5.9Phasor analysis of transformer circuits

Three-phase AC is the dominant mode of electric power generation and distribution in modern societies. Analyzing voltages and currents in these systems can be quite complex, but phasor representation is a tool useful for simplifying that analysis by representing voltage and current quantities in graphical form. This section of the book illustrates the use of phasors as an analytical tool through examples. This section begins with single-phase phasor analysis and progresses to three-phase phasor analysis.

Extensive use of an imaginary instrument I call the “phasometer” helps relate phasor arrow direction to real circuit connections. For a detailed introduction to this instrument, see section 5.5.2 beginning on page 382. Su ce it to say, a “phasometer” is nothing more than a small synchronous AC motor with an arrow-shaped rotor, which when coupled with a strobe light set to flash whenever some reference voltage reaches its positive peak, graphically shows the relative phase of the measured quantity with respect to that reference voltage. In other words, a phasometer/strobe combination shows a “snapshot” view of the waveform’s position at that point in time when the reference wave it at its positive peak. This is precisely how phasors are drawn: arrows with angles displaced from 0o by an amount proportional to the phase shift between that quantity and some reference waveform.

5.9.1Phasors in single-phase transformer circuits

Phasors are to AC circuit quantities as polarity is to DC circuit quantities, and for this reason they are particularly useful in analyzing transformer circuits because transformers by their very nature allow polarities to be mixed and matched.

In the “Basic principles” discussion of the “Transformers” section of this book (subsection 5.4.1 beginning on page 362), we used standard DC polarity marks (+ and −) to denote the polarity of voltage at an instant in time where the sinusoidal waves for each winding reached their peaks. Although it is often important to show the polarity of transformer windings relative to each other, the use of DC notation for this purpose can be misleading. An alternative symbology places a dot, or square, or some other distinguishing mark at one end of each transformer winding to show common polarity at any instant in time. The marked terminal of a transformer winding is called the “polarity” terminal, while the unmarked terminal is called the “nonpolarity” terminal:

Transformer polarity symbols

The marks should be interpreted in terms of voltage polarity, not current. To illustrate using a “test circuit49” feeding a momentary pulse of DC to a transformer:

Pushbutton momentarily pressed

If the battery polarity were reversed, the “polarity” (dot) terminal of each winding would be negative with respect to the “nonpolarity” terminal at the moment in time when the switch is pressed. That is to say, the “polarity” marks merely show which terminals will share the same instantaneous voltage polarity at any given point in time – not which terminal is always positive.

49The battery-and-switch test circuit shown here is not just hypothetical, but may actually be used to test the polarity of an unmarked transformer. Simply connect a DC voltmeter to the secondary winding while pressing and releasing the pushbutton switch: the voltmeter’s polarity indicated while the button is pressed will indicate the relative phasing of the two windings. Note that the voltmeter’s polarity will reverse when the pushbutton switch is released and the magnetic field collapses in the transformer coil, so be sure to pay attention to the voltmeter’s indication only during the instant the switch closes!

A useful way to analyze phase shifts in transformer circuits is to sketch separate phasor diagrams for the primary and secondary windings, and use the polarity marks on each winding to reference each phasor’s position on the diagram. We begin this analysis by first sketching a phasor for any voltage of which the phase angle is given to us. Since transformer windings share common phase shift by virtue of those windings sharing the same magnetic field, the phasor representing the other winding’s voltage must possess the same angle. The only distinction is the length of each phasor (representing primary and secondary voltage magnitudes) and its origin (starting point on the diagram).

Applying this analysis to the in-phase transformer circuit example. We will regard the secondary winding’s voltage as a phasor that is half the length of the primary winding’s voltage given the 2:1 turns ratio, but oriented at the same angle:

2:1

120 VAC

The result is a pair of phasors both at 0o angles. Note the placement of dots on each phasor to mark the relationship of the phasor to each winding polarity. The center of the diagram where the real and imaginary axes intersect is the point representing zero potential (ground). The length of each phasor tells us the magnitude of each waveform, in this case 120 volts and 60 volts AC, respectively. The “nonpolarity” end of each phasor line must be located at the origin because the nonpolarity terminal of each transformer winding is connected to ground.

Now consider the same analysis applied to the out-of-phase transformer circuit:

2:1

120 VAC

The result of this transformer connection is a pair of horizontal phasors, angled 180o apart from each other. Note how the secondary phasor still has its polarity mark on the right-hand end, but now that end of the phasor is anchored at the diagram’s origin because now the “polarity” (dot) terminal of the transformer’s secondary winding is connected to ground. Following our rule of keeping primary and secondary phasors at equal angles, the only di erence made by grounding the opposite terminal of the secondary winding is the reference point of the secondary phasor. The secondary phasor must remain locked at the same angle as the primary phasor, but its position on the diagram is determined by where the transformer winding connects to ground.

Here we have another autotransformer, this time connected to buck the primary voltage. The VP and VS phasors still share common angles, but now they are “stacked” di erently: polarity-end- to-polarity-end. This is challenging to see on the second phasor diagram because the VP and VS overlap each other, sharing a common “polarity” (dot) end point:

"Buck" autotransformer connection

The result of this “buck” configuration is a Vout that is the di erence between VP and VS rather than being the sum of the two as was the case with the “boost” configuration. If the transformer

ratio were 4:1 instead of 2:1, the output of this buck autotransformer would be 90 volts (120 volts − 1204 volts).

5.9.2Phasors in three-phase transformer circuits

In the following three-phase transformer circuit, each transformer has been labeled with a number, and each pair of phasors representing primary and secondary winding voltages for each transformer has that same identifying number written next to it for convenient reference. Numbering the phasors makes it clear which phasor(s) belong to which transformer.

As usual, the primary and secondary phasors for each transformer share the same angle, but the secondary phasors are “stacked” polarity-end-to-nonpolarity-end because that is precisely how the secondary windings are electrically connected to each other. Assuming an ABC phase sequence with VA being the reference phasor defining 0o phase angle:

3-phase transformer with "corner-grounded" Delta secondary