Basic Electronics On The Go - Inductors in Series

From www.electronicshub.org/inductors-in-series/

An inductor is a passive element which is used in electronics circuits for temporary storage of electrical energy in the form of magnetic flux or simply magnetic field. Inductance is the property of any coil which can set up a magnetic flux when current passes through it.

Any device which has the property of inductance can be called an inductor. Usually inductor is built in the form of a coil with copper material around the core of a magnetic (iron) or nonmagnetic medium (like air).

Inductors may be connected in series or parallel configuration depending on the  performance required by the circuit. These combinations are used to design more complex networks. 

Inductors are arranged on the basis of their mutual inductance or magnetic coupling in series or parallel combinations.

Inductors Connected in Series

Assume that inductors connected in the circuit do not have any coupling between them. This implies that there are no flux lines from one inductor linking with another, and hence there will be no mutual flux between the coils.

The end to end connection of two or more inductors is called “series connection of inductors”. In this connection the inductors are connected in series so the effective turns of the inductor increases. 

The inductance of series connected inductors is calculated as the sum of the individual inductances of each coil since the current change through each coil is same.

This series connection is similar to that of the resistors connected in series, except the resistors are replaced by inductors. If the current I is flowing in the series connection and the coils are L1, L2, and so on, the common current in the series inductors is given by

ITotal = IL1 = IL2 = IL3. . . = In

If the individual voltage drops across each coil in this series connection are VL1, VL2, V¬L3, and so on, the total voltage drop between the two terminals VT is given by

VTotal = VL1 + VL2 + VL3…. + Vn

As we know that the voltage drop can be represented in terms of self inductance L, this implies

V = L di/ dt.

This can also be written as

L_T di/dt = L1 di/dt + L2 di/dt + L3 di/dt + . . . + Ln di/dt

Therefore the total inductance is

LTotal = L1 + L2 + L3 + ….. + Ln

Mutually Connected Inductors in Series

Now consider that inductors are connected such that magnetic field of one coil affects the other. When two or more inductors are connected in series, then the inductance of one inductor will be affected by the magnetic field produced by the other coil.

This is called mutual inductance and the coils are called “Mutually connected inductors”. This mutual inductance may increase or decrease the total inductance of the series circuit.

The factor that affects the mutual inductance of a series connected of inductors is the distance between the coils and their orientation.

The mutually connected inductors can be coupled in two ways

1) Cumulatively coupled or Series Aiding

2) Differentially coupled or Series opposing

Cumulatively Coupled Inductors in Series

If the magnetic fluxes produced by the inductors are in the same direction to the flow of current through them, then the coils are known as “Cumulatively coupled”.

In this series aiding or cumulative coupled circuit, the current  that enters or leaves the terminals of coils at any instant of time are in the same direction.

The figure below shows the connection of two inductors in a series aiding arrangement.

If we pass the current through the cumulatively coupled coils (between the nodes A & D) in the same direction, the voltage drop of each individual coil will affect the total inductance of the series.

Let self inductance of the coil-1 is L1, self inductance of the coil-2 is L2 and the mutual inductance is M between coil 1 and coil2.

Self induced emf in coil-1 is

e1 = – L1 di/ dt

Mutual induced emf in coil-1 due to change of current in coil-2 is

eM1 = – M di/ dt

Similarly, Self induced emf in coil-2 is

e2 = – L2 di/ dt

Mutual induced emf in coil-2 due to change of current in coil-1 is

eM2 = – M di/ dt

Therefore, total induced emf in the series aiding circuit is given as

e = – L1 di/ dt– L2 di/ dt– 2M di/ dt

= – (L1+ L2 + 2M) di/ dt

If LT is the total inductance of the circuit, the total induced emf will be equivalent to

e = – LT di/ dt

Substituting in the above equation, we get

– LT di/ dt = – (L1+ L2 + 2M) di/ dt

Therefore, LT = (L1 + L2 + 2M)

Differentially Coupled Inductors in Series

If the magnetic fluxes produced by the inductors are in the opposite direction to each other, then the coils are known as “Differentially coupled”.

