Inductive Circuits | Electricity & Magnetism - Physics PDF Download

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 L₁, therefore, induces voltages across L₁ and across L₂ (as shown in figure below). When the induced voltage v_L2 produces current in L₂ , its varying magnetic field induces voltage in L₁ . Hence, the two coils L₁ and L₂ 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 L₁ and L₂

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

➤ Inductive Reactance
Inductive reactance X_L is the opposition to ac current due to the inductance in the circuit. The unit of inductive reactance is the ohm. The formula for inductive reactance is
X_L =  2πfL
Since 2π = 2 (3.14) = 6.28, X_{L =  6.28fL}
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_T is the sum of the individual inductances, or
Series:  L_T = L₁ + L₂ + L₃ +... + L_n
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_T = L₁ + L₂ ± 2L_M
where L_M is the mutual inductance between the coils. The plus (+) sign is used if the coils are arranged in series-aiding form, while the minus (-) sign is used if the coils are connected in series-opposing form. Series aiding means that the common current produces the same direction of magnetic field for the two coils. The series-opposing connection results in opposite fields.
Three different arrangements for coils L₁ and L₂ are shown both pictorially and schematically as shown in figure below. In figure a coils are spaced too far apart to interact electromagnetically. There is no mutual inductance, so LM is zero. The total inductance is L_T = L₁ + L₂. In figure b, the coils are spaced close together and have windings in the same direction, as indicated by the dots. The coils are series-aiding, so 

 L_T = L₁ + L_{2 +} 2L_M. In figure c, the coil windings are in the opposite direction, so the coils are series-opposing, and L_T = L₁ + L₂ − 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 L₁ and L₂ , 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

1. 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
2. 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 series
   
   Phasor-voltage triangle
   To 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_L to find Z

   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:
   0
3. 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), or
   
   R 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.
4. 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.
5. Power in RL Circuit
   
   Power 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).