§5 Currents after closing and opening the circuit

Consider a circuit consisting of a source current EMF ?, the resistance R, inductance L and the key K. The circuit current flows

At time t = 0 disable EMF, closing the circuit shorted (key in position 2). As soon as the current begins to decrease self-induced emf arises that prevents this decrease.
Current in the circuit satisfies Ohm's Law:


At t = 0

 -time constant determines the rate of decrease (increase) in the circuit

therefore, τ - the time during which the current in the circuit is reduced by a factor e. The more or less L R (ie, the greater τ), the slower the current in the circuit decreases.
2. Current at closure.

After connecting to the EMF until current reaches a steady state value

, in the chain, except EMF E will act self-induced emf, which prevents the instantaneous current rise.
Then Ohm's law can be written as:


inhomogeneous linear differential equation.

His solution is the total solution of the homogeneous differential equation  and any particular solution of the inhomogeneous equations.

The general solution homogeneous equations


Particular solution of the nonhomogeneous

The general solution of the nonhomogeneous

at t = 0 I = 0, hence , I0 = const,then



§6 Mutual induction.


Consider two fixed circuit (1 and 2), located close enough to each other. If the loop current flows one I1, then the magnetic flux generated by this current is proportional to I1. We denote the part of the 21 thread that runs through circuit 2. Then

where L21 - proportionality factor .

If the current I1is changed, the circuit 2 is induced emf Ei2 2

Similarly, during the flow of current in the circuit 2 I2 magnetic flux pervades circuit - 12

If I2is changed, then

The phenomenon emergence the EMF in a circuit when changing current in the other is called mutual induction. Proportionality coefficients L12 and L21are called mutual inductance

L12 = L21.

L12 and L21are dependent on the geometric shape, size, relative position of the contours and the magnetic permeability of the medium surrounding contours

[L12] =√н.

Calculate the mutual inductance of two coils wound on a common toroid. Magnetic induction field generated by the coil 1 to the number of turns N1,the magnetic permeability μ core

where  is the length of the core in the midline

The magnetic flux through one turn of coil 2: :

The total magnetic flux

Consequently, the



2. Transformers (converter) - a device to increase, decrease, electrical isolation, etc. voltage and current.

It is based on the phenomenon of mutual induction. Primary and secondary windings with N1 and N2turns, respectively, mounted on a closed iron core. Change in the flux in the coil 1 is connected to a variable emf is in the secondary appearance EMF mutual induction, and in the primary - EMF of self-induction. Primary current I1is determined by Ohm's law

R1 -the resistance of the primary winding ,

I1R1 small in comparison  and  in variable fields. Consequently,

EMF mutual induction in the secondary


Sign “ - “means that the phases  and opposite .

 - transformation ratio.

k > 1- step-up transformer ;

k < 1- step-down transformer.

Therefore, the currents in the windings are inversely proportional to the number of turns in these coils.

A transformer consists of one winding is called auto-transformer.





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