Three Voltmeter Method and Three Ammeter Method (with Experiments)
The three ammeter method and the three voltmeter method are used for the measurement of singlephase AC power. More specifically, they are used to calculate the power factor of a specific load.
Both these methods are susceptible to accuracy problems caused by small errors in measuring devices. Hence they are rarely used.
Nowadays we have wattmeter methods to measure power, power factor meter and oscilloscope to measure the power factor of AC circuit accurately.
When you don’t have a power factor meter or an oscilloscope on the homebrewer workbench, you can use the three voltmeter method or three ammeter method to measure the power and power factor of the circuit.
Table of Contents
Three voltmeter method
The three voltmeter is used in an inductive circuit to measure the value of power and the power factor. In this method you just need three AC voltmeters and a resistor.
As shown in the figure, one voltmeter is used to measure the voltage of the circuit (V_{i}), the second one measures the voltage on the noninductive resistance (V_{R}) that is connected in the series with the load branch and the third voltmeter is used to measure the voltage of the load (V_{t}).
The power (P) consumed by the load can be determined as the product of the voltage and current of the load (V_{t},I_{t}) and the cosine of the phase angle between these two (the power factor).
P = V_{t} I_{t} cos φ
The formula to find out power in a circuit using three voltmeter method is,
P = (V_{i}^{2} – V_{t}^{2} – V_{R}^{2}) / 2 R
Using the law of cosines, the power factor (cos φ) can be calculated.
The formula used to find power factor using three voltmeter method is,
cos φ = (V_{i}^{2} – V_{R}^{2} – V_{t}^{2}) / 2 V_{R}V_{t}
The right selection of the resistor is crucial. To find out the best value of R, you have to balance two aspects: accuracy and voltage drop. You want a resistor large enough so that your voltmeter (or multimeter) can read it with decent accuracy; but not too large, in any other case the voltage around to the load will be too small.
For optimal accuracy, the noninductive resistance should be large enough so that the voltmeter (or multimeter) can measure it with satisfactory accuracy, but not too large, otherwise, the voltage available to the load would be too small. Ideally, it should be close or equal to the load impedance.
Disadvantages of 3 voltmeter method
The main disadvantage of the three voltmeter method is that it only works fine for linear loads like motors or transformers. It also works quite well with some slightly nonlinear loads like inductive fluorescent tube ballasts or transformerbased arc welders. But it doesn’t work with strongly nonlinear loads such as rectifiers (basically any electronic ballast, switching power supply, motor driven by frequency converters).
Three Voltmeter Method Practical Experiment
To further understand the three voltmeter method, let’s have a look at a couple of examples.
Experiment 1
The first example is the induction motor of a bench grinder. It is nominally rated 230 VAC, 50 Hz, 250 W.
For the series resistor, I used an array of seven power resistors mounted on a heatsink for a total of 8.15 Ω – 150 W that I usually use as a dummy load for testing audio amplifiers. As you can see in the picture below, V_{1} = 239.5 V, V_{2} = 4.630 V and V_{3} = 235.7 V.
By putting all these values in the calculator, we find the power factor cos(φ) = 0.82, the current I = 0.57 A, the apparent power S = 134 VA and the active power P = 110 W. It’s not a surprise that this motor is using less than half its nominal power, because it’s turning idle and it’s just compensating its own losses. Power consumption will increase when grinding something.
To verify these three voltmeters measurements and calculation, the same bench grinder is measured again with an AC power analyzer.
The AC power analyzer reads 109.8 W and 0.579 mA, well confirming our result (the accuracy of all the instruments used here is not better than ± 1%).
Experiment 2
The second example is a large halogen lamp transformer which is rated as 220 V AC, 5060 Hz, 400 VA, with no load connected to its secondary (a transformer on no load is usually quite inductive).
Here, because the transformer has no load and uses much lower power, the series resistor is 165 Ω, 17 W. As you can see in the picture below, V1 = 239.7 V, V2 = 11.10 V and V3 = 232.9 V.
By putting all these values in the calculator, we find the power factor cos(φ) = 0.60, the current I = 67 mA, the apparent power S = 16 VA and the active power P = 9.4 W.
Measuring the active power with an AC power meter gives 9.46 W and 68.7 mA, confirming again our calculation.
Three ammeter method
The three ammeter method is also used in an inductive circuit to measure the value of the power factor, independent of source frequency and waveforms. The disadvantages of measurement of power by three voltmeter method are overcome in this method.
In this method, as seen in the figure, across the inductive circuit load in which the power factor is to be determined, a noninductive resistance is connected parallel with the load branch.
One ammeter is used to measure the total current of the circuit, the second one measures current going through the noninductive resistance and the third ammeter measures the current of the load branch.
Current through the resistive branch (I_{R}) is in phase with source voltage while the current of the load (I_{t}) has its own phase, which will affect the power factor (cos φ).
Total current (Ii) is a vector sum of the other two currents. Using the law of cosines, the power factor can be calculated.
The formula used to find power factor in three ammeter method is,
cos φ = (I_{i}^{2} – I_{R}^{2} – I_{t}^{2}) / 2 I_{R}I_{t}
For optimal accuracy, ideally, the noninductive resistance should be close or equal to load impedance as it should be for the three voltmeter method.
The formula used to measure power in three ammeter method is,
P = R/2 ((I_{i}^{2} – I_{R}^{2} – I_{t}^{2})
Advantage
The advantage of this method is that the value of determined is independent of supply frequency and waveforms.
Conclusion
The three voltmeters method and three ammeter method of determining the power factor is an attractive substitute to the home experimenter with limited instruments. It’s not as userfriendly as a true power analyzer and not as precise as an oscilloscope (properly equipped for high voltage measurements), but needs only a multimeter and a resistor. It will be helpful in many situations where dedicated instruments are not available.
Frequently Asked Questions

Why do we use the three voltmeter method to measure when we have a wattmeter?
A wattmeter measures the REAL power only. We use the three voltmeter method even when we have a wattmeter because the 3 voltmeter method makes it possible for you to determine the REAL, REACTIVE, and APPARENT power in a circuit.
At the same time, you need to be very careful in your measurements, considering that a small error can significantly skew your results.

How to measure power factor with multimeter?
You can measure power factor with multimeters using the three voltmeter method. For this, you need three multimeters set to measure voltage with the same accuracy. Connect the multimeter as shown in the figure.
Take the reading of the multimeter and put it into the equation below.
cos φ = (V_{i}^{2} – V_{R}^{2} – V_{t}^{2}) / 2 V_{R}V_{t}
The solution will give you the power factor of the circuit. 
What is the advantage of the three ammeter method over three voltmeter method?
The advantage of the three ammeter method over three voltmeter method is that the value determined by 3 ammeter method is independent of supply frequency and waveforms.

Why do we use the 3 ammeter method and not the 2 ammeter method for measuring power?
You cannot find out the power factor using 2 ammeter method. So we cannot measure the true power.

Can three ammeter method be used to measure resistive load?
Yes, you can use three ammeter method to measure the power consumed by a resistive load.
Power equation is P = R/2 ((I_{1}^{2} – I_{2}^{2} – I_{3}^{2})
Whatever will be the load (inductive or resistive) the current I3 will flow. Therefore you can use three ammeter method to measure power in a resistive load.