Challenge

Project - Speed Control of a Direct Current (DC) motor

SSmita Dwivedi19 Nov 2022

Answer

1-mathematical modelling of DC motor-

from kirchhoff's voltage law

Vs=Vb+ia*Ra+La*ia_dot (i)

Vb=Km*theta_dot (ii)

torque equation

J*w_dot+B*w=T=Kt*ia (iii)

w=theta_dot (iv)

transfer function of above three equation

[J*s+B]w(s)=Kt*Ia(s) (v)

Vs=Km*w(s)+(La*s+Ra)Ia(s) (vi)

w(s)=s*theta(s)

putting value of Ia(s) from eqn v to vi and rearranging we get

w(s)/Vs=Kt/[J*La*s^2+(J*Ra+B*La)s+Kt*Km+B*Ra]

where J=mass moment of inertia(kg/m^2)

B=viscous friction constant of motor fluid(Nm-s)

La=inductance of armature winding(henry)

Ra=resostance of armaturewinding(ohm)

Ia=armature current(ampere)

Kt=torque constant(Nm/ampere)

Km=back emf constant(volt/rad/sec)

Vs=supply voltage(volt)

ws=speed(rad/s)

%parameters for design of DC motor

J=0.01;%

B=0.1;%

Km=0.01; %

Kt=0.01;

Ra=1;%

La=0.5;%

%% Design of a PID controlller

A=[-B/J Kt/J; -Km/La -Ra/La];

C=[0;1/La];

%% Placement of ploles

f=1;

wn=2*pi*f;

zeta=0.5

p1=-zeta*wn+sqrt(zeta^2-1);

p2=-zeta*wn-sqrt(zeta^2-1);

P=[p1; p2];

K=place(A,C,P);

Kp=K(1)

Kd=K(2)

%% Stability check of system

k_dc=1;

s=tf('s');

G1=k_dc*wn^2/(s^2+2*zeta*wn*s+wn^2);

pzmap(G1);grid on

2-simulink modelling-Block used to build model

clock:to generate time function

to workspace:to generate time variable to workspace

step:to give refence speed and supply voltage

gain:to give gain of Kt Km Kp Kd and Ki

transfer function:to give denominator value

integrator:to integrate

sum:to add and subtract

scope:to get graph

1 speed output

torque

armature current

back emf

supply voltage

Design of dc motor PID control-

kp=23.88, kd=-2.85, ki=0,2,4,8,15

1-when w_ref=2 ki=0

2-when w_ref-10 ki=2

3-when w_rwf=10 ki=4

4=when w_ref=10,ki=8

5-when w_ref=10 ki=15

conclusion-best result can be seen at ki-4

Stability anaylysis-

%% Stability check of system

k_dc=1;

s=tf('s');

G1=k_dc*wn^2/(s^2+2*zeta*wn*s+wn^2);

pzmap(G1);grid on

roots can be seen in negative side of axis hence system is stable

Project - Speed Control of a Direct Current (DC) motor

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Ship work like this
in your own course.