CHAPTER THREE

Alternating Current Circuits

The previous chapter has been devoted to a discussion of circuits and circuit elements upon which is impressed a current consisting of a flow of electrons in one direction. This type of unidirectional current flow is called direct current, abbreviated d. c. Equally as important in radio and communications work, and power practice, is a type of current flow whose direction of electron flow reverses periodically. The reversal of flow may take place at a low rate, in the case of power systems, or it may take place millions of times per second in the case of communications frequencies. This type of current flow is called alternating current, abbreviated a.c.

Alternating Current

Frequency of on An alternating current is

Alternating Current one whose amplitude of current flow periodically rises from zero to a maximum in one direction, decreases to zero, changes its direction, rises to maximum in the opposite direction, and decreases to zero again. This complete process, starting from zero, passing through two maximums in opposite directions, and returning to zero again, is called a cycle. The number of times per second that a current passes through the complete cycle is called the frequency of the current. One and one quarter cycles of an alternating current wave are illustrated diagrammatically in figure 1.

Frequency Spectrum At present the usable frequency range for alternating electrical currents extends over the enormous frequency range from about 15 cycles per second to perhaps 30,000,000,000 cycles per second. It is obviously cumbersome to use a frequency designation in c.p.s. for enormously high frequencies, so three common units which are multiples of one cycle per second have been established.

ir, э\

и о

DIRECT CURRENT

ALTERNATING CURRENT

Figure 1

ALTERNATING CURRENT AND DIRECT CURRENT

Graphical comparison between unldrecllanal (direct} current and alternating current as plotted against time.

These units are:

(1) the kilocycle (abbr., kc), 1000 c.p.s.

(2) the Megacycle (abbr., Mc), 1,000,000 c.p.s. or 1000 kc.

(3) the kilo-Megacycle (abbr., kMc), 1,000,000,000 c.p.s. or 1000 Mc.

With easily handled units such as these we can classify the entire usable frequency range into frequency bands.

The frequencies falling between about 15 and 20,000 c.p.s. are called audio frequencies, abbreviated a.f., since these frequencies are audible to the human ear when converted from electrical to acoustical signals by a loudspeaker or headphone. Frequencies in the vicinity of 60 c.p.s. also are called power frequencies, since they are commonly used to distribute electrical power to the consumer.

The frequencies falling between 10,000 c.p.s. (10 kc) and 30,000,000,000 c.p.s. (30 kMc.) are commonly called radio frequencies, abbreviated r.f., since they are commonly used in radio communication and allied arts. The radio-frequency spectrum is often arbitrarily classified into seven frequency bands, each one of which is ten times as high in frequency as the one just below it in the spectrum (except for the v-l-f band at the bottom end of the spectrum). The present spectrum, with classifications, is given below.

Frequency

10 to 30 kc. 30 to 300 kc 300 to 3000 kc. 3 to 30 Mc. 30 to 300 Mc 300 to 3000 Mc. 3 to 30 kMc. 30 to 300 kMc.

Classification Abbrev.

Very-low frequencies v.l.f

Low frequencies l.f-

Medium frequencies m.f.

High frequencies h.f.

Very-high frequencies v.h.f.

Ultra-high frequencies u.h.f.

Super-high frequencies s.h.f. Extremely-high frequencies

e.h.f.

Generation of Alternoting Current

Faraday discovered that if a conductor which forms part of a closed circuit is moved through a magnetic field so as to cut across the lines of force, a current will flow in the conductor. He also discovered that, if a conductor in a second closed circuit is brought near the first conductor and the current in the first one is varied, a current will flow in the second conductor. This effect is known as induction, and the currents so generated are induced currents. In the latter case it is the lines of force which are moving and cutting the second conductor, due to the varying current strength in the first conductor.

A current is induced in a conductor if there is a relative motion between the conductor and a magnetic field, its direction of flow depending upon the direction of the relative

Figure 2 THE ALTERNATOR

Sem/-schematic representation of the simp/esf form of the alternator.

motion between the conductor and the field, and its strength depends upon the intensity of the field, the rate of cutting lines of force, and the number of turns in the conductor.

