Q3Electronic Measurement & Instrumentation

Question

Explain 3-voltmeter method for power measurement with the circuit and phasor diagram.

Answer 1

This method is used to measure the power and power factor of a single-phase load ( $Z_{L}$ ) without using a wattmeter.

Circuit Arrangement: A purely non-inductive standard resistor ( $R$ ) is connected in series with the unknown inductive load ( $Z_{L}$ ). Three voltmeters are connected to measure: 1. $V_{1}$ : The total supply voltage across the series combination. 2. $V_{2}$ : The voltage drop across the non-inductive resistor $R$ . 3. $V_{3}$ : The voltage drop across the load $Z_{L}$ .

Phasor Derivation: Since $R$ and $Z_{L}$ are in series, the current $I$ is common. $V_{2}$ is in phase with $I$ . $V_{3}$ leads $I$ by angle $ϕ$ . The total voltage phasor is $V_{1} = V_{2} + V_{3}$ . Using the cosine rule on the phasor triangle: $V_{1}^{2} = V_{2}^{2} + V_{3}^{2} + 2 V_{2} V_{3} cos ϕ$

We know the power consumed by the load is $P = V_{3} I cos ϕ$ . Since $I = V_{2} / R$ , we substitute this to find $2 V_{2} V_{3} cos ϕ = 2 R P$ .

Substituting back and rearranging gives the total power: $P = \frac{V _{1}^{2} - V _{2}^{2} - V _{3}^{2}}{2 R}$

Q3Electronic Measurement & Instrumentation

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