Euler-Lagrange models with complex currents of three-phase electrical machines and observability issues

A new Lagrangian formulation with complex currents is developed and yields a direct and simple method for modeling three-phase permanent-magnet and induction machines. The Lagrangian is the sum a mechanical one and of a magnetic one. This magnetic Lagrangian is expressed in terms of rotor angle, complex stator and rotor currents. A complexification procedure widely used in quantum electrodynamic is applied here in order to derive the Euler-Lagrange equations with complex stator and rotor currents. Such complexification process avoids the usual separation into real and imaginary parts and simplifies notably the calculations. Via simple modifications of such magnetic Lagrangians we derive new dynamical models describing permanent-magnet machines with both saturation and saliency, and induction machines with both magnetic saturation and space harmonics. For each model we also provide its Hamiltonian thus its magnetic energy. This energy is also expressed with complex currents and can be directly used in Lyapunov and/or passivity based control. Further, we briefly investigate the observability of this class of Euler-Lagrange models, in the so-called sensorless case when the measured output is the stator current and the load torque is constant but unknown. For all the dynamical models obtained via such variational principles, we prove that their linear tangent systems are unobservable around a one-dimensional family of steady-states attached to the same constant stator voltage and current. This negative result explains why sensorless control of three-phase electrical machines around zero stator frequency remains yet a difficult control problem.Comment: Revised version. Submitted for publicatio

An analysis of the benefits of signal injection for low-speed sensorless control of induction motors

We analyze why low-speed sensorless control of the IM is intrinsically difficult, and what is gained by signal injection. The explanation relies on the control-theoretic concept of observability applied to a general model of the saturated IM. We show that the IM is not observable when the stator speed is zero in the absence of signal injection, but that observability is restored thanks to signal injection and magnetic saturation. The analysis also reveals that existing sensorless algorithms based on signal injection may perform poorly for some IMs under particular operating conditions. The approach is illustrated by simulations and experimental data

Signal injection and averaging for position estimation of Permanent-Magnet Synchronous Motors

Sensorless control of Permanent-Magnet Synchronous Motors at low velocity remains a challenging task. A now well-established method consists in injecting a high-frequency signal and use the rotor saliency, both geometric and magnetic-saturation induced. This paper proposes a clear and original analysis based on second-order averaging of how to recover the position information from signal injection; this analysis blends well with a general model of magnetic saturation. It also experimentally demonstrates the relevance for position estimation of a simple parametric saturation model recently introduced by the authors

Estimation of Saturation of Permanent-Magnet Synchronous Motors Through an Energy-Based Model

We propose a parametric model of the saturated Permanent-Magnet Synchronous Motor (PMSM) together with an estimation method of the magnetic parameters. The model is based on an energy function which simply encompasses the saturation effects. Injection of fast-varying pulsating voltages and measurements of the resulting current ripples then permit to identify the magnetic parameters by linear least squares. Experimental results on a surface-mounted PMSM and an interoir magnet PMSM illustrate the relevance of the approach.Comment: IEMDC-2011 (preliminary version

Energy-based modeling of electric motors

We propose a new approach to model electrical machines based on energy considerations and construction symmetries of the motor. We detail the approach on the Permanent-Magnet Synchronous Motor and show that it can be extended to Synchronous Reluctance Motor and Induction Motor. Thanks to this approach we recover the usual models without any tedious computation. We also consider effects due to non-sinusoidal windings or saturation and provide experimental data

Adding virtual measurements by signal injection

We propose a method to "create" a new measurement output by exciting the system with a high-frequency oscillation. This new "virtual" measurement may be useful to facilitate the design of a suitable control law. The approach is especially interesting when the observability from the actual output degenerates at a steady-state regime of interest. The proposed method is based on second-order averaging and is illustrated by simulations on a simple third-order system

Current Controller for Low Frequency Signal Injection and Rotor Flux Position Tracking at Low Speeds

International audienceRotor flux spatial position can be tracked in an ac machine even at low or zero stator frequency if a low-frequency harmonic current signal is injected into its stator. The harmonic current injection is source of the rotor speed perturbations which induce voltage oscillations in the stator winding at the injected frequency. By analyzing the stator winding voltage response, it is possible to detect the rotor flux position regardless of the stator frequency. This paper presents a stator current controller that is suitable for imposing rotating or pulsating harmonic current injection and a method for tracking the rotor flux position in either induction machines (IMs) or permanent-magnet synchronous machines (PMSMs). The controller contains, in addition to the standard fundamental-frequency-based synchronous reference frame (SRF) current controller, two sets of harmonic current integral controllers placed in respective harmonic SRFs. Such an extended current controller simultaneously performs two important tasks: controlled harmonic current injection with zero steady-state error and separation of particular spectral components in the stator voltage (spectral/sequence decomposition) which contain the rotor flux position information. The theoretical analysis presented, based on perturbation theory and averaging techniques, gives general expressions which link the rotor flux position error in IM and PMSM to the harmonic current controller outputs. Two special cases with the rotational and pulsating harmonic current injections are considered in more detail. The validity of the theoretical analysis and the feasibility of the sensorless rotor flux position detection are experimentally verified

Ostertagia spp., rumen fluke and liver fluke single- and poly-infections in cattle: an abattoir study of prevalence and production impacts in England and Wales

This study aims at investigating the occurrence, risk factors and production impacts on beef carcassparameters of three of the most important cattle helminth infections in England and Wales. Abomasa, reticulorumens and livers from healthy cattle were collected and examined post-mortem quarterly over a one year period in an abattoir in South-West England. Specific viscera from 974 cattle were collected,examined and scored for Ostertagia spp., adult rumen fluke and liver fluke lesions/presence. A total of89%, 25% and 29% of the carcasses had lesions/presence of Ostertagia spp., rumen fluke and liver fluke, respectively, and 39% had presence of helminth co-infection. Animal demographic and carcass parameters associated with helminth infections were investigated using multi level multinomial and multilevel linear mixed models respectively. After adjusting for other factors, significant differences in the distribution of helminth infections were observed among cattle by type of breed, animal category (cow, heifer, steer and young bull), age, season and concurrent helminth infections. Compared to carcasses free of helminths, carcasses presenting solely Ostertagia Spp. lesions or adult rumen fluke had significantly lower cold carcass weight (coef.: −30.58 [−50.92;−10.24] and −50.34 [−88.50;−12.18]) and fat cover-age (coef.: −3.28 [−5.56;−1.00] and −5.49 [−10.28;−0.69]) and carcasses presenting solely liver flukelesions had significantly lower conformation grade (coef.: −3.65 [−6.98;−0.32]). Presence of helminth poly-infections was negatively associated with cold carcass weight

From least action in electrodynamics to magnetomechanical energy -- a review

The equations of motion for electromechanical systems are traced back to the fundamental Lagrangian of particles and electromagnetic fields, via the Darwin Lagrangian. When dissipative forces can be neglected the systems are conservative and one can study them in a Hamiltonian formalism. The central concepts of generalized capacitance and inductance coefficients are introduced and explained. The problem of gauge independence of self-inductance is considered. Our main interest is in magnetomechanics, i.e. the study of systems where there is exchange between mechanical and magnetic energy. This throws light on the concept of magnetic energy, which according to the literature has confusing and peculiar properties. We apply the theory to a few simple examples: the extension of a circular current loop, the force between parallel wires, interacting circular current loops, and the rail gun. These show that the Hamiltonian, phase space, form of magnetic energy has the usual property that an equilibrium configuration corresponds to an energy minimum.Comment: 29 pages, 9 figures, 65 reference