where θr is the rotor angle. The developed torque can be expressed as: d P T = --- ⋅ I ⋅ -------- L ⋅ I dθ r 2 The mech
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where θr is the rotor angle. The developed torque can be expressed as: d P T = --- ⋅ I ⋅ -------- L ⋅ I dθ r 2
The mechanical equations are: dω m J ⋅ ---------- = T em – T load dt dθ r P -------- = --- ⋅ ω m dt 2
2.8.7 Permanent Magnet Synchronous Machine A 3-phase permanent magnet synchronous machine has 3-phase windings on the stator, and permanent magnet on the rotor. The difference between this machine and the brushless dc machine is that the machine back emf is sinusoidal. The image and parameters of the machine are shown as follows. Image: a b
Shaft Node
c n
Attributes:
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Parameters
Description
Rs (stator resistance)
Stator winding resistance, in Ohm
Ld (d-axis ind.)
Stator d-axis inductance, in H
Power Circuit Components
Lq (q-axis ind.)
Stator q-axis inductance, in H. The d-q coordinate is defined such that the d-axis passes through the center of the magnet, and the q-axis is in the middle between two magnets. The q-axis is leading the daxis.
Vpk / krpm
Peak line-to-line back emf constant, in V/krpm (mechanical speed). The value of Vpk/krpm should be available from the machine data sheet. If this data is not available, it can be obtained through an experiment by operating the machine as a generator at 1000 rpm and measuring the peak line-to-line voltage.
No. of Poles P
Number of poles P
Moment of Inertia
Moment of inertia J of the machine, in kg*m2
Mech. Time Constant
Mechanical time constant τmech
Torque Flag
Output flag for internal developed torque Tem
Master/slave Flag
Master/slave flag of the machine (1: master; 0: slave)
The node assignments of the image are: Nodes a, b, and c are the stator winding terminals for Phase a, b, and c, respectively. The stator windings are Y connected, and Node n is the neutral point. The shaft node is the connecting terminal for the mechanical shaft. They are all power nodes and should be connected to the power circuit. For more details on the definition and use of the master/slave flag, refer to Section 2.8.1. The equations of the permanent-magnet synchronous machine are: va vb = vc
λa d 0 R s 0 ⋅ i b + ----- λ b dt ic λc 0 0 Rs
Rs 0 0
ia
where va, vb, vc, and ia, ib, and ic, and λa, λb, λc are the stator phase voltages, currents, and flux linkages, respectively, and Rs is the stator phase resistance. The flux linkages are further defined as:
Motor Drive Module
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λa λb λc
cos ( θ r ) L aa L ab L ac ia 2π⎞ ⎛ -----= L ba L bb L bc ⋅ i b + λ pm ⋅ cos ⎝ θ r – 3 ⎠ L ca L cb L cc
ic
2π cos ⎛ θ r + ------⎞ ⎝ 3⎠
where θr is the rotor electrical angle, and λpm is a coefficient which is defined as: 60 ⋅ V pk ⁄ krpm λ pm = -------------------------------------3 ⋅ π ⋅ P ⋅ 1000
where P is the number of poles. The stator self and mutual inductances are rotor position dependent, and are defined as: L aa = L s + L o + L 2 ⋅ cos ( 2θ r ) 2π L bb = L s + L o + L 2 ⋅ cos ⎛ 2θ r + ------⎞ ⎝ 3⎠ 2π L cc = L s + L o + L 2 ⋅ cos ⎛ 2θ r – ------⎞ ⎝ 3⎠ Lo 2π L ab = L ba = – ----- + L 2 ⋅ cos ⎛ 2θ r – ------⎞ ⎝ 3⎠ 2 Lo 2π L ac = L ca = – ----- + L 2 ⋅ cos ⎛ 2θ r + ------⎞ ⎝ 3⎠ 2 Lo L bc = L cb = – ----- + L 2 ⋅ cos ( 2θ r ) 2
where Ls is the stator leakage inductance. The d-axis and q-axis inductances are associated with the above inductances as follows: 3 3 L d = L s + --- L o + --- L 2 2 2 3 3 L q = L s + --- L o – --- L 2 2 2
The developed torque can be expressed as:
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Power Circuit Components
sin ( 2θ r ) T em
2π 2π sin ⎛ 2θ r – ------⎞ sin ⎛ 2θ r + ------⎞ ⎝ ⎝ 3⎠ 3⎠
P 2π = --- ⋅ L 2 ⋅ i a i b i c ⋅ sin ⎛ 2θ r – 2π ------⎞ sin ⎛ 2θ r + ------⎞ 2 ⎝ ⎝ 3⎠ 3⎠ 2π sin ⎛ 2θ r + ------⎞ ⎝ 3⎠
sin ( 2θ r )
sin ( 2θ r ) 2π sin ⎛ 2θ r – ------⎞ ⎝ 3⎠
ia ⋅ ib – ic
sin ( θ r ) 2π⎞ ⎛ P -----= --- ⋅ λ pm ⋅ i a i b i c ⋅ sin ⎝ θ r – 3 ⎠ 2 2π sin ⎛ θ r + ------⎞ ⎝ 3⎠
The mechanical equations are: dω J ⋅ ---------m- = T em – B ⋅ ω m – T load dt dθ r P -------- = --- ⋅ ω m dt 2
where B is a coefficient, Tload is the load torque, and P is the no. of poles. The coefficient B is calculated from the moment of inertia J and the mechanical time constant τmech as below: J B = -----------τ mech
2.8.8 Permanent Magnet Synchronous Machine with Saturation A 3-phase PMSM machine with saturation differs from that of a linear 3-phase PMSM machine in that the d-axis and q-axis magnetizing inductances Ldm and Lqm can be expressed as a nonlinear function of the d-axis and q-axis currents in the lookup table form. The image and parameters of the machine are shown as follows. Image:
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