Alternating Current

Alternating E.M.F.

When a coil is rotated in a uniform magnetic field an e.m.f. is induced in it. Variation of e.m.f. with time, shown in Fig. 1 is given by

                  E = E₀  sin ωt

Fig. 1. Alternating e.m.f.

     Here E is the instantaneous e.m.f., E₀ is the maximum value of e.m.f. and 'ω' the angular frequency of the variation of e.m.f. . This e.m.f. has following characteristics :

(i) It varies continuously with time. As it clear from the figure that no two consecutive instants of time have same value of e.m.f.

(ii) It reverses periodically in direction. The direction of e.m.f. during the time interval OB and that during the interval BD are opposite to each other. This phenomenon gets repeated after equal intervals of time. Such an e.m.f. is called alternating e.m.f.

        An alternating e.m.f. is one which continuously changes in magnitude and periodically reverses in direction. 

        Graphically, it is represented by sinusoidal curve whose mathematical form is given in above equation. 

Alternating Current

If the external circuit is closed, the alternating e.m.f. produced in the coil rotating in a uniform magnetic field will cause a current to flow. This current, called alternating current, possesses the same characteristics as possessed by alternating e.m.f. and is mathematically represented by the equation. 

                      i = I₀ sin ωt

Fig. 2. Alternating current. 

Where I₀ is the maximum value of the current called its peak value. Graphical representations of such a current is shown in Fig. 2.

     The alternating e.m.f. and currents can also be represented as 

       E = E₀ cos ωt   and   i = I₀ cos ωt

     Graphical representation of E and i as cosine function of time are shown in Fig. 3(i) and Fig. 3(ii) respectively.

Fig. 3. E and i as cosine function of time. 

Some terms connected with alternating currents

     (i) Peak value (I₀). It is the maximum value of electric current in either direction. It has a constant magnitude which depends upon the e.m.f. and the net effective opposition (impedance) offered by the circuit.

  (ii) Time period (T). It is the time interval after which the instantaneous current in the circuit gets repeated (in magnitude and direction).      

     Let 'i' be the instantaneous current

                     i = I₀ sin ωt

     Current 'i' after an interval T (=2π/ω) is given by

                     i' = I₀ sin ω (t + 2π/ω) 

or                 I' = I₀ sin (ωt + 2π) 

     Since     sin (2π + ωt) = sin ωt

∴                                      i' = I₀ sin ωt = i

     Time interval T (=2π/ω) is called the time period of a.c.

   (iii) Cycle of a.c. Variation of current in between two consecutive, similar values of current (in magnitude and direction) is said to constitute one cycle of a.c.

     Thus, variation of current along OABCD in Fig. 2 constitutes one cycle of a.c. It is clear that one cycle of a.c. is completed in one time period. During one cycle, the electric current becomes maximum two times (in opposite directions). Variation between O and C (time interval T/2) constitutes half cycle of a.c.

    (iv) Frequency (F). Number of cycles of a.c. completed in one second is frequency of a.c.

     Alternating currents of different frequencies are being used for different scientific purposes. Electricity for domestic use consists of a frequency of 50 cycles per second, while a.c. of millions of cycles per second is used for communication purposes. 

     Frequency    f = 1/T = 1/2π/ω

or                         f = ω/2π

or                        ω = 2πf

     (v) Phase. There is always a generating element (generally a coil) rotating in uniform magnetic field for production of a.c. Phase of a.c. is measured by the angle θ (= ωt) turned by the generating element with respect to a certain instant of time. 

     In general, a.c. is represented by

               i = I₀ sin (ωt + Φ) 

     The argument (ωt + Φ) of the sine function is called the phase of a.c.

     At t = 0,     i₁ = I₀ sin (ωt × 0 + Φ) = I₀ sin Φ

     Here 'Φ' represents the phase of current initially i.e., at t = 0. It is called its initial phase or epoch. 

     If two currents are represented by equations, 

                     i₁ = I₀ sin (ωt + Φ₁)

and              i₂ = I₀ sin (ωt + Φ₂)

     'Φ₂ - Φ₁' is said to be the phase difference between them.

     If Φ₂ - Φ₁ = 0, the two currents are said to be in phase with each other.

Principles of measurement of A.C.

Average value of alternating current i over a complete cycle is zero whereas it is not so for i². So, the instrument used for measurement of A.C. has to be such that the deflection should be proportional to square of the current. A moving iron type instrument is shown in Fig. 4.

     It consists of a brass cylinder having a solenoid would over it all along the length of the cylinder. A pointer P is pivoted at O on the axis of the solenoid. Two iron rods A and B are placed along the length of the cylinder. A is rigidly attached to the cylinder while B is attached to the pointer. 

     As the current, to be measured is passed through the solenoid iron rods A and B get magnetised. Due to the similar polarity developed by them, they get repelled. The force of repulsion between them is proportional to the pole strengths which in turns depend upon the current. So the force of repulsion varies as the square of the current through the coil. Since A is rigidly clamped, B moves. The restoring force is provided by force of gravity or by spring attached to B. Motion of B can be communicated to the pointer by means of a lever. As the pointer moves over a scale the deflection can be noted. Since the deflection is proportional to the square of current, a linear scale cannot be used. 

Fig. 4. Measurement of A.C.

Comparison of A.C. with D.C.

