Electric Machines

Magnetomotive force drives flux and reluctance opposes it. Read a magnetic circuit like an electric one via Φ = NI/R.

The work to build current is stored in the field as W = ½LI². Inductance L = N²/R and quadratic growth.

The turns ratio raises voltage by a and lowers current by 1/a. Shared flux and power conservation give V₁I₁ = V₂I₂.

Sort the non-idealities by position — a series branch (winding, leakage) the load current passes and a shunt exciting branch (core loss, magnetisation) on the voltage.

Peak efficiency is where variable loss (copper) equals fixed loss (core) at m = √(Pfe/Pcu). Voltage regulation from the series drop is ε ≈ p cosφ + q sinφ.

A spinning coil’s EMF is AC (e ∝ sin θ), but the commutator flips the connection at the zero crossing to give pulsating DC outside. Average EMF E = PΦZN/60a.

The armature cross flux adds vectorially with the main flux, tilting the resultant toward rotation and shifting the neutral axis. Fixed by brush shift, interpoles and a compensating winding.

The back-EMF Eb = kΦN opposes the terminal voltage to set the speed N = (V - IaRa)/(kΦ). Weakening the field raises speed; torque is T = kΦIa.

Feedback from residual magnetism builds the voltage at the intersection of the magnetization curve and the field-resistance line. Above the critical resistance it fails. External characteristic Vt = E - IaRa.

Three phases staggered 120° in time and space make a rotating field of constant 1.5× magnitude. Synchronous speed Ns = 120f/P, θe = (P/2)θm.

Relative motion between field and rotor induces current and torque. At synchronous speed torque is zero, so the rotor always lags by a slip s = (Ns-N)/Ns. f2 = s·f.

Referring the rotor makes the resistance R2/s, splitting as R2/s = R2 + R2(1-s)/s to hold copper loss and mechanical output together. Power Pag : Pcu2 : Pmech = 1 : s : (1-s).

Torque is hill-shaped with maximum-torque slip s = R2/X2, and its value Tmax is independent of R2. Raising R2 shifts the curve along slip (proportional shift), raising the starting torque.

Speed N = (1-s)·120f/P changes by slip, frequency and poles. Since Φ ∝ V/f, keep V/f constant when varying frequency. Starting uses star-delta (1/3), reactor, proportional shift and inverter.

Spinning a DC-excited rotor at synchronous speed makes a three-phase EMF. Frequency is locked to speed as f = PN/120, EMF per phase E = 4.44 f N Φ kw. Excitation sets the voltage size.

The load current’s armature flux demagnetizes (lagging, V↓), magnetizes (leading, V↑) or cross-magnetizes (resistive) the field flux by power factor. Bundled with leakage into Xs: E = V + jIXs.

Torque is set by the load angle, T ∝ (VE/Xs)sinδ. With the load fixed, excitation controls power factor — under-excited lagging, over-excited leading (supplies reactive). No-load over-excited is a synchronous condenser.

Power P = (VE/Xs)sinδ peaks at δ=90°. The synchronizing power dP/dδ ∝ cosδ is zero there, the stability limit (pull-out). The V-curve is the V of armature current at fixed power.

A BLDC turns the DC machine inside out, replacing commutator and brushes with electronic commutation (sensor + inverter). With stepper (step angle 360°/steps) and servo (closed-loop), one family governing excitation by rotor position.

An inverter switches DC fast between +Vdc/-Vdc and modulates the duty sinusoidally (PWM) so the average is a sine. The index m sets voltage, the reference frequency the output frequency (VVVF). V/f is m ∝ f.