbalanced star connection)

**Fig 2.11: unbalanced **

### three phase star connected load

Consider the unbalanced three phase star connected load, where it is assumed that the phase voltages (Vph) are equal. The line currents are obtained from ohms law. Power in phase R, PR = phase voltage x Phase current x cos-, where – is the angle between phase voltage and phase current.

### 2.9 2-WATTMETER AND SINGLE-WATTMETER METHODS OF MEASURING 3-PHASE POWER

**2.9.1 2-Wattmeter Method **

In this method, two wattmeters are connected in two phases and their potential coils are connected to the remaining phase. As shown in fig 2.12, wattmeter W1 is inserted in phase a and wattmeter W2 in phase c. Their potential coils are connected across a and b, and c and b respectively. The current coil (c.c) of each wattmeter measures the line current, while the respective potential coil (p.c) measures the line voltage. The algebraic sum of the two wattmeter readings equals the total power absorbed by the loads.

**Fig 2.12: schematic diagram of a two wattmeter method **

### 2.9.2 Single- Wattmeter Method

In this method, one wattmeter is used to get two readings which are obtained by two wattmeter in the two wattmeter method. The total power is 3 times the reading of that one wattmeter. This method is used only when the load is balanced. Figure 2.13 shows the schematic diagram of a single wattmeter method.

**Fig 2.13: schematic diagram of a single-wattmeter method **

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