Linear and Non-Linear Inductors

An ideal inductor would have zero capacitance and zero resistance.

The Figure below shows a graph of inductive reactance X_{L} versus frequency f. Inductive reactance increases linearly with frequency.

Figure: Inductive reactance X_{L} vs frequency f (ideal inductor)

A real inductor can be modeled by the following elements:

  • a series inductor L
  • a series resistor R_{DC} or R_{S}: The resistance of the inductor winding measured using DC current. The resistance in a component due to the length and diameter of the winding wire used. It represents the DC copper loss (due to DC resistance) of the wire.

R_{DC}=\rho \frac{l}{s}

\rho \ - \ resistivity \ (material \ dependent \ factor), [\Omega m]

l \ - \ length,\ [m]

s \ - \ cross \ section, [m^2].

  • a parallel capacitor C_{P} or C_{d}: It is the distributed capacitance between the turns of the wire and is derived from the Self Resonant Frequency (f_{o}).
  • a parallel resistor R_{P}: It represents the magnetic core loss of the inductor core.

The figure below shows a real-life impedance vs frequency graph.

Figure: Inductive reactance X_{L} vs frequency f (real inductor)

Self-Resonant Frequency (SRF) or f_{o} in Hz: This is the frequency at which the inductance of the inductor L resonates with the inductor’s distributed capacitance C_{P}. Increasing L or C lowers f_{o}. Decreasing L or C raises f_{o}.

Q=2 \pi \frac{maximum \ energy \ stored}{energy \ dissipated \ per \ cycle}=2 \pi \frac{\frac{1}{2}LI_{max}^2}{(\frac{I_{max}}{\sqrt{2}})^2RT}=2 \pi \frac{L}{R \frac{1}{f}}=\frac{2 \pi f L}{R}=\frac{X_{L}}{R}

f_{o} = \frac{1}{2 \pi \sqrt{LC}}

At f_{o}

  • the inductor will act as a pure resistor,
  • the input impedance is at its peak,
  • the Quality factor of the inductor is zero,
  • the reactance of the inductor X_{L} is zero,
  • the capacitance is given by C_{P} = \frac {1}{(2 \pi f_{o})^2L_{o}}

At frequencies below f_{o} the reactance is inductive and increases as the frequency increases.

At frequencies above f_{o} the reactance is capacitive and decreases as the frequency increases.