Excitation system with stabilizing transformer: transfer function

The excitation system supplies DC field current to the synchronous generator and controls it to hold the terminal voltage and share reactive power. A conventional system uses a DC exciter driven by an amplifier, with a stabilizing transformer giving rate feedback to keep the loop stable.

          +---------+   +----------+   +---------+
 Vref --->|comparat.|-->|amplifier |-->| exciter |
          +---------+   +----------+   +---------+
            ^    ^                         |  Efd
            |    |  +-----------------+    |
            |    +--| stabilizing     |<---+
            |       | transformer     |    v
            |       +-----------------+ +----------+
            |                           |generator |
            |   +-------------------+   | field    |
            +---| PT + rectifier    |<--+----------+
                +-------------------+   terminals Vt

Working: the terminal voltage is measured, rectified and compared with the reference. The error is amplified and drives the exciter field; the exciter output $E_{fd}$ feeds the main field. If $V_t$ falls, $E_{fd}$ rises and restores it. The stabilizing transformer feeds back a signal proportional to the rate of change of $E_{fd}$, which damps overshoot and hunting.

Components and their transfer functions

1. Voltage sensor (PT + rectifier + filter): $\dfrac{\Delta V_s}{\Delta V_t} = \dfrac{K_R}{1 + sT_R}$ ($T_R$ is very small, 0.01–0.06 s).
2. Comparator: $\Delta V_e = \Delta V_{ref} - \Delta V_s - \Delta V_F$.
3. Amplifier (magnetic, rotating or thyristor): $\dfrac{\Delta V_R}{\Delta V_e} = \dfrac{K_A}{1 + sT_A}$ ($T_A$ = 0.02–0.1 s).
4. Exciter (separately excited DC exciter): the field winding has resistance $R_{ef}$ and inductance $L_{ef}$; output voltage is proportional to field current. So $\dfrac{\Delta E_{fd}}{\Delta V_R} = \dfrac{K_E}{1 + sT_E}$, with $T_E = L_{ef}/R_{ef}$.
5. Generator field (on no load): $\dfrac{\Delta V_t}{\Delta E_{fd}} = \dfrac{K_G}{1 + sT_G}$, with $T_G = T'_{do}$ (open-circuit field time constant, 1–10 s).

Stabilizing transformer

The primary of the stabilizing transformer is connected across the exciter output ($E_{fd}$) and the secondary is connected in series with the amplifier input. The secondary feeds a high-impedance input, so its current is negligible.

Primary: $E_{fd} = R_1 i_1 + L_1\dfrac{di_1}{dt}$, so $I_1(s) = \dfrac{E_{fd}(s)}{R_1 + sL_1}$.

Secondary: $V_F = M\dfrac{di_1}{dt}$, so $V_F(s) = sM\,I_1(s)$.

$$\frac{\Delta V_F(s)}{\Delta E_{fd}(s)} = \frac{sM}{R_1 + sL_1} = \frac{sK_F}{1 + sT_F}, \qquad K_F = \frac{M}{R_1},\ T_F = \frac{L_1}{R_1}$$

It gives a derivative (rate) feedback: output appears only while $E_{fd}$ is changing and is zero in steady state. So it damps oscillations and allows a high amplifier gain without affecting steady-state accuracy.

Block diagram

 Vref +    Ve  +-------+ VR +-------+ Efd +-------+
 --->(+)------>|  KA   |--->|  KE   |--+->|  KG   |--+-> Vt
     ^-  ^-    |1+sTA  |    |1+sTE  |  |  |1+sTG  |  |
     |   |     +-------+    +-------+  |  +-------+  |
     |   |        VF  +---------+      |             |
     |   +------------|  sKF    |<-----+             |
     |                | 1+sTF   |  (stabilizing      |
     |                +---------+   transformer)     |
     |        Vs      +---------+                    |
     +----------------|  KR     |<-------------------+
                      | 1+sTR   |
                      +---------+

Overall transfer function

Let $G_A = \dfrac{K_A}{1+sT_A}$, $G_E = \dfrac{K_E}{1+sT_E}$, $G_G = \dfrac{K_G}{1+sT_G}$, $H_F = \dfrac{sK_F}{1+sT_F}$, $H_R = \dfrac{K_R}{1+sT_R}$.

Inner (stabilizing) loop:

$$\frac{\Delta E_{fd}}{\Delta V_{ref} - \Delta V_s} = \frac{G_A G_E}{1 + G_A G_E H_F}$$

Outer (voltage) loop:

$$\frac{\Delta V_t(s)}{\Delta V_{ref}(s)} = \frac{G_A G_E G_G}{1 + G_A G_E H_F + G_A G_E G_G H_R}$$

Steady state ($s \to 0$, $H_F(0) = 0$, $K_R = 1$):

$$\frac{\Delta V_t}{\Delta V_{ref}} = \frac{K_A K_E K_G}{1 + K_A K_E K_G}$$

So the stabilizing transformer does not change the steady-state gain; it only improves the transient behaviour.

Effect of the stabilizer: without it, a high gain $K_A$ (needed for small steady-state error) makes the third-order loop oscillatory or unstable. The rate feedback adds damping, so a fast, well-damped voltage response is obtained with a high gain.