6.2  Escap^® Micromotor Energy Flow and Conversion



6.3  Development of the Equivalent Circuit



    V_o = applied terminal voltage
    L_m = rotor inductance
    R_m = rotor resistance
    E_i = motor's induced back Emf
    i = circuit current flow Summation of loop voltage drops yields:
   


Since the Escap^® rotor inductance is 100 to 1,000 times less than that of conventional "iron-core" motors it can be neglected. Therefore, the Lm term vanishes and the simplified voltage loop equation becomes:
      V_o = R_m i + E_i 

6.4 Fundamental Equations of the Ironless Rotor Motor 

6.4.1 Motor Model 
       

Since V = Rl + E we can substitute K_vω for E since the back emf is equal to the angular velocity (ω) multiplied by the motor constant (K).
   Where K_v is the back emf constant and K_M is the torque constant. 
    V = Rl + K_vω and M = K_Ml.

NOTE: K_v = K_M = K 

This equality exists only when K_v and K_M are expressed in metric units.

NOTE: K_v =   
          
Therefore:
      V = Rl + K ω   
      where K K_v = K_M  

This same equation can be derived in another way as follows:
  Air gap power in = air gap power out
(from Figure 9)
       El = Mω 
    Kωl = Mω
   Kl = M or l =
Power ln = dissipation + power out.
(from Figure 10)
       Vl = l²R + El 
       Vl = l²R + Mω  
       Vl = l²R + Klω  
   V = lR + Kω 
Alternate Forms:


6.4.2 Torque Speed



The complete torque expression including mechanical losses is: NOTE: the origin of the torque speed characteristic (Figure 12) is shifted to (M_f, l_NL).
At no load current (l_NL) the no load speed (w₀) is:
   
at a particular load (M_L) the corresponding speed (w) is:
   
In actual applications
     M_f << M_L and l_NL ≅0
where w₀ = (neglecting M_f)
w = w₀ - M_L 
By definition stall torque is computed by: M_STALL =    
6.4.3 Power




By definition: P_OUT = M_Lω (Mechanical Power)
  
at peak power:
    

NOTE: Maximum mechanical power out cannot exceed 1/4 the stalled power input.

6.4.4 Efficiency
By definition: η =
      P_OUT = M_Lω 



Where l_L = load current
             l_S = stall current
             l_NL = no load current


Now P_lN = Vl_L        but V = l_S R 
     P_lN = l_Sl_L R
 
    
Solving for l_L:
     
Substituting in efficiency equation:
    
6.4.5 Thermal Considerations

There is actually only one major criteria that should be taken into account when selecting an ironless rotor DC motor. The final armature temperature must not exceed its maximum rated value so that no separation in the winding occurs under high centrifugal force.
   Given this consideration, it is evident that the ambient temperature (θ₀), the two thermal resistances (R_th1 = rotor to case; R_th2 = case to ambient), and the average power dissipation (P_d), have to be very precisely evaluated in a given application.

         P_d = l²R  (Watts)

         Δθ = P_d(R_Th1 + R_Th2) (ºC)

     Where R_Th1 & R_Th2 are (ºC/Watt)

The equation for the final armature (rotor) temperature is:

        θ_f = θ₀ + Δ θ

      θ_f = θ₀ + (R_Th1 + R_Th2)P_d ºC 

The change in rotor resistance with temperature is expressed as:

        R_f = R[1 + (.004)(θ_f - 20)]

where "R" is that rotor resistance at 20ºC and .004 is the coefficient of thermal resistance for copper at 20ºC, in OHMS/ºC.

R_f is the new resistance at temperature θ_f.
The maximum continuous average current is limited by the following thermal consideration:

    θ_MAX - θ₀ = Δ θ = (R_TH1 + R_TH2) R_f l²_MAX 

   
Where θ_MAX is the maximum permissible rotor temperature (ie 100ºC for a standard motor). The increase in rotor temperature θ versus time with a constant dissipated power is:
  
 
where:    t is elapsed time in seconds
                τ₁ is thermal time constant of coil (seconds).
                τ₂ is thermal time constant of tube (seconds).

6.4.6 Dynamic Performance

6.4.6.1 Starting under load conditions can be expressed as follows:
   
substituting:


The solution for ω(0) = 0 is:
(see figure 14)
Where   
but (mechanical time constant of motor from catalog) 
 

NOTE: Speed of rotation h (RPM):
    
6.4.6.2 Starting an unloaded motor can be stated as follows:

      Where: M_L = 0 and J_L = 0  
                   τ = τ_M and ω_∞ = ω₀  
                  
This means that after infinite time the unloaded motor (assume zero friction torque M_f) will attain the no-load angular velocity corresponding to the power supply voltage. Thus:
   
and ω₀ = ω_∞ under these no load conditions.

For an unloaded motor the initial acceleration would be:
   
Where:    M_d = motor starting torque
                 J_M = rotor inertia
                 l_d = current corresponding to the starting torque

Integration of the function ω(t) with initial conditions of Φ (t = 0) = 0:
  
    Φ (t = 0) = 0
  ω_∞ τ + c = 0 or c = - ω_∞τ 
 
this new function is graphed in figure 15.

7.0 GEARBOXES APPLIED WITH ESCAP^® MOTORS

where g = gear ratio
           
7.1 Efficiency: P_lN = Mω 
                      
                    P_OUT = ηP_lN  

NOTE: Efficiency will change with temperature due to factors such as lubrication, gear mesh, etc.

7.2 Stall Rating of Gearbox:

      M_STALL = gηM          where M_STALL is motor stall torque

       
    This torque (M) must never be permitted to exceed the maximum stall torque rating of the gearbox as stated In the catalog.
    Expressed as a motor current limit:
   
7.3 Inertia Transfer

Acceleration of the load J_L is expressed as:


Where:

(J_LM) is the load inertia referred to the motor shaft


page 4 - Considerations for the Control of DC Micromotors