Inductance in Alternating-Current Circuits
REVIEW OF ELECTROMAGNETISM
Direct Current Fundamentals discussed the following two basic principles of electro- magnetism: (1) a magnetic field surrounds every current-carrying conductor or coil winding, and (2) an increase or decrease in the current causes an increase or decrease in the number of lines of force of this magnetic field.
A changing magnetic field induces a voltage in the conductor, coil, or circuit. This voltage is proportional to the rate of change of the lines of force cutting across the conductor, coil, or circuit. In a direct-current circuit, there will be no inductive effect once the cur- rent reaches the value defined by Ohm’s law and remains constant. However, if the current changes in value, inductance does have a momentary effect. For example, if the current increases, more lines of flux will link the turns of wire in the coil winding. This change in flux linkage will cause a momentary induced voltage that opposes the increase in current. If the current decreases, there will be a decrease in the lines of flux linking the coil, resulting in a momentary induced voltage that attempts to maintain the current. These inductive effects are explained by Lenz’s law, which states:
In all cases of electromagnetic induction, the induced voltage and the resulting current are in such a direction as to oppose the effect producing them.
Inductance in a DC Circuit
The effect of inductance in a direct-current circuit can be shown using the circuit in Figure 3–1.
This circuit consists of a group of lamps connected in series with a coil of wire mounted on a laminated steel core. (This coil has a relatively low resistance.) When the circuit is energized from a 120-V dc supply, a short amount of time passes before the lamps shine at full brightness. Also, the needle of the dc ammeter moves upscale to the Ohm’s law value of current at a rate that is much slower than in a circuit containing only resistance. As the current increases to its Ohm’s law value, more and more lines of force link the turns of the coil to form a magnetic field. This increasing flux causes an induced voltage that opposes the impressed voltage (120-V supply) and limits the current in amperes. This circuit situation is a typical example of Lenz’s law in operation.
Figure 3–2 shows the current increasing to its Ohm’s law value. Also shown is the momentary induced voltage for the circuit of Figure 3–1. At the instant the switch is closed, the current rises rapidly. Note, however, that the rate of current rise decreases with time. When the current reaches its Ohm’s law value, the rate of increase becomes zero. At the same instant when the switch is closed and the circuit is energized, the induced voltage has its maximum value. At this time, there is the greatest rate of increase in the number of lines of force linking the turns of the coil. Thus, the maximum voltage is induced. As the rate at which the current rises decreases with time, the induced voltage also decreases. Finally, when the current reaches a constant (Ohm’s law) value, the lines of force of the magnetic field will reach a maximum, resulting in a maximum field value. The induced voltage will decrease to zero when the current reaches its constant value.
When the circuit switch is opened, there will be a noticeable arc at the switch contacts. As the current decreases to zero (Figure 3–3), the lines of force collapse back into the turns of the coil. The cutting action of the collapsing lines of force is in a direction opposite that of the increasing field (when its switch is closed). As a result, the induced voltage will be in the same direction as the decaying current. This voltage will attempt to maintain the cur- rent, resulting in an arc at the switch contacts.
The decay of current in an inductive circuit is shown in Figure 3–3. The curve profiles for the decay of current and voltage may vary considerably from those shown. The actual curve profiles will depend on the time required for the line switch contacts to open.
A special switching arrangement must be used in the series circuit to observe the decay of the current and voltage. For example, assume that a single-pole switch is connected across the line wire of the series circuits in Figure 3–1. This switch closes at the instant the line switch contacts open. The length of time required for the current to decay to zero will equal the time required for the current to rise to its Ohm’s law value when the circuit is energized.
In Direct Current Fundamentals, it was shown that inductance is really a form of electrical inertia. Inductance is the ability of a circuit component, such as a reactor coil, to store energy in the electromagnetic field. The unit of inductance is the henry. The symbol for the henry is “H.” A henry is defined as follows:
A henry (H) is the inductance of a circuit, or a circuit component, when a current change of one ampere per second induces a voltage of one volt.
