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Direct Vector Control of Induction Machines · Part 3 of 4

Nominal values & open-loop flux estimator

Deriving nominal torque and flux, then estimating the rotor flux vector and its angle ρ.

01

Nominal values

From the nameplate ratings we derive the operating point used later to scale the controllers.

1. Synchronous speed

ωs=2πfn=314.16 rad/s  ⇒  Ωs=ωsnp=157.08 rad/s\omega_s = 2\pi f_n = 314.16\ \text{rad/s} \;\Rightarrow\; \Omega_s = \frac{\omega_s}{n_p} = 157.08\ \text{rad/s}

2. Nominal electromagnetic torque

Cem,n≈PnΩs=1100157.08≈7 N0˘0B7mC_{em,n} \approx \frac{P_n}{\Omega_s} = \frac{1100}{157.08} \approx \mathbf{7\ \text{N\u00B7m}}

3–4. Nominal stator flux / Nominal rotor flux

Φsn=Vsn2ωs≈0.9903 Wb,Φrn=MLsΦsn≈0.931 Wb\Phi_{sn} = \frac{V_{sn}\sqrt{2}}{\omega_s} \approx 0.9903\ \text{Wb}, \qquad \Phi_{rn} = \frac{M}{L_s}\Phi_{sn} \approx \mathbf{0.931\ \text{Wb}}

02

Estimator equations

The rotor flux cannot be measured. The open-loop (current-model) estimator reconstructs it from the measured stator currents and rotor speed, with electrical rotor speed ωr = np Ω:

(9)
dΦrαdt=M isα−ΦrαTr−ωrΦrβ,dΦrβdt=M isβ−ΦrβTr+ωrΦrα\frac{d\Phi_{r\alpha}}{dt} = \frac{M\, i_{s\alpha} - \Phi_{r\alpha}}{T_r} - \omega_r \Phi_{r\beta}, \qquad \frac{d\Phi_{r\beta}}{dt} = \frac{M\, i_{s\beta} - \Phi_{r\beta}}{T_r} + \omega_r \Phi_{r\alpha}

Tr=Lr/Rr≈0.113 sT_r = L_r/R_r \approx 0.113\ \text{s}. Its magnitude and position follow directly:

(10)
Φr=Φrα2+Φrβ2,ρ=atan2⁡(Φrβ,Φrα)\Phi_r = \sqrt{\Phi_{r\alpha}^2 + \Phi_{r\beta}^2}, \qquad \rho = \operatorname{atan2}(\Phi_{r\beta}, \Phi_{r\alpha})
Because this estimator runs open-loop, its accuracy depends on Tr = Lr/Rr. Rotor resistance rises with temperature, so a mismatch slightly mis-orients the d axis.

03

Live estimator

Set the current references and rotor speed. The stator currents follow from the steady-state slip, and the estimator starts from zero flux: the vector spirals out to its steady value M·isd with time constant Tr.

ΦrαΦrβ
0t (s)|Φr| (Wb)
0t (s)ρ (rad)

Slip pulsation: 7.1 rad/s · Steady flux: 0.907 Wb

Left: estimated flux trajectory in the αβ plane. Right: magnitude |Φr| and angle ρ over time.