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
vL = induced voltage across the coil, V
Δi /Δt = 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 vL2 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 LM . Two coils have LM 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 XL 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
XL = 2πfL
Since 2π = 2 (3.14) = 6.28, XL = 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 XL
IL = current through the inductance, A L
VL = voltage across the inductance, V
XL = 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 LT is the sum of the individual inductances, or
Series: LT = L1 + L2 + L3 +... + Ln
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 LT = L1 + L2 ± 2LM
where LM 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 LT = 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
LT = L1 + L2 + 2LM. In figure c, the coil windings are in the opposite direction, so the coils are series-opposing, and LT = L1 + L2 − 2LM .
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 LM has the same sense as L.
If inductors are spaced sufficiently far apart so that their mutual inductance is negligible ( LM = 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 LT 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 LT = L / n = 8 / 2 = 4 mH.
Phasor addition of R and XL to find ZThe 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:
Schematic diagrm of Q of a coil. XL and Ri are distributed uniformly over the length of the coil