Three-Phase Transformers
1
Three-Phase Transformers
Introduction
Transformer power levels range from low-power applications, such as consumer electronics
power supplies to very high power applications, such as power distribution systems. For higherpower applications, three-phase transforms are commonly used. The typical construction of a
three-phase transformer is shown in Figure 1. The detailed analysis of this circuit is not
straightforward since there are numerous combinations of flux paths linking various windings.
For this reason, the three-phase transformer will be modeled as three independent single-phase
transformers herein.
Three-Phase Transformers
2
For practical calculations, it is reasonable to model the three-phase transformer as three ideal
transformers as shown in Figure 2. Since these transformers are ideal, the secondary voltages are
related to the primary voltages by the turns ratio according to
N
Vˆ2 a = 2 Vˆ1a
N1
(1)
N
Vˆ2b = 2 Vˆ1b
N1
(2)
N
Vˆ2 c = 2 Vˆ1c
N1
(3)
Three-Phase Transformers
3
Considering the individual transformers of Figure 2, and that both delta and wye connections are
possible for the primary and secondary windings, there are four possible combinations of
transformer connections. The most straightforward combinations are the wye-to-wye connection
and the delta-to-delta connections shown in Figures 3 and 4 respectively. In these cases, the lineto-line voltages on the secondary side are directly proportional to those on the primary side
through the turns ratio. Therefore, the following relationship holds for both connections.
N
Vˆab = 2 VˆAB
N1
(4)
In Figures 3 and 4, the ideal transforms are schematically rotated by 120o from each other to
represent a 120o electrical displacement in the voltages. The voltage labels on the individual
coils correspond to the labels in Figure 2. An equation similar to (4) can be derived to relate the
primary and secondary currents if desired.
Three-Phase Transformers
4
The delta-to-wye connection is shown in Figure 5. From the ideal transformer equations, it can
be determined that
(
)
(
N
N
Vˆab = 2 Vˆ1a − Vˆ1b = 2 VˆBC + VˆAB
N1
N1
)
(5)
In this analysis A-B-C sequence is assumed. If a source does not have this sequence the input
lines can be relabeled so that it does. With this sequence, (5) is equivalent to
N
Vˆab = 2 3 VˆAB ∠ 30o
N1
As can be seen, the secondary line-to-line voltage leads the primary by 30o. A factor of
also inserted in the magnitude calculation as well.
(6)
3 is
The wye-to-delta connection is shown in Figure 6. With this arrangement, it can be shown that
N 1 ˆ
Vˆab = 2
VAB ∠ − 30o
N1 3
In this case, the turns ratio is divided by
3 and the secondary phase lags by 30o.
(7)
Three-Phase Transformers
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In early power systems, two-phase transmission was used as well as three-phase. To supply a
two-phase system from a three-phase source, two transformers were connected in the Scott-T
connection shown in Figure 7. In this case, the primary windings must have tap locations at 1/2
and 3 /2. As it turns out, many 240-V laboratory transformers have tap settings at 120-V and at
208-V (which is nearly 3 /2). Considering the connection diagram of Figure 7 and the ideal
transformer relationships,
N 2 ˆ

1
V AB + VˆBC 
Vˆa = 2
2
N1 3 

(8)
N
Vˆb = 2 VˆBC
N1
(9)
In terms of the primary voltage VˆAB , the secondary voltages are
N
Vˆa = 2 VˆAB ∠ − 30o
N1
(10)
N
Vˆb = 2 VˆAB ∠ − 120o
N1
(11)
As can be seen, there is a 90o phase relationship between the two secondary voltages which is
necessary in a two-phase system.
Three-Phase Transformers
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Laboratory Software
Figure 8 shows a screen-shot of the program used for the three-phase transformer experiments.
This program displays the voltage, current, and power for the primary side (designated with 1)
and secondary side (designated with 2). The secondary voltage waveforms are shown in the
scope display. When the Connection Type is set to a three-phase type (such as delta-to-delta,
delta-to-wye, wye-to-wye, or wye-to-delta), a three-phase set of line-to-line voltages will be
plotted. When the Connection Type is set to the two-phase type (Scott-T), only two-phase
waveforms will be displayed. Data can be logged by selecting the type of connection and
clicking Add.