Alternators A machine that generates an alternating current is called an alternator or a-c generator. Such a machine in its basic form is shown in figure 2. It consists of two permanent magnets, M, the opposite poles of vdiich face each other and are machined so that they have a common radius. Between these two poles, north (N) and south (S), a substantially constant magnetic field exists. If a conductor in the form of С is suspended so that it can be freely rotated between the two poles, and if the opposite ends of conductor С are brought to collector rings, there will be a flow of alternating current when conductor С is rotated. This current will flow out through the collector rings R and brushes В to the external circuit, X-Y.

The field intensity between the two pole pieces is substantially constant over the entire area of the pole face However, when the conductor is moving parallel to the lines of force at the top or bottom of the pole faces, no lines are being cut. As the conductor moves on across the pole face it cuts more and more lines of force for each unit distance of travel, until it is cutting the maximum number of lines when opposite the center of the pole. Therefore, zero current is induced in the conductor at the instant it is midway between the two poles, and maximum current is induced when it is opposite the center of the pole face. After the conductor has rotated through 180° it can be seen that its position with respect to the pole pieces will be exactly opposite to that when it started. Hence, the second 180° of rotation will produce an alternation of current in the opposite direction to that of the first alternation.

The current does not increase directly as the angle of rotation, but rather as the sine of the angle; hence, such a current has the mathematical form of a sine wave. Although

HANDBOOK

The Sine Wave 43

LINES OF FOBCE

lilllii LINES OF FOBCE (UNIFORM DENSITY )

Figure 3

OUTPUT OF THE ALTERNATOR

Graph showing sine-wavo output current of the alternator of figure 2,

most electrical machinery does not produce a strictly pure sine curve, the departures are usually so slight that the assumption can be regarded as fact for most practical purposes. All that has been said in the foregoing paragraphs concerning alternating current also is applicable to alternating voltage.

The rotating arrow to the left in figure 3 represents a conductor rotating in a constant magnetic field of uniform density. The arrow also can be taken as a vector representing the strength of the magnetic field. This means that the length of the arrow is determined by the strength of the field (number of lines of force), which is constant. Now if the arrow is rotating at a constant rate (that is, with constant angular velocity), then the voltage developed across the conductor will be proportional to the rate at ich it is cutting lines of force, which rate is proportional to the vertical distance between the tip of the arrow and the horizontal base line.

If EO is taken as unity or a voltage of 1, then the voltage (vertical distance from tip of arrow to the horizontal base line) at point С for instance may be determined simply by referring to a table of sines and looking up the sine of the angle which the arrow makes with the horizontal.

When the arrow has traveled from A to point E, it has traveled 90 degrees or one quarter cycle. The other three quadrants are not shown because their complementary or mirror relationship to the first quadrant is obvious.

It is important to note that time units are represented by degrees or quadrants. The fact that AB, ВС, CD, and DE are equal chords (forming equal quadrants) simply means that the arrow (conductor or vector) is traveling at a constant speed, because these points on the radius represent the passage of equal units of time.

The whole picture can be represented in another way, and its derivation from the foregoing is shown in figure 3- The time base is represented by a straight line rather than by

1 CYCLE=

CYCLE = WHERE F = FREQUENCY IN CYCLES

Figure 4 THE SINE WAVE

Illustrating one cycle of a sine wave. One complete cycle of alternation Is broken up Into 360 degrees. Then one-half cycle Is 180 degrees, one-quarter cycle Is 90 degrees, and so on down to the smallest division of the wave. A cosine wave has a shape Identical to a sine wave but Is shifted 90 degrees In phase - In other words the wave begins at full amplitude, the 90-degree point comes at zero amplitude, the 180-degree point comes at full amplitude In the opposite direction of current flow, etc.

angular rotation. Points A, B, C, etc., represent the same units of time as before. When the voltage corresponding to each point is projected to the corresponding time unit, the familiar sine curve is the result.

The frequency of the generated voltage is proportional to the speed of rotation of the alternator, and to the number of magnetic poles in the field. Alternators may be built to produce radio frequencies up to 30 kilocycles, and some such machines are still used fot low frequency communication purposes. By means of multiple windings, three-phase ouфut may be obtained from large industrial alternators.

Radian Notation From figure 1 we see that the value of an a-с wave varies continuously. It is often of importance to know the amplitude of the wave in terms of the total amplitude at any instant or at any time within the cycle. To be able to establish the instant in question we must be able to divide the cycle into parts. We could divide the cycle into eighths, hundredths, or any other ratio that suited our fancy. However, it is much more convenient mathematically to divide the cycle either into electrical degrees (360° represent one cycle) or into radians. A radian is an arc of a circle equal to the radius of the circle; hence there are Itt radians per cycle - or per circle (since there are 7T diameters per circumference, there are Itt radii).