The Series Circuit Connected to an AC Supply
If the series circuit shown in Figure 3–1, consisting of a reactor coil and lamps, is connected to a 120-V, 60-Hz alternating-current supply, the lamps will be very dim. An ac ammeter connected into the circuit will show an effective current value that is lower than that of the current in the dc circuit. This condition applies even though the effective value of the ac voltage is the same as that of the dc source. Also, the resistance of the lamps is almost the same as in the dc circuit. The current reduction indicated by the ammeter reading and the dim lamps is caused by the “choking” effect of inductance in an ac circuit.
The alternating current supplied by the 60-Hz source is changing in magnitude and direction continually. As a result, a countervoltage is induced in the coil. This voltage opposes the impressed voltage and thus limits the current in the series circuit.
To demonstrate that the inductive effect shown in the dc circuit is the cause of the current reduction, the laminated core is slowly removed from the coil. As the core is withdrawn from the coil, the lamps will increase in brightness and the ammeter read- ing will increase. When the core is completely removed, the reluctance of the magnetic circuit of the coil will increase. This means that there is less flux rising and collapsing around the turns of the coil. Thus, the induced voltage decreases. If the core is replaced in the coil, the lamps will dim again and the ac ammeter reading will decrease. If the frequency of the ac source is reduced from 60 Hz to 25 Hz and the effective value of the line voltage is the same, the current will increase. As a result, the ammeter reading will increase and the lamps will be brighter. The lower frequency means that there are fewer cycles per second. Thus, there are fewer changes of current and lines of force per second. The induced or counter voltage in the coil will be less, resulting in an increase in the current.
It can be seen from this discussion that inductance in an alternating-current circuit is just as effective in limiting the current as resistance. This means that both inductance and resistance must be considered in any calculations for ac circuits.
The repeated changes in the direction and magnitude of alternating current give rise to an induced voltage, which limits the current in an inductive circuit. This opposition due to the inductance is called inductive reactance. Inductive reactance is indicated by the symbol X
and is measured in ohms.
Problem in Inductive Reactance
To illustrate the use of the formula for inductive reactance, assume that a coil has an inductance of 0.2652 henry (H) and negligible resistance. This coil is connected across a 60-Hz supply with an effective voltage of 100 V, as shown in Figure 3–4. Determine the inductive reactance of the coil in ohms and also determine the current.
The inductive reactance of the reactor coil in the figure is as follows:
For a frequency of 60 Hz, it is an accepted practice to assume that the product of 2'ITf is 377 rather than 376.8. For a frequency of 25 Hz, the product of 2'ITf is 157. These values are convenient and may be used in the inductive reactance formula.
The current in the reactor coil of Figure 3–4 is found using the expression
The symbol w is the lowercase Greek letter omega. It represents angular velocity. Before angular velocity is defined, it is important to review the meaning of the term radian.
The radian is a unit of angular measurement. This unit is sometimes used in place of electrical time degrees. The definition of the radian is as follows:
One radian is equal to the angle at the center of a circle, subtended by an arc whose length is equal to the radius.
In Figure 3–5, the length of the arc AB is equal to the radius R, or OA. By definition, the angle AOB is equal to one radian and is slightly less than 60°. (Actually, one radian is 57.296°.) The diameter of a circle multiplied by 'IT equals the circumference of the circle. The circumference of a circle can also be obtained by multiplying the radius of the circle by 2'IT. Thus, the actual angle represented by a radian is 360 --: 2'IT = 57.296°.
One cycle is equal to 360 electrical time degrees. The circle shown in Figure 3–5, while illustrating angular measurement in radians, also represents 360°. Recall that the number of cycles per second is the frequency (f). Therefore, 2'IT radians is the angular velocity per cycle of the rotating radius line. The angular velocity per second is represented by 2'ITf. The Y projection of the rotating radius line is the sine wave of frequency, f (see Figure 1–8). If w represents the angular velocity per second, then the formula for inductive reactance may be stated in either of the following forms:
Current in an Inductive Circuit
The reactor coil in Figure 3–4 consists of a 100-0 inductive reactance and negligible resistance. If this coil is connected across an ac potential of 100 V, the current will be one ampere. However, it will be seen that this one-ampere current lags
behind the voltage by 90 electrical degrees. Although this condition may not seem possible, a study of Lenz’s law and the behavior of the induced voltage with an increase of current in a coil connected to a dc source (Figure 3–2) shows that the current may well lag behind the voltage.
