inductance of a coil. |
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inductance of a coil.
UNIT OF INDUCTANCE As stated before, the basic unit of inductance (L) is the HENRY (H), named after Joseph Henry, the co-discoverer with Faraday of the principle of electromagnetic induction. An inductor has an inductance of 1 henry if an emf of 1 volt is induced in the inductor when the current through the inductor is changing at the rate of 1 ampere per second. The relationship between the induced voltage, the inductance, and the rate of change of current with respect to time is stated mathematically as:
where E_{ind} is the induced emf in volts; L is the inductance in henrys; and &ΔI is the change in current in amperes occurring in &Δt seconds. The symbol Δ (Greek letter delta), means "a change in ....". The henry is a large unit of inductance and is used with relatively large inductors. With small inductors, the millihenry is used. (A millihenry is equal to 1 X 10^{-3} henry, and one henry is equal to 1,000 millihenrys.) For still smaller inductors the unit of inductance is the microhenry (μH). (A μH = 1 X 10^{-6}H, and one henry is equal to 1,000,000 microhenrys.)
GROWTH AND DECAY OF CURRENT IN AN LR SERIES CIRCUIT When a battery is connected across a "pure" inductance, the current builds up to its final value at a rate determined by the battery voltage and the internal resistance of the battery. The current buildup is gradual because of the counter emf generated by the self-inductance of the coil. When the current starts to flow, the magnetic lines of force move outward from the coil. These lines cut the turns of wire on the inductor and build up a counter emf that opposes the emf of the battery. This opposition causes a delay in the time it takes the current to build up to a steady value. When the battery is disconnected, the lines of force collapse. Again these lines cut the turns of the inductor and build up an emf that tends to prolong the flow of current.
A voltage divider containing resistance and inductance may be connected in a circuit by means of a special switch, as shown in figure (1-0A). Such a series arrangement is called an LR series circuit.
When switch S_{1} is closed (as shown), a voltage E_{s} appears across the voltage divider. At this instant the current will attempt to increase to its maximum value. However, this instantaneous current change causes coil L to produce a back EMF, which is opposite in polarity and almost equal to the EMF of the source. This back EMF opposes the rapid current change. Figure (10-B) shows that at the instant switch S_{1} is closed, there is no measurable growth current (i_{g}), a minimum voltage drop is across resistor R, and maximum voltage exists across inductor L. As current starts to flow, a voltage (e_{R}) appears across R, and the voltage across the inductor is reduced by the same amount. The fact that the voltage across the inductor (L) is reduced means that the growth current (i_{g}) is increased and consequently e_{R} is increased. Figure (10-B)shows that the voltage across the inductor (e_{L}) finally becomes zero when the growth current (i_{g}) stops increasing, while the voltage across the resistor (e_{R}) builds up to a value equal to the source voltage (E_{S}). Electrical inductance is like mechanical inertia, and the growth of current in an inductive circuit can be likened to the acceleration of a boat on the surface of the water. The boat does not move at the instant a constant force is applied to it. At this instant all the applied force is used to overcome the inertia of the boat. Once the inertia is overcome the boat will start to move. After a while, the speed of the boat reaches its maximum value and the applied force is used up in overcoming the friction of the water against the hull.
When the battery switch (S_{1}) in the LR circuit of figure (10-A) is closed, the rate of the current increase is maximum in the inductive circuit. At this instant all the battery voltage is used in overcoming the emf of self-induction which is a maximum because the rate of change of current is maximum. Thus the battery voltage is equal to the drop across the inductor and the voltage across the resistor is zero. As time goes on more of the battery voltage appears across the resistor and less across the inductor. The rate of change of current is less and the induced emf is less. As the steady-state condition of the current is approached, the drop across the inductor approaches zero and all of the battery voltage is "dropped" across the resistance of the circuit. Thus the voltages across the inductor and the resistor change in magnitude during the period of growth of current the same way the force applied to the boat divides itself between the effects of inertia and friction. In both examples, the force is developed first across the inertia/inductive effect and finally across the friction/resistive effect. Figure (10-C) shows that when switch
S_{2} is closed (source voltage
E_{S}
removed from the circuit), the flux that has been established around the
inductor (L) collapses through the windings. This induces a voltage
e_{L}
in the inductor that has a polarity opposite to E_{S} and is
essentially equal to E_{S} in magnitude. The induced voltage
causes decay current (i_{d}) to flow in resistor R in the same
direction in which current was flowing originally (when S_{ 1}
was closed). A voltage (e_{R}) that is initially equal to source
voltage (E_{S}) is developed across R. The voltage across the
resistor (e_{R}) rapidly falls to zero as the voltage across the
inductor (e_{L}) falls to zero due to the collapsing flux.
Just as the example of the boat was used to explain the growth of current in a circuit, it can also be used to explain the decay of current in a circuit. When the force applied to the boat is removed, the boat still continues to move through the water for a while, eventually coming to a stop. This is because energy was being stored in the inertia of the moving boat. After a period of time the friction of the water overcomes the inertia of the boat, and the boat stops moving. Just as inertia of the boat stored energy, the magnetic field of an inductor stores energy. Because of this, even when the power source is removed, the stored energy of the magnetic field of the inductor tends to keep current flowing in the circuit until the magnetic field collapse. What happens to the voltage across the resistance in an LR circuit during current buildup in the circuit, and during current decay in the circuit? L/R Time Constant
The L/R TIME CONSTANT is a valuable tool for use in determining the time required for current in an inductor to reach a specific value. As shown in figure (11), one L/R time constant is the time required for the current in an inductor to increase to 63 percent (actually 63.2 percent) of the maximum current. Each time constant is equal to the time required for the current to increase by 63.2 percent of the difference in value between the current flowing in the inductor and the maximum current. Maximum current flows in the inductor after five L/R time constants are completed. The following example should clear up any confusion about time constants. Assume that maximum current in an LR circuit is 10 amperes. As you know, when the circuit is energized, it takes time for the current to go from zero to 10 amperes. When the first time constant is completed, the current in the circuit is equal to 63.2% of 10 amperes. Thus the amplitude of current at the end of 1 time constant is 6.32 amperes.
During the second time constant, current again increases by 63.2% (.632) of the difference in value between the current flowing in the inductor and the maximum current. This difference is 10 amperes minus 6.32 amperes and equals 3.68 amperes; 63.2% of 3.68 amperes is 2.32 amperes. This increase in current during the second time constant is added to that of the first time constant. Thus, upon completion of the second time constant, the amount of current in the LR circuit is 6.32 amperes + 2.32 amperes = 8.64 amperes.
During the third constant, current again increases:
During the fourth time constant, current again increases:
During the fifth time constant, current increases as before:
Thus, the current at the end of the fifth time constant is almost equal to 10.0 amperes, the maximum current. For all practical purposes the slight difference in value can be ignored. When an LR circuit is reenergized, the circuit current decreases (decays) to zero in five time constants at the same rate that it previously increased. If the growth and decay of current in an LR circuit are plotted on a graph, the curve appears as shown in figure (11).Notice that current increases and decays at the same rate in five time constants. The value of the time constant in seconds is equal to the inductance in henrys divided by the circuit resistance in ohms. The formula used to calculate one L/R time constant is:
A circuit containing only an inductor and a resistor has a maximum of 12 amperes of applied current flowing in it. After 5 L/R time constants the circuit is opened. How many time constants is required for the current to decay to 1.625 amperes? |
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