Negative amount that the two sine waves are out of phase is alternation indicated by the number of electrical degrees 0° 90° 180° 1 270° 360° between corresponding peaks on the sine waves. In Positive figure 8-169, the current and voltage are 30° out alternation of phase.
1 Second time - Frequency = 2 cycles per sec.
Values of Alternating Current Frequency = 2 cycles per sec.
90,180* 360 270°1 1 Second time
• Frequency = 8 cycles per sec.
FIGURE 8-167. Frequency in cycles per second.
speed of 1,800 r.p.m., substitute in the equation, F = 2 P r.p.m. , as follows:
60 F=½ 1800 60 F = 2 × 30 F = 60 c.p.s.
In addition to frequency and cycle characteristics, alternating voltage and current also have a relationship called "phase." In a circuit that is fed (supplied) by one alternator, there must be a certain phase relationship between voltage and current if the circuit is to function efficiently. In a system fed by two or more alternators, not only must there be a certain phase relationship between voltage and current of one alternator, but there must be a phase relationship between the individual voltages and the individual currents. Also, two separate circuits can be compared by comparing the phase characteristics of one to the phase characteristics of the other.
When two or more sine waves pass through 0° and 180° at the same time and reach their peaks at the same time, an in-phase condition exists, as shown in figure 8-168. The peak values (magnitudes) do not have to be the same for the in-phase condition to exist. When the sine waves pass through 0° and 180° at different times and reach their peaks at different times, an out-of-phase condition exists, as shown in figure 8-169. The There are three values of alternating current which should be considered. They are instantaneous, maximum, and effective.
An instantaneous value of voltage or current is the induced voltage or current flowing at any instant. The sine wave is a series of these values.
The instantaneous value of the voltage varies from zero at 0° to maximum at 90°, back to zero at 180°, to maximum in the opposite direction at 270°, and to zero again at 360°. Any point on the sine wave is considered the instantaneous value of voltage.
The maximum value is the largest instantaneous value. The largest single positive value occurs when the sine wave of voltage is at 90°, and the largest single negative value occurs when it is at 270°. These are called maximum values. Maximum value is 1.41 times the effective value. (See figure 8-170.)
The effective value of alternating current is the same as the value of a direct current which can produce an equal heating effect. The effective value is less than the maximum value, being equal to .707 times the maximum value. Thus, the 110-volt value given for alternating current supplied to homes is only .707 of the maximum voltage of this supply. The maximum voltage is approximately 155 volts (110 X 1.41 = 155 volts maximum).
In the study of alternating current, any values given for current or voltage are assumed to be effective values unless otherwise specified, and in Voltage Current 90° 180° 270° 360° FIGURE 8-168. In-phase condition of current and voltage.
VOLTAGE 180° 270° 360° 90° 150°
CURRENT FICURE 8-169. Out-of-phase condition of current and voltage.
practice, only the effective values of voltage and current are used. Similarly, alternating current voltmeters and ammeters measure the effective value.
cross-sectional area of the coil, and the material in the center of the coil or core. A core of magnetic material greatly increases the inductance of the coil.
INDUCTANCE When an alternating current flows through a coil of wire, the rise and fall of the current flow, first in one direction, and then in another, sets up an expanding and collapsing magnetic field about the coil. A voltage is induced in the coil which is opposite in direction to the applied voltage and which opposes any change in the alternating current (see figure 8-171). The induced voltage is called the counter-electromotive force(abbreviated It must be remembered, however, that even a straight wire has inductance, small though it may be when compared to that of a coil. A.C. motors, relays, and transformers contribute inductance to a circuit, Practically all a.c. circuits contain inductive elements.
The symbol for inductance in formulas is the capital letter "L. Inductance is measured in henrys (abbreviated h). An inductor (coil) has an inductance of 1 henry if an e.m.f. of 1 volt is induced in the inductor when the current through
c.e.m.t.), since it opposes the applied voltage. This property of a coil to oppose any change in the current flowing through it is called inductance.
