Page 306 - The Mechatronics Handbook

0066_frame_C14.fm Page 20 Wednesday, January 9, 2002 1:49 PM

These four nonlinear differential equations are rewritten in the state-space form as

di as
-------- r s RTψ m ( RTθ rm )
dt – ----- 0 0 0 ---------------ω rm sin
L ss L ss
i as
di bs
-------- r s RTψ m
dt = 0 – ----- 0 0 i bs + – ---------------ω rm cos ( RTθ rm )
dω L ss L ss
rm
-------------- B m ω rm RTψ m
dt 0 0 – ------ 0 – --------------- i as sin[ ( RTθ rm ) i bs cos ( RTθ rm )]
–
dθ J θ rm J
rm
------------- 0 0 1 0 0
dt
1
----- 0 0
L ss
0
1
+ 0 ----- u as –
L ss 1 -- T L
u bs J
0 0
0
0 0
The analysis of the torque equation
T e = – RTψ m i as sin ( RTθ rm ) i bs cos ( RTθ rm )]
[
–
guides one to the conclusion that the expressions for a balanced two-phase current sinusoidal set is
i as = – 2i M sin ( RTθ rm ) and i bs = 2i M cos ( RTθ rm )

If these phase currents are fed, the electromagnetic torque is a function of the current magnitude i M , and

T e = 2RTψ m i M

The phase currents needed to be fed are the functions of the rotor angular displacement. Assuming
that the inductances are negligibly small, we have the following phase voltages needed to be supplied:

u as = – 2u M sin ( RTθ rm ) and u bs = 2u M cos ( RTθ rm )

Example 14.5.4: Mathematical Model of Two-Phase Permanent-Magnet
Synchronous Micromotors

Consider two-phase permanent-magnet synchronous micromotors. Using Kirchhoff’s voltage law, we have

u as = r s i as + dψ as
----------
dt
u bs = r s i bs + dψ bs
----------
dt
where the flux linkages are expressed as ψ as = L asas i as + L asbs i bs + ψ asm and ψ bs = L bsas i as + L bsbs i bs + ψ bsm .
The flux linkages are periodic functions of the angular displacement (rotor position), and let

ψ asm = ψ m sin θ rm and ψ bsm = – ψ m cos θ rm

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