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

DC MOTOR

A Machine that converts dc electrical  ower into mechanical power is known as dc motor. Its operation is based on the principle that when a current carrying conductor is placed inside a magnetic field, the conductor experiences a mechanical force. The direction of force is given by Flemings Left Hand Rule and magnitude is given by 

F =B*I*L Newtons

The construction is similar to dc generator, so dc motor can work both as a motor and generator.

                                                 DC MOTOR 

WORKING OF DC MOTOR:

When armature winding is connected to a DC supply, an electric current sets up in the winding.  Permanent magnets or field winding (electromagnetism) provides the magnetic field. In this case, current carrying armature conductors experience a force due to the magnetic field, according to the principle stated above.

The Commutator is made segmented to achieve unidirectional torque. Otherwise, the direction of force would have reversed every time when the direction of movement of the conductor is reversed in the magnetic field. 

When the armature of a dc motor rotates under the influence of driving torque, armature conductors move through the magnetic field and hence emf is induced.This induced emf acts in opposite direction to the applked voltage and is known as back emf or counter emf Eb. The back emf is always less than aplied voltage.

Net voltage across Armature circuit is = V - Eb

If Ra is armature Resistance then Ia = V - Eb/Ra, Since V and Ra are fixed the value of Eb will determine the value of current drawn by the motor. If the speed of motor is high then back emf is large and hence the motor will draw less armature current.

SPEED OF DC MOTOR

Eb = V - IaRa

Eb = P*phi*Z*N/60

So N = K*Eb/phi

SPEED CONTROL OF DC MOTOR

Speed of dc motor can be controlled by three methods

1. Flux Control Method: In this method by varying flux `ϕ`, the motor speed (N`α`1/ϕ) can be changed. A variable resistance is placed in series with shunt field winding. The shunt field rheostat reduces the shunt field current Ish and hence the flux.This method permits to increase the speed in the ratio of 3:1.

2. Armature Control Method: In this the speed control is acheived, by varying the voltage availbale across the armature, the back emf and hence the speed of the motor can be changed. This is acheived by inserting a variable resistance Rc known as controller Resistance in series with the armature.

N is proportional to V - Ia(Ra+Rc)

Due to voltage drop in controller resistance the back emf is decreased and since N'a'Eb the speed of the dc motor is reduced.

3. Voltage Control Method: In this the voltage source supplying the field current is different than which supplies the armature.The shunt field of the motor is connected permanently across a fixed voltage source.The armature is connected across several different voltages by using a suitable switch gear. The speed will be approximately proportional to voltage applied across the armature.

                                                                DC MOTOR SCHEMATIC

As per above schematic a Simulink Model of DC Motor is built.

                                                       MATHEMATICAL MODEL OF DC MOTOR

 Here a Constant block, Sum, Gain, goto,from,Transfer function,to workspace and scope blocks are used.

For modelling DC Motor Parameters are shown below

J = 0.01;        %Moment of inertia of rotor in kgm^2
B = 0.1;         %Motor viscous friction Constant Nms
Km = 0.01;    %Back EMF Constant
Kt = 0.01;      %Motor Torque Constant
Ra = 1;          %Armature Resistance
La = 0.5;        %Armature Inductance

Transfer Function of angular velocity to input voltage is given by

w(s)/V(s) = Kt/[(Ls + R)(Js + B) + Kt*Kb]

A Supply Voltage Vs is applied as Input by using a constant block  and Back EMF Vb is subtracted to get the voltage difference. This Voltage difference is divided by Impedance LaRa which gives Armature Current. Transfer Function Block is used to enter the values of La and Ra. Now when Armature current is multiplied by Torque Constant it gives Motor torque. Motor Torque when divided by Rotor inertia and viscous friction constant gives Motor speed in Radians /Seconds.The motor speed is being converted to RPM by using gain block.

The above model is simulated without using PID Controller.

                                                                mscript File for Motor parameters

                                                                        PLOT SCRIPT FILE 

                                                OUTPUT OF DC MOTOR WITHOUT USING PID

PID CONTROLLER

A Proportional-Integrator-Derivative controller is a control loop mechansim employing feedback that is widely used in industrial control systems. The PID Cobtroller produces an output which is combination of outputs of Proportional-Integrator-Derivative controller. The feedback from the plant which ever is contrallable parameter is given as a input to the PID and processed error is minimized to get desired output.

The  DC moter speed  can be varied or controlled by controlling the input supply voltage. The PID controller is designed in such a way that it controls the Speed of the motor.

Now to design a PID Controller for DC Motor following equations are used

A = [-B/J Kt/J; -Km/La -Ra/La];
B = [0; 1/La];

For finding poles follwing commands are used

fn=1.0;
wn=2*pi*fn;
zeta=0.85;

p1 = -zeta*wn +wn*sqrt(zeta^2-1);
p2 = -zeta*wn -wn*sqrt(zeta^2-1);

SCRIPT FILE FOR CALCULATING PID GAINS AND POLES

REFERENCE SIGNAL PARAMETERS

The Reference signal is set to 1300RPM with step time of 0 and initial value 0.

DC MOTOR MODEL WITH PID CONTROLLER WITH ARMATURE CURRENT FEEDBACK GAIN

In above model PID Controller is designed with Kp, Ki and Kd gains. Kp and Kd are found by using place command. The input is given as reference speed. The output of model ie Speed is converted from rad/s to RPM by using gain block. The actual speed is subtracted from reference speed to generate error and it is fed to PID Controller. The input applied in RPM is 1300.

SCOPE OUTPUT

The speed follows the reference value with some error at start, with minimum settling time & settles at the value of 1300RPM.

The armature current follows the step function & has a value of 1300A

The back emf has the same nature as the reference signal & settles at 1.4V

The Motor Torque has value of 13 Nm.

The Speed output settles quickly at near to 1.5 seconds with settling time less than 2 seconds.

The KP,Kd and Ki values are 16.3221, -0.6593 and 10.

Here the reference speed is set as 1700RPM .The speed follows the reference value with some error at start, with minimum settling time & settles at the value of 1700RPM.

The PID Gains are same as for the earlier case.

The poles has a value of 

-5.34070751110265 + 3.30987324307279i
-5.34070751110265 - 3.30987324307279i

For finding poles location follwing command is used to determine the stability of system.

k_dc = 1;
s = tf('s');
G1 = k_dc*wn^2/(s^2 + 2*zeta*wn*s + wn^2);
pzmap(G1,'r'); grid;

The output is shown below

The poles obtained are marked in red colour 'x'. The poles are at left hand side of the plane which ensures that the system is stable.

DC MOTOR MODEL WITH SPEED ERROR SIGNAL AS FEEDBACK GAIN FOR Kd

Here the  Rest of the model remains same. A derivative block is used for Gain Kd.

From the above graph with values of Kp, Kd and Ki set as 16.3221, -0.6593 and 40 after trial and error run to find best response from system. The system has little overshoot. The RPM Should have reach a value  of more than 1365 RPM Initially but it stays ast 1353 RPM which is less than 5Percent Overshoot and settling time is also 1.5 seconds which is less than 2 seconds. So Above Model can be run with both Armature current as feedback for gain Kd and Speed error signal as input for Kd. The Torque settles at 13Nm after overshoot of less than 5 percent.

CONCLUSION:  In this project a Simulink model for the DC motor is developed and the Proportional Integral and Derivative PID control design is also developed with  knowledge gained in the training and  tested a controller to maintain a desired speed for the motor at 1300RPM and 1700RPM.The model is simulated with both speed error signal is given as feedback for gain Kd and armature current as feedback for gain Kd.