The inductance of a coil is measured in henrys.
the inductor is changing at the rate of 1 ampere per second. However, the henry is a large unit of inductance and is used with relatively large inductors having iron cores. The unit used for In any coil, the inductance depends on several small air-core inductors is the millihenry (mh).
factors, principally the number of turns, the For still smaller air-core inductors the unit of Maximum value = 1.41 effective value 1 Effective value = .707 maximum value G H FIGURE 8-170. Effective and maximum values of voltage.
iron core • Mounted air core- ragon Inductance FIGURE 8-171. A.C. circuit containing inductance.
inductance is the microhenry (uk). Figure 8-172 shows a few of the various types of inductors, together with their symbols.
Inductors may be connected in a circuit in the same manner as resistors. When connected in series, the total inductance is the sum of the inductances of the inductors, or Lp = L1 + La + La, etc.
When two or more inductors are connected in parallel, the total inductance is, like resistances in parallel, less than that of the smallest inductor, or LI = I + Is The total inductance of inductors connected in series-parallel can be computed by combining the parallel inductances and then adding the series values. In all cases, these formulas are valid, providing the magnetic fields of the inductors do not interact.
Inductive Reactance The opposition to the flow of current which inductances put in a circuit is called inductive reactance. The symbol for inductive reactance is Xz, and is measured in ohms, just as resistance is.
In any circuit in which there is only resistance, the expression for the relationship of voltage and current is Ohm's law: I = R• Similarly, when there is inductance in an a.c. circuit, the relationship between voltage and current can be expressed as:
Voltage Reactance or 1 - $ Current = Where:
XL = inductive reactance of the circuit in ohms.
Radio frequency → Radio frequency choke Iron core - ell- Radio frequency 002500 FIGURE 8-172. Various types of inductors.
If all other circuit values remain constant, the greater the inductance in a coil, the greater the effect of self-induction, or opposition to the change in the value of current. As the frequency increases, the inductive reactance increases, since the greater the rate of current change the more the opposition to change by the coil increases. Therefore, inductive reactance is proportional to inductance and frequency or, XI = 2 TfL.
Where:
Xz - inductive reactance in ohms.
= frequency in cycles per second.
* = 3.1416.
In figure 8-173, an a.c. series circuit is assumed in which the inductance is 0.146 henry and the voltage is 110 volts at a frequency of 60 cycles per second. What is the inductive reactance and the current flow? (The symbol) represents an a.c.
generator.)
110V AC B0 cycles FIGURE 8-175. Inductances in parallel.
FIGURE 8-173. A.C. circuit containing inductance.
Solution: CAPACITANCE To find the inductive reactance: Another important property in a.c. circuits, XI = 2= X fX L besides resistance and inductance, is capacitance.
While inductance is represented in a circuit by a X= = 6.28 × 60 × 0.146. coil, capacitance is represented by a capacitor.
To find current:
1 = Xi E 1 = 1º 55 I = 2 amperes.
In a.c. series circuits (figure 8-174), inductive reactances are added like resistances in series in a
d.c. circuit. Thus, the total reactance in the circuit illustrated in figure 8-174 equals the sum of the individual reactances.
The total reactance of inductors connected in parallel (figure 8-175) is found the same way as the total resistance in a parallel circuit. Thus, the total reactance of inductances connected in parallel, as shown, is expressed as, (Xx)r = (Xz)ı Any two conductors separated by a nonconductor, called a dielectric, constitute a capacitor.
In an electrical circuit, a capacitor serves as a reservoir or storehouse for electricity.
When a capacitor is connected across a source of direct current, such as a storage battery in the circuit shown in figure 8-176, and the switch is then closed, the plate marked B becomes positively charged, and the A plate negatively charged.
