Direct torque control in an ac machine, which can be implemented in a number of different ways, is commonly referred to as field-oriented control or vector control. To facilitate our discussion of field-oriented control, it is helpful to return to the discussion of Section 5.6.1. Under this viewpoint, which is formalized in Appendix C, stator quantities (flux, current, voltage, etc.) are resolved into components which rotate in synchronism with the rotor. Direct-axis quantities represent those components which are aligned with the field-winding axis, and quadrature-axis components are aligned perpendicular to the field-winding axis.
Section C.2 of Appendix C derives the basic machine relations in dq0 variables for a synchronous machine consisting of a field winding and a three-phase stator winding. The transformed flux-current relationships are found to be
λ=Ldid+Lafifλ=Ldid+Lafif size 12{λ=L rSub { size 8{d} } i rSub { size 8{d} } +L rSub { size 8{ ital "af"} } i rSub { size 8{f} } } {}(10.15)
λq=Lqiqλq=Lqiq size 12{λ rSub { size 8{q} } =L rSub { size 8{q} } i rSub { size 8{q} } } {}(10.16)
λf=32Lafid+Lffifλf=32Lafid+Lffif size 12{λ rSub { size 8{f} } = { {3} over {2} } L rSub { size 8{ ital "af"} } i rSub { size 8{d} } +L rSub { size 8{ ital "ff"} } i rSub { size 8{f} } } {}(10.17)
where the subscripts d, q, and f refer to armature direct-, quadrature-axis, and fieldwinding quantities respectively. Note that throughout this chapter we will assume balanced operating conditions, in which case zero-sequence quantities will be zero and hence will be ignored.
The corresponding transformed voltage equations are
vd=Raid+dλddt−ωmeλqvd=Raid+dλddt−ωmeλq size 12{v rSub { size 8{d} } =R rSub { size 8{a} } i rSub { size 8{d} } + { {dλ rSub { size 8{d} } } over { ital "dt"} } - ω rSub { size 8{ ital "me"} } λ rSub { size 8{q} } } {}(10.18)
vq=Raiq+dλqdt+ωmeλdvq=Raiq+dλqdt+ωmeλd size 12{v rSub { size 8{q} } =R rSub { size 8{a} } i rSub { size 8{q} } + { {dλ rSub { size 8{q} } } over { ital "dt"} } +ω rSub { size 8{ ital "me"} } λ rSub { size 8{d} } } {}(10.19)
vf=Rfif+dλfdtvf=Rfif+dλfdt size 12{v rSub { size 8{f} } =R rSub { size 8{f} } i rSub { size 8{f} } + { {dλ rSub { size 8{f} } } over { ital "dt"} } } {}(10.20)
where
ωme=(poles/2)ωmωme=(poles/2)ωm size 12{ω rSub { size 8{ ital "me"} } = \( ital "poles"/2 \) ω rSub { size 8{m} } } {} is the electrical angular velocity of the rotor.
Finally, the electromagnetic torque acting on the rotor of a synchronous motor is shown to be
Tmech=32poles2(λdiq−λqid)Tmech=32poles2(λdiq−λqid) size 12{T rSub { size 8{ ital "mech"} } = { {3} over {2} } left [ { { ital "poles"} over {2} } right ] \( λ rSub { size 8{d} } i rSub { size 8{q} } - λ rSub { size 8{q} } i rSub { size 8{d} } \) } {}(10.21)
Under steady-state, balanced-three-phase operating conditions,
ωme=ωeωme=ωe size 12{ω rSub { size 8{ ital "me"} } =ω rSub { size 8{e} } } {} where
ωeωe size 12{ω rSub { size 8{e} } } {} is the electrical frequency of the armature voltage and current in rad/sec. Because the armature-produced mmf and flux waves rotate in synchronism with the rotor and hence with respect to the dq reference frame, under these conditions an observer in the dq reference frame will see constant fluxes, and hence one can set d/dt = 0
Letting the subscripts F, D, and Q indicate the corresponding constant values of field-, direct- and quadrature-axis quantities respectively, the flux-current relationships of Eqs. 10.15 through 10.17 then become
