# Mathematical Modeling and PID Control of Brushless Dc Motor

Updated on June 24, 2017

## BASIC PRINCIPLE OF BRUSHLESS DC MOTOR:

A BLDC has a rotor with permanent magnets and a stator with windings. It is essentially a dc motor turned inside out. The brushes and commutator have been eliminated and the windings are connected to the control electronics. The control electronics replace the function of the commutator and energize the proper winding.

As shown in the fig.1, the winding are energized in a pattern which rotates around the stator. The energized stator winding leads the rotor magnet, and switches just as the rotor aligns with the stator i.e. rotor magnetic field chases the rotating stator magnetic field, never quite catching up.

## PERFORMANCE WITHOUT PID CONTROLLER:

Once the transfer function is known, the next step is to check the parameters of the motor by applying a step input to the motor using a MATLAB code. The MATLAB code and the resulting parameters are as follows

## CODE:

```num=[.01]
den=[.005 .06 .0121]
p=tf(num,den);
figure
step(p)
stepinfo(p)
figure
rlocus(p)
```

## PERFORMANCE WITH PID CONTROLLER:

The system is stable because the root locus of the poles are in the left half plane but the parameters of the system are not at all desired, so there is need of a PID controller. So a controller is designed and its gains i.e. (kp, ki ,kd) are adjusted using the PID tuner application in MATLAB to optimize the performance of the system, the MATLAB code for step response, performance parameters and root locus is given below.

## CODE:

```num=[.01]
den=[.005 .06 .0121]
p=tf(num,den);
q=1
step(feedback(p,q))
kp=24.0076847682402
ki=10.2836856520437
kd=0
x=pid(kp,ki,kd)
stepinfo(p)
y=feedback(x*p,q)
figure
step(y)
stepinfo(y)
figure
rlocus(y)
```

## RESULTS COMPARISON:

The values of kp, ki, kd for which the system is optimized are given in Table.3.

The resulting values of the parameters form both with and without PID controller are compared in the Table.4.

## CONCLUSION:

Without the PID controller the system i.e. BLDC motor had very poor response towards a step input. It had high settling and high rise time. Once the PID is introduced in the system using the PID tuner app in MATLAB, the system become more stable as the root locus of its roots is changed and its response towards a step input is optimized. This shows that how much PID is important in control systems.

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