Page 636 - The Mechatronics Handbook

0066_Frame_C20.fm Page 106 Wednesday, January 9, 2002 1:46 PM

The micromachine parameters are the stator resistance r s , the magnetizing inductances in the quadra-
, the leakage inductance L ls , the
ture and direct axes L mq and L md , the average magnetizing inductance L m
moment of inertia J, and the viscous friction coefficient B m .
The expressions for inductances are nonlinear functions of the electrical angular displacement θ r .
Hence, the torsional-mechanical dynamics must be used. Taking note of the Newton’s second law of
rotational motion, and using ω r and θ r (electrical angular velocity and displacement) as the state variables
(mechanical variables), one obtains
2 2 dw r dq r
T e – B m ---w r – T L = J --- ---------, -------- = w r
P P dt dt
where T e and T L are the electromagnetic and load torques.
Torque Production Analysis—Using the coenergy, the electromagnetic torque, which is a nonlinear
function of the micromotor variables (phase currents and electrical angular position) and micromotor
parameters (number of poles P and inductance L ∆m ), is found to be [8],



T e = P 2 2q r + 2i as i bs sin 2 q r – 1 --p + 2i as i cs sin 2 q r + 1 
--p
---L ∆m i as sin
2  3   3 



2
--p +
2
+ i bs sin 2 q r – 2  2i bs i cs sin 2q r + i cs sin 2 q r + 2 
--p
3 
3 


To control the angular velocity, the electromagnetic torque must be regulated. To maximize the
electromagnetic torque, ICs must feed the following phase currents as functions of the angular displace-
ment measuring or observing (sensorless control) θ r
i as = 2i M sin q r + 1 --j i p
3
i bs = 2i M sin q r – 1 ( –
-- 2 j i )p
3
i cs = 2i M sin q r + 1 -- 2 +( j i )p
3
Thus, for ϕ i = 0.3245, one obtains
2
T e = 2PL ∆m i M
That is, T e is maximized and controlled by changing the magnitude of the phase currents i M . Furthermore,
it is no torque ripple (in practice, based upon the experimental results, and performing the high-fidelity
modeling integrating nonlinear electromagnetics using Maxwell’s equations, one finds that there exists
the torque ripple which is due to the cogging torque, eccentricity, bearing, pulse-width-modulation, and
other phenomena).
The majority of ICs are designed to control the phase voltages u as , u bs , and u cs . Therefore, the three-
phase balance voltage set is important. We have

u as = 2u M sin q r + 1
--j i p
3
u bs = 2u M sin q r – 1 -- 2 j i )p
(
–
3
-- 2 +(
u cs = 2u M sin q r + 1 j i )p
3
where u M is the magnitude of the supplied voltages.

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