λD=LdiD+LafiFλD=LdiD+LafiF size 12{λ rSub { size 8{D} } =L rSub { size 8{d} } i rSub { size 8{D} } +L rSub { size 8{ ital "af"} } i rSub { size 8{F} } } {}(10.22)
λQ=LqiQλQ=LqiQ size 12{λ rSub { size 8{Q} } =L rSub { size 8{q} } i rSub { size 8{Q} } } {}(10.23)
λF=32LafiD+LffiFλF=32LafiD+LffiF size 12{λ rSub { size 8{F} } = { {3} over {2} } L rSub { size 8{ ital "af"} } i rSub { size 8{D} } +L rSub { size 8{ ital "ff"} } i rSub { size 8{F} } } {}(10.24)
Armature resistance is typically quite small, and, if we neglect it, the steady-state voltage equations (Eqs. 10.18 through 10.20) then become
vD=−ωeλQvD=−ωeλQ size 12{v rSub { size 8{D} } = - ω rSub { size 8{e} } λ rSub { size 8{Q} } } {}(10.25)
vQ=ωeλDvQ=ωeλD size 12{v rSub { size 8{Q} } =ω rSub { size 8{e} } λ rSub { size 8{D} } } {}(10.26)
vF=RfiFvF=RfiF size 12{v rSub { size 8{F} } =R rSub { size 8{f} } i rSub { size 8{F} } } {}(10.27)
Finally, we can write Eq. 10.21 as
Tmech=32poles2(λDiQ−λQiD)Tmech=32poles2(λDiQ−λQiD) size 12{T rSub { size 8{ ital "mech"} } = { {3} over {2} } left [ { { ital "poles"} over {2} } right ] \( λ rSub { size 8{D} } i rSub { size 8{Q} } - λ rSub { size 8{Q} } i rSub { size 8{D} } \) } {}(10.28)
From this point on, we will focus our attention on machines in which the effects of saliency can be neglected. In this case, the direct- and quadrature-axis synchronous inductances are equal and we can write
Ld=Lq=LsLd=Lq=Ls size 12{L rSub { size 8{d} } =L rSub { size 8{q} } =L rSub { size 8{s} } } {}(10.29)
where
LsLs size 12{L rSub { size 8{s} } } {} is the synchronous inductance. Substitution into Eqs. 10.22 and 10.23 and then into Eq. 10.28 gives
T
mech
=
3
2
poles
2
[
(
L
s
i
D
+
L
af
i
F
)
i
Q
−
L
s
i
Q
i
D
]
T
mech
=
3
2
poles
2
[
(
L
s
i
D
+
L
af
i
F
)
i
Q
−
L
s
i
Q
i
D
]
size 12{T rSub { size 8{ ital "mech"} } = { {3} over {2} } left [ { { ital "poles"} over {2} } right ] \[ \( L rSub { size 8{s} } i rSub { size 8{D} } +L rSub { size 8{ ital "af"} } i rSub { size 8{F} } \) i rSub { size 8{Q} } - L rSub { size 8{s} } i rSub { size 8{Q} } i rSub { size 8{D} } \] } {}
=32poles2LafiFiQ=32poles2LafiFiQ size 12{ {}= { {3} over {2} } left [ { { ital "poles"} over {2} } right ]L rSub { size 8{ ital "af"} } i rSub { size 8{F} } i rSub { size 8{Q} } } {}(10.30)
Equation 10.30 shows that torque is produced by the interaction of the field flux (proportional to the field current) and the quadrature-axis component of the armature current, in other words the component of armature current that is orthogonal to the field flux. By analogy, we see that the direct-axis component of armature current, which is aligned with the field flux, produces no torque.
This result is fully consistent with the generalized torque expressions which are derived in Chapter 4. Consider for example the equation which expresses the torque in terms of the product of the stator and rotor mmfs (
FsFs size 12{F rSub { size 8{s} } } {} and
FrFr size 12{F rSub { size 8{r} } } {} respectively) and the sine of the angle between them.
T=−poles2μ0πDl2gFsFrsinδsrT=−poles2μ0πDl2gFsFrsinδsr size 12{T= - left [ { { ital "poles"} over {2} } right ] left [ { {μ rSub { size 8{0} } π ital "Dl"} over {2g} } right ]F rSub { size 8{s} } F rSub { size 8{r} } "sin"δ rSub { size 8{ ital "sr"} } } {}(10.31)
where
δrδr size 12{δ rSub { size 8{r} } } {} is the electrical space angle between the stator and rotor mmfs. This shows clearly that no torque will be produced by the direct-axis component of the armature mmf which, by definition, is that component of the stator mmf which is aligned with that of the field winding on the rotor.
