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**Introduction**

The ability of a conductor to induce voltage in itself when the current changes is its self-inductance or simply inductance. The symbol for inductance is L, and its unit is the henry (H). One henry is the amount of inductance that permits one volt to be induced when the current changes at the rate of one ampere per second. The formula for inductance is

where L = inductance, H

v_{L} = induced voltage across the coil, V

Δ_{i} /Δ_{t} = rate of change of current, A/s

The inductance of a coil is 1H when a change of 1 A/s induces IV across the coil.

The self-induced voltage vL from) is When the current in a conductor or coil changes, the varying flux can cut across any other conductor or coil located nearby, thus inducing voltages in both. A varying current in L_{1}, therefore, induces voltages across L_{1} and across L_{2} (as shown in figure below). When the induced voltage v_{L}_{2} produces current in L_{2} , its varying magnetic field induces voltage in L_{1} . Hence, the two coils L_{1} and L_{2} have mutual inductance because current change in one coil can induce voltage in the other. The unit of mutual inductance is the henry, and the symbol is L_{M} . Two coils have L_{M} of 1H when a current change of 1 A/s in one coil induces 1 V in the other coil.

Mutual inductance Between L_{1 }and L_{2}

The schematic symbol for two coils with mutual inductance is shown in figure below.

**➤ Inductive Reactance**Inductive reactance X

X

Since 2π = 2 (3.14) = 6.28, X

where = inductive reactance, Ω, f = frequency Hz, L = inductance, H

If any two quantities are known the third can be found

In a circuit containing only inductance (as shown in figure below, Ohm’s law can be used to find current and voltage by substituting X L for R.

Circuit with only X_{L}

where

I_{L} = current through the inductance, A L

V_{L} = voltage across the inductance, V

X_{L} = inductive reactance, Ω

**➤ Inductors in Series or Parallel**If inductors are spaced sufficiently far apart so that they do not interact electromagnetically with each other; their values can be combined just like resistors when connected together. If a number of inductors are connected in series (as shown in figure below), the total inductance L

Series: L

Inductances in series without mutual coupling If two series-connected coils are spaced close together so that their magnetic field lines interlink, their mutual inductance will have an effect on the circuit. In that case the total inductance is L

where L

Three different arrangements for coils L

L_{T} = L_{1} + L_{2 +} 2L_{M}. In figure c, the coil windings are in the opposite direction, so the coils are series-opposing, and L_{T} = L_{1} + L_{2} − 2L_{M} .

The large dots above the coil (figure b and c) are used to indicate the polarity of the windings without having to show the actual physical construction. Coils with dots at the same end (figure b) have the same polarity or same direction of winding. When current enters the dotted ends for L_{1} and L_{2} , their fields are aiding and L_{M} has the same sense as L.

If inductors are spaced sufficiently far apart so that their mutual inductance is negligible ( L_{M }= 0), the rules for combining inductors in parallel are the same as for resistors. If a number of inductors are connected in parallel (as shown in figure below), their total inductance L_{T} is

Inductances in parallel without mutual coupling

The total inductance of two coils connected in parallel is

All inductances must be given as the same units. The shortcuts for calculating parallel R can be used with parallel L. For example, if two 8-mH inductors are in parallel, the total inductance is L_{T} = L / n = 8 / 2 = 4 mH.

**Inductive Circuits**

**Inductance Only**

If an ac voltage v is applied across a circuit having only inductance (figure a), the resulting ac current through the inductance, i_{L}, will lag the voltage across the inductance, v_{L}, by 90° (figure b and c). Voltage v and v_{L}are the same because the total applied voltage is dropped only across the inductance. Both i_{L}and v_{L}are sine waves with the same frequency. Lowercase letters such as i and v indicate instantaneous values; capital letters such as I and V show dc or ac rms values.Circuit with L only**RL in Series**