In an ac circuit containing only inductance, the current is opposed by the induced volt- age and thus cannot increase immediately with the impressed voltage. In Figure 3–6, the line voltage was omitted to make it easier to see the relationship between the alternating current and the induced voltage. The current is shown with its negative maximum value at the start of the cycle. At this point, and for a brief instant, there is no change in the current. Hence, there is no induced voltage. As the current decreases toward zero, the rate of change of current is increasing and the induced voltage increases. This voltage is represented by the dashed line.
The greatest rate of change of current for a given time occurs as it passes through zero. This means that the induced voltage has a maximum negative value at this same time. When the current passes through zero (at 90°) and increases to its positive maximum value, the rate of change of current is decreasing. As a result, the change in the lines of force in a given time period decreases proportionally and the induced voltage is reduced.
When the current reaches its positive maximum value at 180°, there will be a very brief period of time when there is no change in the current. For this period, the induced voltage is again zero. As the current decreases and passes through zero at 270° in a negative direction, the induced voltage reaches its positive maximum value. By developing the induced wave pattern for the remainder of the cycle, a sine wave of induced voltage is formed that lags behind the sine wave of current by 90 electrical degrees. To maintain the current in this coil, the line voltage applied must be equal and opposite to the induced volt- age. Obviously, if the applied line voltage is directly opposite to the induced voltage, then the two voltages are 180° out of phase with each other.
Figure 3–7 is similar to Figure 3–6, except that the impressed line voltage is also shown. It can be seen that the line voltage and the induced voltage are 180° out of phase. Note also that the current lags the line voltage by 90°.
When the current and the line voltage are both positive, energy is being stored temporarily in the magnetic field of the coil. Energy is also being stored in the magnetic field when both the current and the line voltage are negative. However, when the current is positive and the line voltage is negative, energy is returned from the inductor coil to the supply. This return of energy occurs because the current and voltage are acting in opposition. In those parts of the cycle where the line voltage is positive and the current is negative, energy is also released from the magnetic field of the inductor coil and returned to the supply. This energy released from the electromagnetic field of an inductor coil maintains the current when the voltage and current are in opposition.
The inductor coil circuit in Figure 3–4 has an inductive reactance of 100 0. The line voltage is 100 V and the current is one ampere. The line voltage can be plotted as a sine wave with a maximum value of 141.4 V. The line current lags the line voltage by 90 electrical degrees. The RMS value of current is one ampere, and the maximum current value is 1.414 A.
The power in watts at any instant is equal to the product of the volts and amperes at the same instant. Table 3–1 lists the instantaneous values of voltage, current, and power at 15° intervals for one complete cycle (360°) for this inductive circuit.
The wave patterns in Figure 3–8 were plotted using the values given in Table 3–1. Note that the current lags the line voltage by 90 electrical degrees. It is assumed that the reactor coil has no resistance and the circuit consists of pure inductance. Between 0° and 90°, the current is negative and the voltage is positive. Therefore, the product of volts and amperes for this part of the cycle gives negative power.
Between 90° and 180°, both the current and the voltage are positive. This means that there is a pulse of power above the zero reference line. This positive power indicates that the supply is feeding energy to the electromagnetic field of the coil.
In the period between 180° and 270°, there is a second pulse of negative power. This is due to the negative voltage and the positive current. The negative power pulse indicates that
energy stored in the magnetic field of the coil is released to the supply. This energy, which is stored momentarily in the coil, maintains the current in opposition to the line voltage.
Between 270° and 360°, both the current and the voltage are negative and are acting together. The product of these negative values results in positive power, as shown by the last power pulse for the cycle in Figure 3–8. Because the power pulse is above zero, energy is being supplied by the source to the reactor coil.
The power waveform in Figure 3–8 shows that the areas of the two positive pulses of power are equal to the areas of the two negative pulses of power. Keep in mind that the pulses of power above the zero reference line are positive power that is fed by the source to the load. The pulses of power below the zero reference line are called negative power. These pulses represent power that is returned from the load to the source. The assumption is made that the areas of the two positive pulses of power are
equal to the areas of the two negative pulses of power. This means that the net power taken by the inductor coil at the end of one complete cycle, or any number of cycles, is zero.
The actual power in watts taken by this circuit is zero. It should be noted that the product of the effective voltage and the effective current in amperes may not equal the power in watts in any circuit containing inductive reactance.