Both radian notation and electrical degree

notation are used in discussions of alternating current circuits. However, trigonometric tables are much more readily available in terms of degrees than radians, so the following simple conversions are useful.

277 radians = 1 cycle = 360°

77 radians = \ cycle = 180°

- radians 2

- radians 3

- radians 4

\ cycle = 90°

\ cycle = 60°

% cycle = 45° 1

1 radian =-cycle = 57.3° In

When the conductor in the simple alternator of figure 2 has made one complete revolution it has generated one cycle and has rotated through In radians. The expression 2JTf then represents the number of radians in one cycle multiplied by the number of cycles per second (the frequency) of the alternating voltage or current. The expression then represents the number of radians per second through which the conductor has rotated. Hence 277f represents the angular velocity of the rotating conductor, or of the rotating vector which represents any alternating current or voltage, expressed in radians per second.

In technical literature the expression 2ni is often replaced by tu, the lower-case Greek letter omega. Velocity multiplied by time gives the distance travelled, so 277ft (or oit) represents the angular distance through which the rotating conductor or the rotating vector has travelled since the reference time t = 0. In the case of a sine wave the reference time t = 0 represents that instant when the voltage or the current, whichever is under discussion, also is equal to zero.

Instantaneous Value The instantaneous volt-of Voltoge or age Or current is propor-

Current tional to the sine of the

angle through which the rotating vector has travelled since reference time t = 0, Hence, when the peak value of the a-c wave amplitude (either voltage or current amplitude) is known, and the angle through which the rotating vector has travelled is established, the amplitude of the wave at this instant can be determined through use of the following expression;

e = Emax sin 2n-ft,

where:

6 (tHETa) phase angle = £7rFT

a = radians oh 90°

B=n- radians or 1 BO

с = radians oh 270-

d = гГГ radians or 3B0 =

1 radian = 57.324 degrees

Figure 5

ILLUSTRATING RADIAN NOTATION The radian is о unit of phase angle, equal to 57.324 degrees. It Is commonly used In mathematical relationships involving phase angles since such relationships are simplified when radian notation Is used.

where e = the instantaneous voltage

E = maximum crest value of voltage,

f = frequency in cycles per second, and

t = period of time which has elapsed since t = 0 expressed as a fraction of one second.

The instantaneous current can be found from the same expression by substituting i for e and lma for Emax.

It is often easier to visualize the process of determining the instantaneous amplitude by ignoring the frequency and considering only one cycle of the a-c wave. In this case, for a sine wave, the expression becomes:

e = Emax sin в

where в represents the angle through which the vector has rotated since time (and amplitude) were zero. As examples:

when 0 = 30° sin 9 = 0.5 so e = 0.5 Emax

when (9 = 60° sin в = 0.866 so e = 0.866 Emax

when 0 = 90° sin e= 1.0

so e = Emax

when в = I radian sin 0 = 0.8415 so e -0.8415 Emax

HANDBOOK

A-C Relationships 45

Effective Value The instantaneous value of on of an alternating current

Alternating Current or voltage varies continuously throughout the cycle. So some value of an a-c wave must be chosen to establish a relationship between the effectiveness of an a-c and a d-c voltage or current/ The heating value of an alternating current has been chosen to establish the reference between the effective values of a.c. and d. c. Thus an alternating current will have an effective value of 1 ampere when it produces the same heat in a resistor as does I ampere of direct current.

The effective value is derived by taking the instantaneous values of current over a cycle of alternating current, squaring these values, taking an average of the squares, and then taking the square root of the average. By this procedure, the effective value becomes known as the root mean square or r.m.s. value. This is the value that is read on a-c voltmeters and a-c ammeters. The r.m.s. value is 70.7 (for sine waves only) per cent of the peak or maximum instantaneous value and is expressed as follows:

Eeff- or Et.m.s. = 0.707 X Emax Or leff. or Ir.m.s. = 0.707 X Imax.

The following relations are extremely useful in radio and power work:

Ei.m.s. = 0.707 X Emax, and

Emax = 1.414 X Er.m.s.