In early days, we used d.c. everywhere. If somebody happened to touch a live wire, it gave a rude shock and the person got thrown away and saved. Now a days everywhere a.c. is being used. It is dangerous if somebody touches a live wire. He may not survive. If the person touching the live wire is connected to the earth, his body acts as a conductor and the current flows through him. His body (along with his brain) starts vibrating with the frequency of a.c. (generally 50 Hz). Brain, when in state of vibrations becomes incapable of relaying signals to different parts of body. The person is not capable to break his contact with the live wire. If another person tries to pull him apart, same thing shall happen to him. In such an eventually the connection with the live wire should be broken with the help of an insulating (wooden, glass etc.) rod. 

     In spite of being dangerous to human life a.c. is being extensively used in our daily life. Naturally, it much have some advantages over d.c. The two types of electricity can be compared with respect to each other. 

A.C

D.C

Its magnitude changes continuously.  It's direction reverses after equal interval of time.  It has a frequency which may change depending upon its use.  Polarity of the source changes after equal interval of time.  Opposition offered by an inductor / capacitor depends upon its frequency.  Power loss depends upon the frequency the relative component.  Power factor is always less than one.  Highly economical when transmitted from one place to another at high voltage.  Not suitable for some purposes like electroplating.

Its magnitude may or may not change.  Its direction always remains the same.                   Its frequency is zero.                                                                          Polarity of the source always remains the same.  Opposition offered by an inductor is zero while that by a capacitor is infinite.            Power loss is I²R.                                                                                Power factor is one.                      High power loss takes place during transmission. So, it is not so economical.  Suitable for electroplating.

Mean value of A.C. or "D.C. value of A.C." ('Im') 

Mean value of a.c. is that value of steady current which sends the same amount of charge through a circuit in same time as is done by a.c. in half its cycle. 

     i - t curve for a.c. during half its cycle is shown in Fig. 5. 

Fiɡ. 5. Half cycle if a.c.

Let the interval T/2 be divided into n very small and equal intervals, each having value T/2n. Let i₁, i₂, ..., in be the values of instantaneous currents durinɡ these intervals. If q₁, q₂, ..., qn are the  amount of charɡes transferred across any section of the conductor durinɡ these intervals, 

q₁ = i₁ × T/2n,   q₂ = i₂ × T/2n, …, qn = in × T/2n

Total charɡe Q transferred in half cycle is 

           Q = q₁ + q₂ + … + qn

or       Q = (i₁ + i₂ + … + in) T/2n          …(1)

If        Im = mean value of a.c.

        

   Q = Im × T/2                              …(2)

From equation (1) and (2), we ɡet

          Im T/2 = (i₁ + i₂ + … + in) T/2n

or             Im = i₁ + i₂ + … + in / n

     

     Thus, mean value of alternatinɡ current is equal to the arithmetic mean of the instantaneous currents.

Average value of A.C. over a complete cycle (Iav) 

Average value of alternating current is defined as 

              Iav = 1/T T0∫ i dt

Since       i = I0 sin ωt

              Iav = 1/T T0∫ I0 sin ωt dt

   

                  = I0/T [- (cos ωt/ω)]T0

                 = -I0/ωt [cos ωt]T0 

                 = -I0/ωt [cos ωT - cos 0°]

                 = -I0/ωt [cos 2π - cos 0] 

                 = -I0/ωt (1 - 1) = 0

     Thus, the average value of a.c. taken over the complete cycle of a.c. is zero. 

Root mean square value of a.c. Or 'Virtual value of a.c.' or 'Effective value of a.c./a.c. value of a.c.'

Root mean square value of alternating current is defined as the value of steady current which produces same heating effect, in a resistance, in a certain time as is produced by the alternating current in same resistance in same time. The r.m.s. value of a.c. is also called its virtual value. 

     Let the time for one cycle (T) be divided into n small and equal intervals each equal to T/n. Let i₁, i₂, ... , in be the instantaneous current during these intervals. If H₁, H₂, ..., Hn are the amounts of heat produced during these intervals, then

          H₁ = i₁² R (T/n), H₂ = i₂² R(T/n), Hn = in² R (T/n) 

   If H is the amount of heat produced in one cycle of alternating current. 

         H = H₁ + H₂ + ... + Hn

or     H = i₁² R (T/n) + i₂² R(T/n) + ... + in² R (T/n) 

or     H = (i₁² +  i₂² + .... + in²) R (T/n)         ... (3) 

Let Iv = virtual value of alternating current

          H = Iv² RT                                          ... (4) 

From equations (3) and (4), we get

     Therefore, root mean square value of alternating current is the square root of the mean of the squares of instantaneous currents. 

Important Notes

Greater the frequency of a.c. smaller will be the duration of cycle of a.c. As a result the consecutive maximas will be situated close to each other.  During one cycle current becomes zero twice. Thus, an electric bulb, fed with a.c., shall be put off twice in one cycle. But this is not visually observed. This is due to the property of persistence of vision. The time interval between two consecutive brightnesses is much less than 1/16th of a second. Therefore, the two consecutive impressions of brightness merge into one another, thus, eliminating the region of darkness.  Alternating current is dangerous to human life. This is due to the high frequency (50 cs-1) of a.c. When a person, having communication with the ground, touches a live wire, 50 cs-1 a.c. passes through his body. All parts of his body including his brain start vibrating with a frequency of 50 cs-1 . Therefore, his brain is rendered helpless to relay orders to his different parts of body with the result that body remains paralysed in same position. If the live wire is not removed away within seconds it may prove fatal. It should be kept in mind that the wire should be removed away only with an insulating material.