Current flows in the external circuit during the time the electrons are moving from B to A. The current flow in the circuit is maximum the instant the switch is closed, but continually decreases thereafter until it reaches zero. The current becomes zero as soon as the difference in voltage of A and B becomes the same as the voltage of the battery. If the switch is opened, the plates remain charged. However, the capacitor quickly discharges when it is short circuited.
The amount of electricity a capacitor can store depends on several factors, including the type of material of the dielectric. It is directly proportional to the plate area and inversely proportional to the distance between the plates.
B ++++++ FIGURE 8-174. Inductances in series. FIGURE 8-176. Capacitor in a d.c. circuit.
Battery SW position Capacitor position Meter FIGURE 8-177. A basic capacitor (condenser) circuit.
In figure 8-177, two flat metal plates are placed close to each other (but not touching). Usually the plates are electrically neutral; that is, no electrical charge will be evident on either plate. At the instant the switch is closed to the battery position, the meter will show a definite current surge in one direction, but almost instantly will return to zero.
If the battery is then taken out of the circuit and the switch closed in the capacitor position, the meter again shows a momentary current surge, but this time in an opposite direction. From this experiment it is apparent that the two plates store • energy when connected to a voltage source, and release the energy when short-circuited. The two plates make up a simple electrical capacitor, or condenser, and possess the property of storing electricity. The energy is actually stored in the electric, or dielectric, field between the plates.
Also, it should be clear that during the time the capacitor is being charged or discharged there is current in the circuit, even though the circuit is broken by the gap between the capacitor plates.
However, there is current only during the time of charge and discharge, and this period of time is very short. There can be no continuous movement of direct current through a capacitor. A good capacitor will block direct current (not pulsating
d.c.) and will pass the effects of alternating current.
The charge of electricity that can be placed on a capacitor is proportional to the applied voltage and to the capacitance of the capacitor (condenser).
Capacitance depends upon the total area of the plates, the thickness of the dielectric and the composition of the dielectric.
If a thin sheet of bakelite (mica-filled) is substituted for air between the plates of a capacitor, for example, the capacitance will be increased about five times.
Any electric charge produced by applied voltage and kept in bounds by an insulator (dielectric) creates a dielectric field. Once the field is created, it tends to oppose any voltage change which would affect its original position. All circuits contain some capacitance, but unless they contain a unit called a capacitor, the capacitance, for all practical purposes, is disregarded. Two conductors, called electrodes or plates, separated by a nonconductor (dielectric) make up a simple capacitor. The plates may be made of copper, tin, or aluminum.
Frequently, they are made of foil (metals compressed into thin sheets and capable of being rolled). The dielectric may be air, glass, mica, or an electrolyte made by an oxide film, but the type used will determine the amount of voltage that may be applied and the quantity of energy that will be stored, The dielectric materials have different atomic structures and present different quantities of atoms to the electrostatic field. All dielectric materials are compared to a vacuum and are given a numerical value according to the capacity ratio between them. The number given to a material is based on the same area and thickness as used in the vacuum. The numbers used to express this ratio are called dielectric constants and are expressed as the letter "K." The chart in figure 8-178 gives the K-value of some materials used.
If a source of alternating current is substituted for the battery, the capacitor acts quite differently than it does with direct current. When an 347
have approximately constant capacitance, may K (Dielectric Material constant) then be further divided, according to the type of dielectric used, into the following classes: paper, Air 1.0 oil, mica, and electrolytic capacitors. Ceramic Resin_- 2.5 capacitors are also used in some circuits.
Asbestos paper_. 2.7 When connecting electrolytic capacitors in a Hard rubber. 2.8 circuit, the proper polarity must be observed.
Isolantite_ _ Dry paper-- 3.5 3.5 "ground," which means that this terminal connects Paper capacitors may have one terminal marked Common glass _ 4.2 to the outside foil. Polarity does not ordinarily Quartz___ 4.5 have to be observed in connecting paper, oil, mica, Mica_ 4.5-7.5 or ceramic capacitors.
