# Determining Phase and Line Voltages and Currents in Wye-connected Generators

## A generator that produces three separate sinewave output voltages with 120° phase differences is called a three-phase generator. The three generator coils may be connected in one of two possible configurations: Y-connection and Δ-connection. In this article, you’ll learn how to determine phase and line voltages and currents in wye-connected generators.

# To complement?? this article, we provide you with the Delta-Wye Calculator.

The output voltage levels produced by a three-phase generator depend on the connection arrangement of the three individual generator loops (or coils). The two possible connection methods are wye (Y) and delta (Δ) because of circuit resemblances to those symbols.

### Phase Voltages and Currents

Figure 1 represents a Y-connected three-phase generator, in which the coil terminals a, b, and c are connected to a single neutral terminal (N). There are four output terminals: A, B, C, and N.

The system illustrated in Figure 1 is termed a three-phase four-wire supply. In some cases, the neutral conductor may not be connected, and the system becomes a three-phase three-wire supply.

**Figure 1.** For a Y-connected three-phase generator, the phase voltage and phase current are the output voltage and current from each coil. The line voltage is the voltage measured between any two output lines (conductors). The line current is the current flowing in a line. Image used courtesy of Amna Ahmad

**Figure 1.**For a Y-connected three-phase generator, the phase voltage and phase current are the output voltage and current from each coil. The line voltage is the voltage measured between any two output lines (conductors). The line current is the current flowing in a line. Image used courtesy of Amna Ahmad

All three coils in a three-phase generator are identical, producing the same level of rms output voltage. This is termed the phase voltage(V_{p}). Figure 1 shows E_{Aa}, E_{Bb}, and E_{Cc,} now identified as E_{AN}, E_{BN}, and E_{CN}, respectively. All three are equal to V_{p}.

The phase current (I_{p}) is the current level supplied by each of the three generator coils. The directions of the phase currents (I_{a}, I_{b}, and I_{c}) shown in Figure 1 are those that occur when each coil has a positive output voltage. Thus, the phase current for coil Aa is shown flowing out of terminal A and into a.

### Double Subscript Notation

The double subscript notation is used with the voltage symbols to indicate the direction of current flow when each coil (or loop) output is positive. Subscript Aa used with voltage E_{Aa} identifies terminal A as positive concerning terminal a when this coil output is positive. E_{Aa} is rewritten as E_{AN} in Figure 1, indicating that terminal A is positive concerning the neutral terminal while the coil output is positive.

Similarly, when the output of coil Bb is a positive quantity, terminal B is positive concerning b, and the internal current direction is from b to B. The voltage is identified as E_{Bb} or E_{BN} in Figure 1. The voltage labeled E_{Cc,} or E_{CN,} shows that terminal C is positive concerning c when coil Cc has a positive output. In a different subscript system, the order of the double subscripts indicates the current direction instead of the voltage polarity.

### Line Currents and Voltages

The conductors that connect a three-phase generator to a load are referred to as lines, and the current carried by each of these conductors is known as the line current(I_{L}). Figure 1 shows that the phase and line currents in a Y-connected generator are the same quantities.

The neutral conductor is the return conductor for all three individual coils in a Y-connected, four-wire system. So, the neutral current (I_{N}) is the phasor sum of all three line currents. This would seem to make I_{N} larger than I_{L} and therefore require a neutral conductor that is thicker than the line conductors. However, in Example 1, it is demonstrated that when all three coils have identical loads (a balanced load), I_{N} is zero. When a balanced load condition does not exist, current flows in the neutral conductor.

In a Y-connected generator, the phase voltage is the voltage measured between any line and neutral. The line voltage is the voltage measured between any two lines. Thus, the line voltage is the phasor difference of two phase voltages. Figure 2 demonstrates how the line voltages are calculated.

*(a) Phasor diagram for voltages*

*(b) Line voltage E*_{AB} is the phasor difference E_{AN} - E_{BN}

_{AB}is the phasor difference E

_{AN}- E

_{BN}

*(c) Line voltage E*_{BC} = E_{BN} - E_{CN} and E_{CA} = E_{CN} - E_{AN}

_{BC}= E

_{BN}- E

_{CN}and E

_{CA}= E

_{CN}- E

_{AN}

**Figure 2.** For a Y-connected three-phase generator, the line voltages are the phasor sums of pairs of phase voltages. This makes each line voltage equal to 3 (phase voltage). Image used courtesy of Amna Ahmad

**Figure 2.**For a Y-connected three-phase generator, the line voltages are the phasor sums of pairs of phase voltages. This makes each line voltage equal to 3 (phase voltage). Image used courtesy of Amna Ahmad

Figure 2(a) shows the individual coil voltage phasors. Line voltage E_{AB} in Figure 1 is the phasor difference between E_{AN} and E_{BN}. To determine E_{AB}, -E_{BN} is first drawn equal and opposite to E_{BN} [see Figure 2(b)]. Because of the 120° phase difference between E_{AN} and E_{BN}, there is a 60° angle between E_{AN} and -E_{BN}. Voltages E_{AN} and E_{BN} are equal in magnitude; therefore, phasors E_{AN} and –E_{BN} are equal in length. Because of this equality, phasor E_{AB }(representing the sum of E_{AN} and -E_{BN}) is situated 30° from E_{AN} and 30° from -E_{BN}, as illustrated in Figure 2(b).

