TRANSFORMER, Its losses and Domestic power supply

Transformer

Different electrical instruments require different voltages for their operation. An X-ray tube requires thousands of volts while filaments of diodes require only 6 volt for their operation. Transformer is a device used for changing the voltage of an a.c. power source to a higher to lower value. 

 

     Principle. It is based upon the principle of mutual induction, i.e., an e.m.f. is induced in a circuit due to a current changing in a neighbouring circuit. 

     Construction. It consists of two coils, namely primary and secondary coils. Both the coils are wound over a soft iron core. The number of turns and the thickness of primary and secondary depends upon the job to be performed by the transformer, i.e., whether the output voltage is to be increased or decreased, the core is made of soft iron in the form of laminations properly insulated with a thin coating of insulation (Fig. 1). 

Fig. 1. Transformer. 

It is taken in the form of laminations to avoid setting up of of eddy currents. Input source of a.c. is connected across the free terminals of primary while the output is taken across the free terminals of secondary. 

     Action. When a.c. is connected in primary, the current and hence the magnetic field around primary and secondary changes continuously. Due to the phenomenon of electro-magnetic induction, a changing magnetic flux through secondary results in induction of a continuous e.m.f. in it. 

     Theory. If the primary and secondary coils are wound close to each other, it can be assumed, to a fair approximation, that magnetic flux 'ΦB' linked with each turn of primary and secondary is same. If 'E' is the e.m.f. induced in each turns. 

                      E = dΦB/dt

     Let          nw = number of turns of primary coil 

and              ns = number of turns of secondary coil

    ∴    E.M.F. induced in a primary coil = 

                E = np (dΦB/dt) 

     At any instant, the e.m.f. induced in primary must be same as that applied in it. 

    ∴   Ep  = - np (dΦB/dt)                             ... (1) 

     If 'Es' is the e.m.f. induced in the secondary, 

              Es = - ns (dΦB/dt)                          ... (2) 

          Es/Ep = - ns(dΦB/dt)- np (dΦB/dt) 

                  = ns/np = K

      where 'K' is called 'transformer ratio'. 

Case (i) Step-up transformer

      If                              ns > np  , i.e., K > 1

     In such a case        Es > Ep 

     A transformer in which voltage across secondary is greater than that across primary is called a step-up transformer. 

Case (ii) Step-down transformer

     If                             ns < np  , i.e., K < 1  

     In such case         Es < Ep

     A transformer in which voltage across secondary is less than that across primary is called a step-down transformer. 

     Let 'Is' and 'Ip' be the currents flowing through primary and secondary (when closed). 

       Power input = Ep Ip       

     Power output = Es Is

     Under ideal conditions, when no loss of energy takes place, 

                  Es Is  = Ep Ip

 ∴            Es / Ep = Ip / Is

     If Es > Ep , then Is < Ip

     Thus, a gain in voltage across secondary results in a consequent loss in electric current through it and vice versa. 

Losses in transformer

     Following are the various sources of loss of energy in case of a transform. 

1. Iron loss. As a result in change in magnetic flux through the core currents known as 'eddy currents' are set up in closed circuits. Energy lost in the form of heat energy as a result of these currents is called 'iron loss'. 
2. Copper loss. Amount of energy lost as a result of production of heat due to currents flowing through primary and secondary coils is called 'copper loss'. 
3. Hysteresis loss. The soft iron core undergoes a complete cycle of magnetisation due to a.c. It also results in some loss of energy, known as 'hysteresis loss'. 
4. Flux leakage. Due to improper alignment it is possible that the flux linked with the secondary may be a little less. 

Long distance transmission of A.C. by transformer

Apart from its natural use of increasing or decreasing the voltage, transformers are used in long distance transmission of a.c.

  

     Generally, the distance between the stations where electric power is generated and that where it is consumed is very large. The wires connecting these two stations have a large length and hence a high resistance. Following losses are likely to take place. 

     (i) Fall of potential across the wire. When a current 'i' flows through a wire of resistance 'R', potential difference 'V' across it is, 

                      V = i R

     This results in a lesser potential reaching the consuming station. 

    (ii) Loss of energy in the form of heat. Heat 'H' produced in the wire is

                     H = i² Rt

     Both the losses are large due to a greater 'R'. Amount of losses can be decreased if transmission is done at a low current and high potential so that the amount of energy transmitted is not affected. 

     A transformer (step-up) converts low  voltage and high current electric energy at generating Station (Fig. 2), into a high voltage, low current energy. This high voltage electricity is transmitted to distant consuming station. At the consuming station this voltage is stepped down in three stages by making use of step-down transformers. 

