--- title: "DC motor with tachometer feedback and P-controller" subject: "Control Systems" chapter: 2 key: "semester_3/control_systems:bd-dc-motor-tacho-p" source: https://www.bctengineer.com/q/semester_3/control_systems/bd-dc-motor-tacho-p site: BCT Engineering --- # DC motor with tachometer feedback and P-controller Control Systems · Chapter 2: Component Modeling - Asked in: 2081 Bhadra, 4 marks - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/control_systems/bd-dc-motor-tacho-p - Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-2 ## Question Draw block diagram of a general armature-controlled dc motor with feedback taken via a tachometer using P-controller and describe each variable at different points before or after the blocks used. ## Short answer Tachometer voltage Vt=KT ω is subtracted from reference Vr; error e amplified by Kp gives armature voltage Va. Motor with back emb: Gm(s)=Kt/[(Ra+sLa)(Js+B)+KtKb]. Closed-loop speed control: ω(s)/Vr(s)=Kp Gm/(1+Kp KT Gm). Variables: Vr, e, Va, Ia, Tm, ω, Eb, Vt. ## Answer A tachometer is a small DC generator on the motor shaft whose output voltage is proportional to speed, $v_t = K_T\,\omega$. Feeding it back and comparing with a reference voltage gives a closed-loop **speed control** system with a proportional (P) controller. #### Block diagram ```text Vr + e ┌──┐Va + ┌──────┐Ia┌──┐Tm┌────┐ ω ──►(Σ)───►│Kp├─►(Σ)─►│ 1 ├─►│Kt├─►│ 1 ├─┬──► ω ▲ - └──┘ ▲ - │Ra+sLa│ └──┘ │Js+B│ │ │ │ └──────┘ └────┘ │ │ │ Eb ┌──┐ │ │ └─────────┤Kb├◄────────────┤ │ └──┘ │ │ Vt ┌──┐ │ └──────────────┤KT├◄─────────────────────┘ └──┘ ``` #### Variables at each point | Point | Variable | Meaning | |---|---|---| | Reference input | $V_r$ | voltage set for the desired speed | | After first summer | $e = V_r - V_t$ | speed error voltage | | After $K_p$ | $V_a = K_pe$ | armature voltage from the P-controller/power amplifier | | After second summer | $V_a - E_b$ | net voltage across armature impedance | | After $\frac{1}{R_a + sL_a}$ | $I_a$ | armature current | | After $K_t$ | $T_m = K_tI_a$ | motor torque | | After $\frac{1}{Js + B}$ | $\omega$ | shaft speed (output) | | Inner feedback $K_b$ | $E_b = K_b\omega$ | back emf | | Outer feedback $K_T$ | $V_t = K_T\omega$ | tachometer voltage | #### Closed-loop transfer function With $G_m(s) = \dfrac{K_t}{(R_a + sL_a)(Js + B) + K_tK_b}$ (motor with back emf): $$ \frac{\Omega(s)}{V_r(s)} = \frac{K_pG_m(s)}{1 + K_pK_TG_m(s)} $$ Increasing $K_p$ reduces the steady-state speed error and the effect of load-torque disturbances, but a P-controller alone always leaves a small steady-state error for a step speed command. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What does the tachometer provide? A. A voltage proportional to speed B. A voltage proportional to armature current C. A reference voltage D. Torque Answer: A. A tachometer is a small generator giving v_t = K_T ω, used as speed feedback. (Key point: Tachometer) 2. What is the error voltage after the first summer? A. V_r + V_t B. V_r - V_t C. V_a - E_b D. K_p V_r Answer: B. The reference is compared (subtracted) with the tachometer feedback to give the error e = Vr - Vt. (Key point: Error signal) 3. What is the output of the P-controller block? A. V_a = K_p e B. I_a = K_p e C. T_m = K_p e D. E_b = K_p e Answer: A. The proportional block multiplies the speed error to produce the armature voltage. (Key point: P-controller output) 4. What is a limitation of using only a P-controller? A. It always leaves a small steady-state error for a step command B. It makes the system unstable C. It removes feedback D. It cannot reduce disturbance Answer: A. Raising K_p reduces the error but a P-controller alone cannot eliminate it. 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