Sequence network of loaded generator with Zn

Significance of sequence components

Sequence components let an unbalanced three-phase system be analysed as three independent balanced single-phase networks (positive, negative, zero). For balanced equipment the sequence impedances are uncoupled, so unbalanced faults are solved by connecting the sequence networks only at the fault point.

Sequence networks of a synchronous generator grounded through $Z_n$

A generator produces only balanced positive sequence EMFs $E_a$, $E_b = a^2E_a$, $E_c = aE_a$. Let $Z_1$ and $Z_2$ be its positive and negative sequence impedances and $Z_{g0}$ its zero sequence impedance per phase. The neutral is grounded through $Z_n$.

            Ia -->
    +-(Ea)--Zg--o a
    |
 n  +-(Eb)--Zg--o b     Va, Vb, Vc measured
    |                   to ground
    +-(Ec)--Zg--o c
    |
   Zn  In = Ia+Ib+Ic
    |
  ground

Here $Z_g$ is the winding impedance of each phase.

Positive sequence: the EMF is purely positive sequence, and positive sequence currents are balanced, so no current flows in $Z_n$:

$$V_{a1} = E_a - I_{a1}Z_1$$

Negative sequence: there is no negative sequence EMF, and the currents are balanced, so no current flows in $Z_n$:

$$V_{a2} = -I_{a2}Z_2$$

Zero sequence: there is no zero sequence EMF. The neutral current is $I_n = 3I_{a0}$, so the drop in $Z_n$ is $3I_{a0}Z_n$:

$$V_{a0} = -I_{a0}Z_{g0} - 3I_{a0}Z_n = -I_{a0}(Z_{g0} + 3Z_n) = -I_{a0}Z_0$$

In matrix form:

$$\begin{bmatrix} V_{a0}\\V_{a1}\\V_{a2} \end{bmatrix} = \begin{bmatrix} 0\\E_a\\0 \end{bmatrix} - \begin{bmatrix} Z_0&0&0\\0&Z_1&0\\0&0&Z_2 \end{bmatrix}\begin{bmatrix} I_{a0}\\I_{a1}\\I_{a2} \end{bmatrix}$$

 Positive          Negative          Zero
 F1 o              F2 o              F0 o
    | Ia1 ^           | Ia2 ^           | Ia0 ^
   Z1                Z2               Zg0
    |                 |                 |
  (Ea)                |                3Zn
    |                 |                 |
 N1 o (ref)        N2 o (ref)        N0 o (ground)

Loaded generator: when the generator supplies a balanced load before the fault, the load appears in the positive sequence network (and in the negative and zero sequence networks with its own sequence impedances). Using Thevenin's theorem at the fault point, $E_a$ is replaced by the prefault voltage $V_f$ at that point, and $Z_1$ by the Thevenin impedance of the generator in parallel with the load. The zero sequence network still contains $3Z_n$.