Porcelain_ 5.5 Flint glass. 7.0 Paper Capacitors Crown glass. 7.9 The plates of paper capacitors are strips of metal FIGURE 8-178. Dielectric constants.
alternating current is impressed on the circuit (figure 8-179), the charge on the plates constantly changes. This means that electricity must flow first from Y clockwise around to X, then from X counterclockwise around to Y, then from Y clockwise around to X, and so on. Although no current flows through the insulator between the plates of the capacitor, it constantly flows in the remainder of the circuit between X and Y. In a circuit in which there is only capacitance, current leads the impressed voltage as contrasted with a circuit in which there is inductance, where the current lags the voltage.
The unit of measurement of capacitance is the farad, for which the symbol is the letter "f." The farad is too large for practical use, and the units generally used are the microfarad (uf.), one millionth of a farad, and the micromicrofarad (uuf.), one millionth of a microfarad.
Types of Capacitors Capacitors may be divided into two groups:
fixed and variable. The fixed capacitors, which foil separated by waxed paper (figure 8-180). The capacitance of paper capacitors ranges from about 200 uf. to several uf. The strips of foil and paper are rolled together to form a cylindrical cartridge, which is then sealed in wax to keep out moisture and to prevent corrosion and leakage. Two metal leads are soldered to the plates, one extending from each end of the cylinder. The assembly is enclosed either in a cardboard cover or in a hard, molded plastic covering Bathtub-type capacitors consist of papercapacitor cartridges hermetically sealed in metal containers. The container often serves as a common terminal for several enclosed capacitors, but when not a terminal, the cover serves as a shield against electrical interference (figure 8-181).
Oil Capacitors In radio and radar transmitters, voltages high enough to cause arcing, or breakdown, of paper dielectrics are often employed. Consequently, in these applications capacitors that use oil or oilimpregnated paper for the dielectric material are preferred. Capacitors of this type are considerably more expensive than ordinary paper capacitors, and their use is generally restricted to radio and radar transmitting equipment (figure 8-182).
Mica Capacitors The fixed mica capacitor is made of metal foil AC Generator plates that are separated by sheets of mica, which form the dielectric. The whole assembly is covered in molded plastic, which keeps out moisture. Mica is an excellent dielectric and will withstand higher voltages than paper without allowing arcing FIGURE 8-179. Capacitor in an a.c. circuit. between the plates. Common values of mica 348
Negative Lead Waxed paper Tinfoil Lead 5 mfd 600 wvdc Tinfoil FIGURE 8-180. Paper capacitors.
capacitors range from approximately 50 uf, to about 0.02 uf. Mica capacitors are shown in figure 8-183.
Electrolytic Capacitors For capacitances greater than a few microfarads, the plate areas of paper or mica capacitors must become very large; thus, electrolytic capacitors are usually employed instead. These units provide large capacitance in small physical sizes. Their values range from 1 to about 1,500 microfarads.
Unlike the other types, electrolytic capacitors are generally polarized, and should be subjected to direct voltage, or pulsating direct voltage only;
however, a special type of electrolytic capacitor is made for use in motors.
The electrolytic capacitor is widely used in electronic circuits and consists of two metal plates separated by an electrolyte. The electrolyte in contact with the negative terminal, either in paste or liquid form, comprises the negative electrode.
The dielectric is an exceedingly thin film of oxide deposited on the positive electrode of the capacitor.
The positive electrode, which is an aluminum sheet, is folded to achieve maximum area.
The capacitor is subjected to a forming process during manufacture, in which current is passed through it. The flow of current results in the deposit of the thin coating of oxide on the aluminum plate.
The close spacing of the negative and positive electrodes gives rise to the comparatively high capacitance value, but allows greater possibility of voltage breakdown and leakage of electrons from one electrode to the other.