Three-Phase Transformers
7
Laboratory Transformers
Figure 9 shows the laboratory transformer and a corresponding electrical connection diagram.
The transformer has multiple primary and secondary windings which may be connected in series
or parallel for different voltage ratings. Internally, the primary windings have been connected in
series and the secondary has been connected in parallel yielding the ratings show in the table
below.
Laboratory transformer ratings.
Vˆ = 240 V
Vˆ = 120 V
1
Iˆ1 = 4.17 A
2
Iˆ2 = 8.33 A
N1
= 2 . The connection diagram
N2
also shows the primary taps which are accessible through the connectors.
From the voltage ratings, it can be seen that the turns ratio is
As can be seen, primary taps at 120-V and 208-V are available using terminals H2-H3 and H4
respectively. In commercial applications, 120-V is a common voltage level. This is typically
obtained by splitting a 240-V winding. Alternatively, 120-V can be obtained from the line-toneutral voltage of a 208-V three-phase system. For these reasons, 120-V and 208-V tap settings
are commonly available on many 240-V transformers.
Three-Phase Transformers
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Laboratory Work
Connect the transformers in the delta-to-delta connection as shown in Figure 10. The load box
should be set to use two resistors in series in each phase by setting the switches as shown in
Figure 11. Note: it will be simpler to connect the source and load cables to the transformer first
and place the interconnecting wires on top since these wires will be changed to make the other
types of connections. Switch on the load box fans. Energize the circuit and increase the
primary voltage to 100% (approximately 208-V line-to-line). Note the voltage ratio corresponds
to the turns ratio according to (4). Select Delta-Delta as a Connection Type and add the data to
the log. Reduce the voltage to zero and switch off the source power switch.
Re-configure the transformer to the delta-to-wye configuration as shown in Figure 12. Note that
this is very similar to the delta-to-delta configuration and only the interconnecting wires between
the transformers need be changed. Repeat the above test with this configuration. Be sure to
change the Connection Type to Delta-Wye and log the data.
Repeat the above test for the wye-to-wye and then the wye-to-delta configurations as shown in
Figures 13 and 14 respectively. Log the data for each case; labeling appropriate connection type.
Connect the transformers in the Scott-T connection as shown in Figure 15. Before energizing the
circuit, change the Connection Type to Scott-T so that the waveforms display properly. Switch
on the source circuit breaker and increase the voltage to 100%. Notice the 90o phase relationship
between the secondary voltages as predicted by (10-11) verifying two-phase output. Print the
waveforms on the screen, lower the source voltage to zero and switch off the source circuit
breaker.
Three-Phase Transformers
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Three-Phase Transformers 10
Three-Phase Transformers 11
Three-Phase Transformers 12
Three-Phase Transformers 13
Three-Phase Transformers 14
Three-Phase Transformers 15
Calculations and Questions
1. Derive the wye-to-delta transformer equation (7) from the connection diagram of Figure 6 and
the ideal transformer relationships of equations (1-3). Note that with A-B-C sequence, the
transformer primary voltages can be expressed as
Vˆ1a = Vrms ∠0
Vˆ1b = Vrms ∠ − 120 o
Vˆ1c = Vrms ∠120 o
2. Sketch the phasor diagram for the Scott-T connection. Sketch the three-phase primary line-toline voltages
VˆAB = 3 Vrms ∠ 30 o
VˆBC = 3 Vrms ∠ − 90 o
VˆCA = 3 Vrms ∠150 o
as well as the voltages Vˆa and Vˆb according to (8-9). Include phasors for both terms on the
right-hand side of (8) as well.
3. Using the lab measurements from the delta-to-delta and wye-to-wye configurations, verify
equation (4). Verify equations (6) and (7) using the delta-to-wye and wye-to-delta
measurements respectively.