In this differential coupled or series opposition connection, the current enters or leaves the terminals of coils at any instant of time are in the opposite direction.

The figure below shows the connection of two inductors in series opposition arrangement.

 total induced emf in the series opposing circuit is given as

e = – L1 di/ dt– L2 di/ dt + 2M di/ dt

= – (L1+ L2 – 2M) di/ dt

If LT is the total inductance of the circuit, the total induced emf will be equivalent to

e = – LT di/ dt

Substituting in the above equation, we get

– LT di/ dt = – (L1+ L2 – 2M) di/ dt

Therefore, LT = (L1 + L2 – 2M)

 Summary

• An inductor is a passive element which is used in electronics circuits for storing energy as magnetic flux. Inductance is measured in Henry.
• The dissipation amount of actual power with the current flow in the circuit is called “Inductive reactance”. It is measured in ohms. XL = 2  f L

• Self inductance is the property of an electric circuit or a loop in which its own magnetic field opposes any change in current
• Mutual inductance is the ability of an inductor that causes to induce emf in another inductor placed very close to it when current in first inductor changes.
• The end to end connection of two or more inductors is called “series connection of inductors”. The formula for total inductance in series is LT = L1 + L2
• The total inductance of the series connected inductors is always greater than the largest inductor in that series.
• If the magnetic fluxes produced by the inductors are in the same direction to the flow of current through them, then the coils are known as “Cumulatively coupled”. LT = L1 + L2 + 2M
• If the magnetic fluxes produced by the inductors are in the opposite direction to each other, then the coils are known as “Differentially coupled”. LT = L1 + L2 – 2M

Basic Electronics On The Go - Inductance of an Inductor

From www.electronicshub.org/inductance-of-an-indcutor/

Like capacitors and resistors, an inductor is also a passive element. Simply, an Inductor is a twisted wire or coil of electroconducting material. Inductance is the property of an electric conductor or a circuit that opposes the change to a flow of current.

An electric conductor or a circuit element with the property of Inductance is called an Inductor. When there is a change of current in a coil or a twisted wire (inductor), it opposes this change by generating or inducing an electromotive force (EMF) in itself and nearby conducting materials.

Capacitance is the measure of the ability of a conductor to store electric charge i.e. electric field energy. In contrast, Inductance of an electrical conductor is the measure of its ability to store magnetic charge i.e. magnetic field energy.

An inductor stores the energy in the form of magnetic field. As magnetic field is associated with flow of current, inductance is associated with current carrying material. The inductance of a coil is proportional to the number of turns of the coil.

Di-electric materials like plastic, wood and glass have least inductance. But the Ferro magnetic substances (iron, Alnico, chromium ferroxide) will have high inductance.

The unit for inductance is Henry, micro Henry, milli Henry etc. It can also be measured in Weber/ ampere. The relation between Weber and Henry is, 1H = 1 Wb/A.

To understand the inductance of a coil, we should know about Lenz law, which explains us how the emf will induce in an inductor. Lenz’s law states an induced electromotive force that generates a current that induces a counter magnetic field opposing the magnetic field generating the current. 

Another definition of Inductance is “The electromagnetic force produced in a coil by applying the voltage of 1 volt, and is exactly equal to one Henry or 1 ampere/ second”.

In other words, for 1 volt of voltage VL and the rate of flow of current is 1 amp/ sec then the inductance of the coil is L, measuring 1 Henry. 

The induced voltage in the inductor (coil) is given as

V_L = -L di/dt (volts)

The negative sign indicates the opposing voltage in the coil per unit time (di /dt).

The inductance in a coil is of 2 types, they are

• Self inductance
• Mutual inductance
• 
  

Self Inductance

Inductance or self inductance is the property of a current carrying conductor where an EMF is induced in it when there is a change in flow of current.