Two kinds of electrolytic capacitors are in use:
(1) Wet-electrolytic and (2) dry-electrolytic capacitors. In the former, the electrolyte is a liquid and the container must be leakproof. This type should always be mounted in a vertical position.
The electrolyte of the dry-electrolytic unit is a paste contained in a separator made of an absorbent material such as gauze or paper. The separator not only holds the electrolyte in place but also prevents short-circuiting the plates.
Dry-electrolytic capacitors are made in both cylindrical and rectangular-block form and may be contained either within cardboard or metal covers. Since the electrolyte cannot spill, the dry capacitor may be mounted in any convenient position. Electrolytic capacitors are shown in figure 8-184.
Capacitors in Parallel and in Series Capacitors may be combined in parallel or series to give equivalent values, which may be either the FIGURE 8-181. Bathtub-case paper capacitor. FIGURE 8-182. Oil capacitor.
FIGURE 8-183. Mica capacitors. A Parallel sum of the individual values (in parallel) or a value less than that of the smallest capacitance (in series). Figure 8-185 shows the parallel and series connections.
Two units used in the measurement of capacitance are the farad and the coulomb. As previously defined, the farad is the amount of capacitance present in a capacitor when one coulomb of electrical energy is stored on the plates and one volt is applied across the capacitor. One coulomb is the electrical charge of 6.28 billion billion 1 1 electrons. From this it can be seen that + a Q (in coulombs) C (in farads) = E (in volts) In A of figure 8-185 the voltage, E, is the same for all the capacitors. The total charge, Q« is the sum of all the individual charges, Qu Q2, and Q3.
B Series FIGURE 8-185. Capacitors in parallel and in series.
Using the basic equation for the capacitor, C== The total charge is Q, = CE, where C is the total capacitance. Since the total charge on capacitors in parallel is the sum of the individual capacitor charges, Using both equations for total charge develops the equation GIE = GIE + CrE + CE.
Dividing both sides of this equation by & gives Gi = CI + Cz + C3.
This formula is used to determine the total capacitance of any number of capacitors in parallel.
In the series arrangement, (B of figure 8-185), FIGURE 8-184. Electrolytic capacitors. the current is the same in all parts of the circuit.
Each capacitor develops a voltage during charge, and the sum of the voltages of all the capacitors Dielectric K (volts per .001 inch) Dielectric Strength must equal the applied voltage, E. By the capacitor equation, the applied voltage, E, is equal Air 1.0 80 to the total charge divided by the total capacitance, or Paper (1) Paraffined 2.2 1,200
(2) Beeswaxed 3.1 1,800 B = Glass 4.2 200 Castor oil 4.7 380 The total charge, Q« is equal to the charge on Bakelite 6.0 500 any one of the capacitors because the same Mica 6.0 2,000 current flows in all for the same length of time, and Fiber 6.5 50 because the charge equals current multiplied by time in seconds (Q:=1 xT). Therefore, FIGURE 8-186. Strength of some dielectric Q: = 01 = 02 = 03 , materials.
and, since in a circuit with capacitors in series Ei = Es + Er + E3 , where E1, E2, and E, are the voltages of the three capacitors. Then Dividing both sides of the equation by Q, gives = + = + Cs The reciprocal of the total capacitance of any number of capacitors in series is equal to the sum of the reciprocals of the individual values.
Parallel capacitors combine by a rule similar to that used to combine resistors in series. Series capacitors combine by a rule similar to that for combining parallel resistors.
In the series arrangement of two capacitors, Ci and C2 the total capacitance is given by the G + Ci Voltage Rating of Capacitors In selecting or substituting a capacitor for use in a particular circuit, the following must be considered: (1) The value of capacitance desired and (2) the amount of voltage to which the capacitor is to be subjected. If the voltage applied across the plates is too great, the dielectric will break down and arcing will occur between the plates. The capacitor is then short-circuited, and the possible flow of direct current through it can cause damage to other parts of the equipment.
