5.0 REVIEW OF PHYSICAL LAWS GOVERNING ESCAP® MOTOR APPLICATIONS

5.1 Rectilinear Motion

5.1.1 Velocity, sometimes referred to as speed, is equal to distance (displacement) traveled per unit time (second).




5.1.2 Acceleration is defined as the change in velocity per unit time. Therefore:



5.1.3 Displacement is the distance traveled after time (t). Thus:

and with zero initial velocity (when v₀ = 0 and x₀ = 0)


5.1.4 Force is defined as a mass (m) multiplied by the acceleration (a) which it is experiencing.
Thus: F = ma

Conversely, the mass of a body is represented by the force per unit acceleration.

5.1.5 The concept of inertia comes into play when describing the amount of force that is required to accelerate a body. A small inertia requires less force to achieve a given acceleration than does a body with a larger inertia. Thus, load inertia is a critical consideration when making a motor application.

5.1.6 When a force is applied to a body which is free to rotate about some axis, the product of the magnitude of the force and its force arm is called the moment of the force about the axis.

The force arm is defined as the perpendicular distance between the line of action of the force and the axis of rotation. It is this property of a moment of a force which produces torque and thus rotation of a body (load).

5.1.7 Work and energy
Work is defined as the product of displacement and the component of force in the direction of displacement. Thus:

Thus, the work done in accelerating a body equals the increase in its kinetic energy.

5.1.8 Mechanical power

Average power:

and instantaneous power:

and under constant force: P = Fv

5.2 Rotational Motion

5.2.1 Angular velocity


NOTE: 2 π radians = 1 rotation (360°)
By definition a radian is an angle whose arc length is equal to the radius of the arc.
Also, "speed" should not be used to describe angular velocity. Rotational speed can only be expressed as RPM or RPS.

5.2.2 Angular acceleration 
      
Under acceleration angular velocity changes as follows:
     w = w₀ + α t 

5.2.3 Angular displacement f after time (t) is defined as:
          
5.2.4 Handy equations of angular rotation physics with constant angular velocity  
     (ω = constant and Φ_o = 0)
      Φ  = ω t  
     alternate forms: ω =      
with constant acceleration (α = constant):
        
where ω = ω₀ + α t   
    α =     
with constant acceleration (a) and with zero initial velocity (ω_o = 0):
          
angular acceleration of a load (J) with constant torque (M) is:
      α =    
5.2.5 Moment of inertia. The moment of Inertia (J) of a body referred to an axis of rotation is the product of the mass of the body and the square of the distance between the center of mass and the axis of rotation.
       J = Σmr²

For a flywheel:
    (J) = mr² 
and for a homogeneous disc:
  
5.2.6 Energy, work and power
Rotational kinetic energy is expressed by:
    
And work (W) is defined as:
    dW = MdΦ 

where M is the torque causing the displacement (Φ).
   Under constant torque the work done over a given displacement is stated as:
       W = M (Φ₂ -Φ₁)

Now 
but is rate of doing work (Power)
and 

Hence, power P = M ω (under constant torque)
Net work = change in kinetic energy 
      
and
      
M = J = J α which is the rotational analogue of Newton's second law, F = ma in linear motion.

5.2.7 Momentum
The law of conservation of momentum states that the total momentum of a system can only be changed by external forces acting on the system.
   In rectilinear motion the system momentum is given as the product of mass and velocity (mv).
   Whereas angular momentum is:

   L = J ω


5.3 Basics of DC Circuits 
5.3.1 OHM'S law variations:
        E = lR and l =  
5.3.2 Power 
     P = power (watts)
Power input to a network is computed as:
     P = El 
and the power (heat) dissipated in a resistance is given by:
     p = l² R  
5.3.3 Kirchhoff's rules 
Point rule: The algebraic sum of the currents toward any point of a network is zero.
     Si = 0 
loop rule: The algebraic sum of the voltage sources and the lR products in any loop of a network equals zero.
     SE + SiR = 0  
5.3.4 Series and parallel resistance 
The equivalent resistance of resistors in series is the sum of the individual resistances. 
      Thus: R = R₁ + R₂ + R₃ + ... R_N  
The equivalent resistance of resistors in parallel is computed as:
   
A special solution for only 2 parallel resistors is;
   

6.0 BASIC MOTOR PHYSICS

6.1 Simplified Coil In a Magnetic Flux Generator (Tacho)

Action:     
   displacement (d)

Reaction:  
   induced Emf (E)        












Motor  
Action:
        current (i)

Reaction:
        Force (F)         







β = 4500 Gauss

Force = f (β, l, i)
Torque = f (β, l, i)
Emf = f (β, l, ω)

Torque is current (i)
dependent
Emf is speed (ω)
dependent

βl is a constant as determined by the winding length (l) and the magnetic flux density (β).
β l = K_T torque constant (motor)
β l = K_v voltage constant (Tacho)

page 3 - Escap® Micromotor Energy Flow and Conversion