80 Power Circuit Components

where θr is the rotor angle.

The developed torque can be expressed as:

The mechanical equations are:

2.8.7 Permanent Magnet Synchronous MachineA 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 thebrushless dc machine is that the machine back emf is sinusoidal.

The image and parameters of the machine are shown as follows.

Image:

Attributes:

Parameters Description

Rs (stator resistance) Stator winding resistance, in Ohm

Ld (d-axis ind.) Stator d-axis inductance, in H

T P2--- I

ddθr-------- L I⋅ ⋅ ⋅=

Jdωm

dt----------⋅ Tem Tload–=

dθr

dt-------- P

2--- ωm⋅=

a

b

c

Shaft Node

n

Motor Drive Module 81

The node assignments of the image are: Nodes a, b, and c are the stator windingterminals for Phase a, b, and c, respectively. The stator windings are Y connected, andNode n is the neutral point. The shaft node is the connecting terminal for the mechanicalshaft. 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:

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 linkagesare further defined as:

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 d-axis.

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)

va

vb

vc

Rs 0 00 Rs 00 0 Rs

ia

ib

ic

ddt-----

λa

λb

λc

+⋅=

82 Power Circuit Components

where θr is the rotor electrical angle, and λpm is a coefficient which is defined as:

where P is the number of poles.

The stator self and mutual inductances are rotor position dependent, and are defined as:

where Ls is the stator leakage inductance. The d-axis and q-axis inductances areassociated with the above inductances as follows:

The developed torque can be expressed as:

λa

λb

λc

Laa Lab Lac

Lba Lbb Lbc

Lca Lcb Lcc

ia

ib

ic

λpm

θr( )cos

θr2π3

------–⎝ ⎠⎛ ⎞cos

θr2π3

------+⎝ ⎠⎛ ⎞cos

⋅+⋅=

λpm60 Vpk krpm⁄⋅

3 π P 1000⋅ ⋅ ⋅---------------------------------------=

Laa Ls Lo L2 2θr( )cos⋅+ +=

Lbb Ls Lo L2 2θr2π3

------+⎝ ⎠⎛ ⎞cos⋅+ +=

Lcc Ls Lo L2 2θr2π3

------–⎝ ⎠⎛ ⎞cos⋅+ +=

Lab LbaLo

2-----– L2 2θr

2π3

------–⎝ ⎠⎛ ⎞cos⋅+= =

Lac LcaLo

2-----– L2 2θr

2π3

------+⎝ ⎠⎛ ⎞cos⋅+= =

Lbc LcbLo

2-----– L2 2θr( )cos⋅+= =

Ld Ls32---Lo

32---L2+ +=

Lq Ls32---Lo

32---L2–+=

Motor Drive Module 83

The mechanical equations are:

where B is a coefficient, Tload is the load torque, and P is the no. of poles. Thecoefficient B is calculated from the moment of inertia J and the mechanical timeconstant τmech as below:

2.8.8 Permanent Magnet Synchronous Machine with SaturationA 3-phase PMSM machine with saturation differs from that of a linear 3-phase PMSMmachine in that the d-axis and q-axis magnetizing inductances Ldm and Lqm can beexpressed as a nonlinear function of the d-axis and q-axis currents in the lookup tableform.

The image and parameters of the machine are shown as follows.

Image:

TemP2--- L2 ia ib ic

2θr( )sin 2θr2π3

------–⎝ ⎠⎛ ⎞sin 2θr

2π3

------+⎝ ⎠⎛ ⎞sin

2θr2π3

------–⎝ ⎠⎛ ⎞sin 2θr

2π3

------+⎝ ⎠⎛ ⎞sin 2θr( )sin

2θr2π3

------+⎝ ⎠⎛ ⎞sin 2θr( )sin 2θr

2π3

------–⎝ ⎠⎛ ⎞sin

ia

ib

ic

⋅ ⋅ ⋅ –⋅=

P2--- λpm ia ib ic

θr( )sin

θr2π3

------–⎝ ⎠⎛ ⎞sin

θr2π3

------+⎝ ⎠⎛ ⎞sin

⋅ ⋅⋅=

Jdωm

dt----------⋅ Tem B ωm Tload–⋅–=

dθr

dt-------- P

2--- ωm⋅=

B Jτmech------------=