Function of excitation system and its transfer function

Function of excitation system

The excitation system supplies and controls the DC field current of a synchronous generator. Its functions are:

• Supply DC current to the rotor field winding to produce the main flux.
• Keep the terminal voltage constant as load changes (automatic voltage regulation).
• Control the reactive power (VAr) output and share it correctly among parallel generators.
• Improve transient and steady-state stability by fast field forcing during faults.
• Provide protective limits (over/under-excitation limiters, V/Hz limiter).

Transfer function with stabilizing transformer

The system has four blocks: amplifier, exciter, generator field and a stabilizing transformer that feeds back the rate of change of exciter voltage to damp oscillations.

Vref +  e   +   +-----------+    +---------+  Efd
 --->(Σ)-->(Σ)-->| Amplifier |--->| Exciter |--+--+
      ^-    ^-   +-----------+    +---------+  |  |
      |     |     +---------------+            |  |
      |     +-----| Stab. transf. |<-----------+  |
      |           +---------------+               |
      |   Vt      +-----------+                   |
      +-----------| Generator |<------------------+
                  +-----------+

1. Amplifier: $\dfrac{V_R(s)}{V_e(s)} = \dfrac{K_A}{1+sT_A}$

2. Exciter (field $R_e$, $L_e$; $E_{fd} = K i_e$): from $v_R = R_e i_e + L_e \frac{di_e}{dt}$,

$$\frac{E_{fd}(s)}{V_R(s)} = \frac{K/R_e}{1+s(L_e/R_e)} = \frac{K_E}{1+sT_E}$$

3. Generator field (open circuit): $\dfrac{V_t(s)}{E_{fd}(s)} = \dfrac{K_G}{1+sT_G}$

4. Stabilizing transformer: primary ($R_1$, $L_1$) is across the exciter output; secondary (mutual inductance $M$) is practically open-circuited by the high-impedance amplifier input.

$$\begin{aligned} E_{fd} &= R_1 i_1 + L_1 \frac{di_1}{dt} \;\Rightarrow\; I_1(s) = \frac{E_{fd}(s)}{R_1(1+sT_F)} \\ V_F &= M\frac{di_1}{dt} \;\Rightarrow\; \frac{V_F(s)}{E_{fd}(s)} = \frac{sK_F}{1+sT_F} \end{aligned}$$

where $K_F = M/R_1$ and $T_F = L_1/R_1$.

5. Closed loop. Let $G_1 = \dfrac{K_A K_E}{(1+sT_A)(1+sT_E)}$, $H_F = \dfrac{sK_F}{1+sT_F}$, $G_2 = \dfrac{K_G}{1+sT_G}$. The minor loop gives $\dfrac{G_1}{1+G_1H_F}$; with unity voltage feedback:

$$\frac{V_t(s)}{V_{ref}(s)} = \frac{G_1G_2}{1+G_1H_F+G_1G_2}$$

Substituting and clearing fractions:

$$\frac{V_t(s)}{V_{ref}(s)} = \frac{K_AK_EK_G(1+sT_F)}{(1+sT_A)(1+sT_E)(1+sT_G)(1+sT_F) + sK_FK_AK_E(1+sT_G) + K_AK_EK_G(1+sT_F)}$$

The $sK_F$ term adds damping, so a high loop gain (small steady-state error) can be used without oscillation.