When an alternating varying current flows through the inductor coil, the magnetic flux in the coil will also vary, to produce the induced emf. This process is called “Self induction” and the inductance achieved by the coil is called “Self-inductance”.

The concept of self inductance can be understood by assuming a current carrying circuit element or an inductor coil of N turns. When current flows through the coil, a magnetic field is produced in and out of the coil.

There is a magnetic flux introduced because of this magnetic field. Then, the self inductance of the coil is the magnetic flux linkage per unit current. When the inductor coil intercepts the magnetic flux lines caused by an electric field, the self emf will be induced in the coil itself.

In other words, the self-inductance means, the ability of a coil to oppose the current’s change. It is measured in Henry. The magnetic properties or the magnetic nature of a coil affects the self-inductance of the coil.

This is the reason why Ferromagnetic materials are used to increase the inductance of the coil, by increasing the magnetic flux in it.

The expression to find the Self-inductance of a coil, is

L = N Φ /I

Where   N represents the number of turns in the coil

Φ is the magnetic flux

I is the current due to the produced emf

L means the inductance value in Henries.

Self Induced EMF and Coefficient of Self Inductance

We know that the current flowing through the inductor is represented by I and Φ is the magnetic flux. They both are directly proportional to each other. So it can be represented as I ∝ Φ.

The number of turns in the inductor is also proportional to the current in the coil. We can derive the relation between current and the emf induced in it as

(dΦ )/dt = L (di )/dt

e = – (dΦ )/dt

e = – L (di )/dt

The value of inductance depends on the geometry or shape of the coil. That is value is called “coefficient of self-inductance”.

We can design the inductor coils as per our need by using the high or low permeability materials and using coils having different number of turns. The magnetic flux produced inside of a inductor core is given as

Φ = B x A

Here B is the flux density and A is the area occupied by the coil.

Self-inductance in a long solenoid

If we consider a long hollow solenoid having its cross sectional area A and length l with n number of turns, then its magnetic field due to the flow of current I is given as

B = μ_0 H = μ_0(N.I )/l

The total flux in the solenoid is given as N Φ = LI

Substituting this in the above equation, (N.B. B= Φ/ A)

L = N Φ /I

L = (μ_0 N2 A )/ l

Where   L is self-inductance in Henry

μ_0 is the permeability of air or hollow space

N represents the number of turns in the coil i.e. inductor

A is the inner cross sectional area of the solenoid

l is the length of the coil in meters.

This is the self-inductance of the long length hollow solenoid. μ represents the absolute permeability of the material with which the solenoid is filled. Here in this case, we calculated the self-inductance for the hollow solenoid, hence we use μ_0.

To have the high permeability or to produce high magnetic flux we fill the solenoid with ferromagnetic substances like soft iron.

Self Inductance of a circular coil

Let’s find the self-inductance of a circular shaped inductor. Consider a circular coil with the area of cross section A = π r^2, with N number of turns in it. Then the magnetic flux is given as

B = μ0 (N.I )/2r

The total flux in the circular conductor is given as N Φ = LI.

Substituting this in the above equation, (N.B. B= Φ/ A)

L = N Φ /I

L = (μ_0 N2 A )/2r

We know the area of circle is A = π r2, so self-inductance of a circular inductor is also given as

L = (μ0 N‍2 π r )/2

Factors Affecting the Self Inductance

Observing the above equation of inductance, we can say there are 4 factors that affect the self-inductance of a coil, they are

1. Number of turns in the coil (N)
2. Area of the inductor coil (A)
3. Length of the coil (l)
4. Material of the coil

• Number of turns

The inductance of the coil will depend up on the number of turns of the coil. Number of turns or twists in a coil and the inductance are in proportional to each other. N ∝ L

Higher the number of turns means greater the value of inductance.

Lower the number of turns means lower the value of inductance.

• Cross sectional area

The inductance of a coil will increase with increase in the cross sectional area of the inductor. L∝ A. If the area of the coil is high, it will produce more number of magnetic flux lines, this results in formation of more magnetic flux. Hence the, inductance will be high.

