Inductance is a fundamental property of an electrical conductor that causes it to oppose any change in the electric current flowing through it. This opposition arises because the conductor stores energy temporarily in a surrounding magnetic field. The stored magnetic energy resists the fluctuation, creating a brief counter-electromotive force, which is the defining characteristic of inductance. This property is quantified and measured in the Henry (H).
Calculating Inductance from Physical Dimensions
The most direct way to determine an inductor’s value is by analyzing its physical construction, which is relevant for component design. An inductor is typically constructed as a coil or solenoid, and its inductance depends on four physical characteristics: the number of wire turns, the coil’s cross-sectional area, its length, and the magnetic permeability of the core material.
The relationship between these factors for a long solenoid is defined by a specific equation. The inductance (\(L\)) is directly proportional to the square of the number of turns (\(N^2\)) and the cross-sectional area (\(A\)). Doubling the number of turns increases the inductance fourfold, and a larger area increases the inductance linearly by allowing a greater magnetic field to be contained.
Inductance is inversely proportional to the length (\(l\)) of the coil; a longer coil results in a lower inductance value. The fourth variable is the absolute permeability (\(\mu\)) of the core material, which represents how easily the material can form a magnetic field. This absolute permeability is the product of the permeability of free space (\(\mu_0\)) and the material’s relative permeability (\(\mu_r\)).
The core material has a dramatic effect on the final inductance value. An air-core inductor has a relative permeability (\(\mu_r\)) of approximately one, providing a baseline for calculation. Introducing a magnetic core material, such as iron or ferrite, significantly increases the total inductance because these materials concentrate magnetic field lines much more efficiently than air.
Soft iron can have a relative permeability in the thousands, while specialized ferrite materials can range from 10 to over 20,000. By substituting a high-permeability core for an air core, engineers can achieve a much higher inductance value using fewer turns and smaller physical dimensions. Therefore, the formula for a long solenoid is expressed as \(L = \mu \frac{N^2 A}{l}\), where \(\mu\) incorporates the effect of the core material.
Calculating Inductance Using Measured Reactance
When the physical dimensions of an inductor are unknown, or when checking its value within an existing circuit, inductance can be calculated by measuring its opposition to an Alternating Current (AC) signal. This opposition is called Inductive Reactance (\(X_L\)), and it is measured in Ohms. Unlike simple resistance, inductive reactance is dependent on the frequency of the applied AC signal.
The relationship between inductance (\(L\)), frequency (\(f\)), and inductive reactance (\(X_L\)) is given by the formula \(X_L = 2\pi f L\). This equation shows that the opposition to current flow increases linearly as either the inductance or the frequency of the AC signal increases. To calculate the unknown inductance, this formula is rearranged to solve for \(L\): \(L = \frac{X_L}{2\pi f}\).
A practical method for determining inductance involves applying a known AC voltage (\(V\)) at a known frequency (\(f\)) to the inductor. By measuring the current (\(I\)) flowing through the inductor with an ammeter, the inductive reactance (\(X_L\)) can be found using a variation of Ohm’s Law for AC circuits: \(X_L = V/I\). This calculated reactance value and the test frequency are then used in the derived formula to find the inductance.
For instance, if a 12-volt AC signal at 60 Hertz results in a measured current of 0.1 Amperes, the reactance is calculated as 120 Ohms. Plugging this 120-Ohm reactance and the 60-Hertz frequency into the rearranged formula yields the inductance value. This approach is practical because it only requires common laboratory equipment, such as a signal generator and a multimeter capable of AC measurements.
Combining Inductors in Circuits
In complex electronic designs, multiple inductors may be connected together, requiring the calculation of the total effective inductance. The rules for combining inductors are straightforward and mirror those used for combining resistors in electrical circuits.
When inductors are connected in series (end-to-end along a single path), their individual inductance values simply add up to determine the total inductance (\(L_{total}\)). This is represented by the formula \(L_{total} = L_1 + L_2 + L_3 + \dots\). The total value is always greater than any single inductor because connecting them in series is effectively like increasing the total number of turns in a single coil.
For inductors connected in parallel, where the current splits, the calculation uses a reciprocal relationship, similar to that for parallel resistors. The reciprocal of the total inductance equals the sum of the reciprocals of the individual inductances: \(\frac{1}{L_{total}} = \frac{1}{L_1} + \frac{1}{L_2} + \frac{1}{L_3} + \dots\). The resulting total inductance is always smaller than the smallest individual inductance value. These rules are only valid when the inductors are physically separated enough so their magnetic fields do not interact (no mutual inductance).