When a coil has series resistance (figure a shown below), the rms current I is limited by both X_{L}and R . I is the same in X_{L}and R since they are in series. The voltage drop across R is V_{R }=IR, and the voltage drop across X_{L}is V_{L}= IX_{L}. The current I through X_{L}must lag V_{L}by 90° because this is the phase angle between current through an inductance and its self-induced voltage (figure b). The current I through R and its IR voltage drop are in phase so the phase angle is 0°.R and X_{L}in seriesPhasor-voltage triangleTo combine two waveforms out of phase, we add their equivalent phasors. The method is to add the tail of one phasor to the arrowhead of the other, using the angle to show their relative phase. The sum of the phasors is a resultant phasors from the start of one phasor to the end of the other phasor. Since V_{R}and V_{L}phasors form a right angle, the resultant phasor is the hypotenuse of a right triangle. From the geometry of a right triangle, the resultant is where the total voltage V_{T}is the phasor sum of the two voltages V_{R}and V_{L}that are 90° out of phase. All the voltages must be in the same units-rms values, peak values, or instantaneous values. For example, when V_{T}is an rms value, V_{R}and V_{L}also are rms values. Most of the ac calculations will be made in rms units.

The phase angle θ between V_{T}and V_{R}is**➤ Impedance in Series RL**Phasor addition of R and X

The resultant of the phasor addition of R and X L is called impedance. The symbol for impedance is Z. Impedance is the total opposition to the flow of current, expressed in ohms. The impedance triangle (as shown in figure) corresponds to the voltage triangle, but the common factor I cancels. The equations for impedance and phase angle are derived as follows:_{L }to find Z

0**RL in Parallel**

For parallel circuits with R and X_{L}(figure a shown below), the same applied voltage V_{T }is across R_{ }and XL since both are in parallel with V_{T}. There is no phase difference between these voltages. Therefore V_{T}will be used as the reference phasor. The resistive branch current I_{R}= V_{T}/ R is in phase with V_{T}. The inductive branch current I_{L}= V_{T}/ X_{L}lags V_{T}by 90° (figure b) because the current in an inductance lags the voltage across it by 90° . The phasor sum of I_{R}and I_{L}equals the total line current I_{T}(figure c), orR and X_{L}in parallel**➤ Impedance in Parallel R**_{L}

For the general case of calculating the total impedance Z_{T}of R and X_{L}in parallel, assume any number for the applied voltage V_{T}because in the calculation of Z_{T}in terms of the branch currents the value of V_{T}cancels. A convenient value to assume for VT is the value of either R or X_{L}, whichever is the higher number. This is only one method among others for calculating Z_{T}.**Q of a Coil**Schematic diagrm of Q of a coil. X

_{L }and R_{i}are distributed uniformly over the length of the coil

The quality or merit Q of a coil is indicated by the equation

where R_{i}is the internal resistance of the coil equal to the resistance of the wire in the coil (figure a). Q is a numerical value without any units since the ohms cancel in the ratio of reactance to resistance. If the Q of a coil is 200, it means that the X_{L}of the coil is 200 times more than its R_{i}. The Q of a coil may range in the value from less than 10 for a low-Q coil up to 1000 for a very high Q coil. Radio frequency (RF) coils have a Q of about 30 to 300.

As an example, a coil with an X_{L}of 300 Ω and a R_{i}of 3Ω has a Q of 300/3 =100.- Power in RL CircuitPower triangle for RL circuit

In an ac circuit with inductive reactance, the line current I lags the applied voltage V. The real power P is equal to the voltage multiplied by one that portion of the line current which is in phase with the voltage.

Therefore, Real power P = V(I cosθ) = VI cosθ

where θ is the phase angle between V and I, and cosθ is the power factor (PF) of the circuit. Also, Real power P = I 2 R

where R is the total resistive component of the circuit.

Reactive power Q in voltamperes reactive (VAR), is expressed as follows: Reactive power Q = VI sinθ

Apparent power S is the product of V x I. The unit is voltamperes (VA).

In formula form, Apparent power S = VI In all the power formulas, the V and I are in rms values. The relationships of real, reactive and apparent power can be illustrated by the phasor diagram of power (as shown in above figure).

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