The advanced physics of motor

Motor simulator

Theory

Simulations of the motor timing

Summary

Theory

This simulator is made to study the motor timing and performance of the acceleration and deceleration of the motor, which means the unsteady state revolutions (non-constant rev.), unlike the steady state motor theory shown previously. For the purpose, the following time dependent physical equations are fully resolved. 

dJ/dt=A*(V-w*B-Rm*J)      -----(1)
dw/dt=C*(J*B-K*w-a*w*|w|) -----(2)
dx/dt=w                   -----(3)

t:time(sec),   J:current(A),    w:phase angle speed (rpm),    
x:phase angle (180>x>-180degs),    V:input voltage, 
Rm:coil resistance,    B=Bo*cos(x):magnetic field by magnet
K: dissipation coefficient(for viscoity and friction loss)
A:motor cosntant 1,  C:motor constant 2,   a:propella size parameter
These equations' main parts are based on the Lorentz force which is taught in high school and university. I don't know your country's education system, but probably, university level's acknowledgement will be required to understand fully these equations. But, don't worry ! I believe that this similator gives you some easy and important charcteristics of the motor.

A and C are motor constants. A is related to the self inductance of the motor and C is the inverse of the motor (rev. part) inertia effect. For simplicity, let us consider a simple two-pole motor shown in the next figure.

Don't try to apply this theory for real motors ! In principle, it is possible but extremely hard. You will need some master course level's techiques in university for it. Because, A, C and so on for usual real motors are very hard to be specified (measured). If you try to know those values for your motors, you may have to exactly consider the magnetic field and electric field generated in the motor. It requires the electromagnetic analysis in addition to this present theory. If you knew every value by any way, those values are NOT constant but depend on the voltage, load and so on ! Hence, this present theory is available to only gives us some characteristics tendencys of the unsteady state motor performance, as shown below. It should be sufficiently informative for us.

The image of motor studied here.

The voltage V is added following the next equation. "x" is the phase angle for the motor rev. and varies between 180deg and -180deg. Note that 180deg is equal to -180deg. When x becomes larger than 180deg, it is always replaced in -180 

V=Vo  for 90deg+ang>x>-90deg+ang             (4a)
V=-Vo   for x>90deg+ang or -90deg+ang>x      (4)b
The energy equation is as follows. 

0.5*d(C*J*J+A*w*w)/dt=A*C*(V*J-Rm*J*J-K*w*w-a*w*w*|w|)    ----(5)
The left hand side is the time derivatives of energy stored in the motor. The first term is for the electro-magnetic energy and the second is for the mechanical energy. The right hand side consists of the output energy derived from the motor and input energy supplyed into the motor. The first term is input power J*V. The second is loss energy in the coil resistance. The third is the other loss energy, including the heat loss generated in magnet, iron loss in the motor can, air viscocity loss in the rev. part, mechanical and electric noises and so on. Hence, this term should have more complicated configuration, but now, a simple mechanical loss is modeled and assumed here. Finally, the forth is the output energy to propella or main rotor via the motor axis shaft.

The motor efficiency is (the right forth term) / (the right first term).

The so-called steady state motor theory based on Rm, KV(Ke), Kt and Io corresponds to when the case of d/dt=0 in this page. Note that the no load current Io cannot be simply shown here. Io is un-directly related to the constant K. The relation should have not an explicit formulation.

Resolving these equations (a set of (1)(2)(3)(4a) and (4b)) is not so easy. Mathematically, these can be explicitly resolved as infinite series of (1/n)*period waves. Rather, these equations are numerically resolved by a personal computer. For the purpose, this equations is re-written in a C-langrage program and easily evaluated in usual PC.

Simulations of the motor timing

An appropriate prop. is attached. I don't know the real prop. size ! In this section, the motor timing is changed. Also, an appropriate Rm and so on are given. Note that KV and the no-load current Io cannot be positively controlled in this present theory. Rather, they are naturally determined in this simulation run. Hence, the motor rpm W and current J are also unknown, until you simulate the motor.

Fig.1: Case of zero motor timing (ang=0).

The top graph shows the time variations of the input voltage V. This V profile is directly given by eqs.(4a) and (4b). From the next top, the simulated response of the phase angle x, rev. speed W and current J are shown. These values are not constant due to the motor switching and magnetic induction effect.

In the phase angle x profile, when x reaches 180deg, x is changed to -180deg. It is repeated every rev. cycle. So, the motor is continuously revolving in a direction.

In the motor axis rpm(=w) profile, w is negative, so that the rev. direction is minus. The profile is not constant. It means that the rev. is not so smooth. This effect is related to the cogging force.

In the current J, inductive current is generated and hence the profile is distorted. Exactly, the J time variation is slightly delayed for the voltage time variation. This effect is caused by the electron's and motor's itself inertia effects.

The efficiency rate is also calculated. 82.7% is not so bad ?

Fig.2: Case of forward motor timing (15deg=ang).

I set ang=+15deg for the motor timing. The rpm (=w) became higher. The current profile is largely changed. The efficiency rate (=84%) was improved ! Good !

Fig.3: further forward timing (ang=30deg).

The rpm further increased but the efficiency rate goes down. This motor timing is too forward.

Summary

As a result, it was revealed that this motor's suitable timing is around 15deg. However, this best timing should be modified depending on the load and voltage.

I tried ang=90, also. The motor could not run. It fails to revolve, due to too bad timing. In this situation, the staedy state motor theory is not applicable.

Today, this is end. The next topic is :

How does the best timing change depending on given voltage and load ?

Is the no-load current Io really constant for voltage ?

Coming soon ?