Network Analysis Parallel Networks
InductancePrepared byMd. Amirul IslamLecturerDepartment of Applied Physics & ElectronicsBangabandhu Sheikh Mujibur Rahman Science & Technology University, Gopalganj 8100 Self-inductance2Self Induction and Back emf:Self-inductance IReference: Physics II by Robert Resnick and David Halliday, Topic 32.1, Page 1015Consider the circuit consisting of a switch, a resistor, and a source of emf, as shown in figure. When the switch is thrown to its closed position, the source current does not immediately jump from zero to its maximum value /R. Faradays law of electromagnetic induction can be used to describe this effect as follows: As the source current
increases with time, the magnetic flux through the circuit loop due to this current also increases with time. This increasing flux creates an induced emf in the circuit. The direction of the induced emf is such that it would cause an induced current in the loop, which would establish a magnetic field that would oppose the change in the source magnetic field.Self-inductance IReference: Physics II by Robert Resnick and David Halliday, Topic 32.1, Page 1015Thus, the direction of the induced emf is opposite the direction of the source emf; this results in a gradual rather than instantaneous increase in the source current to its final equilibrium value. This effect is called self-induction because the changing flux through the circuit
and the resultant induced emf arise from the circuit itself. The emf L set up in this case is called a self-induced emf. It is also often called a back emf.Inductor5Inductor or Solenoid:Self-inductance IIReference: Physics II by Robert Resnick and David Halliday, Topic 32.1, Page 1015Consider a coil wound on a cylindrical iron core as shown in figure.
(a) A current in the coil produces a magnetic field directed to the left (Screw rule) (b) If the current increases, the increasing magnetic flux creates an induced emf having the polarity shown by the dashed battery (Lenzs Law) (c) The polarity of the induced emf reverses if the current decreases.Quantitative Analysis:Self-inductance IIReference: Physics II by Robert Resnick and David Halliday, Topic 32.1, Page 1015From Faradays law we know that the induced emf L is equal to the negative time rate of change of the magnetic flux ( dB/dt). Again, a self-induced emf L is always proportional to the time rate of change of the source current (dI/dt).Thus, we can write,
where L is a proportionality constant called the inductance of the coil that depends on the geometry of the circuit and other physical characteristics. From this expression, we see that the inductance of a coil containing N turns is,
Unit of Inductance:Self-inductance IIReference: Physics II by Robert Resnick and David Halliday, Topic 32.1, Page 1015From the equation of induced emf, we can write that,
Thus the SI unit of inductance is Henry and can be written as,
Energy Stored in a Magnetic Field9Energy Stored in a Magnetic FieldReference: Physics II by Robert Resnick and David Halliday, Topic 32.3, Page 1021Because the emf induced in an inductor prevents a battery from establishing an instantaneous current, the battery must do work against the inductor to create a current. Part of the energy supplied by the battery appears as internal energy in the resistor, while the remaining energy is stored in the magnetic field of the inductor.
Applying KVL, we get,
I is the energy supplied from the source, I2R is the energy delivered to the resistor and thus, LI(dI/dt) is the energy stored in the inductor.Energy Stored in a Magnetic FieldReference: Physics II by Robert Resnick and David Halliday, Topic 32.3, Page 1021If we let U denote the energy stored in the inductor at any time, then we can write the rate dU/dt at which energy is stored as,
To find the total energy stored in the inductor, we can rewrite this expression as dU = LIdI and integrate over the limit 0 to I:This is the equation of energy stored in an inductor.Mutual Inductance12Mutual InductanceReference: Physics II by Robert Resnick and David Halliday, Topic 32.3, Page 1021Very often, the magnetic flux through the area enclosed by a circuit varies with time because of time-varying currents in nearby circuits. This condition induces an emf through a process known as mutual induction, so called because it depends on the interaction of two circuits.
Consider the two closely wound coils of wire in cross-sectional view in figure. The current I1 in coil 1, which has N1 turns, creates magnetic field lines, some of which pass through coil 2, which has N2 turns. The magnetic flux caused by the current in coil 1 and passing through coil 2 is represented by 12.Mutual InductanceReference: Physics II by Robert Resnick and David Halliday, Topic 32.3, Page 1021
In analogy to equation L = N/I, we define the mutual inductance M12 of coil 2 with respect to coil 1:
Induced emf in coil 2 is,
Mutual InductanceReference: Physics II by Robert Resnick and David Halliday, Topic 32.3, Page 1021
In the preceding discussion, we assumed that the source current is in coil 1. We can also imagine a source current I2 in coil 2. The preceding discussion can be repeated to show that there is a mutual inductance M21 . If the current I2 varies with time, the emf induced by coil 2 in coil 1 is,
It can be experimentally shown that,M12 = M21 = M and thus,
The unit of mutual inductance is also Henry.Ideal Transformer16Ideal TransformerReference: Physics II by Robert Resnick and David Halliday, Topic 33.8, Page 1060When electric power is transmitted over great distances, it is economical to use a high voltage and a low current to minimize the I2R loss in the transmission lines. Consequently, 33,000V lines are common. At the receiving end of such lines, the consumer requires power at a low voltage. Therefore, a device is required that can change the alternating voltage and current without causing appreciable changes in the power delivered. The ac transformer is that device.Construction:The ac transformer consists of two coils of wire wound around a core of iron, as illustrated in figure. The coil on the left, which is connected to the input alternating voltage source and has N1 turns, is called the primary winding (or the primary).
Ideal TransformerReference: Physics II by Robert Resnick and David Halliday, Topic 33.8, Page 1060The coil on the right, consisting of N2 turns and connected to a load resistor R, is called the secondary winding (or the secondary). The purpose of the iron core is to increase the magnetic flux through the coil and to provide a medium in which nearly all the flux through one coil passes through the other coil. Eddy current losses are reduced by using a laminated core. Iron is used as the core material because it is a soft ferromagnetic substance and hence reduces hysteresis losses. Although practical transformer have some power loss due to the resistance of the coil wire, as we assumed an ideal transformer, so energy losses in the windings and core are zero.
Working Principle:According to Faradays law the voltage V1 across the primary is,
Ideal TransformerReference: Physics II by Robert Resnick and David Halliday, Topic 33.8, Page 1060where B is the magnetic flux through each turn. If we assume that all magnetic field lines remain within the iron core, the flux through each turn of the primary equals the flux through each turn of the secondary. Hence, the voltage across the secondary is:
Dividing and then rearranging these two equations, we get,
When, N2 > N1, then V2 > V1. This setup is referred to as a step-up transformer. When N2 < N1 , the output voltage is less than the input voltage, and we have a step-down transformer.Ideal TransformerReference: Physics II by Robert Resnick and David Halliday, Topic 33.8, Page 1060For an ideal transformer, there is no power loss on primary or secondary winding. Thus,
When a resistive load RL is connected to the secondary, then current I2 on the secondary will be,
Furthermore, the current in the primary is,
Ideal TransformerReference: Physics II by Robert Resnick and David Halliday, Topic 33.8, Page 1060The above equation relates to the input resistance to the output resistance. Req is the equivalent resistance of the load resistance when viewed from the primary side. From this analysis we see that a transformer may be used to match resistances (impedance matching) between the primary circuit and the load. In this manner, maximum power transfer can be achieved between a given power source and the load resistance.