TIME

Figure 6 FULL-WAVE RECTIFIED SINE WAVE

Waveform obtained at the output of a fullwave rectifier being fed with a sine wave and having 100 per cent rectification efficiency. Each pulse has the same shape as one-half cycle of a sine wave. This type of current is fcnown as pulsating direct current.

It is thus seen that the average value is 63-6 per cent of the peak value.

Relotionship Between Peak, R.M.S. or Effective, and Average Values

To summarize the three most significant values of an a-c sine wave: the peak value is equal to 1.41 times the r.m.s. or effective, and the r.m.s. value is equal to 0.707 times the peak value; the average value of a full-wave rectified a-c wave is 0.636 times the peak value, and the average value of a rectified wave is equal to 0.9 times the r.m.s. value.

R.M.S. = 0.707 X Peak Average = 0.636 x Peak

Rectified Alternating If an alternating current Current Or Pulsat- is passed through a recti-ing Direct Current fier, it emerges in the form of a current of varying amplitude which flows in ояе direction only. Such a current is known as rectified a.c. or pulsating d.c. A typical wave form of a pulsating direct current as would be obtained from the output of a full-wave rectifier is shown in figure 6.

Measuring instruments designed for d-c operation will not read the peak for instantaneous maximum value of the pulsating d-c output from the rectifier; they will read only the average value. This can be explained by assuming that it could be possible to cut off some of the peaks of the waves, using the cutoff portions to fill in the spaces that are open, thereby obtaining an average d-c value. A milliammeter and voltmeter connected to the adjoining circuit, or across the output of the rectifier, will read this average value. It is related to peak value by the following expres-

sion;

Average =0-9 R.M.S. =1.11

x R.M.S. X Average

Peak Peak

= 1.414 X R.M.S. = 1-57 X Average

Eavg = 0.636 X Emax

Applying Ohms Law Ohms law applies to Alternating Current equally to direct or alternating current, provided the circuits under consideration are purely resistive, that is, circuits which have neither inductance (coils) nor capacitance (capacitors). Problems which involve tube filaments, drop resistors, electric lamps, heaters or similar resistive devices can be solved from Ohms law, regardless of whether the current is direct or alternating. When a capacitor or coil is made a part of the circuit, a property common to either, called reactance, must be taken into consideration. Ohms law still applies to a-c circuits containing reactance, but additional considerations are involved; these will be discussed in a later paragraph.

CURRENT LAGGING VOLTAGE BY 90

(circuit CONTAIMING pure INDUCTANCE ONLY )

Figure 7

LAGGING PHASE ANGLE

Showing fhe manner In which the current lags the voltage In an a-c circuit confofn/ng pore Inductance only. The lag Is equal to one-quarter cycle or 90 degrees.

CURRENT LEADING VOLTAGE BY 90°

(circuit containing pure capacitance only )

Figure 8 LEADING PHASE ANGLE

Showing the manner In which fhe current leads the voltage In an a-c circuit containing pure capacitance only. The lead Is equal to one-quarter cycle or 90 degrees.

Inductive As was stated in Chapter Two, Reactance when a changing current flows through an inductor a back- or counter-electromotive force is developed; opposing any change in the initial current. This property of an inductor causes it to offer opposition or impedance to a change in current. The measure of impedance offered by an inductor to an alternating current of a given frequency is known as its inductive reactance. This is expressed as Xl.

Xl =2rfL,

= inductive reactance expressed in

;t = 3-1416 (27г = 6.283), f = frequency in cycles, L = inductance in henrys.

where xl ohms.

Inductive Reactance at Radio Frequencies

It is very often necessary to compute inductive reactance at radio frequencies. The same formula may be used, but to make it less cumbersome the inductance is expressed in millihenrys and the frequency in kilocycles. For higher frequencies and smaller values of inductance, frequency is expressed in megacycles and inductance in microhenrys. The basic equation need not be changed, since the multiplying factors for inductance and frequency appear in numerator and denominator, and hence are cancelled out. However, it is not possible in the same equation to express Lin millihenrys and f in cycles without conversion factors.

Copacitive It has been explained that induc-Reactance tive reactance is the measure of the ability of an inductor to offer impedance to the flow of an alternating current.

