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Inductance

2007 Schools Wikipedia Selection. Related subjects: Electricity and
Electronics

   Inductance (or electric inductance) is a measure of the amount of
   magnetic flux produced for a given electric current. The term was
   coined by Oliver Heaviside in February 1886. The SI unit of inductance
   is the henry (symbol: H), in honour of Joseph Henry. The symbol L is
   used for inductance, possibly in honour of the physicist Heinrich Lenz.

   The inductance has the following relationship:

          L= \frac{\Phi}{i}

   where

          L is the inductance in henrys,
          i is the current in amperes,
          Φ is the magnetic flux in webers

   Strictly speaking, the quantity just defined is called self-inductance,
   because the magnetic field is created solely by the conductor that
   carries the current.

   When a conductor is coiled upon itself N number of times around the
   same axis (forming a solenoid), the current required to produce a given
   amount of flux is reduced by a factor of N compared to a single turn of
   wire. Thus, the inductance of a coil of wire of N turns is given by:

          L= \frac{\lambda}{i} = N\frac{\Phi}{i}

   where

          λ is the total 'flux linkage'.

Inductance of a solenoid

   The amount of magnetic flux produced by a current flowing through a
   coil depends upon the permeability of the medium surrounded by the
   current, the area inside the coil, and the number of turns. The greater
   the permeability, the greater the magnetic flux generated by a given
   current. Certain ( ferromagnetic) materials have much higher
   permeability than air. If a conductor (wire) is wound around such a
   material, the magnetic flux becomes much greater and the inductance
   becomes much greater than the inductance of an identical coil wound in
   air. The self-inductance L of such a solenoid can be calculated from

          L = {\mu_0 \mu_r N^2 A \over l} = \frac{N \Phi}{i}

   where

          μ[0] is the permeability of free space (4π × 10^-7 henrys per
          metre)
          μ[r] is the relative permeability of the core (dimensionless)
          N is the number of turns.
          A is the cross sectional area of the coil in square metres.
          l is the length of the coil (NOT the wire) in metres.
          Φ = BA is the flux in webers (B is the flux density, A is the
          area).
          i is the current in amperes

   This, and the inductance of more complicated shapes, can be derived
   from Maxwell's equations. For rigid air-core coils, inductance is a
   function of coil geometry and number of turns, and is independent of
   current. However, since the permeability of ferromagnetic materials
   changes with applied magnetic flux, the inductance of a coil with a
   ferromagnetic core will generally vary with current.

Inductance of a circular loop

   The inductance of a circular conductive loop made of a circular
   conductor can be determined using

          L = {r \mu_0 \mu_r \left( \ln{ \frac {8 r}{a}} - 2 + Y\right) }

   where

          μ[0] and μ[r] are the same as above
          r is the radius of the loop
          a is the radius of the conductor
          Y is a constant. Y=0 when the current flows in the surface of
          the wire ( skin effect), Y=1/4 when the current is homogeneous
          across the wire.

Inductance for any shaped loop

   Consider a current loop δS with current i(t). According to Biot-Savart
   law, current i(t) sets up a magnetic flux density at r:

          \mathbf{B}(\mathbf{r},t)= \frac{\mu_{0}\mu_{r} i(t)}{4\pi}
          \int_{\delta S}{\frac{d\mathbf{l} \times \mathbf{\hat r}}{r^2}}

   Now magnetic flux through the surface S the loop encircles is:

          \Phi(t) = \int_S\mathbf{B}(\mathbf{r},t) \cdot d\mathbf{A} =
          \frac{\mu_{0}\mu_{r} i(t)}{4\pi} \int_S \int_{\delta
          S}{\frac{d\mathbf{l} \times \mathbf{\hat r}}{r^2}} \cdot
          d\mathbf{A} = Li(t)

   From where we get the expression for inductance of the current loop:

          L = \frac{\mu_{0}\mu_{r} }{4\pi} \int_S \int_{\delta
          S}{\frac{d\mathbf{l} \times \mathbf{\hat r}}{r^2}} \cdot
          d\mathbf{A}

   where

          μ[0] and μ[r] are the same as above
          d\mathbf{l} is the differential length vector of the current
          loop element
          \mathbf{\hat r} is the unit displacement vector from the current
          element to the field point r
          r is the distance from the current element to the field point r
          d\mathbf{A} differential vector element of surface area A, with
          infinitesimally small magnitude and direction normal to surface
          S

   As we see here, the geometry and material properties (if material
   properties are same in surface S and the material is linear) of the
   current loop can be expressed with single scalar quantity L.