• Coil length

The magnetic flux induced in a longer coil is less than that of the flux induced in short coil. As the induced magnetic flux reduces, the inductance of the coil also decreases. So the induction of the coil is inversely proportional to the length of the coil. L∝ 1/l

• Material of the coil

The permeability of the material with which the coil is wrapped, will have an effect on the induced emf and inductance. The materials with high permeability can produce low inductance.

L∝μ0.

We know μ = μ0 μr

So L∝ 1 / μr

Mutual Inductance

The phenomenon of inducing an emf in a coil as a result of change in current flow of its coupled or adjacent coil is called “Mutual induction”. Here, the two coils are in the influence of same magnetic field.

As we discussed in the self-inductance concept, the emf produced due to mutual inductance can be explained by Faraday’s law and the direction of the emf can be described by Lenz’s law.

The direction of the emf is always opposite to the change in the magnetic field. The emf induced in the second coil is due to the change in the current of the first coil.

The emf induced in the second coil can be given as

EMF2 = – N_2  A  ΔB/Δt = -M (ΔI_1)/Δt

Where M is the mutual inductance, which is the proportionality between the generated emf in the second coil and the current change in the first coil.

To understand the concept of mutual inductance, observe the above picture. In that we connect two inductors are wound around a single conductor. Let’s call them  loop 1 and loop 2. If the current in the loop 1 is varying then magnetic flux is induced.

When the loop 2 intercepts the magnetic flux then without any current flowing directly into the second coil, there will be some emf induced. That is called Mutual inductance and this phenomenon is called “Mutual induction”.

Mutually Induced EMF and Coefficient of Mutual Inductance

Whenever we keep the 2 coils in the current varying field, there will be an emf induced because of the current flow. As the current in the loop varies, the magnetic flux also varies.

In this case the mutual induction is a vector quantity because it may induce in 2nd coil due to the current flow in 1st coil, or may be induced in the 1st coil due to the magnetic flux (B) produced by the 2nd coil

When the current flowing in the inductor 1 varies, magnetic flux will be generated around it (according to Lenz’s law and Faraday’s law). Then, the mutually induced emf in the second coil due to current in 1st coil will be given as

M12 = (N2 Φ12)/I1

Where   M12 is the mutual inductance in coil 2

N is the number of turns in loop

Φ12 is the magnetic flux generated in the coil 2

I1 is the current in loop 1

In the same manner, when we vary the current flow in inductor 1, magnetic flux will generate around it. Then the mutually induced emf in the 1st coil due to current in 2nd coil will be given as

M21 = ( N2 Φ21)/I2

Where   M21 is the mutual inductance in coil 1

N is the number of turns in loop

Φ21 is the magnetic flux generated in the coil 1

I2 is the current in loop 2

The important thing we need to remember is M21 = M12 = M, irrespective of relative position of the two coils, size and the number of turns in them. This is called the ‘Coefficient of mutual inductance’.

The formula for self-inductance of each coil is

L1 = (μ 0 μ r N12 A)/l and L2 = (μ 0 μr N22 A)/l

From the above equations, we can write M^2 = L1 L2. This is the relation between self inductances of each coil and the mutual inductance.

It can also be written as M = √(L1 L2 ) Henry. The above equation represents the ideal condition that there is no leakage of flux. But in reality, there is always some flux leakage due to the position and geometry of the coil.

Magnetic Coupling Coefficient or Coefficient of Coupling

The amount of inductive coupling between two coils is denoted by ‘Coefficient of coupling’. The value of the coefficient of coupling will be less than 1 and always greater than 0 i.e. it lies between 0 and 1. This is represented by ‘k’.