Capacitors have a similar property although in this case the opposition is to any change in the voltage across the capacitor. This property is called capacitive reactance and is expressed as follows:

where Xc = capacitive reactance in ohms, ff= 3.1416

f = frequency in cycles, С = capacitance in farads.

Capacitive Reactance at Radio Frequencies

Here again, as in the case of inductive reactance, the units of capacitance and frequency can be converted into smaller units for practical problems encountered in radio work. The equation may be written:

1,000,000

where f = frequency in megacycles,

С = capacitance in micro-microfarads. In the audio range it is often convenient to express frequency (f) in cycles and cac-itance (C) in microfarads, in which event the same formula applies.

Phase When an alternating current flows through a purely resistive circuit, it will be found that the current will go through maximum and minimum in perfect step with the voltage. In this case the current is said to be in step or in phase with the voltage. For this reason, Ohms law will apply equally well for AC. or d.c. where pure resistances ate concerned, provided that the same values of the

wave (either peak or r.m.s.) for both voltage and current are used in the calculations.

However, in calculations involving alternating currents the voltage and current are not necessarily in phase. The current through the circuit may lag behind the voltage, in which case the current is said to have lagging phase. Lagging phase is caused by inductive reactance. If the current reaches its maximum value ahead of the voltage (figure 8) the current is said to have a leading phase. A leading phase angle is caused by capacitive reactance.

In an electrical circuit containing reactance only, the current will either lead or lag the voltage by 90°. If the circuit contains inductive reactance only, the current will lag the voltage by 90°. If only capacitive reactance is in the circuit, the current will lead the voltage by 90°.

Reactances Inductive and capacitive rein Combination actance have exactly opposite effects on the phase relation between current and voltage in a circuit. Hence when they are used in combination their effects tend to neutralize. The combined effect of a capacitive and an inductive reactance is often called the net reactance of a circuit. The net reactance (X) is found by subtracting the capacitive reactance from the inductive reactance, X = Xl - Xq.

The result of such a combination of pure reactances may be either positive, in which case the positive reactance is greater so that the net reactance is inductive, or it may be negative in which case the capacitive reactance is greater so that the net reactance is capacitive. The net reactance may also be zero in which case the circuit is said to be resonant- The condition of resonance will be discussed in a later section. Note that inductive reactance is always taken as being positive while capacitive reactance is always taken as being negative.

Impedance; Circuits Pure reactances intro-Containing Reactance duce a phase angle of and Resistance 90° between voltage and

current; pure resistance introduces no phase shift between voltage and current. Hence we cannot add a reactance and a resistance directly. When a reactance and a resistance are used in combination the resulting phase angle of current flow with respect to the impressed voltage lies somewhere between plus or minus 90° and 0° depending upon the relative magnitudes of the reactance and the resistance.

The term impedance is a general term which can be applied to any electrical entity which impedes the flow of current. Hence the term may be used to designate a resistance, a pure

Y-AXiS

Figure 9

Operation on the vector (fA by the quantity (~ 1) causes vector to rotate through 780 degrees.

reactance, or a complex combination of both reactance and resistance. The designation for impedance is Z. An impedance must be defined in such a manner that both its magnitude and its phase angle are established. The designation may be accomplished in either of two ways - one of which is convertible into the other by simple mathematical operations.

The J Operator The first method of designating an impedance is actually to specify both the resistive and the reactive component in the form R + jX. In this form R represents the resistive component in ohms and X represents the reactive component. The j merely means that the X component is reactive and thus cannot be added directly to the R component. Plus jX means that the reactance is positive or inductive, while if minus jX were given it would mean that the reactive component was negative or capacitive.

In figure 9 we have a vector (+A) lying along the positive X-axis of the usual X-Y coordinate system. If this vector is multiplied by the quantity (-1), it becomes (-A) and its position now lies along the X-axis in the negative direction. The operator (-1) has caused the vector to rotate through an angle of 180 degrees. Since (-1) is equal to {y/-l x \T), the same result may be obtained by operating on the vector with the operator (-1 x However if the vector is operated on but once

by the operator (л/-1), it is caused to rotate only 90 degrees (figure 10). Thus the operator (i;/-!)rotates a vector by 90 degrees. For convenience, this operator is called the / operator. In like fashion, the operator (-j) rotates the vector of figure 9 through an angle of 270 degrees, so that the resulting vector (-jA) falls on the (-Y) axis of the coordinate system.

