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Direct Vector Control of Induction Machines · Part 4 of 4

PI controller workbench

Tuning the current, flux and speed loops interactively, and comparing RFOC with V/f control.

01

Current loops

The current loops see the stator as an equivalent RL circuit. Placing the PI zero on the plant pole (pole-zero cancellation) leaves a first-order closed loop with bandwidth ωc.

Gi(s)=1Req+σLss,σLs≈0.04463 H,Req=Rs+Rr(MLr)2≈8.818 ΩG_i(s) = \frac{1}{R_{eq} + \sigma L_s s}, \quad \sigma L_s \approx 0.04463\ \text{H}, \quad R_{eq} = R_s + R_r\left(\tfrac{M}{L_r}\right)^2 \approx 8.818\ \Omega
Kp,i=ωc,i σLs≈420.6 Ω,Ti=σLsReq≈5.06 ms,Ki,i=Kp,iTi≈83,110 Ω/sK_{p,i} = \omega_{c,i}\,\sigma L_s \approx \mathbf{420.6\ \Omega}, \qquad T_i = \frac{\sigma L_s}{R_{eq}} \approx 5.06\ \text{ms}, \qquad K_{i,i} = \frac{K_{p,i}}{T_i} \approx \mathbf{83,110\ \Omega/\text{s}}

02

Rotor flux loop

The flux plant is a first-order lag. Identifying the closed-loop characteristic polynomial with a standard second-order system gives both gains.

GΦ(s)=M1+Trs,s2+1+MKp,ΦTr s+MKi,ΦTr=s2+2ζωns+ωn2G_\Phi(s) = \frac{M}{1 + T_r s}, \qquad s^2 + \frac{1 + M K_{p,\Phi}}{T_r}\, s + \frac{M K_{i,\Phi}}{T_r} = s^2 + 2\zeta\omega_n s + \omega_n^2
ts=100 ms, ζ=0.707⇒ωn≈42.43,Kp,Φ=2ζωnTr−1M≈15.9,Ki,Φ=ωn2TrM≈559.7t_s = 100\ \text{ms},\ \zeta = 0.707 \Rightarrow \omega_n \approx 42.43, \quad K_{p,\Phi} = \frac{2\zeta\omega_n T_r - 1}{M} \approx \mathbf{15.9}, \quad K_{i,\Phi} = \frac{\omega_n^2 T_r}{M} \approx \mathbf{559.7}

03

Speed loop

With the flux held at its nominal value, isq drives the mechanical plant through the torque constant KT.

KT=npMLrΦrn≈1.752 N0˘0B7m/A,GΩ(s)=KTJs+f0K_T = n_p \frac{M}{L_r}\Phi_{rn} \approx 1.752\ \text{N\u00B7m/A}, \qquad G_\Omega(s) = \frac{K_T}{J s + f_0}
ts=200 ms⇒ωn≈21.22,Kp,Ω=2ζωnJ−f0KT≈0.44,Ki,Ω=ωn2JKT≈6.86t_s = 200\ \text{ms} \Rightarrow \omega_n \approx 21.22, \quad K_{p,\Omega} = \frac{2\zeta\omega_n J - f_0}{K_T} \approx \mathbf{0.44}, \quad K_{i,\Omega} = \frac{\omega_n^2 J}{K_T} \approx \mathbf{6.86}

04

Workbench

Pick a loop, adjust the specification and watch the gains and simulated step response update. Note that the PI zero adds some overshoot beyond the pure second-order prediction.

Gains

Kp
420.6
Ki
83,110
Overshoot
0.0 %
2% settling
0.42 ms
0t (ms)y / y*
Normalized closed-loop step response (reference = 1).

05

RFOC vs V/f

Why go to all this trouble? The comparison with scalar V/f control:

FeatureDirect vector control (RFOC)Scalar control (V/f)
Control conceptDecouples flux (isd) and torque (isq) via the Park transformation.Regulates the Vs/fs ratio without vector orientation.
Dynamic responseFast torque response (< 5 ms) without overshoot.Sluggish torque response, oscillations under fast load changes.
Low-speed performanceFull nominal torque down to zero speed.Torque degrades at low frequency due to stator resistance drop (needs V0 boost).
ComplexityHigh: real-time flux estimation (Φr, ρ).Low: simple open-loop voltage/frequency law.