Properties of inductance

   The equation relating inductance and flux linkages can be rearranged as
   follows:

          \lambda = Li \,

   Taking the time derivative of both sides of the equation yields:

          \frac{d\lambda}{dt} = L \frac{di}{dt} + i \frac{dL}{dt} \,

   In most physical cases, the inductance is constant with time and so

          \frac{d\lambda}{dt} = L \frac{di}{dt}

   By Faraday's Law of Induction we have:

          \frac{d\lambda}{dt} = -\mathcal{E} = v

   where \mathcal{E} is the Electromotive force (emf) and v is the induced
   voltage. Note that the emf is opposite to the induced voltage. Thus:

          \frac{di}{dt} = \frac{v}{L}

   or

          i(t) = \frac{1}{L} \int_0^tv(\tau) d\tau + i(0)

   These equations together state that, for a steady applied voltage v,
   the current changes in a linear manner, at a rate proportional to the
   applied voltage, but inversely proportional to the inductance.
   Conversely, if the current through the inductor is changing at a
   constant rate, the induced voltage is constant.

   The effect of inductance can be understood using a single loop of wire
   as an example. If a voltage is suddenly applied between the ends of the
   loop of wire, the current must change from zero to non-zero. However, a
   non-zero current induces a magnetic field by Ampere's law. This change
   in the magnetic field induces an emf that is in the opposite direction
   of the change in current. The strength of this emf is proportional to
   the change in current and the inductance. When these opposing forces
   are in balance, the result is a current that increases linearly with
   time where the rate of this change is determined by the applied voltage
   and the inductance.

Phasor circuit analysis and impedance

   Using phasors, the equivalent impedance of an inductance is given by:

          Z_L = V / I = j L \omega \,

   where

          X_L = L \omega \, is the inductive reactance,
          \omega = 2 \pi f \, is the angular frequency,
          L is the inductance,
          f is the frequency, and
          j is the imaginary unit.

Coupled inductors

   When the magnetic flux produced by an inductor links another inductor,
   these inductors are said to be coupled. Coupling is often undesired but
   in many cases, this coupling is intentional and is the basis of the
   transformer. When inductors are coupled, there exists a mutual
   inductance that relates the current in one inductor to the flux linkage
   in the other inductor. Thus, there are three inductances defined for
   coupled inductors:

          L[11] - the self inductance of inductor 1
          L[22] - the self inductance of inductor 2
          L[12] = L[21] - the mutual inductance associated with both
          inductors

   When either side of the transformer is a tuned circuit, the amount of
   mutual inductance between the two windings determines the shape of the
   frequency response curve. Although no boundaries are defined, this is
   often referred to as loose-, critical-, and over-coupling. When two
   tuned circuits are loosely coupled through mutual inductance, the
   bandwidth will be narrow. As the amount of mutual inductance increases,
   the bandwidth continues to grow. When the mutual inductance is
   increased beyond a critical point, the peak in the response curve
   begins to drop, and the centre frequency will be attenuated more
   strongly than its direct sidebands. This is known as overcoupling.

Vector field theory derivations

Mutual inductance

   The circuit diagram representation of mutually inducting inductors. The
   two vertical lines between the inductors indicate a solid core that the
   wires of the inductor are wrapped around. "n:m" shows the ratio between
   the number of windings of the left inductor to windings of the right
   inductor. This picture also shows the dot convention.
   The circuit diagram representation of mutually inducting inductors.
   The two vertical lines between the inductors indicate a solid core that
   the wires of the inductor are wrapped around. "n:m" shows the ratio
   between the number of windings of the left inductor to windings of the
   right inductor. This picture also shows the dot convention.

   Mutual inductance is the concept that the current through one inductor
   can induce a voltage in another nearby inductor. It is important as the
   mechanism by which transformers work, but it can also cause unwanted
   coupling between conductors in a circuit.

   The mutual inductance, M, is also a measure of the coupling between two
   inductors. The mutual inductance by circuit i on circuit j is given by
   the double integral Neumann formula

          M_{ij} = \frac{\mu_0}{4\pi} \oint_{C_i}\oint_{C_j}
          \frac{\mathbf{ds}_i\cdot\mathbf{ds}_j}{|\mathbf{R}_{ij}|}

   See a derivation of this equation.

