## Formulas for Inductance

Formulas for inductance are moderately complicated and are also somewhat approximate. Equations 3-15 and 3-16 give generally accepted formulas for a wire in air and for a microstrip trace:

Wire in air:

where

L |
= inductance in µH |

a |
= radius (in) |

b |
= length (in) |

Ln |
= natural log (base e) |

For example, let's assume a typical through hole wire is .018 in in diameter. Then a 1.0 in length would have an inductance of:

Microstrip trace:

Where:

L |
= inductance in nH/in |

w |
= trace width (mils) |

h |
= height above the plane (mils) |

For example, a 10-mil-wide trace 1.0 in long and 10 mils above the plane would have an approximate inductance of

Note that these component lead and trace inductances are pretty small, on the order of 9 to 12 nH per inch. But if the current is rising at, say 10 mA per ns, (which might be representative of a 1-ns rise time device) the voltages generated can be surprisingly large:

This gives an idea of why stray inductance is a signal integrity problem in high-speed designs. Ordinarily 9 to 12 nH is an inductance that would be considered negligible (unless we were talking about RF or microwave circuits). But when we talk about nanosecond rise times, such inductances can be a problem, indeed.

High-speed digital circuits don't usually have very many inductors in them, but inductors are an integral part of electromechanical relays and transformers. They may be used in power supply filtering circuits because of their ability to pass DC power supply voltages while "blocking" (thereby filtering out) higher frequency noise. Sometimes a simple ferrite bead is used to provide localized inductance for this purpose. Inductors sometimes are used to set frequencies or as part of filtering circuits, but there are often better ways to accomplish these functions with capacitors and active circuits.

It is the stray inductance that causes problems in high-speed designs. There is enough inductance in a short length of component lead or PCB trace to cause really significant potential problems in a circuit. We will see later that (just like capacitive coupling) inductive coupling between traces must often be carefully controlled or minimized.