Equation 10.31 shows the torque in a nonsalient synchronous motor is proportional to the product of the field current and the quadrature-axis component of the armature current. This is directly analogous to torque production in a dc machine for which the equations can be combined to show that the torque is proportional to the product of the field current and the armature current.
The analogy between a nonsalient synchronous machine and dc machine can be further reinforced. Consider the equation, which expresses the rms value of the line-toneutral generated voltage of a synchronous generator as
Eaf=ωeLafiF2Eaf=ωeLafiF2 size 12{E rSub { size 8{ ital "af"} } = { {ω rSub { size 8{e} } L rSub { size 8{ ital "af"} } i rSub { size 8{F} } } over { sqrt {2} } } } {}(10.32)
Substitution into Eq. 10.30 gives
Tmech=32poles2EafiQωeTmech=32poles2EafiQωe size 12{T rSub { size 8{ ital "mech"} } = { {3} over {2} } left [ { { ital "poles"} over { sqrt {2} } } right ] { {E rSub { size 8{ ital "af"} } i rSub { size 8{Q} } } over {ω rSub { size 8{e} } } } } {}(10.33)
This is directly analogous to Eq.
Tmech=EaIa/ωmTmech=EaIa/ωm size 12{T rSub { size 8{ ital "mech"} } =E rSub { size 8{a} } I rSub { size 8{a} } /ω rSub { size 8{m} } } {} for a dc machine in which the torque is proportional to the product of the generated voltage and the armature current.
The brushes and commutator of a dc machine force the commutated armature current and armature flux along the quadrature axis such that
IdId size 12{I rSub { size 8{d} } } {} = 0 and it is the interaction of this quadrature-axis current with the direct-axis field flux that produces the torque. A field-oriented controller which senses the position of the rotor and controls the quadrature-axis component of armature current produces the same effect in a synchronous machine.
Although the direct-axis component of armature current does not play a role in torque production, it does play a role in determining the resultant stator flux and hence the machine terminal voltage, as can be readily shown. Specifically, from the transformation equations of Appendix C,
va=vDcos(ωet)−vQsin(ωet)va=vDcos(ωet)−vQsin(ωet) size 12{v rSub { size 8{a} } =v rSub { size 8{D} } "cos" \( ω rSub { size 8{e} } t \) - v rSub { size 8{Q} } "sin" \( ω rSub { size 8{e} } t \) } {}(10.34)
And thus the rms amplitude of the armature voltage is equal to
V
a
=
v
D
2
+
v
Q
2
2
=
ω
e
λ
D
2
+
λ
Q
2
2
V
a
=
v
D
2
+
v
Q
2
2
=
ω
e
λ
D
2
+
λ
Q
2
2
size 12{V rSub { size 8{a} } = sqrt { { {v rSub { size 8{D} } rSup { size 8{2} } +v rSub { size 8{Q} } rSup { size 8{2} } } over {2} } } =ω rSub { size 8{e} } sqrt { { {λ rSub { size 8{D} } rSup { size 8{2} } +λ rSub { size 8{Q} } rSup { size 8{2} } } over {2} } } } {}
=ωe(LsiD+LafiF)2+(LsiQ)22=ωe(LsiD+LafiF)2+(LsiQ)22 size 12{ {}=ω rSub { size 8{e} } sqrt { { { \( L rSub { size 8{s} } i rSub { size 8{D} } +L rSub { size 8{ ital "af"} } i rSub { size 8{F} } \) rSup { size 8{2} } + \( L rSub { size 8{s} } i rSub { size 8{Q} } \) rSup { size 8{2} } } over {2} } } } {}(10.35)
Dividing
VaVa size 12{V rSub { size 8{a} } } {} by the electrical frequency
ωeωe size 12{ω rSub { size 8{e} } } {}, we get an expression for rms armature flux linkages
(λa)rms=λD2+λQ22=(LsiD+LafiF)2+(LsiQ)22(λa)rms=λD2+λQ22=(LsiD+LafiF)2+(LsiQ)22 size 12{ \( λ rSub { size 8{a} } \) rSub { size 8{ ital "rms"} } = sqrt { { {λ rSub { size 8{D} } rSup { size 8{2} } +λ rSub { size 8{Q} } rSup { size 8{2} } } over {2} } } = sqrt { { { \( L rSub { size 8{s} } i rSub { size 8{D} } +L rSub { size 8{ ital "af"} } i rSub { size 8{F} } \) rSup { size 8{2} } + \( L rSub { size 8{s} } i rSub { size 8{Q} } \) rSup { size 8{2} } } over {2} } } } {}(10.36)