This gives,

\[E_{AB}=E_{AN}cos30°+{\large(}-E_{BN}cos30°{\large)}\]

which is the same as,

\[E_{AB}=V_{p}cos30°+V_{p}cos30°=2{\Large(}V_{p}cos30°{\Large)}=1.732V_{p}\]

or

\[E_{AB}=\sqrt{3V_{p}}\]

So, for a Y-connected generator,

\[Line\,Voltage=\sqrt{3}\times Phase\,Voltage\]

that is,

\(V_{L}=\sqrt{3}V_{p}\) (1)

Also,

Line Current = Phase Current

or

\(I_{L}=I_{p}\) (2)

Figure 2(b) also shows that line voltage E_{AB} leads phase voltage E_{AN} by 30°. Figure 2(c) shows the determination of E_{BC} and E_{CA} as the phasor differences E_{BN}-E_{CN} and E_{CN}-E_{AN}, respectively. Note that for line voltage E_{AB}, the subscript indicates that the generator terminal A is positive with respect to terminal B when E_{AB} is a positive quantity (see Figure 1).

### Y-Connected Loads

The circuit of a Y-connected generator with Y-connected load resistors is shown in Figure 3 (a). Note that the generator phase voltages are once again identified as E_{AN}, E_{BN}, and E_{CN}, and the load voltages are similarly identified. The reasoning that developed Equations 1 and 2 for a Y-connected generator can also be applied to determine the voltage and current relationships for the Y-connected load.

Referring to Figure 3 (a), the voltages across the individual loads are obviously equal to the generator phase voltages. Therefore, Equation 1 can be rewritten as

\[Line\,Voltage=\sqrt{3}\times Load\,Voltage\]

Also, the individual load currents are clearly the same currents that flow in the lines.

Line Current = Load Current

*(a) Y-connected three-phase generator with Y-connected resistive loads*

*(b) Phasor diagram of load voltages and currents for purely resistive loads*

*(c) When I*_{1} = I_{2} = I_{3}, I_{1} + I_{2} = -I_{3}, giving I_{1} + I_{2} + I_{3} = 0

_{1}= I

_{2}= I

_{3}, I

_{1}+ I

_{2}= -I

_{3}, giving I

_{1}+ I

_{2}+ I

_{3}= 0

**Figure 3.** Circuit and phasor diagrams for a Y-connected three-phase four-wire generator with a Y-connected balanced load. In this case, the load voltages and currents are equal to the generator phase voltage and phase current. Images used courtesy of Amna Ahmad

**Figure 3.**Circuit and phasor diagrams for a Y-connected three-phase four-wire generator with a Y-connected balanced load. In this case, the load voltages and currents are equal to the generator phase voltage and phase current. Images used courtesy of Amna Ahmad

**Example 1**

Load resistors, R_{1}, R_{2}, and R_{3} in Figure 3(a) are each 100 Ω, and the phase voltage is V_{p}=100 V. Determine: (a) the line current, (b) the neutral current, and (c) the line voltage.

**Solution**

**(a)** The line current

\[E_{AN}=E_{BN}=E_{CN}=100V\]

\[I_{L}=I_{A}=I_{B}=I_{C}\]

\[I_{L}=\frac{V_{P}}{R_{1}}=\frac{100V}{100\Omega}=1A\]

**(b)** The phasor diagram for the currents is drawn in Figure 3(b). Because the loads are resistive, each load current is in phase with the phase voltages. Figure 3(c) shows that the phasor sum of I_{1 }and I_{2 }equals -I_{3}. So,

\[I_{N}=I_{A}+I_{B}+I_{C}=0\]

**(c)** The line voltage

\[V_{L}=\sqrt{3}V_{p}=\sqrt{3}\times100V=173.2V\]

**Key Takeaways**

- A three-phase wye–connected generator produces three separate sinusoidal alternating voltages from three output terminals.
- The terminals are identified as A, B, and C; sometimes, a neutral terminal (N) is also available.
- The three waveforms have 120° phase differences from each other.
- The generator output voltages can be measured as phase voltage and line voltage.
- The phase and line currents in a Y-connected generator are the same quantities.

*Featured image used courtesy of Adobe Stock*

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