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Because of the effect of capacitance, an electrical circuit can store energy, even after being de-energized.



Electrical devices that are constructed of two metal plates separated by an insulating material, called a dielectric, are known as capacitors (Figure 1a). Schematic symbols shown in Figures 1b and 1c apply to all capacitors.

Figure 1 Capacitor and Symbols

The two conductor plates of the capacitor, shown in Figure 2a, are electrically neutral, because there are as many positive as negative charges on each plate. The capacitor, therefore, has no charge.

Now, we connect a battery across the plates (Figure 2b). When the switch is closed (Figure 2c), the negative charges on Plate A are attracted to the positive side of the battery, while the positive charges on Plate B are attracted to the negative side of the battery. This movement of charges will continue until the difference in charge between Plate A and Plate B is equal to the voltage of the battery. This is now a "charged capacitor." Capacitors store energy as an electric field between the two plates.

Figure 2 Charging a Capacitor

Because very few of the charges can cross between the plates, the capacitor will remain in the charged state even if the battery is removed. Because the charges on the opposing plates are attracted by one another, they will tend to oppose any changes in charge. In this manner, a capacitor will oppose any change in voltage felt across it.

If we place a conductor across the plates, electrons will find a path back to Plate A, and the charges will be neutralized again. This is now a "discharged" capacitor (Figure 3).

Figure 3 Discharging a Capacitor


Capacitance is the ability to store an electrical charge. Capacitance is equal to the amount of charge that can be stored divided by the applied voltage, as shown in Equation (3-7).

The unit of capacitance is the farad (F). A farad is the capacitance that will store one coulomb of charge when one volt is applied across the plates of the capacitor.

The dielectric constant (K) describes the ability of the dielectric to store electrical energy. Air is used as a reference and is given a dielectric constant of 1. Therefore, the dielectric constant is unitless. Some other dielectric materials are paper, teflon, bakelite, mia, and ceramic.

The capacitance of a capacitor depends on three things.

  1. Area of conductor plates
  2. Separation between the plates
  3. Dielectric constant of insulation material

Equation (3-8) illustrates the formula to find the capacitance of a capacitor with two parallel plates.

Example 1: Find the capacitance of a capacitor that stores 8 C of charge at 4 V.

Example 2: What is the charge taken on by a 5F capacitor at 2 volts?

Example 3: What is the capacitance if the area of a two plate mica capacitor is 0.0050 m2 and the separation between the plates is 0.04 m? The dielectric constant for mica is 7.

Types of Capacitors

All commercial capacitors are named according to their dielectrics. The most common are air, mica, paper, and ceramic capacitors, plus the electrolytic type. These types of capacitors are compared in Table 1.

Capacitors in Series and Parallel

Capacitors in series are combined like resistors in parallel. The total capacitance, CT, of capacitors connected in series (Figure 4), is shown in Equation (3-9).

Figure 4 Capacitors Connected in Series

When only two capacitors are in series, Equation (3-9) may be simplified as given in Equation (3-10). As shown in Equation (3-10), this is valid when there are only two capacitors in series.

When all the capacitors in series are the same value, the total capacitance can be found by dividing the capacitors value by the number of capacitors in series as given in Equation (3-11).

Capacitors in parallel are combined like resistors in series. When capacitors are connected in parallel (Figure 5), the total capacitance, CT, is the sum of the individual capacitances as given in Equation (3-12).

Figure 5 Capacitors Connected in Parallel

Example 1: Find the total capacitance of 3µF, 6µF, and 12µF capacitors connected in series (Figure 6).

Figure 6 Example 1-Capacitors Connected in Series

Example 2: Find the total capacitance and working voltage of two capacitors in series, when both have a value of 150 µF, 120 V (Figure 7).

Figure 7 Example 2-Capacitors Connected in Series

Total voltage that can be applied across a group of capacitors in series is equal to the sum of the working voltages of the individual capacitors. working voltage = 120 V + 120 V = 240 volts

Example 3: Find the total capacitance of three capacitors in parallel, if the values are 15 µF-50 V, 10 µF-100 V, and 3 µF-150 V (Figure 8). What would be the working voltage?

Figure 8 Example 3-CapacitorsConnected in Parallel

The working voltage of a group of capacitors in parallel is only as high as the lowest working voltage of an individual capacitor. Therefore, the working voltage of this combination is only 50 volts.

Capacitive Time Constant

When a capacitor is connected to a DC voltage source, it charges very rapidly. If no resistance was present in the charging circuit, the capacitor would become charged almost instantaneously. Resistance in a circuit will cause a delay in the time for charging a capacitor. The exact time required to charge a capacitor depends on the resistance (R) and the capacitance (C) in the charging circuit. Equation (3-13) illustrates this relationship.

The capacitive time constant is the time required for the capacitor to charge to 63.2 percent of its fully charged voltage. In the following time constants, the capacitor will charge an additional 63.2 percent of the remaining voltage. The capacitor is considered fully charged after a period of five time constants (Figure 9).

Figure 9 Capacitive Time Constant for Charging Capacitor

The capacitive time constant also shows that it requires five time constants for the voltage across a discharging capacitor to drop to its minimum value (Figure 10).

Figure 10 Capacitive Time Constant for Charging Capacitor

Example: Find the time constant of a 100 µF capacitor in series with a 100Ω resistor(Figure 11).

Figure 11 Example-Capacitive Time Constant


Experiments investigating the unique behavioral characteristics of inductance led to the invention of the transformer.


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 12, 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.

Figure 12 Induced EMF

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.

This is an example of Faradays 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 Lenzs Law, which states that the induced EMF opposes the EMF that caused it.

The three requirements for inducing an EMF are:

  1. a conductor,
  2. a magnetic field,and
  3. relative motionbetween the two.

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 13.

Figure 13 Induced EMF in Coils

Self-induced EMF is another phenomenon of induction. The circuit shown in Figure 14 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 14 Induced EMF in Coils

The polarity of self-induced EMF is given to us by Lenzs 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 inductors 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).

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 15) is given by Equation (3-2). Leq = L1 + L2 + ... Ln (3-2)

Inductors in parallel are combined like resistors in parallel as given by Equation (3-3).

Figure 15 Inductors in Series

When only two inductors are in parallel, as shown in Figure 16, 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 16 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 17).

Figure 17 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 18).

Figure 18 Time Constant

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).

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).

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 19).

Figure 19 Time Constant

  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 10v/100Ω = 0.1A 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.
  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 theswitch 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 1Ω resistor (Figure 20).

Figure 20 Inductor and Resistor in Parallel

  1. While the switch is closed, a current of 20 v/1Ω = 20 ampsflows through the inductor. This causes a very large magnetic field around the inductor.
  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 101Ω ofresistance. The CEMF = (101)(20) = 2020 volts.
  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: