## Sco

When a circuit contains resistance, capacitance and inductance, all three in series, the value of reactance will be the difference between that of the coil and that of the condenser. Since for a given coil and condenser the inductive reactance increases with frequency and capacitive reactance decreases with frequency, Xji is conventionally considered positive and Xc negative.

.In JSnding_the current flow thro_ugh_a con-denser in an alternating current circuit wcaji

usually assume that_/ = — (Xc being the capacitive reactance of the conciftnsgj). The use of the term Z (impedance) is, in such cases, made unnecessary because the resistance of the usual good condenser is not high enough to warrant consideration. When there is a resistance in series with the condenser, however, it can be taken into account in exactly the same

.Vo/tage

7h? A. C Msters read the effective va/ues of current unci va/ta\$e(T~rn s 707 of njax value ofs/zje wave.) a TTme

FIG. 306 — REPRESENTING SINE-WAVE ALTERNATING VOLTAGE AND CURRENT

7h? A. C Msters read the effective va/ues of current unci va/ta\$e(T~rn s 707 of njax value ofs/zje wave.)

a TTme

FIG. 306 — REPRESENTING SINE-WAVE ALTERNATING VOLTAGE AND CURRENT

manner as was the resistance of the coil in the example just given. The impedance of the condenser-resistance combination is then computed and used as the Z term in the Ohm's Law formulas.

### The Sine Wave

In Fig. 306 a curve describing the voltage developed by an alternating-current generator during one complete cycle is shown. This curve is actually a graph of the instantaneous values of the voltage amplitude, plotted against time, assuming a theoretically perfect generator. It is known as a sine curve, since it represents the equation e - iimax sin at, where e is the instantaneous voltage, Emax is the maximum voltage and t is the time from the beginning of the cycle. The term o>, or 2ir/, represents the angular velocity, there being 2v radians in each complete cycle and / cycles per second. All the formulas given for alternating current circuits have been derived with the assumption that any alternating voltage under consideration would follow such a curve.

It is evident that both the voltage and current are swinging continuously between their positive maximum and negative maximum values, and it might be wondered how one can speak of so many amperes of alternating current when the value is changing continuously. The problem is simplified in practical work by considering that an alternating current has an effective value of one amvere when it produces "heat at the same average rate as one ampere of continuous direct current flowing ' through a gvvenj-esistor. This effective value is the square root of the mean value of the instantaneous current squared. For the sine-wave form, FIG. 307 — A COMPLEX WAVE AND ITS SINE-WAVE COMPONENTS

FIG. 307 — A COMPLEX WAVE AND ITS SINE-WAVE COMPONENTS

For this reason, the effective value of an alternating current, or voltage, is also known as the root-mean-square or r.m.s. value. Hence, the effective value is the square root of Vi or 0.707 of the maximum value — practically considered, 70% of the maximum value.

Another important value, involved where alternating current is rectified to direct current, is the average. This is equal to 0.636 of the maximum (or peak) value of either current or voltage. The three terms maximum (or peak), effective (or r.m.s.) and average are so important and are encountered so frequently in radio work that they should be fixed firmly in mind right at the start.

They are related to each other as follows:

Bmax = Em X 1.414 = #ave X 1.57 Em = EmX . 707 = Em X 1.11

The relationships for current are the same as those given above for voltage. The usual alternating current ammeter or voltmeter gives a direct reading of the effective or r.m.s. (root mean square) value of current or voltage. A direct current ammeter in the plate circuit of a vacuum tube approximates the average value of rectified plate current. Maximum values can be measured by a peak vacuum-tube voltmeter. Instruments for making such measurements are treated in Chapter Seventeen.

### Complex Waves

• Alternating currents having the ideal sine-wave form just described are practically never found in actual radio circuits, although waves closely approximating the perfectly sinusoidal can be generated with laboratory-type equipment. Even the current in power mains is somewhat non-sinusoidal, although it can be considered sinusoidal for most practical purposes. In the usual case, such a current actually has components of two or more frequencies integrally related, as shown in Fig. 307. The lowest and principal frequency is the fundamental. The additional frequencies are whole-number multiples of the fundamental frequency (twice, three times, etc.), and are called harmonics. One of double frequency is the second harmonic, one of triple frequency the third harmonic, etc. Although the wave resulting from the combination is non-sinusoidal the wave-form of each component taken separately has the sine-wave form.

