--- title: "Excitation system with stabilizing transformer: transfer function" subject: "Power Plant Equipment" chapter: 3 key: "semester_7/power_plant_equipment:excitation-stabilizing-transformer-model" source: https://www.bctengineer.com/q/semester_7/power_plant_equipment/excitation-stabilizing-transformer-model site: BCT Engineering --- # Excitation system with stabilizing transformer: transfer function Power Plant Equipment · Chapter 3: Var/Voltage Control in Hydrogenerating Systems - Asked in: 2076 Asoj, 8 marks; 2081 Baisakh, 6 marks; 2080 Baisakh, 6 marks; 2079 Bhadra, 10 marks; 2070 Asar, 8 marks; 2072 Kartik, 8 marks; 2081 Bhadra, 6 marks; 2079 Baisakh, 8 marks; 2075 Chaitra, 8 marks; 2073 Shrawan, 7 marks; 2071 Chaitra, 5 marks; 2070 Chaitra, 6 marks; 2080 Bhadra, 8 marks (asked 13 times) - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_7/power_plant_equipment/excitation-stabilizing-transformer-model - Whole chapter: https://www.bctengineer.com/md/semester_7/power_plant_equipment/chapter-3 ## Question Describe the excitation system of a synchronous generator with stabilizing transformer. Derive mathematical model of the system in term of transfer function of each component of the system. ## Short answer The excitation system supplies DC field current and regulates terminal voltage. Transfer functions are: sensor KR/(1+sTR), amplifier KA/(1+sTA), exciter KE/(1+sTE), generator KG/(1+sTG), and stabilizing transformer sKF/(1+sTF), which provides rate feedback to damp oscillations. ## Answer 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. ```text +---------+ +----------+ +---------+ 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 ```text 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. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What does the stabilizing transformer provide? A. Voltage step-up for the field B. Rate (derivative) feedback of exciter voltage to damp oscillations C. Reactive power compensation D. Isolation of the PT Answer: B. It feeds back a signal proportional to the rate of change of Efd, damping overshoot and hunting. (Key point: Purpose of stabilizing transformer) 2. The stabilizing transformer transfer function dVF/dEfd is: A. K_F/(1 + sT_F) B. sK_F/(1 + sT_F) C. K_F s^2 D. 1/(1 + sT_F) Answer: B. VF = M di1/dt with i1 = Efd/(R1 + sL1), giving sKF/(1+sTF) with KF = M/R1, TF = L1/R1. (Key point: Stabilizer transfer function) 3. For a separately excited DC exciter, TE equals: A. R_ef / L_ef B. L_ef / R_ef C. L_ef x R_ef D. K_E / R_ef Answer: B. The field circuit gives KE/(1+sTE) with TE = Lef/Ref. (Key point: Exciter time constant) 4. What is the effect of the stabilizer on steady-state gain? A. It reduces it to zero B. It doubles it C. None, since H_F(0) = 0, so Vt/Vref = K/(1+K) D. It makes the error infinite Answer: C. The derivative feedback vanishes as s goes to 0, so only transient behaviour improves. (Key point: Steady-state effect) ## Related in Power Plant Equipment - [Function of excitation system and its transfer function](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_plant_equipment%3Aexcitation-function-tf) — Power Plant Equipment, chapter 3, asked 2× - [Dynamic response of excitation system](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_plant_equipment%3Aexcitation-dynamic-response) — Power Plant Equipment, chapter 3, asked 3× - [Water turbine transfer function and characteristics](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_plant_equipment%3Awater-turbine-tf-characteristics) — Power Plant Equipment, chapter 2, asked 2× - [Brush excitation system](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_plant_equipment%3Abrush-excitation-system) — Power Plant Equipment, chapter 3 - [Static excitation system](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_plant_equipment%3Astatic-excitation-system) — Power Plant Equipment, chapter 3 - [Functional block diagram of excitation control](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_plant_equipment%3Aexcitation-control-block-diagram) — Power Plant Equipment, chapter 3 ## Related in other subjects - [Transfer function from state model [0 1; −5 −6]](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Ass-tf-0-1-5-6) — Control Systems, chapter 8, asked 3× - [Transfer function from symbolic mass-spring-damper state model](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Ass-tf-mbk-symbolic) — Control Systems, chapter 8 - [Pulse transfer function from a 3×3 state model](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Apulse-tf-third-order-numeric) — Digital Control System, chapter 5 - [Vertical two-mass system transfer function X1/F](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Atf-vertical-m1-m2-k1-b1) — Control Systems, chapter 2