     A data from a hydro-electric power station shows that electricity is generated at 11,000 V. It is stepped up to 1,32,000 V before transmission. At the receiving station, it is stepped down in three different stages. 

Fig. 2. Long distance transmission of a.c.

Auto-Transformer (Variac) 

It is a device used to obtain a continuous supply of a.c. at a desired voltage. It makes use of a single winding of insulated copper wire that serves as both a primary and a secondary. [Fig. 3(i)] illustrates the principle of a variac. 

Fig. 3. Auto-transformer. 

The input is connected in between the terminals A and B while the output is taken from terminals A and C. Terminal 'C' is a variable contact and can be moved all along the length of coil. If A to C contains more turns than A to B, the device serves as a step-up transformer. Conversely, if 'C' is moved to the left of B, so that A to C has fewer number of turns than A to B, it serves as a step-down transformer. 

     The actual construction of a variac is shown in [Fig. 3(ii)]. It consists of a large number of turns of copper wire wound over a soft iron ring which serves the purpose of iron core. Terminal G is connected to a sliding contact 'S' which is moved along the windings by the turn of a knob K. The insulation on the wire is removed on the very top of the widening and a small block of graphite, under spring pressure serves as a sliding contact. A pointer P attached with the knob K moves over a circular scale from which the output voltage can be read directly. 

Domestic Power Supply

For operating heavy electrical machines, we need three phase a.c. supply in our homes. Supply of three phase a.c. can be done in any of the following two ways. 

     (i) Star connection. One terminal of each of the three coils constituting the armature of an a.c. generator is connected at O. A wire connected at O is called neutral wire. Three wires coming from terminals A, B and C are named as wire 1, 2 and 3 respectively and are known as phase wires [Fig. 4].

Fig. 4. Star connection for three phase a.c. supply. 

     Potential difference between the neutral wire and any of the phase wires is 230 V, while the potential difference between any two phase wires is 300 √3 V Or nearly 400 V. Supply from the former is given to households while that from the letter is given to factories, etc., for operating heavy machines. 

     

    (ii) Delta connection. In this type of connection, the free terminals from coil in a.c. generator are connected together at A, B and C as shown in [Fig. 5]. Wire numbers 1, 2 and 3 taken from A, B and C are known as phase wires. There is no neutral wire in this case. The potential difference between any to wires is 230 V. This arrangement is useful in transmission of small potential difference generally used for domestic purpose. 

Fig. 5. Delta connection for three phase a.c. supply. 

Important Notes

A transformer is effective only in varying current. If a source of d.c. is connected in the primary there will be no e.m.f. in the secondary except at make or break.  In case of a step-down transformer, current in secondary is very large. So the secondary has to be made of a thick wire while a thin wire can be used in secondary of a step-up transformer.  Core of the transformer is taken in laminated form in order to avoid loss of energy due to eddy current.  If the core of the transformer is packed loose, it may give a humming sound. This is due to vibrations of air trapped in between the gaps of the constitutents of metallic core.  Long distance transmission of electric energy is done at high voltage and low current to minimise energy losses.

Some questions with answer

1. Why is a transformer so named? 

Ans. Due to the property of mutual induction a transformer is capable of transforming energy from one circuit to the second. Secondly it is also capable of bring a transformation in the nature of electrical energy i.e., one from a low voltage to one at high voltage and vice versa. Hence it is known as a transformer. 

2. Why should the core of the transformer be laminated ? 

Ans. Eddy currents which get generated within the core of the transformer are undesirable since they result in wear and tear of transformer and also in a loss of energy. So these have to be minimised. If the core is laminated with insulation in between the eddy currents are likely to be terminated, thus, saving some amount of energy. 

3. Some time we hear a humming sound when a transformer is working. What is the reason for this ?

Ans. The laminated core of the transformer should held very tightly so that its constituents are very close to each other. If the core is loose, some air will get trapped in between its constituents. While the transformer is working, some heat is produced due to eddy currents. The air in between starts vibrating with a high frequency, thus, producing a loud sound. 

4. Would you use a thick or a thin wire in the secondary of a step-down transformer ? 

Ans. A step-down transformer converts electrical energy from a high voltage to one at a low low voltage. Accordingly the current in the secondary will be larger than that in the primary. In order to have lesser production of heat in the secondary we shall have to use a wire of lesser resistance i.e., a thick wire. 

5. What should be the characteristics of the material which forms the core of a transformer. 

Ans. 

  (i) It should have a lesser co-ericivity. 

 (ii) It should have a greater retentivity. 

(iii) It should have a greater permeability. 

(iv) It should have a narrow hysteresis loop so that there is a lesser loss of energy for each cycle of magnetisation. 

