In the buk it is discribed that if a dc source is been applied across the L.R circuit..the potential across the inductor=-Ldi/dt but... if the source ac the potential is = iXL ....where XL is the inductive reactance... why does not self induction plays role...? ac current's dir is continuosly changin'....

In the buk it is discribed that if a dc source is been applied across the L.R circuit..the potential across the inductor=-Ldi/dt
if the source ac the potential is = iXL ....where XL is the inductive reactance...
why does not self induction plays role...?
ac current's dir is continuosly changin'....

Grade:upto college level

1 Answers

askIITians Faculty 829 Points
2 years ago
A current{\displaystyle i} [i] flowing through a conductor generates amagnetic fieldaround the conductor, which is described byAmpere's circuital law. The totalmagnetic fluxthrough a circuit{\displaystyle \Phi } [\Phi] is equal to the product of the perpendicular component of the magnetic flux density and the area of the surface spanning the current path. If the current varies, themagnetic flux{\displaystyle \Phi } [\Phi] through the circuit changes. ByFaraday's law of induction, any change in flux through a circuit induces anelectromotive force(EMF) or voltage{\displaystyle v} [v] in the circuit, proportional to the rate of change of flux

{\displaystyle v(t)=-{\frac {\text{d}}{{\text{d}}t}}\,\Phi (t)} [{\displaystyle v(t)=-{\frac {\text{d}}{{\text{d}}t}}\,\Phi (t)}]

The negative sign in the equation indicates that the induced voltage is in a direction which opposes the change in current that created it; this is calledLenz's law. The potential is therefore called aback EMF. If the current is increasing, the voltage is positive at the end of the conductor through which the current enters and negative at the end through which it leaves, tending to reduce the current. If the current is decreasing, the voltage is positive at the end through which the current leaves the conductor, tending to maintain the current. Self-inductance, usually just called inductance,{\displaystyle L} [L] is the ratio between the induced voltage and the rate of change of the current

{\displaystyle v(t)=L\,{\frac {{\text{d}}i}{{\text{d}}t}}\qquad \qquad \qquad (1)\;} [{\displaystyle v(t)=L\,{\frac {{\text{d}}i}{{\text{d}}t}}\qquad \qquad \qquad (1)\;}]

Thus, inductance is a property of a conductor or circuit, due to its magnetic field, which tends to oppose changes in current through the circuit. The unit of inductance in theSIsystem is thehenry(H), named after American scientistJoseph Henry, which is the amount of inductance which generates a voltage of onevoltwhen the current is changing at a rate of oneampereper second.

All conductors have some inductance, which may have either desirable or detrimental effects in practical electrical devices. The inductance of a circuit depends on the geometry of the current path, and on themagnetic permeabilityof nearby materials;ferromagneticmaterials with a higher permeability likeironnear a conductor tend to increase the magnetic field and inductance. Any alteration to a circuit which increases the flux (total magnetic field) through the circuit produced by a given current increases the inductance, because inductance is also equal to the ratio ofmagnetic fluxto current[11][12][13][14]

{\displaystyle L={\Phi (i) \over i}} [{\displaystyle L={\Phi (i) \over i}}]

Aninductoris anelectrical componentconsisting of a conductor shaped to increase the magnetic flux, to add inductance to a circuit. Typically it consists of a wire wound into acoilorhelix. A coiled wire has a higher inductance than a straight wire of the same length, because the magnetic field lines pass through the circuit multiple times, it has multipleflux linkages. The inductance is proportional to the square of the number of turns in the coil, assuming full flux linkage.

The inductance of a coil can be increased by placing amagnetic coreofferromagneticmaterial in the hole in the center. The magnetic field of the coil magnetizes the material of the core, aligning itsmagnetic domains, and the magnetic field of the core adds to that of the coil, increasing the flux through the coil. This is called aferromagnetic core inductor. A magnetic core can increase the inductance of a coil by thousands of times.