Y-AXlS

+ ja

- (-i-a)x [V ) rotates -jf vector through 90

к±А-x-AXIS

Figure 10

Operation on fhe vector (+A) by the quantity (j) causes vecfor fo rotate through 90 degrees.

Polar Notation The second method of representing an impedance is to specify its absolute magnitude and the phase angle of current with respect to voltage, in the form ZLd. Figure 11 shows graphically the relationship between the two common ways of representing an impedance.

The construction of figure 11 is called an impedance diagram. Through the use of such a diagram we can add graphically a resistance and a reactance to obtain a value for the resulting impedance in the scalar form. With zero at the origin, resistances are plotted to the right, positive values of reactance (inductive) in the upward direction, and negative values of reactance (capacitive) in the downward direction.

Note that the resistance and reactance are drawn as the two sides of a right triangle, with the hypotenuse representing the resulting impedance. Hence it is possible to determine mathematically the value of a resultant impedance through the familiar right-triangle relationship - the square of the hypotenuse is equal to the sum of the squares of the other two sides:

or zl = yjR +

Note also that the angle в included between R and Z can be determined from any of the following trigonometric relationships:

sin t> =

cos в =

tan в = - R

One common problem is that of determining the scalar magnitude of the impedance, Z,

i +j3

z= 4+j 3

1z= 5 J tan 0.75

szl= 5/38.95

r= 4 ohms

Figure 11 THE IMPEDANCE TRIANGLE

Showing the graphical construction of a triangle for obtaining the net (scalar) impedance resulting from the connection of a resistance and a reactance in series. Shown also alongside Is the alternative mathematical procedure for obtaining the values associated with the triangle.

and the phase angle в, when resistance and reactance are known; hence, of converting from the Z = R + jX to the \Z\ Ld form. In this case we use two of the expressions just given:

\z\ = VR +

tan в = -, (от в = tan - ) R R

The inverse problem, that of converting from the \Z\ Ld to the R + jX form is done with the following relationships, both of which are obtainable by simple division from the trigonometric expressions just given for determining the angle в:

R = jz] cos в

jX = \Z\ j sin в

By simple addition these two expressions may be combined to give the relationship between the two most common methods of indicating an impedance:

R + jX = \Z\ (cos e + ) sin в)

In the case of impedance, resistance, or reactance, the unit of measurement is the ohm; hence, the ohm may be thought of as a unit of opposition to current flow, without reference to the relative phase angle between the applied voltage and the current which flows.

Further, since both capacitive and inductive reactance are functions of frequency, impedance will vary with frequency. Figure 12 shows the manner in which \Z\ will vary with frequency in an RL series circuit and in an RC series circuit.

Series RLC Circuits In a series circuit containing R, L, and C, the im-

HANDBOOK

Impedance 49

pedance is determined as discussed before except that the reactive component in the expressions becomes: (The net reactance - the difference between Xl and Xc-) Hence (Xl -Xc) may be substituted for X in the equations. Thus:

izl = vr+(Xl - Xc)

= tan

..(Xl-Xc)

A series RLC circuit thus may present an impedance which is capacitively reactive if the net reactance is capacitive, inductively reactive if the net reactance is inductive, or resistive if the capacitive and inductive reactances are equal.

Addition of The addition of complex

Complex Quantities quantities (for example, impedances in series) is quite simple if the quantities are in the rectangular form. If they are in the polar form they only can be added graphically, unless they are converted to the rectangular form by the relationships previously given. As an example of the addition of complex quantities in the rectangular form, the equation for the addition of impedances is:

(R, + JXJ + (R, + jXj)= (R, + R,) + j(X, + X,)

For example if we wish to add the impedances (10-I-J50) and (20 - j30) we obtain:

(10 + j50) + (20 - j30)

= (10 + 20) +j(50 + (-30)) = 30 + j(50-30) = 30 + j20

Multiplication ond It is often necessary in Division of solving certain types of

Complex Quontittes circuits to multiply or divide two complex quantities. It is a much simplier mathematical operation to multiply or divide complex quantities if they are expressed in the polar form. Hence if they are given in the rectangular form they should be converted to the polar form before multiplication or division is begun. Then the multiplication is accomplished by multiplying the \Z\ terms together and adding algebraically the I. в terms, as:

(jz.i /.e,)( \z,\ Le,) = z. z,i {Le, + Le;)