6. Transformer (step-down). A transformer which decreases the voltage of a.c. source. 

7. Transformer (step-up). A transformer which increases the voltage of a.c. source. 

8. Transformer ratio. Ratio between number of turns of secondary so that of primary coil. 

9. Transformer. Device to change the voltage of an a.c. source. 

 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.

Self Induction:

 Consider a coil 'L' wound over a hollow wooden cylinder [in Fig. 1]. Soft iron is packed into a hollow region to increase magnetic flux linked with 'L' (since 'μ' for iron is greater than one). The two free terminals of the coil are connected to a source of e.m.f. 'E' through a tap key 'K'. 

Fig. 1. Self induced e.m.f. in a coil. 

     As key 'K' is pressed, current flowing through the circuit starts increasing. As it increases (at make) from zero to maximum, an induced e.m.f. is set up in it due to the phenomenon of electromagnetic induction. 

     According to Lenz's law, the induced e.m.f. opposes the change (increasing current) which produces it. Therefore, its direction is opposite to that of current. When current achieves its maximum value, magnetic flux linked with the coil becomes maximum (constant). No induced e.m.f. is there in the coil at this stage. When key is released (at break) current decreases from maximum to zero. This results in decrease of magnetic flux linked with the coil. Again an e.m.f. is set up. The direction of e.m.f., according to Lenz's law, should be same as that of the current. Thus, it is clear, that induced e.m.f. induced e.m.f., whenever it appears in the circuit, opposes a change in the strength of current flowing through the circuit. This property of the circuit is called self induction. 

     Self induction of a circuit is defined as the property of the circuit, by virtue of which it tends to oppose a change in the strength of current, through it, by inducing an e.m.f. in itself. 

Experimental demonstration of self induction

     Property of self induction can be demonstrated by connecting a bulb 'B' across the two terminals of the coil. The e.m.f. induced in the coil comes directly across the bulb and causes a strong current to flow through it. Therefore, the bulb gives a flash of light, whenever induced e.m.f. is present. A bright flash in the bulb at make and at break of the circuit testifies the property of self induction. 

Co- efficient of Self Induction

Let 'ΦB' be the magnetic flux linked with a circuit due to a current 'I' flowing through it. 

         ΦB ∝ I           or      ΦB = LI       .... (1) 

Where 'L' us called the 'Co-efficient of self induction' of the circuit. It depends upon :

(i) area of cross-section of coil

(ii) number of turns of the coil

(iii) nature of material packed within the coil. 

If          I = 1, ΦB = L

     Co-efficient of self induction of a circuit is defined as the magnetic flux linked with it when a unit current flows through it. 

     Differentiating equation (1) with respect to 't', we get

      ΦB /dt = d/dt (LI) = L dI/dt    [∵  L does not depends upon time]

     According to Faraday's law of electromagnetic induction, 

               E = -  ΦB/dt

Where E = e.m.f. induced in the circuit due to a current changing at the rate dI/dt, 

          - E = L dI/dt       

or        E = - L dI/dt                   ......(2) 

If         dI/dt = 1, L = [E]

     Therefore, co-efficient of self induction is also defined as the e.m.f. induced in the circuit, due to a current changing at a unit rate in itself. 

Unit of 'L'

     (i) In S.I. co-efficient of self induction of a circuit is measured in 'henry'. 

     In equation (2) 

     If     dI/dt = 1 A s-1, E = 1 V, then L = 1 H. 

     Co-efficient os self induction of a circuit is said to be 1 henry if an e.m.f. of 1 V is induced in it due to a current changing at the rate of 1 amp s-1 in itself. 

       1 henry= 1 V/1 A s-1 = 1 Vs A-1  

     (ii) In C.G.S. (e.m.u.) system. In the C.G.S. system (e.m.u.) co-efficient of self induction is measured in 'e.m.u. of inductance' or 'abhenry'. 

            |L| = E/dI/dt

∴   1 e.m.u. of inductance = 

       1 e.m.u. of potential difference/1 e.m.u. of current/1 second

     Co-efficient of self induction of a circuit is said to be 1 e.m.u. of inductance if an e.m.f. of 1 e.m.u. is induced in it due to a current changing at the rate 1 e.m.u. s-1.

Dimensions of L

From equation (2) 

         [L] = E/dI/dt

               = [M L2 T-3 A-1]/[A1 T-1]

∴      [L] = [M1 L2 T-2 A-2]

So, the dimension of L are 1, 2, -2, -2 in mass, length, time and electric current respectively. 

Relation between 'henry' and e.m.u. of inductance

      1 henry = 1 volt/1 amp/sec

                     = 10⁸ e.m.u. of potential difference/1/10 e.m.u. of current/sec

∴    1 henry = 10⁹ e.m.u. of inductance. 

