Mathematical Intuition behind Electric Motors

Explore the dc motor's stator and rotor, brushes, coils, and the derivation of excitation voltage, flux, and torque, governed by geometry-dependent constants case a b.

Derive the excitation and armature circuit equations for a dc motor, linking flux, inductance, and emf with current, brushes, and magnetic gaps.

Derive six constitutive equations for the dc motor, linking excitation flux, armature current, back emf, torque, and voltages across excitation and armature circuits, including resistance and inductance.

Analyze the dc motor in stationary conditions, derive i_a = E/R_e and linear torque relation T = C1 − C2 ω, and identify ω_not = C1/C2 as the steady-state speed.

Examine the mathematical modelling of a three-phase asynchronous motor with windings in the stator and rotor, highlighting 60-degree phase spacing and the TAO parameter.

Examine how a stator phase generates magnetic fields in the air gaps of a three-phase motor, linking electrical and mechanical angles with conductor cavities and divergence-free field concepts.

Derive the stator magnetic field from the first harmonic of a Fourier series for the phases, detailing coefficient calculation, periodic extension, and a complex-exponential representation.

Learn how the total magnetic field from a three-phase stator arises from phase contributions. Represent HST via a complex vector with alpha powers.

Explore the sinusoidal regime of a 3-phase motor by modeling phase currents with i_sm cos(ωs t + φ) and complex exponentials, revealing the rotating magnetic field in the air gap.

Explore how to express electrical power P = V·I in a 3-phase asynchronous motor by transforming three-phase currents and voltages into complex vectors, using dot products and conjugates.

Show why a zero-term vanishes in a three-phase motor by analyzing triangle and star configurations, applying Kirchhoff's laws to voltages and currents, and simplifying the power expression.

Derive per-phase circuit equations for a stator in star or delta, assuming R_s is isotropic, relating phase voltage, current, and flux derivative, using vector forms for flux and current.

Compute total magnetic field and flux in the air gap by integrating the magnetic field over the air gap surface, combining stator and rotor contributions via phase currents and geometry.

Compute the stator flux by evaluating the integral of the real part of H-bar e^-j s, deriving phase flux components and relating results to dispersion and mutual inductance.

Derive the stator circuit equation in vector space by relating v_s, i_s, and flux derivatives to r_s and l_s, covering star and triangle configurations.

Derive rotor equations using symmetry, transform stator equations to the rotor frame, express the total field and flux, and link these to rotor energy and mechanical power.

explains the energy flow and power terms in a synchronous motor, detailing stator and rotor contributions, self and mutual inductance, resistance, mechanical power, and complex-exponential representations.

Recap the power and energy equations for a three-phase motor, linking grid input to resistive losses, inductive and mutual energy storage, and mechanical power tied to angular velocity.

Explore the sinusoidal regime of a three-phase asynchronous motor, deriving rms current, phase dissipation in stator and rotor, and the steady-state energy balance using phasor methods.

Derive mechanical power as torque times angular velocity in a three-phase asynchronous motor, relate omega to theta dot and rotor speed, and compare star and delta voltage configurations.

Derive the torque expression for a three-phase motor from stator and rotor equations, showing its dependence on rotor velocity and the stator and rotor electrical frequencies.

Rewrite the motor equations to derive an equivalent three-phase circuit, defining A, s, and the rotor frequency relation for torque expressions in terms of slip.

Express torque in terms of angular frequencies by transforming the motor circuit equations, substituting currents with input voltages, taking magnitudes, and applying simplifying approximations.

Derive a simplified torque expression for electric motors by applying approximations to the equivalent circuit, neglecting resistance and leakage inductance, and confirming the result through two algebraic approaches.

Derive the torque versus angular velocity curve for a three-phase synchronous motor, showing zero torque at omega equals omega s, a peak in between, and a symmetric decline beyond.