   The mutual inductance also has the relationship:

          M_{21} = N_1 N_2 P_{21} \!

   where

          M[21] is the mutual inductance, and the subscript specifies the
          relationship of the voltage induced in coil 2 to the current in
          coil 1.
          N[1] is the number of turns in coil 1,
          N[2] is the number of turns in coil 2,
          P[21] is the permeance of the space occupied by the flux.

   The mutual inductance also has a relationship with the coefficient of
   coupling. The coefficient of coupling is always between 1 and 0, and is
   a convenient way to specify the relationship between a certain
   orientation of inductor with arbitrary inductance:

          M = k \sqrt{L_1 L_2} \!

   where

          k is the coefficient of coupling and 0 ≤ k ≤ 1,
          L[1] is the inductance of the first coil, and
          L[2] is the inductance of the second coil.

   Once this mutual inductance factor M is determined, it can be used to
   predict the behaviour of a circuit:

          V = L_1 \frac{dI_1}{dt} + M \frac{dI_2}{dt}

   where

          V is the voltage across the inductor of interest,
          L[1] is the inductance of the inductor of interest,
          dI[1] / dt is the derivative, with respect to time, of the
          current through the inductor of interest,
          M is the mutual inductance and
          dI[2] / dt is the derivative, with respect to time, of the
          current through the inductor that is coupled to the first
          inductor.}}

   When one inductor is closely coupled to another inductor through mutual
   inductance, such as in a transformer, the voltages, currents, and
   number of turns can be related in the following way:

          V_s = V_p \frac{N_s}{N_p}

   where

          V[s] is the voltage across the secondary inductor,
          V[p] is the voltage across the primary inductor (the one
          connected to a power source),
          N[s] is the number of turns in the secondary inductor, and
          N[p] is the number of turns in the primary inductor.

   Conversely the current:

          I_s = I_p \frac{N_p}{N_s}

   where

          I[s] is the current through the secondary inductor,
          I[p] is the current through the primary inductor (the one
          connected to a power source),
          N[s] is the number of turns in the secondary inductor, and
          N[p] is the number of turns in the primary inductor.

   Note that the power through one inductor is the same as the power
   through the other. Also note that these equations don't work if both
   transformers are forced (with power sources).

Self-inductance

   Self-inductance, denoted L, is the usual inductance one talks about
   with an inductor. Formally the self-inductance of a wire loop would be
   given by the above equation with i =j. However, 1/|\mathbf{R}| now gets
   singular and the finite radius a and the distribution of the current in
   the wire must be taken into account. There remain the contribution from
   the integral over all points where |\mathbf{R}| \ge a/2 and a
   correction term,

          M_{ij} = M_{jj} = L_{jj} = L_j = L = \left (\frac{\mu_0}{4\pi}
          \oint_{C}\oint_{C'}
          \frac{\mathbf{ds}\cdot\mathbf{ds}'}{|\mathbf{R}|}\right
          )_{|\mathbf{R}| \ge a/2} + \frac{\mu_0}{2\pi}lY

   Here a and l denote radius and length of the wire, and Y is a constant
   that depends on the distribution of the current in the wire: Y = 0 when
   the current flows in the surface of the wire ( skin effect), Y = 1 / 4
   when the current is homogenuous across the wire.

   Physically, the self-inductance of a circuit represents the back-emf
   described by Faraday's law of induction.

Usage

   The flux \Phi_i\ \! through the i-th circuit in a set is given by:

          \Phi_i = \sum_{j} M_{ij}I_j = L_i I_i + \sum_{j\ne i} M_{ij}I_j
          \,

   so that the induced emf, \mathcal{E} , of a specific circuit, i, in any
   given set can be given directly by:

          E = -\frac{d\Phi_i}{dt} = -\frac{d}{dt} \left (L_i I_i +
          \sum_{j\ne i} M_{ij}I_j \right ) = -\left(\frac{dL_i}{dt}I_i
          +\frac{dI_i}{dt}L_i \right) -\sum_{j\ne i} \left
          (\frac{dM_{ij}}{dt}I_j + \frac{dI_j}{dt}M_{ij} \right)

   Retrieved from " http://en.wikipedia.org/wiki/Inductance"
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   with only minor checks and changes (see www.wikipedia.org for details
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