The effective value of the current or voltage for such a complex wave will not be the same as for a pure sine wave of the same maximum value. Instead, the effective value for the complex wave will be equal to the square root of the sum of the squares of the effective values of the individual frequency components. That is,

E = VEi2+Ef+W, where E is the effective value for the complex

E = VEi2+Ef+W, where E is the effective value for the complex FIG. 308 — PULSATING CURRENT COMPOSED OF ALTERNATING CURRENT SUPERIMPOSED ON DIRECT CURRENT

of the fundamental and harmonics. The same relation also applies where currents of different frequencies not harmonically related flow in the same circuit. Further aspects of complex waves are discussed in connection with distortion in the following chapter. The subject is of particular importance in 'phone transmission, as shown in Chapters Eleven and Twelve.

# There are many practical instances of simultaneous flow of alternating and direct current in a circuit. When this occurs there is a pulsating current and it is said that an alternating current is superimposed on a direct current. As shown in Fig. 308, the maximum value is equal to the d.c. value plus the a.c. maximum, while the minimum value (on the negative a.c. cycle) is the difference between the d.c. and the maximum a.c. values. If a d.c. ammeter is used to measure the current, only the average or direct-current component will be indicated. An a.c. meter, however, will show the effective value of the combination. But this effective value is not the simple arithmetical sum of the effective value of the a.c. and the d.c., but is equal to the square root of the sum of the effective a.c. squared and the d.c. squared.

where /ao is the effective value of the a.c. component, I is the effective value of the combination and /d0 is the average (d.c.) value of the combination. If the a.c. component is of sine-wave form, its maximum value will be its effective value, as determined above, multiplied by 1.414. If the a.c. component is not sinusoidal the maximum value will have a different ratio to the effective value, of course, depending on its wave form, as discussed in the preceding section.

### Power With Pulsating Current

• In a resistance circuit, the power developed by a pulsating current will be I2R watts, I being the effective or r.m.s. value of the current and R the resistance of the circuit in ohms. In the special case of sine-wave a.c. having maximum value equal to the d.c., which represents 100% modulation of the d.c. by the a.c., the effective value of the a.c. component is 0.707 (70%) of its maximum a.c. value and likewise of the d.c. value. If the two maximum values are each 1 ampere,

Hence, when sine-wave alternating current is superimposed on direct current in a resistance circuit the average power is increased 50% if the maximum value of the a.c. component is equal to the d.c. component. If the a.c. is not sinusoidal, the power increase will be greater or less, depending on the alternating-current wave form. This point is discussed further in connection with speech modulation in Chapter Eleven.

Phase

0 It has been mentioned that in a circuit containing inductance, the rise of current is.

(p) QrnjiUjfndVolUge "nphase'with Pure (ts/stanot (b) Current iaflmf Yo/tafe mttv Pure Inductanc*

(cj Current 'It^BnfWUjt with Pun CapocJtane

FIG. 309 —VOLTAGE AND CURRENT PHASE RELATIONS WITH RESISTANCE AND REACTANCE CIRCUITS

(p) QrnjiUjfndVolUge "nphase'with Pure (ts/stanot (b) Current iaflmf Yo/tafe mttv Pure Inductanc*

(cj Current 'It^BnfWUjt with Pun CapocJtane

FIG. 309 —VOLTAGE AND CURRENT PHASE RELATIONS WITH RESISTANCE AND REACTANCE CIRCUITS

delayed by the effect of electrical inertia presented by the inductance. Both increases and decreases of current are similarly delayed. It is also true that a current must flow into a condenser before its elements can be charged and so provide a voltage difference between its terminals. Because of these facts, we say that a current "lags" behind the voltage in a circuit which has a preponderance of inductance and that the current "leads" the voltage in a circuit where capacity predominates. Fig. 309 shows three possible conditions in an alternating current circuit. In the first, when the load is a pure resistance, both voltage and current rise to the maximum values simultaneously. In this case the voltage and current are said to be in phase. In the second instance, the existence of inductance in the circuit has caused the current to lag behind the voltage. In the diagram, the current is lagging one quarter cycle behind the voltage. The current is therefore said to be 90 degrees out of phase with the voltage (360 degrees being the complete cycle). In the third example, with a capacitive load, the voltage is lagging one quarter cycle behind the current. The phase difference is again 90 degrees. These, of course, are theoretical examples in which it is assumed that the inductance and the condenser have