Similarly, the transformation equations of Appendix C can be used to show that the rms amplitude of the armature current is equal to
Ia=iD2+iQ22Ia=iD2+iQ22 size 12{I rSub { size 8{a} } = sqrt { { {i rSub { size 8{D} } rSup { size 8{2} } +i rSub { size 8{Q} } rSup { size 8{2} } } over {2} } } } {}(10.37)
From Eq.10.33 we see that torque is controlled by the product
iFiQiFiQ size 12{i rSub { size 8{F} } i rSub { size 8{Q} } } {} of the field current and thee quadrature-axis component of the armature current. Thus, simply specifying a desired torque is not sufficient to uniquely determine either
iFiF size 12{i rSub { size 8{F} } } {} or
iQiQ size 12{i rSub { size 8{Q} } } {}.
In fact, under the field-oriented-control viewpoint presented here, there are actually three independent variables,
iF,iQiF,iQ size 12{i rSub { size 8{F} } ,i rSub { size 8{Q} } } {} and
iDiD size 12{i rSub { size 8{D} } } {}, and , in general, three constraints will be required to uniquely determine them. In addition to specifying the desired torque, a typical controller will implement additional contraints flux-linkages and current using the basic relationships found in Eqs.10.36 and 10.37.
Figure 10.12a shows a typical field-oriented torque-control system in block-diagram form. The torque-controller block has two inputs,
TrefTref size 12{T rSub { size 8{ ital "ref"} } } {}, the reference value or set point for torque and
(iF)ref(iF)ref size 12{ \( i rSub { size 8{F} } \) rSub { size 8{ ital "ref"} } } {}, the reference value or set point for the field current, which is also supplied to the power supply which supplies the current
iFiF size 12{i rSub { size 8{F} } } {} to the motor field winding.
(iF)ref(iF)ref size 12{ \( i rSub { size 8{F} } \) rSub { size 8{ ital "ref"} } } {} is determined by an auxiliary controller which also determines
Figure 10.12 (a) Block diagram of a field-oriented torque-control system for
a synchronous motor. (b) Block diagram of a synchronous-motor speed-control
loop built around a field-oriented torque control system.
the reference value
(iD)ref(iD)ref size 12{ \( i rSub { size 8{D} } \) rSub { size 8{ ital "ref"} } } {} of the direct-axis current, based upon desired values for the armature current and voltage. The torque controller calculates the desired quadrature axis current from Eq. 10.30 based upon
TrefTref size 12{T rSub { size 8{ ital "ref"} } } {} and
(iF)ref(iF)ref size 12{ \( i rSub { size 8{F} } \) rSub { size 8{ ital "ref"} } } {}
(iQ)ref=232polesTrefLaf(iF)ref(iQ)ref=232polesTrefLaf(iF)ref size 12{ \( i rSub { size 8{Q} } \) rSub { size 8{ ital "ref"} } = { {2} over {3} } left [ { {2} over { ital "poles"} } right ] { {T rSub { size 8{ ital "ref"} } } over {L rSub { size 8{ ital "af"} } \( i rSub { size 8{F} } \) rSub { size 8{ ital "ref"} } } } } {}(10.38)
Note that a position sensor is required to determine the angular position of the rotor in order to implement the dq0 to abc transformation.
In a typical application, the ultimate control objective is not to control motor torque but to control speed or position. Figure 10.12a shows how the torque-control system of Fig. 10.12b can be used as a component of a speed-control loop, with speed feedback forming an outer control loop around the inner torque-control loop.