Derivation of coupling coefficient

Consider two inductor coils of length L1 and L2 having N1 and N2 turns respectively. The currents in coils 1 and 2 are I1 and I2. Assume that the flux produced in the second coil due to current flow I1 is Φ21. Then the mutual inductance will be given as M = N1 Φ21/ I1

Φ21 can be described as the part of flux Φ1 linked with 2nd coil. I.e. Φ21 = k1 Φ1

… M = N1 ( k1 Φ1) / i1 . . . . . . . . . . (1)

Similarly, the flux produced in the first coil due to current flow I2 is Φ12. Then the mutual inductance will be given as M = N2 Φ12/ I2

Φ21 can be described as the part of flux Φ1 linked with 2nd coil. I.e. Φ12 = k2 Φ2

… M = N2 ( k2 Φ2) / i2 . . . . . . . . . . (2)

Multiplying the equations (1) and (2), we get

M^2 = k1 k2 [(N1 Φ1 )/I_1 ]. [(N2 Φ2 )/I_2 ]

Now we know that the self-inductance of coil 1 is L1 = N1 Φ1 / i1

Self-inductance of coil 1 is L2 = N2 Φ2 / i2

Substituting L1 and L2 in the above equation we get

M^2 = (k1 k2) x (L1 L2)

… M = √(k1 k2) x √(L1 L2)

Let k = √ (k1k2)

…M = k √(L1L2)

Where k is the coefficient of coupling

K = M/((√(L1 L2 )) )

We can describe the magnetic coupling of two coils by using the magnetic coupling coefficient. When the magnetic flux of one coil is completely links with the other, then the coefficient of coupling will be high.

The maximum range of the coupling coefficient is 1, while the minimum is 0. When the value of coupling coefficient is 1, then the coils are called “Perfectly coupled coils”. If the value is 0, the coils are called “Loosely coupled coils”.

Summary of self-inductance and Mutual inductance

• ‘Inductance’ is a phenomenon that a twisted coil experiences a magnetic force on it when it is applied with an electric voltage. An inductor stores the energy in the form of magnetic field. It is measured in Henry.
• The induction in an inductor can be explained by Lenz’s law and Faraday’s law. Lenz’s law states an induced electromotive force that generates a current that induces a counter magnetic field opposing the magnetic field generating the current. 
• The inductance in a coil is of 2 types, they are

1. Self inductance
2. Mutual inductance
3. Definition of self-inductance: The self inductance of a coil is the induction of electromotive force in the coil when it is placed in a current varying circuit. This phenomenon of self-inductance is called “Self-induction”. Represented by L.     L = N Φ /I
4. 
   
5. Self-inductance of a long solenoid is L = (μ0 N2 A )/l
6. Self-inductance of a circular core is L = (μ0 N2Πr )/2
7. 
   
8. Self-inductance will depend on 4 factors - Number of turns in the coil (N), Area of the inductor coil (A), Length of the coil (l), Material of the coil.
9. 
   
10. Definition of mutual induction: The phenomenon of inducing the emf in a coil as a result of change in current flow of its coupled coil is called “Mutual inductance”. M = √(L1 L2 )

• Definition of coupling factor: The amount of inductive coupling between two coils is denoted by ‘Coefficient of coupling’.
• The value of coefficient of coupling will be less than 1 and always greater than 0. This is represented with ‘k’. K = M/((√(L1 L2 )) )

Basic Electronics On The Go - Inductive Reactance

From http://www.electronicshub.org/inductive-reactance/

When equal values of direct voltage and alternating voltage are applied to the same circuit which has inductor in series with the load, more current would flow in a DC circuit than in an AC circuit.

This is because, only induced voltage opposes the current flow in DC circuit when the current approaches to its maximum value and once it reaches to a steady state value, there will be no more inductive effect.

In the case of AC circuits, current is continuously changing, therefore inductive effect is present at all time. Consider the following DC and AC circuits to understand this concept.

DC Inductive Circuit

In the above figure, if the switch is operated from node A to node B and immediately from node B to node A, a change in current flows through the circuit.

This change in current induces an emf in the inductor proportional to the rate of change of current and this emf opposes the applied voltage (which is the cause for the production of current). This is called self induction.

