Tag Archives: Inductor

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.

Inductor specification

  • Inductance L (\mu H) (tested at a certain test conditions for example 100 KHz, 0.1 V_{rms}, 0 A DC)
  • Inductive tolerance: It is the allowed amount of variation from the nominal value specified by the manufacturer (e.g. ±20%).
  • Rated operating voltage (across inductor)
  • DC Resistance (DCR)
  • AC Resistance (ACR)
    • AC loss that comes from losses in the core as the magnetic field transitions. This includes eddy current losses and hysteresis losses.
    • The AC resistance of the wire due to the skin effect. It can be important at very high frequencies.
  • Maximum DC current I_{DC}: Maximum DC current is the DC current at which the inductance falls to 90% of its nominal value or until its temperature rise reaches 30 °C.

Figure: Inductance vs DC Bias Load (or DC Bias Characteristic)

DC Bias current relates to a constant current element that is added to the AC signal.

  • Incremental Current Rating: The DC bias current that causes an inductance drop of 5% from the initial zero DC bias inductance value.
  • I_{rms} or RMS current:
    • I_{rms} for a 20°C rise above 25°C ambient temperature
    • I_{rms} for a 40°C rise above 25°C ambient temperature
  • Saturation current I_{SAT}: The DC bias current that causes the inductor to drop by a specified percentage (e.g. 10% or 20%) from its value without current. See Figure Inductance vs DC Bias Load (or DC Bias Characteristic)
  • Q factor or Quality factor: Q=\frac{2 \pi f L}{R}=\frac{X_{L}}{R}
  • Self-Resonant Frequency (SRF) or f_{o} in Hz
  • Curie temperature T_{C} (in degrees Celsius): It is the temperature at which the core material start to lose its magnetic properties.
  • Inductance temperature coefficient: The change in inductance per unit temperature change. Measured under zero bias conditions and expressed in parts per million (ppm).
  • Resistance temperature coefficient: The change in DC wire resistance per unit temperature change. Measured at low DC Bias (<1 VDC) and expressed in parts per million (ppm).
  • Magnetic saturation flux density B_{SAT}: At this value of flux density, all magnetic domains within the core are magnetized and aligned.
  • Shielding
    • with shield
    • without shield
  • Electromagnetic interference (EMI): It refers to the magnetic field radiated away from the inductor into space. The magnetic field may cause interference with other magnetically sensitive components.
  • Core material
    • Ferrite cores
    • Iron powder cores
  • Storage temperature range
  • Operating temperature range
    • Ambient temperature range not including self-temperature rise
    • Product temperature range including self-temperature rise. The operating temperature T_{Op} is equal to the ambient temperature T_{Amb} plus component’s self-heating \Delta T. The maximum allowable temperature for an inductor is the maximum ambient temperature plus the maximum temperature rise.
  • Moisture Sensitivity Level (MSL)

Inductor equations

If we assume that a constant DC current has been flowing through the inductor for some time, then \frac{dI_{L}}{dt} is zero and thus V_{L} is zero.

Under DC conditions the inductor acts like a short circuit.

Inductive reactance is measured in Ohms and it tells as “how bad” the inductor passes the current.

Inductive susceptance is measured in Siemens and it tells as “how well” the inductor passes the current.

Practical inductor, RL in series:

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}