As we have discussed, a practical field-oriented control must determine values for all three currents
iFiF size 12{i rSub { size 8{F} } } {},
iDiD size 12{i rSub { size 8{D} } } {}, and
iQiQ size 12{i rSub { size 8{Q} } } {}. In Example 10.8 two of these values were chosen relatively arbitrarily (
iFiF size 12{i rSub { size 8{F} } } {} = 2.84 and
iDiD size 12{i rSub { size 8{D} } } {} = 0) and the result was a control that achieved the desired torque but with a terminal voltage 30 percent in excess of the motorrated voltage. In a practical system, additional constraints are required to achieve an acceptable control algorithm.
One such algorithm would be to require that the motor operate at rated flux and at unity terminal power factor. Such an algorithm canbe derived with reference to the
Figure 10.13 Phasor diagram for unity-powerfactor
field-oriented-control algorithm
phasor diagram of Fig 10.13 and can be implemented using the following steps:
Step 1. Calculate the line-to-neutral armature voltage corresponding to rated flux as
Va=(Va)ratedωm(ωm)ratedVa=(Va)ratedωm(ωm)rated size 12{V rSub { size 8{a} } = \( V rSub { size 8{a} } \) rSub { size 8{ ital "rated"} } left [ { {ω rSub { size 8{m} } } over { \( ω rSub { size 8{m} } \) rSub { size 8{ ital "rated"} } } } right ]} {}(10.39)
where
***SORRY, THIS MEDIA TYPE IS NOT SUPPORTED.***
is the rated line-to-neutral armature voltage at rated motor speed,
ωmωm size 12{ω rSub { size 8{m} } } {} is the desired motor speed, and
(ωm)rated(ωm)rated size 12{ \( ω rSub { size 8{m} } \) rSub { size 8{ ital "rated"} } } {} is its rated speed.
Step 2. Calculate the rms armature current from the desired torque
TrefTref size 12{T rSub { size 8{ ital "ref"} } } {} as
Ia=Pref3Va=Trefωm3VaIa=Pref3Va=Trefωm3Va size 12{I rSub { size 8{a} } = { {P rSub { size 8{ ital "ref"} } } over {3V rSub { size 8{a} } } } = { {T rSub { size 8{ ital "ref"} } ω rSub { size 8{m} } } over {3V rSub { size 8{a} } } } } {}(10.40)
where
PrefPref size 12{P rSub { size 8{ ital "ref"} } } {} is the mechanical power corresponding to the desired torque.
Step 3. Calculate the angle
δδ size 12{δ} {} based upon the phasor diagram of Fig 11.14
δ=−tan−1ωeLsIaVaδ=−tan−1ωeLsIaVa size 12{δ= - "tan" rSup { size 8{ - 1} } left [ { {ω rSub { size 8{e} } L rSub { size 8{s} } I rSub { size 8{a} } } over {V rSub { size 8{a} } } } right ]} {}(10.41)
where
ωe=ωme=(poles/2)ωmωe=ωme=(poles/2)ωm size 12{ω rSub { size 8{e} } =ω rSub { size 8{ ital "me"} } = \( ital "poles"/2 \) ω rSub { size 8{m} } } {} is the electrical frequency corresponding to the desired motor speed.
Step 4. Calculate
(iQ)ref(iQ)ref size 12{ \( i rSub { size 8{Q} } \) rSub { size 8{ ital "ref"} } } {} and
(iD)ref(iD)ref size 12{ \( i rSub { size 8{D} } \) rSub { size 8{ ital "ref"} } } {}
(iQ)ref=2Iacosδ(iQ)ref=2Iacosδ size 12{ \( i rSub { size 8{Q} } \) rSub { size 8{ ital "ref"} } = sqrt {2} I rSub { size 8{a} } "cos"δ} {}(10.42)
(iD)ref=2Iasinδ(iD)ref=2Iasinδ size 12{ \( i rSub { size 8{D} } \) rSub { size 8{ ital "ref"} } = sqrt {2} I rSub { size 8{a} } "sin"δ} {}(10.43)
Step 5. Calculate
(iF)ref(iF)ref size 12{ \( i rSub { size 8{F} } \) rSub { size 8{ ital "ref"} } } {}
(iF)ref=232polesTrefLaf(iQ)ref(iF)ref=232polesTrefLaf(iQ)ref size 12{ \( i rSub { size 8{F} } \) rSub { size 8{ ital "ref"} } = { {2} over {3} } left [ { {2} over { ital "poles"} } right ] { {T rSub { size 8{ ital "ref"} } } over {L rSub { size 8{ ital "af"} } \( i rSub { size 8{Q} } \) rSub { size 8{ ital "ref"} } } } } {}(10.44)
The discussion of this section has focused upon synchronous machines with field windings and the corresponding capability to control the field excitation. The basic concept, of course, also applies to synchronous machines with permanent magnets on the rotor. However, in the case of permanent-magnet synchronous machines, the effective field excitation is fixed and, as a result, there is one less degree of freedom for the field-oriented control algorithm.
