Inductors

An inductor is a passive element designed to store energy in its magnetic field.

An inductor consists of a coil of conducting wire.

Inductance \(L\) is the property whereby an inductor exhibits opposition to the change of current flowing through it, measured in henrys (H).

\(\boxed{L = \frac{\mu N^2 A}{l}}\)

\(\boxed{\mu = \mu_0 \mu_r}\)

where:

  • \(N\) is the number of turns (a dimentionless quantity).
  • \(l\) is the length (in meters).
  • \(A\) is the cross-sectional area (in square meters).
  • \(\mu\) is the permeability (in henrys per meter) of the core.
  • \(\mu_0\) is the vacuum (magnetic) permeability or permeability of free space, \(4\pi \times 10^{-7} H/m\) (henrys per meter).
  • \(\mu_r\) is the relative permeability (a dimensionless quantity).

If current is allowed to pass through an inductor, it is found that the voltage across the inductor is directly proportional to the time rate of change of the current. Using the passive sign convention,

\(\boxed{v = L\,\frac{di}{dt}}\)

where \(L\) is the constant of proportionality called the inductance of the inductor, measured in henrys (H).

\(\boxed{1\,henry = 1\,\frac{volt \cdot second}{ampere}}\)

For an inductor to have voltage across its terminals, its current must vary with time. Hence, \(v = 0\) for constant current through the inductor.

The current-voltage relation can be obtained by integrating the voltage-current relation of the inductor.

\(\displaystyle v\,dt = L\,di\)

Integrating both sides

\(\displaystyle \int_{-\infty}^t v\left(\tau\right)\,d\tau = L \int_{-\infty}^t di\)

\(\displaystyle i\left(t\right) - i\left(-\infty\right) = \frac{1}{L} \int_{-\infty}^t v\left(\tau\right)\,d\tau\)

\(\displaystyle i\left(t\right) = \frac{1}{L} \int_{-\infty}^t v\left(\tau\right)\,d\tau + i\left(-\infty\right)\)

where \(i\left(-\infty\right) = 0\)

\(\boxed{i\left(t\right) = \frac{1}{L} \int_{-\infty}^t v\left(\tau\right)\,d\tau}\)

or

\(\boxed{i\left(t\right) = \frac{1}{L} \int_{t_0}^t v\left(\tau\right)\,d\tau + i\left(t_0\right)}\)

The inductor is designed to store energy in its magnetic field. The power delivered to the inductor is

\(\displaystyle p = vi = \left(L\,\frac{di}{dt}\right)i\)

The energy stored \(w\) is

\(\displaystyle w = \int_{-\infty}^t p\left(\tau\right)\,d\tau = L \int_{-\infty}^t \frac{di}{d\tau}\,i\,d\tau = L \int_{-\infty}^t i\,di\)

\(\displaystyle w = \frac{1}{2}L\left[i^2\left(t\right) - i^2\left(-\infty\right)\right]\)

where \(i\left(-\infty\right) = 0\)

\(\boxed{w = \frac{1}{2}Li^2}\)

Important properties of an inductor:

  • The voltage across an inductor is zero when the current is constant.

    An inductor acts like a short circuit to DC.

  • The current through an inductor cannot change instantaneously.

    A discontinuous change in the current through an inductor requires an infinite voltage, which is not physically possible. Thus, an inductor opposes an abrupt change in the current through it. However, the voltage across an inductor can change abruptly.

  • The ideal inductor does not dissipate. The energy stored in it can be retrieved at a later time. The inductor takes power from the circuit when storing energy and delivers the power to the circuit when returning previously stored energy.

  • A practical, non-ideal inductor has a significant resistive component. This resistance is called the winding resistance \(R_w\), and it appears in series with the inductance of the inductor. The presence of \(R_w\) makes it both an energy storage device and an energy dissipation device. Since \(R_w\) is usually very small, it is ignored in most cases.

    The non-ideal inductor also has a winding capacitance \(C_w\) due to the capacitive coupling between the conducting coils. \(C_w\) is very small and can be ignored in most cases, except at high frequencies.

The wye-delta transformation for resistors can be extended to inductors.

Series Inductors

The equivalent inductance \(L_{eq}\) of \(N\) series-connected inductors is the sum of the individual inductances.

\(\boxed{L_{eq} = \sum_{n=1}^N L_n}\)

Parallel Inductors

The equivalent inductance \(L_{eq}\) of \(N\) parallel-connected inductors is the reciprocal of the sum of the reciprocals of the individual inductances.

\(\boxed{L_{eq} = \frac{1}{\displaystyle \sum_{n=1}^N \frac{1}{L_n}}}\)

Links to this page
  • Step Response of an RL Circuit

    Since the inductor current cannot change instantaneously,

    Use the fact that the inductor acts like a short circuit to DC at steady state.

    where \(R\) is the thevenin resistance at the inductor terminals and \(L\) is the equivalent inductance.

