ALTERNATING CURRENTS If the potential difference applied to a circuit is a periodic function of the time, the current in the circuit is also a periodic function of the time. If the time average of the current is zero, the current is called an alternating current. For example if the current I is equal to Acospt., where t de notes the time, and A and p are constants, then the current is an alternating current. In fig. 21 let APB be a circle with centre at 0, and let AOB be a diameter. Suppose that the radius OP moves round the circle with uniform angular velocity p, so that the angle POB is equal to pt. Let PN be a perpendicular from P on to the diameter AOB. If AOB remains fixed while P goes round, we have ON =OP cosPOB, or ON = OBcospt. Thus ON may be taken to represent the alternating current
if OB = A. The current is positive when N is between 0 and B, and negative when N is between 0 and A. The time average of the current is clearly zero for times which are multiples of the time of one revolution of P, or of 27r/p. The number of revolutions per second made by P or p/2ir is called the frequency of the alternating current. Each complete revolution of P corresponds to a cycle of the current, and the frequency is equal to the number of cycles per second. If n denotes the frequency, then p= 27rn.
The average value of the square of the current which we may r
denote by I2 is given by I'2 =
dt =
for, since
I, and the average value of
is clearly
equal to that of
we see that the average value of either
is equal to 1. The instruments which are used for measuring alternating currents are so constructed that they indicate the square root of 12, which is called the root mean square current or simply the alternating current. Thus if an alternating am meter indicates 25 amperes, this means that ,I J2 is equal to 25 when I is in amperes (see INSTRUMENTS, ELECTRICAL). In general, if the current I in a circuit is given by /=f(t), where f(t) denotes a function of the time t, and if f (t) = f (nT +t), where n is an in t+T teger and T a constant, and if
f (t)dt = o, then the cur T rent is an alternating current of period T and frequency r/ T.
t+T Also
I Jt (f
and an alternating ammeter will give _ the value of
T. Any alternating current of frequency n = 1/T can be regarded as the sum of currents
... , where the
and a's are constants; that is to say, as the sum of currents with fre quencies n, 2n, 3n, ... , each of which is a simple harmonic function of the time. The components with frequencies which are multiples of the fundamental frequency n are called the harmonics of the alternating current. In what follows we shall consider only alternating currents which are represented by the simple formula I= Acospt; i.e., currents without harmonics.
P = Bcospt is ap plied to a circuit of resistance R and self induction S. The current I will be given by the equation IR = Bcospf —SdI/dt. If the alternating potential difference has been acting for some time, the current will be an alternating current given by I=A cos(pt — a) , where A and a are constants. By substituting this value of I in the differential equation, we find that A =
and tans =Sp/R, so that
This solution of the equation Bcospt = dt may be obtained graphically as follows: Assuming I = A cos(pt — a) , we get Bcospt = A Rcos(pt—a) —ASpsin(pt—a).
22 let the angle BOE be made equal to pt and OB = B, also let the angle BOA be made equal to a; from B drop a perpen dicular BA to OA ; draw BD and 4E perpendicular to OE, and draw AC perpendicular to BD; then we have OD =OE —DE=OE—CA, the angle CBA is equal to the angle AOD = pt— a; hence OB cos pt = OA cos (pt — a) — A Bsi n (pt — a) . But OB = B, so that, comparing this with the equation Bcospt =
— ASpsin(pl —a), we see that OA = AR and AB= ASp. Hence tana =BA/0A, and AB=ASp and
AB'
so that
as before.
R is small compared with Sp, the current is approximately equal to Bsinpi/Sp, since cos(pt— 2) =sinpt.
lags behind the potential difference. The angle of lag, a, is called the phase difference between the potential difference and the current.
The capacity of a condenser in electromagnetic units is defined, as in electrostatics, to be the ratio of the charge on one of the plates to the potential difference between the plates. The practical unit of capacity of a condenser is called a farad, and is a capacity such that, when the charge on one of the plates is one coulomb, the potential difference between the plates is one volt. Since one coulomb is one tenth of an electromagnetic unit of quantity and one volt is
electro magnetic units of potential difference, it follows that one farad is equal to io a of an electromagnetic unit of capacity. The microfarad is one millionth part of a farad and so is io 15 of an electromagnetic unit.
