Power & Energy Electromagnetics

Ideal transformer coupling a source to a load. — The transformer is a direct engineering application of magnetic flux linkage and Faraday induction. Image source: Wikimedia Commons, CC0.

30-second intuitionPower conversion is controlled electromagnetism: electric current creates magnetic field, changing magnetic flux creates voltage, and magnetic fields exchange force/torque with current-carrying conductors and magnetized matter.

1. Faraday induction

\[\mathcal E=-\frac{d\Phi_B}{dt}.\]

The minus sign encodes Lenz’s law: induced current opposes the change in flux that produced it.

2. Transformers

For an ideal transformer,

\[\frac{V_2}{V_1}=\frac{N_2}{N_1},\qquad \frac{I_2}{I_1}=\frac{N_1}{N_2}.\]

A real transformer additionally has winding resistance, leakage inductance, finite magnetizing inductance, core loss, interwinding capacitance and insulation constraints.

3. Magnetic energy

For a linear inductor,

\[W_m=\frac12LI^2.\]

In field form, magnetic energy density for a linear medium is

\[u_m=\frac12\mathbf B\cdot\mathbf H.\]

4. Motors

Force on current density is described by

\[\mathbf f=\mathbf J\times\mathbf B.\]

At system level, motors engineer the spatial and temporal relation between stator and rotor fields to create torque.

5. Generators

Mechanical motion changes magnetic flux linkage and induces voltage. Motors and generators are therefore reciprocal viewpoints of electromagnetic energy conversion.

6. Magnetic materials

Important properties include permeability, saturation, coercivity, hysteresis, remanence and frequency-dependent loss. Soft magnetic materials favor low coercivity for transformers/inductors; hard magnetic materials retain magnetization for permanent magnets.

7. Core loss

Core loss typically combines hysteresis, eddy-current and additional dynamic loss terms. Lamination/ferrites reduce eddy currents by increasing electrical resistance or interrupting current loops.

8. Conductor loss

At high frequency, skin and proximity effects crowd current and increase AC resistance. The skin depth is

\[\delta=\frac{1}{\sqrt{\pi f\mu\sigma}}.\]

Litz wire can reduce AC resistance in suitable frequency/geometry ranges.

9. Wireless power transfer

Inductive/resonant systems use magnetic coupling between coils. A simplified coupling coefficient is

\[k=\frac{M}{\sqrt{L_1L_2}},\]

where $M$ is mutual inductance. Efficiency depends on coupling, Q, matching, alignment, load and parasitic loss.

10. Permanent magnets and Halbach structures

Permanent-magnet machines and compact field sources use remanent magnetization. Halbach arrays deliberately rotate magnetization direction to enhance field on one side or inside a bore while reducing external flux. See Ground-State Magnetometry — Halbach arrays.

Worked example — transformer turns ratio

A 120 V primary with $N_1=600$ turns and $N_2=60$ turns gives an ideal secondary voltage

\[V_2=120\times\frac{60}{600}=12\ \text{V}.\]

A real design must then account for regulation, copper loss, core flux density and thermal rise.

11. Measurement

oscilloscope + differential voltage probe;
current probe / shunt;
power analyzer for real/reactive/apparent power and harmonics;
LCR/impedance analyzer;
B-H loop tracer;
Hall/fluxgate/search-coil magnetic probes;
thermal imaging/thermocouples for loss validation.

12. Engineering reality

What limits power magneticsSaturation, temperature rise, insulation, copper loss, core loss, leakage, parasitic capacitance, acoustic noise, EMI and manufacturing tolerance define real performance.

References

S. J. Chapman, Electric Machinery Fundamentals.
A. E. Fitzgerald, C. Kingsley and S. Umans, Electric Machinery.
R. W. Erickson and D. Maksimović, Fundamentals of Power Electronics.

Related: Lorentz Force · EMC · Measurements

Advanced topic — magnetic co-energy and force extraction

In a quasi-static electromagnetic actuator, magnetic co-energy (W_m’(i,x)) determines mechanical force at fixed current:

\[F_x=\left.\frac{\partial W_m'}{\partial x}\right|_i.\]

For a linear single-winding inductor (W_m’=\frac12 L(x)i^2), giving (F_x=\frac12i^2\,dL/dx). This expression assumes linear magnetics and should not be applied uncritically to saturated or hysteretic systems. With nonlinear flux linkage (\lambda(i,x)), integrate the measured (\lambda)–(i) relation to evaluate co-energy.

For coupled-resonator wireless power, the magnetic coupling coefficient is (k=M/\sqrt{L_1L_2}). Efficiency depends not just on coupling but on loaded (Q), detuning, matching, rectifier loss and thermal behavior.

Advanced exercise: compare a linear FEM force estimate with a co-energy calculation from nonlinear magnetic material data; quantify error caused by using constant permeability.

Further reading: S. J. Chapman, Electric Machinery Fundamentals; MIT Electromagnetics and Applications.