Capacitors have a voltage rating that should not be exceeded The working voltage of the capacitor is the maximum voltage that can be steadily applied without danger of arc-over. The working voltage depends on (1) the type of material used as the dielectric and (2) the thickness of the dielectric.
The voltage rating of the capacitor is a factor in determining the capacitance because capacitance decreases as the thickness of the dielectric increases. A high-voltage capacitor that has a thick dielectric must have a larger plate area in order to have the same capacitance as a similar low-voltage capacitor having a thin dielectric. The strength of some commonly used dielectric materials is listed in figure 8-186. The voltage rating also depends on frequency because the losses, and the resultant heating effect, increase as the frequency increases.
A capacitor that can be safely charged to 500 volts d.c. cannot be safely subjected to a.c. or pulsating d.c. whose effective values are 500 volts.
An alternating voltage of 500 volts (r.m.s.) has a peak voltage of 707 volts, and a capacitor to which it is applied should have a working voltage of at least 750 volts. The capacitor should be selected so that its working voltage is at least 50 percent greater than the highest voltage to be applied to it.
Capacitance Reactance Capacitance, like inductance, offers opposition to the flow of current. This opposition is called capacitive reactance and is measured in ohms. The symbol for capacitive reactance is X.. The 351
equation, reactances. Thus, Current = Voltage , Or Xei = (Xc) 1 + (Xc)2.
Capacitive reactance The total reactance of capacitors connected in parallel is found in the same way total resistance is computed in a parallel circuit:
is similar to Ohm's law and the equation for current in an inductive circuit. The greater the frequency, the less the reactance. Hence, the capacitive reactance, where: f = frequency in c.p.s.
C = capacity in farads.
2 т = 6.28.
Problem:
A series circuit is assumed in which the impressed voltage is 110 volts at 60 c.p.s., and the capacitance of a condenser is 80 uf. Find the capacitive reactance and the current flow.
Solution:
To find capacitive reactance, the equation X. = 27 fc is used. First, the capacitance, 30 f., is changed to farads by dividing 80 by 1,000,000, since 1 million microfarads is equal to 1 farad. This quotient equals .000080 farad. This is substituted in the equation and X: = 6.28 × 60 × .000080 X. = 33.2 ohms reactance.
Find the current flow:
1 = 110 33.2 I = 3.31 amperes.
(X): = (Хc)1 Phase of Current and Voltage in Reactive Circuits When current and voltage pass through zero and reach maximum value at the same time, the current and voltage are said to be in phase (A of figure 8-187). If the current and voltage pass through zero and reach the maximum values at different times, the current and voltage are said to be out of phase. In a circuit containing only inductance, the current reaches a maximum value later than the voltage, lagging the voltage by 90°, or one-fourth cycle (B of figure 8-187). In a circuit containing only capacitance, the current reaches its maximum value ahead of the voltage and the current leads the voltage by 90°, or one-fourth cycle (C of figure 8-187). The amount the current lags or leads the voltage in a circuit depends on the relative amounts of resistance, inductance, and capacitance in the circuit.
OHM'S LAW FOR A. C. CIRCUITS The rules and equations for d.c. circuits apply to a.c. circuits only when the circuits contain resistance alone, as in the case of lamps and heating elements. In order to use effective values of voltage and current in a.c. circuits, the effect of inductance and capacitance with resistance must be considered.
The combined effect of resistance, inductive reactance, and capacitive reactance makes up the total opposition to current flow in an a.c. circuit.
This total opposition is called impedance and is represented by the letter "Z." The unit for the measurement of impedance is the ohm.