  • Step Response of a Series RLC Circuit

    Since the inductor current cannot change instantaneously,

    where roots \(s_1\) and \(s_2\) are called natural frequencies, measured in nepers per second \((\textrm{Np}/\textrm{s})\); \(\omega_0\) is known as the resonant frequency or strictly as the undamped natural frequency, expressed in radians per second \((\textrm{rad}/\textrm{s})\); \(\alpha\) is the neper frequency or the damping factor, expressed in nepers per second \((\textrm{Np}/\textrm{s})\); \(R\) is the equivalent resistance; \(L\) is the equivalent inductance; and \(C\) is the equivalent capacitance.

  • Step Response of a Parallel RLC Circuit

    Since the inductor current cannot change instantaneously,

    Use the fact that the inductor acts like a short circuit to DC at steady state.

    where roots \(s_1\) and \(s_2\) are called natural frequencies, measured in nepers per second \((\textrm{Np}/\textrm{s})\); \(\omega_0\) is known as the resonant frequency or strictly as the undamped natural frequency, expressed in radians per second \((\textrm{rad}/\textrm{s})\); \(\alpha\) is the neper frequency or the damping factor, expressed in nepers per second \((\textrm{Np}/\textrm{s})\); \(R\) is the equivalent resistance; \(L\) is the equivalent inductance; and \(C\) is the equivalent capacitance.

  • Source-Free Series RLC Circuit

    Since the inductor current cannot change instantaneously,

    where roots \(s_1\) and \(s_2\) are called natural frequencies, measured in nepers per second \((\textrm{Np}/\textrm{s})\); \(\omega_0\) is known as the resonant frequency or strictly as the undamped natural frequency, expressed in radians per second \((\textrm{rad}/\textrm{s})\); \(\alpha\) is the neper frequency or the damping factor, expressed in nepers per second \((\textrm{Np}/\textrm{s})\); \(R\) is the equivalent resistance; \(L\) is the equivalent inductance; and \(C\) is the equivalent capacitance.

  • Source-Free RL Circuit

    A source-free RL circuit occurs when its DC source is suddenly disconnected. The energy already stored in the inductor is released to the resistors.

    Assume \(i\left(t\right)\) is the current through the inductor. At \(t = 0\), the inductor has an initial current \(I_0\)

    Since the inductor current cannot change instantaneously,

    The energy that was initially stored in the inductor is eventually dissipated in the resistor.

    where \(R\) is the thevenin resistance at the inductor terminals and \(L\) is the equivalent inductance.

  • Source-Free Parallel RLC Circuit

    Since the inductor current cannot change instantaneously,

    where roots \(s_1\) and \(s_2\) are called natural frequencies, measured in nepers per second \((\textrm{Np}/\textrm{s})\); \(\omega_0\) is known as the resonant frequency or strictly as the undamped natural frequency, expressed in radians per second \((\textrm{rad}/\textrm{s})\); \(\alpha\) is the neper frequency or the damping factor, expressed in nepers per second \((\textrm{Np}/\textrm{s})\); \(R\) is the equivalent resistance; \(L\) is the equivalent inductance; and \(C\) is the equivalent capacitance.

  • Second-Order Circuits

    Typical examples of second-order circuits are RLC circuits, in which the three kinds of passive elements (resistors, inductors, and capacitors) are present. Other examples are RL and RC circuits.

  • Power Factor Correction

    An inductive load is modeled as a series combination of an inductor and a resistor.

    It is also possible that the load is capacitive; that is, the load is operating at a leading power factor. In this case, an inductor should be connected across (in parallel with) the load for power factor correction. The value of the required shunt inductance \(L\) is determined as

  • Phasor Relationships for Circuit Elements
    Inductor \(L\)

    The voltage and current of inductor are \(90^{\circ}\) out of phase. Specifically, the current lags the voltage by \(90^{\circ}\).

    For the inductor \(L\)

  • Instantaneous and Average Power

    When \(p(t)\) is positive, power is absorbed by the circuit. When \(p(t)\) is negative, power is absorbed by the source; that is, power is transferred from the circuit to the source. This is possible because of the storage elements (capacitors and inductors) in the circuit.

  • Impedance and Admittance
    Inductor \(L\)
  • First-Order Circuits

    A circuit comprising a resistor and capacitor and a circuit comprising a resistor and an inductor are called RC and RL circuits, respectively. The differential equations resulting from analyzing RC and RL circuits are of the first-order.

  • Circuit Element Models in s-Domain

    Since the inductor current cannot change instantaneously, \(i(0^-) = i(0^+) = i(0)\)

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