C is charged, and that then the plates are connected together by a wire of resistance R and self-induction S. If P denotes the potential difference between the plates and I the current in the wire, after a time t, we have I= (P— SdI/dt)/R, or SdI/dt+RI — P = o. If E is the charge on one of the plates, then E= PC and I= —dE/dt so that
-I- RdE/dt -}- E/C = o. Putting E = Aeat, we get
= o. Denoting the two roots of this equation by a and 11, we see that the solution of the differential equation is E= A eat+, Befit where A and B are constants. When t=o we have I= — dE/dt = o, so that A a + B/3 = o, and if
denotes the value of E at t=o, then
Hence A= EGO/ (I3—a), and B=
(Li —a), so that E = E0 ((3eat— aest). The roots of 0—a
o are given by a = — — —
2 —
and
4S
(3 = — R -{- — I • If
is greater than I/SC,
S C both roots are real and negative, so that the charge E gradually diminishes and finally becomes zero. The current -Bt /2s is given by I = 0 e —
—
4SC) so that if
is less than I/SC, we get Eo e-rft/2s 2 1 _-
r
—
2E0
t/28 I
I= — 2 2 sin ( C
R C) S 4 since
e-.i(-1't = 21 1( I) sin x. It appears therefore that when
is less than 1/SC the current from the condenser is an alternating current which gradually diminishes. The fre quency, or number of oscillations per second, is given by 2 n = I I — If R/S is very small, n is approximately 27 S C43' equal to 1/(27r- SC), so that the time of one oscillation is 27r s,/ SC. In this case I = E0 sin(t/,I SC), so .that, since
S C I= —dE/dt, we have E =
(tHSC) . The charge E is there fore alternately positive and negative, or we may say that the electricity os cillates backwards and forwards from one plate of the condenser to the other through the wire. The number of oscil lations per second, when an ordinary Ley den jar is discharged through a short wire, is very large. For example, if the capaci ty of the jar is 2,000 electrostatic units, or 2,222X i o 12 farads, and the self induction of the wire used is 1/222,200 henry, then T= 27r-s1(10 ") = 6.28X io 7 sec., so that n =
R • S9 X ios oscillations per second.
Containing a Condenser.—Now consider the case of a con denser of capacity C, with its plates connected by a wire of self induction S and resistance R, when an alternating electromotive force Bcospt acts in the wire. If E is the charge on one of the plates of the condenser, it is easy to show that Bcospt/R, which is the same as the current due to an electro motive force Bcospt in a wire of resistance R and zero self-induc tion. The effect of the self-induction in the circuit can therefore be eliminated by the introduction of a condenser. If the fre quency p/2ir of the applied electromotive force is varied, the current obtained is comparatively small until the frequency ap proaches I/(27r ,ISC), at which value the current is a maximum. The oscillations of electricity in circuits containing a condenser may be compared with the oscillations of a body hung on a spiral spring. If x denotes the displacement of the body from its equilibrium position then we have m d 2 + r r dtµ
= F, where dt m is the mass of the body, r the viscous resistance to its motion per unit velocity, µ the restoring force for unit displacement and F the external applied force (see MECHANICS). If we put x = E, rn = S, r = R, and µ = 1/C this equation becomes the same as that for the condenser and circuit, with F standing for the applied electromotive force. The maximum current obtained when the frequency n = T/(27r II SC) is thus analogous to resonance in dynamics. When the applied force F has the frequency I/(27rV m/µ), with which the body oscillates freely when F and r are zero, the amplitude of the vibration set up is a maximum.
If an alternating potential difference P = Bcospt is applied to any circuit, and the current I produced is equal to
the electrical work done in time dt is equal to PIdt. The rate of doing work, or the power, is therefore, at any instant, W = PI = BA cos pt cos (pt — a) . We have
The average values of cosipt is I/ 2, and that of sin 2 pt is zero, so that we see that the average value of W is equal to IBA cosa. If the root mean square values of the potential difference and current are denoted by P' and I', then P' = B/ 1V 2 and I' = A/'I2 ; so that P'I' = AB/2 and the average value of the power is P'I'cosa; Cosa is called the power factor.
In the case of a potential difference, Bcospt, sending a current I = Bcos(pt _ a)/
through a wire, we have P' ` B/1,12, I'=BH all
and, since tana=Sp/R,
so that
which is equal
In this case all the energy expended is converted into heat, which is equal per unit time to the mean square of the current times the re sistance, just as in the case of a steady current C it is equal to
per unit time.
In the case of a potential difference Bcospt acting on a circuit of inductance S, connected to a condenser of capacity C, if the resistance of the circuit is negligible, we have for the current I =
where tang= oo , so that a
The power P'I'cosa is therefore zero, since cos(7r/2) =o. The work done in charging the condenser is given out again when the condenser is discharged, so that no energy is used up. In the transmission of power by means of alternating currents, it is important that the power factor should be nearly equal to unity since, if it is small, large currents are required to transmit a small amount of power.
When a con stant potential difference P acts on a circuit of resistance R and inductance S then as we have seen above P — SdI/dt = I R, where I denotes the current. The electrical energy expended in a time dt is PIdt, so that denoting this by dW, we have dW =
dt.
dt Now
dt is the heat energy produced, so that SIdI must rep resent energy stored up in some form. The current produces a magnetic field, and the electromotive force SdI/dt is due to the increase of this field, so that it is natural to suppose that the energy Sidi is the energy of the magnetic field. If the current increases from zero to I, then the energy stored up is
which we therefore suppose to be the energy of the
field of the current I. In fig. 23 let ABC be the circuit, and let