If multipleelectric circuitsare located close to each other, the magnetic field of one can pass through the other; in this case the circuits are said to beinductively coupled. Due toFaraday's law of induction, a change in current in one circuit can cause a change in magnetic flux in another circuit and thus induce a voltage in another circuit. The concept of inductance can be generalized in this case by defining themutual inductance{\displaystyle M_{k,\ell }} [{\displaystyle M_{k,\ell }}] of circuit{\displaystyle k} [k] and circuit{\displaystyle \ell } [\ell] as the ratio of voltage induced in circuit{\displaystyle \ell } [\ell] to the rate of change of current in circuit{\displaystyle k} [k] . This is the principle behind atransformer.The property describing the effect of one conductor on itself is more precisely calledself-inductance, and the properties describing the effects of one conductor with changing current on nearby conductors is calledmutual inductance.[15]

Self-inductance and magnetic energy[edit]

If the current through a conductor with inductance is increasing, a voltage{\displaystyle v(t)} [v(t)] is induced across the conductor with a polarity that opposes the current—in addition to any voltage drop caused by the conductor's resistance. The charges flowing through the circuit lose potential energy. The energy from the external circuit required to overcome this "potential hill" is stored in the increased magnetic field around the conductor. Therefore, an inductor stores energy in its magnetic field. At any given time{\displaystyle t} [t] the power{\displaystyle p(t)} [p(t)] flowing into the magnetic field, which is equal to the rate of change of the stored energy{\displaystyle U} [U] , is the product of the current{\displaystyle i(t)} [i(t)] and voltage{\displaystyle v(t)} [v(t)] across the conductor[16][17][18]

{\displaystyle p(t)={\frac {{\text{d}}U}{{\text{d}}t}}=v(t)\,i(t)} [{\displaystyle p(t)={\frac {{\text{d}}U}{{\text{d}}t}}=v(t)\,i(t)}]

From (1) above

{\displaystyle {\frac {{\text{d}}U}{{\text{d}}t}}=L(i)\,i\,{\frac {{\text{d}}i}{{\text{d}}t}}} [{\displaystyle {\frac {{\text{d}}U}{{\text{d}}t}}=L(i)\,i\,{\frac {{\text{d}}i}{{\text{d}}t}}}]

{\displaystyle {\text{d}}U=L(i)\,i\,{\text{d}}i\,} [{\displaystyle {\text{d}}U=L(i)\,i\,{\text{d}}i\,}]

When there is no current, there is no magnetic field and the stored energy is zero. Neglecting resistive losses, theenergy{\displaystyle U} [U] (measured injoules, inSI) stored by an inductance with a current{\displaystyle I} [I] through it is equal to the amount of work required to establish the current through the inductance from zero, and therefore the magnetic field. This is given by:

{\displaystyle U=\int _{0}^{I}L(i)\,i\,{\text{d}}i\,} [{\displaystyle U=\int _{0}^{I}L(i)\,i\,{\text{d}}i\,}]

If the inductance{\displaystyle L(i)} [{\displaystyle L(i)}] is constant over the current range, the stored energy is[16][17][18]

{\displaystyle U=L\int _{0}^{I}\,i\,{\text{d}}i} [{\displaystyle U=L\int _{0}^{I}\,i\,{\text{d}}i}]

{\displaystyle U={\tfrac {1}{2}}L\,I^{2}} [{\displaystyle U={\tfrac {1}{2}}L\,I^{2}}]

Inductance is therefore also proportional to the energy stored in the magnetic field for a given current. This energy is stored as long as the current remains constant. If the current decreases, the magnetic field decreases, inducing a voltage in the conductor in the opposite direction, negative at the end through which current enters and positive at the end through which it leaves. This returns stored magnetic energy to the external circuit.

Ifferromagneticmaterials are located near the conductor, such as in an inductor with amagnetic core, the constant inductance equation above is only valid forlinearregions of the magnetic flux, at currents below the level at which the ferromagnetic materialsaturates, where the inductance is approximately constant. If the magnetic field in the inductor approaches the level at which the core saturates, the inductance begins to change with current, and the integral equation must be used.

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