Self Induction of Solenoid

A coil wound over an insulating cylinder and having some self induction can be treated as  a solenoid carrying current. Magnetic field at any point, on the axis of a solenoid is

        B = μ0μr  N/l I

     Therefore, magnetic flux linked with one turn of coil 

                  = B × A = μ0μr N/l I × A

Where 'A' is the cross-sectional area of the coil. Total magnetic flux ΦB linked with whole of coil (of N-turns), 

                 ΦB = μ0μr N/l  I × A × N

or             ΦB = μ0μr N² I A/l

     Therefore, induced e.m.f. E is 

             E = - dΦB/dt = - μ0μr N² A/l  dI/dt    ... (3) 

     Comparing equations (2) and (3), 

       L = μ0μr N² A/l = μ0μr n NA

Where 'n' is the number of turns per unit length. 

     Thus, co-efficient of self induction of a coil depends upon following factors :

1. Area of the coil. Greater the area if coil, greater is the co-efficient of self induction. 
2. Number of turns. Greater the number of turns, greater is co-efficient of self induction. 
3. Relative permeability of the core. A coil having soft iron as its core has a greater co-efficient of self induction than that having air core.

IMPORTANT NOTES Resistance 'R' is a measure of opposition to the electric current (constant) while self inductance 'L' is a measure of opposition to a change in current in a circuit.  Self induction is often termed as 'electrical inertia' since it is analogous to inertia in mechanics. Inertia in mechanics opposes a change in velocity while self induction (electrical current).  An ideal inductor is considered to be a coil of zero resistance while an ideal resistor (straight wire) is considered to be possessing zero self induction. Both, ideal inductor and ideal resistor cannot be obtained practically.  An ideal inductor is equivalent to a closed key in an electrical circuit.  Wires used in resistance boxes have a special type of winding called non-inductive winding. The wire is first of all doubled in itself (Fig. 2) and then it is wound over a mica sheet. When current flows through it, the direction of current in two windings situated close to each other will be opposite to each other. Thus, the magnetic field produced by them will get neutralised. Since there is no resultant magnetic field in the region, there will no induced e.m.f. in the wire when current in the wire is changing. Thus, the only opposition will be of resistive nature.  Fig. 2.

Inductances in series and parallel

Like resistors, inductors can also be connected in series and parallel. While connecting them we shall suppose that the two are so spaced apart that changing current in one does not affect the second. 

 

(i) Inductances in series

     Consider two inductances L₁ and L₂ connected in series as shown in Fiɡ. 3. Let I be the instantaneous current flowinɡ throuɡh each of them . Let E₁ and E₂ be the values of e.m.fs. induced in the first and second coil respectively. 

Fig. 3. Inductances in series. 

          E₁ = - L₁ dI/dt   and    E₂ = - L₂ dI/dt

   

     If 'E' is the net e.m.f. induced in the combination

                E = E₁ + E₂

     ∴         E = (- L₁ dI/dt) + (- L₂ dI/dt) 

or            E = - (L₁ + L₂) dI/dt                      ... 4

     If L = net inductance of the combination, 

                E = - L dI/dt                                 ... 5

     Comparing equations (4) and (5), we get

                L = L₁ + L₂

     Thus, the resultant inductance of two inductors in series with each other is equal to the sum of their individual inductances. 

(ii) Inductances in parallel

     When the two inductances are connected in parallel with each other, they carry different electric currents but have same potential difference across their ends. 

Fig. 4. Inductances in parallel. 

     If I₁ + I₂ are the instantaneous currents flowing through them (Fig. 4), the e.m.f.'s induced in them are given by

               E₁ = - L₁ dI₁/dt

and        E₂ = - L₂ dI₂/dt

     Since    E₁ = E₂ = E          (Resultant e.m.f.) 

     ∴             E = L₁ dI₁/dt = - L₂ dI₂/dt        ... 6

     Since the inductances are connected in parallel, 

                     I = I₁ + I₂

     ∴       dI/dt = dI₁/dt + dI₂/dt                  ... 7

     From equation (6), 

             dI₁/dt = - E/L₁    and    dI₂/dt = - E/L₂

     Substituting in equation (7), 

            dI/dt = (- E/L₁) + (- E/L₂) 

or        dI/dt = - E (1/L₁) + (1/L₂)              ... 8

     If L is the resultant inductance of the combination

                  E = - L dI/dt

or        dI/dt = - E × 1/L                            ... 9

     Comparing equations (8) and (9), 

   

                 1/L = 1/L₁ + 1/L₂.

     Thus, when two inductances are connected in parallel with each other, the reciprocal of the resultant inductance is equal to the sum of the reciprocal of individual inductances. 

     It may be noted that the inductances combine in manner similar to the resistance.