--- title: "Dynamic response of excitation system" subject: "Power Plant Equipment" chapter: 3 key: "semester_7/power_plant_equipment:excitation-dynamic-response" source: https://www.bctengineer.com/q/semester_7/power_plant_equipment/excitation-dynamic-response site: BCT Engineering --- # Dynamic response of excitation system Power Plant Equipment · Chapter 3: Var/Voltage Control in Hydrogenerating Systems - Asked in: 2082 Baisakh, 6 marks; 2074 Chaitra, 8 marks; 2076 Chaitra, 8 marks (asked 3 times) - Video: [11. Excitation System (7) Dynamic Performance Measures (1)](https://www.youtube.com/watch?v=0YBSbfxTMgs) — Saher Albatran - 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-dynamic-response - Whole chapter: https://www.bctengineer.com/md/semester_7/power_plant_equipment/chapter-3 ## Question Explain the dynamic response of excitation system with suitable mathematical deduction. ## Short answer Dynamic response is governed by loop gain and time constants. Closed loop is G/(1+G) with G=K/[(1+sTA)(1+sTE)(1+sTG)]. Steady-state error is 1/(1+K); high K improves accuracy but may cause instability, so stabilizing feedback adds damping. ## Answer The **dynamic response of an excitation system** is how fast and how smoothly the generator terminal voltage follows a change in reference voltage (or recovers after a load change). It depends on the gains and time constants of the amplifier, exciter and generator field, and on stabilizing feedback. #### Mathematical model Basic AVR loop (sensor time constant neglected, unity feedback): ```text Vref + Ve +-------+ +-------+ +-------+ --->(+)----->| KA |-->| KE |-->| KG |--+--> Vt ^ - |1+sTA | |1+sTE | |1+sTG | | | +-------+ +-------+ +-------+ | +-------------------------------------------+ ``` Open-loop transfer function: $$ G(s) = \frac{K}{(1 + sT_A)(1 + sT_E)(1 + sT_G)}, \qquad K = K_A K_E K_G $$ Closed loop: $$ \frac{\Delta V_t(s)}{\Delta V_{ref}(s)} = \frac{G(s)}{1 + G(s)} $$ #### Steady-state response For a step $\Delta V_{ref}$, using the final value theorem: $$ \begin{aligned} \Delta V_t(\infty) &= \frac{K}{1 + K}\,\Delta V_{ref} \\ e_{ss} &= \frac{1}{1 + K}\,\Delta V_{ref} \end{aligned} $$ To keep the voltage error small (e.g. below 1 %), $K$ must be large (about 100 or more). #### Transient response and stability Characteristic equation: $$ (1 + sT_A)(1 + sT_E)(1 + sT_G) + K = 0 $$ $$ T_A T_E T_G\, s^3 + (T_A T_E + T_E T_G + T_A T_G)\, s^2 + (T_A + T_E + T_G)\, s + (1 + K) = 0 $$ By the Routh criterion, the system is stable only if $$ (T_A T_E + T_E T_G + T_A T_G)(T_A + T_E + T_G) > T_A T_E T_G\,(1 + K) $$ So a large $K$ (good accuracy) makes the response more oscillatory and can make it unstable. This is the basic conflict in AVR design. ```text Vt high K: fast but oscillatory | .-. | / '-.__.------------- | / ___.-------------- low K | / _.-' (slower, larger error) |/.' +---------------------------> t ``` #### Improving the dynamic response - **Stabilizing (rate) feedback** from exciter output to amplifier input, $\dfrac{sK_F}{1 + sT_F}$: adds damping only during transients, leaving steady-state gain unchanged. Closed loop becomes $$ \frac{\Delta V_t}{\Delta V_{ref}} = \frac{G_A G_E G_G}{1 + G_A G_E H_F + G_A G_E G_G} $$ - **Fast exciters** (static thyristor or brushless) with small $T_E$, and high **ceiling voltage** and **response ratio**, so the field current can be forced quickly. - Lead-lag compensation or a power system stabilizer (PSS) for damping rotor oscillations. A good excitation system gives a fast rise, small overshoot, short settling time and a small steady-state voltage error. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. For the basic AVR loop with loop gain K, the steady-state error for a step in Vref is: A. K/(1+K) B. 1/(1+K) C. 1/K^2 D. K Answer: B. Vt(inf) = K/(1+K) dVref, so the error is 1/(1+K) of dVref. (Key point: Steady-state error) 2. What happens when the loop gain K is increased? A. Error falls but response becomes more oscillatory and may become unstable B. Error rises and stability improves C. Both error and oscillation fall D. Nothing changes Answer: A. Routh stability requires the time-constant product to exceed T_A T_E T_G (1+K), so large K risks instability. (Key point: Gain vs stability) 3. The characteristic equation of the basic AVR loop is: A. (1+sTA)(1+sTE)(1+sTG) + K = 0 B. (1+sTA)(1+sTE)(1+sTG) - K = 0 C. s + K = 0 D. (sTA)(sTE)(sTG) = K Answer: A. 1 + G(s) = 0 with G = K/[(1+sTA)(1+sTE)(1+sTG)]. (Key point: Characteristic equation) 4. Which measure adds damping only during transients without changing steady-state gain? A. Increasing K_A B. Rate feedback sK_F/(1+sT_F) from exciter output C. Raising the exciter time constant D. Removing the amplifier Answer: B. Rate feedback is zero at steady state, so it only affects transients. (Key point: Improving response) ## Related in Power Plant Equipment - [Need for excitation and improving dynamic response](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_plant_equipment%3Aexcitation-dynamic-improvement) — Power Plant Equipment, chapter 3 - [Excitation system with stabilizing transformer: transfer function](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_plant_equipment%3Aexcitation-stabilizing-transformer-model) — Power Plant Equipment, chapter 3, asked 13× - [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× - [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 ## Related in other subjects - [Static vs dynamic mathematical model](https://www.bctengineer.com/question.md?key=semester_8%2Fsimulation_and_modeling%3Astatic-vs-dynamic-math-model) — Simulation and Modeling, chapter 2, asked 3× - [Dynamic mathematical model with example](https://www.bctengineer.com/question.md?key=semester_8%2Fsimulation_and_modeling%3Adynamic-mathematical-model) — Simulation and Modeling, chapter 2, asked 2× - [Dynamic characteristics: rise time, delay, response](https://www.bctengineer.com/question.md?key=semester_4%2Finstrumentation_and_measurement%3Adynamic-characteristics-rise-time-delay) — Instrumentation and Measurement, chapter 4 - [Effect of feedback on dynamic response](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Afeedback-dynamic-response) — Control Systems, chapter 1