Once the current reaches a steady state value, there will be no self induction in the inductor and hence no opposition to the current flow.

AC Inductive Circuit

We know that, when AC current is applied to the circuit, current continuously changes at a supply frequency rate and hence the back emf will change accordingly.

This back emf opposes the supply voltage and hence the flow of current is limited. Therefore, the actual opposition to the current flow created by an inductor in an AC circuit is referred as the inductive reactance.

Inductive reactance in an Inductor

In an inductive circuit by observing the self-inductance and its effect within the circuit, we can define the inductive reactance. The magnetic field induces the voltage in the inductor which is always opposite in polarity to the voltage that produces it, i.e., applied voltage.

This opposing voltage limits the current flowing through an inductor and it is called reactance (X). Since this reactance is caused due to inductance, it is called inductive reactance (XL). It is measured in Ohms.

The amount of inductive reactance offered by an inductor is proportional to the inductance and frequency of applied voltage. This reactance can be determined by the following formula.

XL = 2 π fL

Where XL= inductive reactance in ohms

π = 3.14

f = frequency in Hertz (Hz)

L = inductance in Henrys (H)

According to the ohm’s law, inductive reactance is directly proportional to the applied voltage and inversely proportional to the current. It can be expressed as

I = V/XL

From the above equation it is clear that increasing voltage or decreasing inductive reactance causes an increase in current. Likewise, current decreases with increase in inductive reactance and decrease in voltage.

Any practical inductor must be made with wound wire which consists of some resistance so it is not possible to obtain a purely inductive coil.

Therefore, there are two factors that oppose the current flow in an inductor, namely resistance associated with the coil (which is considered as separate resistor R in series with inductor) and inductive reactance offered by the  inductance property.

Thus the total current limiting property of an inductor in AC circuit is the combination of resistance and reactance which is called the  impedance, Z.

This impedance value is calculated by Ohm’s law and is given as

Z = V / I

Where Z = total opposition offered by inductor to the current flow, in ohms

V = applied voltage

I = current flowing through the circuit

Impedance Triangle

Another method of determining the impedance is the use of impedance triangle method when inductive reactance and resistance values are known. Below diagram shows an impedance triangle which consists of resistance and reactance vectors.

In the above figure, resistance vector is along with horizontal line (because resistance does not offer any phase shift) and inductive reactance vector is along with vertical line (because pure inductance offers 900 phase shift).

By connecting the ends of these two vectors, impedance, Z is obtained. Therefore, the total opposition to current or impedance can be calculated by

Z = √[(R)2+(XL)2 ]

Where

Z = impedance in ohms

R = resistance in ohms

XL = inductive reactance in ohms

Also, from above diagram,

tan∅= XL/R

sin∅= XL/R

cos∅= R/Z

RL Circuits and Inductive Reactance

The figure below shows the relationship between applied voltage and current through an inductive circuit. In a pure inductive circuit, the current lags the source voltage by 900 . It can also be stated as source voltage leads the current by 900 in an inductive circuit.

When an inductor is connected in series with the resistor RL, series circuit is obtained as shown below. This can also be considered as inductance consisting of some resistance 

Thus, the current and voltage  not exactly maintains 900 phase shift, but less than that of purely inductive case as shown below.

The figure below shows the vector diagram of RL series circuit consisting of voltage drop vectors across resistor and inductor. AE represents the current reference line. AB represents the voltage drop across the resistance which is in phase with current line.

AD represents the inductive voltage drop which leads the current by 900. The resultant of these vectors gives the total voltage across the circuit.

By applying the Pythagoras’s theorem to above voltage triangle, we get

Vtotal= √(VL2+VR2)

tan∅= VL/VR

We know that VR=I×R and VL=I×XL

By these equations we can rewrite the Vtotal as

Vtotal= √((I×R)2+(I×XL)2 )

I = V/ √((R)2+(XL)2) = V/Z (Amps)