Skip to main content

DC motor with tachometer feedback and P-controller

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.

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.

A tachometer is a small DC generator on the motor shaft whose output voltage is proportional to speed, vt=KT ωv_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

 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

PointVariableMeaning
Reference inputVrV_rvoltage set for the desired speed
After first summere=Vr−Vte = V_r - V_tspeed error voltage
After KpK_pVa=KpeV_a = K_pearmature voltage from the P-controller/power amplifier
After second summerVa−EbV_a - E_bnet voltage across armature impedance
After 1Ra+sLa\frac{1}{R_a + sL_a}IaI_aarmature current
After KtK_tTm=KtIaT_m = K_tI_amotor torque
After 1Js+B\frac{1}{Js + B}ω\omegashaft speed (output)
Inner feedback KbK_bEb=KbωE_b = K_b\omegaback emf
Outer feedback KTK_TVt=KTωV_t = K_T\omegatachometer voltage

Closed-loop transfer function

With Gm(s)=Kt(Ra+sLa)(Js+B)+KtKbG_m(s) = \dfrac{K_t}{(R_a + sL_a)(Js + B) + K_tK_b} (motor with back emf):

Ω(s)Vr(s)=KpGm(s)1+KpKTGm(s)\frac{\Omega(s)}{V_r(s)} = \frac{K_pG_m(s)}{1 + K_pK_TG_m(s)}

Increasing KpK_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.

✓ Answer checked by Claude Opus 5.5 (high effort)

Related questions

All Control Systems chapter 2 questions →