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In a practical implementation of the technique which we have derived, the directand quadrature-axis currents i D and i Q size 12{i rSub { size 8{D} } " and "i rSub { size 8{Q} } } {} must be transformed into the three motor phase currents i a ( t ) , i b ( t ) , and i c ( t ) size 12{i rSub { size 8{a} } \( t \) ,i rSub { size 8{b} } \( t \) ,"and "i rSub { size 8{c} } \( t \) } {} . This can be done using the inverse dq0 transformation of Eq. C.48 which requires knowledge of θ S size 12{θ rSub { size 8{S} } } {} , the electrical angle between the axis of phase a, and the direct-axis of the synchronously rotating reference frame.

Since it is not possible to measure the axis of the rotor flux directly, it is necessary to calculate θ S size 12{θ rSub { size 8{S} } } {} , where θ S = ω e t + θ 0 size 12{θ rSub { size 8{S} } =ω rSub { size 8{e} } t+θ rSub { size 8{0} } } {} as given by Eq. C.46. Solving Eq. 10.70 for ω e size 12{ω rSub { size 8{e} } } {} gives ω e = ω me R aR i QR λ DR size 12{ω rSub { size 8{e} } =ω rSub { size 8{ ital "me"} } - R rSub { size 8{ ital "aR"} } left [ { {i rSub { size 8{ ital "QR"} } } over {λ rSub { size 8{ ital "DR"} } } } right ]} {} (10.79)

From Eq. 10.63 with λ QR size 12{λ rSub { size 8{ ital "QR"} } } {} = 0 we see that

i QR = L m L R i Q size 12{i rSub { size 8{ ital "QR"} } = - left [ { {L rSub { size 8{m} } } over {L rSub { size 8{R} } } } right ]i rSub { size 8{Q} } } {} (10.80)

Eq. 10.80 in combination with Eq. 10.77 then gives

ω e = ω me + R aR L R i Q i D = ω me + 1 τ R i Q i D size 12{ω rSub { size 8{e} } =ω rSub { size 8{ ital "me"} } + { {R rSub { size 8{ ital "aR"} } } over {L rSub { size 8{R} } } } left [ { {i rSub { size 8{Q} } } over {i rSub { size 8{D} } } } right ]=ω rSub { size 8{ ital "me"} } + { {1} over {τ rSub { size 8{R} } } } left [ { {i rSub { size 8{Q} } } over {i rSub { size 8{D} } } } right ]} {} (10.81)

where τ R = L R / R aR size 12{τ rSub { size 8{R} } =L rSub { size 8{R} } /R rSub { size 8{ ital "aR"} } } {} is the rotor time constant. We can now integrate Eq. 10.81 to find

θ ˆ S = ω me + 1 τ R i Q i D t + θ 0 size 12{ { hat {θ}} rSub { size 8{S} } = left [ω rSub { size 8{ ital "me"} } + { {1} over {τ rSub { size 8{R} } } } left [ { {i rSub { size 8{Q} } } over {i rSub { size 8{D} } } } right ] right ]t+θ rSub { size 8{0} } } {} (11.82)

where θ ˆ S size 12{ { hat {θ}} rSub { size 8{S} } } {} indicates the calculated value of θ S size 12{θ rSub { size 8{S} } } {} (often referred to as the estimated value of θ S size 12{θ rSub { size 8{S} } } {} ). In the more general dynamic sense

θ ˆ S = o t ω me + 1 τ R i Q i D d t ' + θ 0 size 12{ { hat {θ}} rSub { size 8{S} } = Int rSub { size 8{o} } rSup { size 8{t} } { left [ω rSub { size 8{ ital "me"} } + { {1} over {τ rSub { size 8{R} } } } left [ { {i rSub { size 8{Q} } } over {i rSub { size 8{D} } } } right ] right ]} d { {t}} sup { ' }+θ rSub { size 8{0} } } {} (10.83)

Note that both Eqs. 10.82 and 10.83 require knowledge of θ 0 size 12{θ rSub { size 8{0} } } {} , the value of θ S size 12{θ rSub { size 8{S} } } {} at t = 0. Although we will not prove it here, it turns out that in a practical implementation, the effects of an error in this initial angle decay to zero with time, and hence it can be set to zero without any loss of generality.

Figure 10.20 a shows a block diagram of a field-oriented torque-control system for an induction machine. The block labeled "Estimator" represents the calculation of Eq. 10.83 which calculates the estimate of θ S size 12{θ rSub { size 8{S} } } {} required by the transformation from dq0 to abc variables.

Note that a speed sensor is required to provide the rotor speed measurement required by the estimator. Also notice that the estimator requires knowledge of the rotor time constant τ R = L R / R aR size 12{τ rSub { size 8{R} } =L rSub { size 8{R} } /R rSub { size 8{ ital "aR"} } } {} . In general, this will not be known exactly, both due to uncertainty in the machine parameters as well as due to the fact that the rotor

Figure 10.20 (a) Block diagram of a field-oriented torque-control system

for an induction motor. (b) Block diagram of an induction-motor speed-control

loop built around a field-oriented torque control system.

resistance R aR size 12{R rSub { size 8{ ital "aR"} } } {} will undoubtedly change with temperature as the motor is operated. It can be shown that errors in τ R size 12{τ rSub { size 8{R} } } {} result in an offset in the estimate of θ S size 12{θ rSub { size 8{S} } } {} , which in turn will result in an error in the estimate for the position of the rotor flux with the result that the applied armature currents will not be exactly aligned with the direct- and quadrature-axes. The torque controller will still work basically as expected, although there will be corresponding errors in the torque and rotor flux.

As with the synchronous motor, the rms armature flux-linkages can be found from Eq. 10.36 as

( λ a ) rms = λ D 2 + λ Q 2 2 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} } } } {} (10.84)

Combining Eqs. 10.61 and 10.80 gives

λ Q = L S i Q + L m i QR = L S L m 2 L R i Q size 12{λ rSub { size 8{Q} } =L rSub { size 8{S} } i rSub { size 8{Q} } +L rSub { size 8{m} } i rSub { size 8{ ital "QR"} } = left [L rSub { size 8{S} } - { {L rSub { size 8{m} } rSup { size 8{2} } } over {L rSub { size 8{R} } } } right ]i rSub { size 8{Q} } } {} (10.85)

Substituting Eqs. 10.78 and 10.85 into Eq. 11.84 gives

( λ a ) rms = L S 2 i D 2 + L s L m 2 L R 2 i Q 2 2 size 12{ \( λ rSub { size 8{a} } \) rSub { size 8{ ital "rms"} } = sqrt { { {L rSub { size 8{S} } rSup { size 8{2} } i rSub { size 8{D} } rSup { size 8{2} } + left [L rSub { size 8{s} } - { {L rSub { size 8{m} } rSup { size 8{2} } } over {L rSub { size 8{R} } } } right ] rSup { size 8{2} } i rSub { size 8{Q} } rSup { size 8{2} } } over {2} } } } {} (10.86)

Finally, as discussed in the footnote to Eq. 10.35, the rms line-to-neutral armature voltage can be found as

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Source:  OpenStax, Electrical machines. OpenStax CNX. Jul 29, 2009 Download for free at http://cnx.org/content/col10767/1.1
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