Overview of Inductance Equations and Applications in Circuit Design

Key Takeaways

• “Henries” is the measurement of inductance—or the opposition to changes in flow.

• The ratio of the induced electromagnetic force or voltage to the rate of change of the current equals the inductance.

• An inductor influences the flow of current, but the total effect depends on the size of the inductor and the frequency at that point.

Although Sir Isaac Newton’s First Law of Motion is called The Law of Inertia, Galileo originally developed the concept of inertia. During his studies of the impact of friction, Galileo placed two ramps facing each other and then watched as a ball rolled down one plane and up the other to nearly the same height. The smooth surface of the ramp allowed the ball to roll higher. Given what he observed, Galileo concluded that eliminating friction would allow the ball to reach its original height.

Newton expanded on Galileo’s theory by showing that an object does not require a force to remain in motion. Without friction serving as a force working against the object, motion would continue with the same speed and direction.

To my knowledge, neither Galileo nor Newton became electrical engineers. If they had, the world might be very different today. Nevertheless, their work set the foundation for our studies on inductance equations.

Measuring Inductance

In mechanical engineering, inertia is defined by a mass that opposes a change in velocity. Inductance opposes a change in current in an electrical circuit. The amount of force required to start or stop current is greater than the amount of energy expended to keep the current flowing.

Every conducting path in a circuit has some amount of inductance and opposing current flow. We measure the amount of inductance—or the opposition to changes in flow—in “henries.” Named after Joseph Henry, the henry(h) is a Standard International (SI) derived unit of inductance. In terms of base SI units, one henry equals one-kilogram meter squared per second squared per ampere squared or:

kg . m/s2 x A2

One kilogram meter per second equals one newton-second or the SI unit of impulse. In electronics, inductances measure in millihenrys (mH) or microhenries (µH).

Inductance Equations

An inductor possesses the property of inductance and stores energy in a magnetic field. Alternating current flowing through a loop--or coil--induces a voltage or EMF in one wire of the coil. Because the amount of current increases and decreases, the magnetic field also increases and decreases. As the magnetic field forms concentric loops around the wire, the loops join to create larger loops that surround the coil. Increasing the current in one loop of the coil causes the magnetic field to expand and cut across other loops of wire and induce voltage in the loops.

Cutting a circular loop of conducting wire and injecting current into the gap produces a magnetic flux density(Φ) that penetrates the surface surrounded by current. Faraday’s Law of Induction describes the effect of loop inductance by showing that any change in the magnetic field induces a voltage in the conductors. The value of the inductance equals the ratio of the induced electromagnetic force or voltage to the rate of change of the current. The voltage has its greatest value with the change in time (t).

Utilizing Faraday’s Law may not be a conscious choice anymore, but

Because the EMF is directly proportional to the number of turns, a coil that has more than one loop produces a voltage N times greater than the amount of voltage produced by a single coil. In equation form, the EMF equals:

N = ΔΦ/Δt

Increasing the frequency or increasing the number of turns in the coil increases the amount of induced voltage. Another form of the equation looks at the value from the perspective of the induced voltage:

Induced voltage (VL) = the value of inductance (L) in henries multiplied by the rate of change in amperes per second (di/dt) or VL = L x di/dt

When we look at a henry in terms of electricity, one henry equals the value of inductance needed to induce one volt when the current(i) in a coil changes at a rate (t) of one ampere per second. The equation tells us that the current changes and causes a force to build across the inductor, working against the inertia of the inductor. Without the steady change in current, the force cannot increase.

Adding a minus sign to the original equation shows that the EMF creates a current and a magnetic field that opposes any change in the flux:

-N x ΔΦ/Δt

Because Heinrich Lenz stated this aspect of induction, we refer to it as Lenz’s law. Induction opposes and slows any change. Lenz tells us that an induced current has a direction that causes the magnetic field and opposes the change in the magnetic field that induces the current. Self-inductance—or the self-induction of a voltage—occurs when the current in current-carrying wire changes. The magnetic field created by a changing current self-induces a voltage in the circuit.

Applying a voltage across a conductor causes a magnetic field to build around the conductor. The increase in the magnetic field prevents current from flowing during the same instant that the voltage is applied. As the magnetic field stabilizes, current flows. Removing the source of the current causes the collapse of the magnetic field and the generation of a force that maintains the current flow in the same direction.

While an inductor influences the flow of current, the total effect depends on the size of the inductor and the frequency at that point. Reductions in current occur with a large inductor or high frequencies. More current flow occurs with smaller inductors or lower frequencies. The letter L represents inductors in equations and schematic diagrams.

Placing two inductors in series yields an equivalent inductance (Leq) that equals the sum of the two inductances:

Leq - L1 + L2

With two inductors placed in series, the current is reduced to the value that the smaller inductor will allow to flow. Because of this factor, the equivalent inductance changes to:

Leq = L1 + L2 ≈ L2

Placing two inductors in parallel produces an equivalent inductor that equals the product of the two inductors divided by the sum of the two inductors:

Leq = L1L2 / L1 + L2

Because inductance changes current flow, it also causes power dissipation that we measure as inductive reactance in ohms. The net result of inductive reactance is reduced current flow. As the next equation shows, inductive reactance (XL) varies with frequency (f) and the value of the inductor (L):

 XL = 2πfL

Mutual Inductance

Magnetic flux moving through a circuit influences the current in that circuit along with the currents in circuits located near the first circuit. As the magnetic field produced by the first circuit intersects a conductor in the next circuit, it creates current flow. One magnetic field exists for the original circuit and another magnetic field exists for the next circuit. Both fields remain proportional to the current that produced each one.

As current flows, the two magnetic fields interact and—through mutual inductance—become one field that has one part influenced by the first current, and another part influenced by the second current. The amount of mutual inductance depends on the geometrical arrangement of the circuits. Placing the circuits further apart reduces the amount of mutual inductance.

There’s a Trace Amount

Parasitic inductance can create different types of problems in high-speed circuits. Problems with stray inductance can occur through component leads and PCB traces, causing problems with signal integrity. Those problems appear as crosstalk, noise coupling, and electromagnetic interference caused by induced currents.

Even though leads or traces have small physical areas and produce small amounts of stray inductance, changes in current can generate high voltages. The parasitic inductance that resides along a PCB trace increases the impact of any voltage spike induced by switching power supplies.

The next equation shows the amount of inductance in microhenries for a component lead that has a diameter of 0.15 inches and a length of 1/4 inch. In this equation, we will use the natural log base e (ln) as a multiplier. The natural log base e is an irrational number e ≈ 2.718.

µh = .00508 x length {[ln x 2 x length / radius] - .75} or

µh = .00508 x .25 x {[2.718268237 (2 x .25 / 0.075] - .75} or

µh = 0.00127 x 17.86

µh = 0.022

A one-inch long, 10-mil-wide trace that extends 10 mils above the plane has an even smaller amount of inductance. In equation form, the trace inductance in nanohenries appears as:

While both amounts of parasitic inductance seem small, a rising current impacts the circuit by generating a significant voltage spike. Parasitic inductance also affects signal integrity. As an example, trace inductance found at the non-inverting input of a high-speed op-amp can cause low-level oscillation. In addition, parasitic induction at the connection of a filter capacitor to a PCB can decrease the filtering effect of the capacitor at high frequencies, or allow impedance mismatches to form and cause reflections.

Reducing parasitic induction in a circuit design begins with decreasing the equivalent loop area covered by traces. Routing sensitive traces between the power and ground planes prevents EMI from signals at one layer from inducing signals in another layer of the PCB.

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