For a permanent-magnet synchronous machine, since the effective equivalent field current is fixed by the permanent magnet, the quadrature-axis current is determined directly by the desired torque. Consider a three-phase permanent-magnet motor whose rated rms, line-to-neutral open-circuit voltage is
(Eaf)rated(Eaf)rated size 12{ \( E rSub { size 8{ ital "af"} } \) rSub { size 8{ ital "rated"} } } {} at electrical frequency
(ωe)rated(ωe)rated size 12{ \( ω rSub { size 8{e} } \) rSub { size 8{ ital "rated"} } } {}. From Eq. 10.32 we see that the equivalent
LafIfLafIf size 12{L rSub { size 8{ ital "af"} } I rSub { size 8{f} } } {} product for this motor, which we will refer to by the symbol
ΛPMΛPM size 12{Λ rSub { size 8{ ital "PM"} } } {}, is
ΛPM=2(Eaf)rated(ωe)ratedΛPM=2(Eaf)rated(ωe)rated size 12{Λ rSub { size 8{ ital "PM"} } = { { sqrt {2} \( E rSub { size 8{ ital "af"} } \) rSub { size 8{ ital "rated"} } } over { \( ω rSub { size 8{e} } \) rSub { size 8{ ital "rated"} } } } } {} (10.45)
Thus, the direct-axis flux-current relationship for this motor, corresponding to Eq. 10.22, becomes
λD=LdiD+ΛPMλD=LdiD+ΛPM size 12{λ rSub { size 8{D} } =L rSub { size 8{d} } i rSub { size 8{D} } +Λ rSub { size 8{ ital "PM"} } } {}(10.46)
and the torque expression becomes
Tmech=32poles2ΛPMiQTmech=32poles2ΛPMiQ size 12{T rSub { size 8{ ital "mech"} } = { {3} over {2} } left [ { { ital "poles"} over {2} } right ]Λ rSub { size 8{ ital "PM"} } i rSub { size 8{Q} } } {}(10.47)
From Eq. 10.47 we see that, for a permanent-magnet sychronous machine under field-oriented control, the quadrature-axis current is uniquely determined by the desired torque and Eq. 10.38 becomes
(iQ)ref=232polesTrefΛPM(iQ)ref=232polesTrefΛPM size 12{ \( i rSub { size 8{Q} } \) rSub { size 8{ ital "ref"} } = { {2} over {3} } left [ { {2} over { ital "poles"} } right ] { {T rSub { size 8{ ital "ref"} } } over {Λ rSub { size 8{ ital "PM"} } } } } {}(10.48)
Once
(iQ)ref(iQ)ref size 12{ \( i rSub { size 8{Q} } \) rSub { size 8{ ital "ref"} } } {} has been specified, the only remaining control choice remains to determine the desired value for the direct-axis current,
(iD)ref(iD)ref size 12{ \( i rSub { size 8{D} } \) rSub { size 8{ ital "ref"} } } {}.One possibility is simply to set
(iD)ref=0(iD)ref=0 size 12{ \( i rSub { size 8{D} } \) rSub { size 8{ ital "ref"} } =0} {}.
Figure 10.14 Block diagram of a field-oriented torque-control system
for a permanent-magnet synchronous motor.
This will clearly result in the lowest possible armature current for a given torque. However, as we have seen in Example 10.8, this is likely to result in terminal voltages in excess of the rated voltage of the machine. As a result, it is common to supply direct-axis current so as to reduce the direct-axis flux linkage of Eq. 10.22, which will in turn result in reduced terminal voltage. This technique is commonly referred to as flux weakening and comes at the expense of increased armature current. In practice, the chosen operating point is determined by a trade-off between reducing the armature voltage and an increase in armature current. Figure 10.14 shows the block diagram for a field-oriented-control system for use with a permanent-magnet motor.
