Sequence Circuits for Transformers

Section V: Sequence Circuits for Transformers

• Y-Y Connected Transformer
• Δ - Δ Connected Transformer
• Y- Δ Connected Transformer

In this section we shall discuss the sequence circuits of transformers. As we have seen earlier that the sequence circuits are different for Y- and Δ -connected loads, the sequence circuits are also different for Y and Δ connected transformers. We shall therefore treat different transformer connections separately.

Y-Y Connected Transformer

Fig. 7.10 shows the schematic diagram of a Y-Y connected transformer in which both the neutrals are grounded. The primary and secondary side quantities are denoted by subscripts in uppercase letters and lowercase letters respectively. The turns ratio of the transformer is given by α = N₁ : N₂ .

Fig. 7.10 Schematic diagram of a grounded neutral Y-Y connected transformer.

The voltage of phase-a of the primary side is

Expanding V_A and V_AN in terms of their positive, negative and zero sequence components, the above equation can be rewritten as

(7.53)

Noting that the direction of the neutral current I_n is opposite to that of I_N , we can write an equation similar to that of (7.53) for the secondary side as

(7.54)

Now since the turns ratio of the transformer is α = N₁ : N₂ we can write

Substituting in (7.54) we get

Multiplying both sides of the above equation by a results in

(7.55)

Finally combining (7.53) with (7.55) we get

(7.56)

Separating out the positive, negative and zero sequence components we can write

	(7.57)
	(7.58)
	(7.59)

Fig. 7.11 Zero sequence equivalent circuit of grounded neutral Y-Y connected transformer.

From (7.57) and (7.58) we see that the positive and negative sequence relations are the same as that we have used for representing transformer circuits given in Fig. 1.18. Hence the positive and negative sequence impedances are the same as the transformer leakage impedance Z . The zero sequence equivalent circuit is shown in Fig. 7.11.

The total zero sequence impedance is given by

(7.60)

The zero sequence diagram of the grounded neutral Y-Y connected transformer is shown in Fig. 7.12 (a) in which the impedance Z₀ is as given in (7.60). If both the neutrals are solidly grounded, i.e., Z_n = Z_N = 0, then Z₀ is equal to Z . The single line diagram is still the same as that shown in Fig. 7.12 (a). If however one of the two neutrals or both neutrals are ungrounded, then we have either Z_n = ∞ or Z_N = ∞ or both. The zero sequence diagram is then as shown in Fig. 7.12 (b) where the value of Z₀ will depend on which neutral is kept ungrounded.

Fig. 7.12 Zero sequence diagram of (a) grounded neutral and (b) ungrounded neutral Y-Y connected transformer.

Δ - Δ Connected Transformer

The schematic diagram of a Δ - Δ connected transformer is shown in Fig. 7.13. Now we have

(7.61)

Again

Fig. 7.13 Schematic diagram of a Δ - Δ connected transformer.

Therefore from (7.61) we get

(7.62)

The sequence components of the line-to-line voltage V_AB can be written in terms of the sequence com ponents of the line-to-neutral voltage as

(7.63)

(7.64)

Therefore combining (7.62)-(7.64) we get

(7.65)

Hence we get

(7.66)

Thus the positive and negative sequence equivalent circuits are represented by a series impedance that is equal to the leakage impedance of the transformer. Since the Δ -connected winding does not provide any path for the zero sequence current to flow we have

However the zero sequence current can sometimes circulate within the Δ windings. We can then draw the zero sequence equivalent circuit as shown in Fig. 7.14.

Fig. 7.14 Zero sequence diagram of Δ - Δ connected transformer.

Y- Δ Connected Transformer

The schematic diagram of a Y- Δ connected transformer is shown in Fig. 7.15. It is assumed that the Y-connected side is grounded with the impedance Z_N . Even though the zero sequence current in the primary Y-connected side has a path to the ground, the zero sequence current flowing in the Δ -connected secondary winding has no path to flow in the line. Hence we have I_a0 = 0. However the circulating zero sequence current in the Δ winding magnetically balances the zero sequence current of the primary winding.

Fig. 7.15 Schematic diagram of a Y- Δ connected transformer.

The voltage in phase-a of both sides of the transformer is related by

Also we know that

We therefore have

(7.67)

Separating zero, positive and negative sequence components we can write

(7.68)

(7.69)

(7.70)

The positive sequence equivalent circuit is shown in Fig. 7.16 (a). The negative sequence circuit is the same as that of the positive sequence circuit except for the phase shift in the induced emf. This is shown in Fig. 7.16 (b). The zero sequence equivalent circuit is shown in Fig. 7.16 (c) where Z₀ = Z + 3Z_N . Note that the primary and the secondary sides are not connected and hence there is an open circuit between them. However since the zero sequence current flows through primary windings, a return path is provided through the ground. If however, the neutral in the primary side is not grounded, i.e., Z_N = ∞ , then the zero sequence current cannot flow in the primary side as well. The sequence diagram is then as shown in Fig. 7.16 (d) where Z₀ = Z .

Fig. 7.16 Sequence diagram of a Y- Δ connected transformer: (a) positive sequence, (b) negative sequence, (c) zero sequence with grounded Y-connection and (d) zero sequence with ungrounded Y-connection.

Section VI: Sequence Networks

The sequence circuits developed in the previous sections are combined to form the sequence networks. The sequence networks for the positive, negative and zero sequences are formed separately by combining the sequence circuits of all the individual elements. Certain assumptions are made while forming the sequence networks. These are listed below.

1. Apart from synchronous machines, the network is made of static elements.
   
2. The voltage drop caused by the current in a particular sequence depends only on the impedance of that part of the network.
   
3. The positive and negative sequence impedances are equal for all static circuit components, while the zero sequence component need not be the same as them. Furthermore subtransient positive and negative sequence impedances of a synchronous machine are equal.
   
4. Voltage sources are connected to the positive sequence circuits of the rotating machines.
   
5. No positive or negative sequence current flows between neutral and ground.