For example, suppose that the two impedances 20j/i43° and l32l /.-23° are to be multiplied. Then:

( 20 /.43°) (132 Z-23°) = I 20-32]

(Z43° + Z-23°) = 640 L 20°

FREQUENCY

Figure 12

IMPEDANCE AGAINST FREQUENCY FOR R-L AND R-C CIRCUITS

The Impedance of an R-C cireulf approaches infinity as the frequency approaches zero (d.c), while the impedance of a series R-L circuit approaches Infinity as the frequency approaches infinity. The impedance of an R-C circuit approaches the Impedance of the series resistor as the frequency approaches Infinity, while the Impedance of a series R-L circuit approaches the impedance of the resistor as the frequency approaches zero.

Division is accomplished by dividing the denominator into the numerator, and subtracting the angle of the denominator from that of the numerator, as:

For example, suppose that an impedance of 50 Z67°is to be divided by an impedance of 1101 Z45°. Then:

50Z.67° 50

llOl Z45

.(/.67°-Z45°) = 5 1(Z22°)

Ohms Low for Complex Quantities

The simple form of Ohms Law used for d-c circuits may be stated in a more general form for application to a-c circuits involving either complex quantities or simple resistive elements. The form is:

in which, in the general case, /, E, and Z are complex (vector) quantities. In the simple case where the impedance is a pure resistance with an a-c voltage applied, the equation

simplifies to the familiar I = E/R. In any case

the applied voltage may be expressed either as peak, r.m.s or average; the resulting

=p -J 300

Figure 13 SERIES R-L-C CIRCUIT

current always will be in the same type of units as used to define the voltage.

In the more general case vector algebra must be used to solve the equation. And, since either division or multiplication is involved, the complex quantities should be expressed in the polar form. As an example, take the case of the series circuit shown in figure 13 with 100 volts applied. The impedance of the series circuit can best be obtained first in the rectangular form, as:

200 + j (100-300) = 200-j200

Now, to obtain the current we must convert this impedance to the polar form.

IZl = 200 +(-200) = V 40,000 + 40,000

= V 80,000 = 282 n

-200

= tan- -1

= -45° Therefore Z = 282 L-45°

Note that in a series circuit the resulting impedance takes the sign of the largest reactance in the series combination.

Where a slide-rule is being used to make the computations, the impedance may be found without any addition or subtraction operations by finding the angle в first, and then using the trigonometric equation below for obtaining the impedance. Thus:

i = tan - = tan

Then Zl =

. -200

-= tan- -1

cos -45°= 0.707

, , 200

Z =-=2820

0.707

Since the applied voltage will be the reference for the currents and voltages within the circuit, we may define it as having a zero phase angle: E = 100 jLO°. Then:

100 LO °

= 0.354 0°-(-45°)

282 -45° = 0.35 4 Л45° amperes.

This same current must flow through all three elements of the circuit, since they are in series and the current through one must already have passed through the other two. Hence the voltage drop across the resistor (whose phase angle of course is 0°) is:

Б = I R

e =(0.354 /-45 ) (200 Lq°)

= 70.8 45° volts

The voltage drop across the inductive reactance is:

E = /Xl

E = (0.354 /.45°) (100 Z.90°)

= 35.4 /- 135° volts

Similarly, the voltage drop across the capacitive reactance is:

E = f Xc

e = (0.354 /- 45°) (300 /-90°)

= 106.2 /.-45°

Note that the voltage drop across the capacitive reactance is greater than the supply voltage. This condition often occurs in a series RLC circuit, and is explained by the fact that the drop across the capacitive reactance is cancelled to a lesser or greater extent by the drop across the inductive reactance.

It is often desirable in a problem such as the above to check the validity of the answer by adding vectorially the voltage drops across the components of the series circuit to make sure that they add up to the supply voltage - or to use the terminology of Kirchhoffs Second Law, to make sure that the voltage drops across all elements of the circuit, including the source taken as negative, is equal to zero.

In the general case of the addition of a number of voltage vectors in series it is best to resolve the voltages into their in-phase and out-of-phase components with respect to the supply voltage. Then these components may be added directly. Hence:

Er = 70.8 /-45°

= 70.8 (cos 45°+ j sin 45°) = 70.8 (0.707 + jO.707) = 50 + j50