Capacitive Reactances in Series and in Parallel When capacitors are connected in series, the total reactance is equal to the sum of the individual Series A. C. Circuits If an a.c. circuit consists of resistance only, the value of the impedance is the same as the 352
A for the EMF EMF EMF Current and voltage in phase Effect of inductance Effect of capacitance B FIGURE 8-187. Phase of current and voltage.
resistance, and Ohm's law for an a.c. circuit, 1 - is exacily the same as for a de. sirenit. In figure 8-188 a series circuit containing a lamp with Il ohms resistance connected across a source is illustrated. To find how much current will flow if 110 volts d.c. are applied and how much current will flow if 110 volts a.c. are applied, the following examples are solved:
I = R E I - → (where Z = R) = 115 110V = 10 amperes d.c. = 10 amperes a.c.
When a.c. circuits contain resistance and either inductance or capacitance, the impedance, L, is not the same as the resistance, R. The impedance of a circuit is the circuit's total opposition to the flow of current. In an a.c. circuit, this opposition consists of resistance and reactance, either inductive or capacitive, or elements of both.
Resistance and reactance cannot be added directly, but they can be considered as two forces acting at right angles to each other. Thus, the relation between resistance, reactance, and impedance may be illustrated by a right triangle, as shown in figure 8-189.
Since these quantities may be related to the sides of a right triangle, the formula for finding the impedance, or total opposition to current flow in an a.c. circuit, can be found by using the law of right triangles. This theorem, called the Pythagorean theorem, applies to any right triangle. It states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Thus, the value of any side of a right triangle can be found if the other two sides are known. If an
a.c. circuit contains resistance and inductance, as shown in figure 8-190, the relation between the sides can be stated as:
22 = R + XL.
Reactance Impedance 110V - Xc Z R Resistance FIGURE 8-188. Applying d.c. and a.c. to a circuit. FIGURE 8-189. Impedance triangle.
Then the current flow, 110V AC 62 1 = Z E 60 cycles 0.021 henrys 10 FIGURE 8-190. A circuit containing resistance and inductance.
The square root of both sides of the equation gives Z = VR + X2.
This formula can be used to determine the impedance when the values of inductive reactance and resistance are known. It can be modified to solve for impedance in circuits containing capacitive reactance and resistance by substituting Xc in the formula in place of Xr. In circuits containing resistance with both inductive and capacitive reactance, the reactances can be combined, but because their effects in the circuit are exactly opposite, they are combined by subtraction:
X = X1 - Xc or X = Xc - Xz (the smaller is always subtracted from the larger).
In figure 8-190, a series circuit consisting of resistance and inductance connected in series is connected to a source of 110 volts at 60 cycles per second. The resistive element is a lamp with 6 ohms resistance, and the inductive element is a coil with an inductance of 0.021 benry. What is the value of the impedance and the current through the lamp and the coil?
Solution:
First, the inductive reactance of the coil is computed:
I = 11 amperes current.
The voltage drop across the resistance (En) is ER = IXR ER - 11 X 6 = 66 volts.
The voltage drop across the inductance (Ext) is Ex: = IX XI Exz = 11 × 8 = 88 volts.
The sum of the two voltages is greater than the impressed voltage. This results from the fact that the two voltages are out of phase and, as such, represent the maximum voltage. If the voltage in the circuit is measured by a voltmeter, it will be approximately 110 volts, the impressed voltage.
This can be proved by the equation E = V(ER)? + (ExL)2 E = V662 + 882 E = V4356 + 7744 E = V12100 E = 110 volts.
In figure 8-191, a series circuit is illustrated in which a capacitor of 200 uf. is connected in series with a 10-ohm lamp. What is the value of the impedance, the current flow, and the voltage drop across the lamp?
XI = 2=XfXL Xx = 6.28 × 60 × 0.021 Xz = 8 ohms inductive reactance. 110V AC 1022 Next, the total impedance is computed: 60 cycles 200 ut ;
Z VRE + XE Z 62 ÷ 82 Z 36 + 64 Z · 100 FIGURE 8-191. A circuit containing resistance Z = 10 ohms impedance. and capacitance.