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Experiments investigating the unique behavioral characteristics of inductance led to the invention of the transformer. EO 1.1 DESCRIBE how current flow, magnetic field, and stored energy in an inductor relate to one another. EO 1.2 DESCRIBE how an inductor opposes a change in current flow. EO 1.3 Given a circuit containing inductors, CALCULATE total inductance for series and parallel circuits. EO 1.4 Given an inductive resistive circuit, CALCULATE the time constant for the circuit. Inductors An inductor is a circuit element that will store electrical energy in the form of a magnetic field. It is usually a coil of wire wrapped around a core of permeable material. The magnetic field is generated when current is flowing through the wire. If two circuits are arranged as in Figure 1, a magnetic field is generated around Wire A, but there is no electromotive force (EMF) induced into Wire B because there is no relative motion between the magnetic field and Wire B. If we now open the switch, the current stops flowing in Wire A, and the magnetic field collapses. As the field collapses, it moves relative to Wire B. When this occurs, an EMF is induced in Wire B.
Figure 1 Induced EMF This is an example of Faraday's Law, which states that a voltage is induced in a conductor when that conductor is moved through a magnetic field, or when the magnetic field moves past the conductor. When the EMF is induced in Wire B, a current will flow whose magnetic field opposes the change in the magnetic field that produced it. For this reason, an induced EMF is sometimes called counter EMF or CEMF. This is an example of Lenz's Law, which states that the induced EMF opposes the EMF that caused it. The three requirements inducing an EMF are: 1. a conductor, 2. a magnetic field, and 3. relative motion between the two.
Figure 2 Induced EMF in Coils The faster the conductor moves, or the faster the magnetic field collapses or expands, the greater the induced EMF. The induction can also be increased by coiling the wire in either Circuit A or Circuit B, or both, as shown in Figure 2. Self-induced EMF is another phenomenon of induction. The circuit shown in Figure 3 contains a coil of wire called an inductor (L). As current flows through the circuit, a large magnetic field is set up around the coil. Since the current is not changing, there is no EMF produced. If we open the switch, the field around the inductor collapses. This collapsing magnetic field produces a voltage in the coil. This is called self-induced EMF.
Figure 3 Self-Induced EMF The polarity of self-induced EMF is given to us by Lenz's Law. The polarity is in the direction that opposes the change in the magnetic field that induced the EMF. The result is that the current caused by the induced EMF tends to maintain the same current that existed in the circuit before the switch was opened. It is commonly said that an inductor tends to oppose a change in current flow. The induced EMF, or counter EMF, is proportional to the time rate of change of the current. The proportionality constant is called the "inductance" (L). Inductance is a measure of an inductor's ability to induce CEMF. It is measured in henries (H). An inductor has an inductance of one henry if one amp per second change in current produces one volt of CEMF, as shown in Equation (3-1).
where
The minus sign shows that the CEMF is opposite in polarity to the applied voltage. Example: A 4-henry inductor is in series with a variable resistor. The resistance is increased so that the current drops from 6 amps to 2 amps in 2 seconds. What is the CEMF induced?
Inductors in series are combined like resistors in series. Equivalent inductance (Leq) of two inductors in series (Figure 4) is given by Equation (3-2). Figure 4 Inductors in Series Leq = Ll + L2 + ... Ln (3-2)
Inductors in parallel are combined like resistors in parallel as given by Equation (3-3).
When only two inductors are in parallel, as shown in Figure 5, Equation (3-3) may be simplified as given in Equation (3-4). As shown in Equation (3-4), this is valid when there are only two inductors in parallel.
Figure 5 Inductors in Parallel
Inductors will store energy in the form of a magnetic field. Circuits containing inductors will behave differently from a simple resistance circuit. In circuits with elements that store energy, it is common for current and voltage to exhibit exponential increase and decay (Figure 6).
Figure 6 DC Current Through an Inductor The relationship between values of current reached and the time it takes to reach them is called a time constant. The time constant for an inductor is defined as the time required for the current either to increase to 63.2 percent of its maximum value or to decrease by 63.2 percent of its maximum value (Figure 7).
The value of the time constant is directly proportional to the inductance and inversely proportional to the resistance. If these two values are known, the time constant can be found using Equation (3-5).
where TL = time constant (seconds) L = inductance (henries) R = resistance (ohms) The voltage drop across an inductor is directly proportional to the product of the inductance and the time rate of change of current through the inductor, as shown in Equation (3-6).
where
After five time constants, circuit parameters normally reach their final value. Circuits that contain both inductors and resistors are called RL circuits. The following example will illustrate how an RL circuit reacts to changes in the circuit (Figure 8). 1. Initially, the switch is in Position 1, and no current flows through the inductor. 2. When we move the switch to Position 2,
the battery attempts to force a current of
Figure 8 Voltage Applied to an Inductor 3. When the switch is returned to Position 1, the magnetic field collapses, inducing an EMF that tends to maintain current flow in the same direction through the inductor. Its polarity will be opposite to that induced when the switch was placed in Position 2. The example that follows shows how a circuit with an
inductor in parallel with a resistor reacts to changes in the circuit.
Inductors have some small resistance, and this is shown schematically as a 1 1. While the
switch is closed, a current of 20 v/1 2. When we
open the switch, there is no longer a current through the inductor. As the
magnetic field begins to collapse, a voltage is induced in the inductor. The
change in applied voltage is instantaneous; the counter EMF is of exactly the
right magnitude to prevent the current from changing initially. In order to
maintain the current at 20 amps flowing through the inductor, the self-induced
voltage in the inductor must be enough to push 20 amps through the 101 3. With the switch open, the circuit looks like a series RL circuit without a battery. The CEMF induced falls off, as does the current, with a time constant TL Of:
Figure 9 Inductor and Resistor in Parallel
Summary The important information on inductors is summarized below. Inductance Summary When an inductor has a DC current flowing through it, the inductor will store energy in the form of a magnetic field. An inductor will oppose a change in current flow by the CEMF induced when the field collapses or expands. Inductors in series are combined like resistors in series. Inductors in parallel are combined like resistors in parallel. The time constant for an inductor is defined as the required time for the current either to increase to 63.2 percent of its maximum value or to decrease by 63.2 percent of its maximum value.
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O.1A through the inductor. But as current begins to flow, the inductor
generates a magnetic field. As the field increases, a counter EMF is induced
that opposes the battery voltage. As a steady state is reached, the counter EMF
goes to zero exponentially.
resistor (Figure 9).
