Inductor
mathematics
Updated
June 7, 2008
Click
here to go to our main inductor page
For July 2008, we've added
new material on airbridge inductance
and wirebond inductance on
separate pages.
For December
2005, we've added a a new formula contributed by a reader, for
wire spiral inductors! We have to admit, we haven't personally tested
some formulas on this page for accuracy against measured data. Also,
note that any inductor model that doesn't consider parasitic capacitance
and resistance will have limited accuracy at microwave frequencies.
Below is an index of our mathematical
discussion of inductors:
Inductive
reactance
Inductance
of a transmission line (separate page)
Solenoid
inductance
Spiral inductor
(wire)
Toroid inductance
Wirebond
inductance (now on a separate page)
Airbridge
inductance (separate page)
Via
hole inductance (separate page)
DC and
RF resistance of inductors
Resonances
Inductive
reactance
Use our reactance
calculator if you are interested in this topic!
The well-known equation for inductive
reactance is shown below. Note that inductive reactance is positive,
the opposite polarity of capacitive reactance. On the Smith
Chart, this means that series inductance tends to move a reflection
coefficient in a clockwise direction.
A more useful form
of the inductive reactance equation is given below, where frequency
is in GHz and inductance is in nanohenries. Luckily all of those
decimal places just cancel each other out!
Solenoid
inductance
A solenoid is a cylindrical shape
that is wound with wire to create inductance. It can have a single
layer of windings, or multilayer, and it can use an air core or
a core with high magnetic permeability for increased inductance.
The most useful (read that "highest Q") solenoids for
microwave applications are miniature, single-layer air-core inductors.
The graphic below was contributed by Sebastiaan. Many thanks!
The classic formula for single-layer
inductance (air core) is called Wheeler's formula, which dates back
to the radio days of the 1920s:
where:
L = inductance in micro-Henries
(not nano-Henries!)
N= number of turns of wire
R= radius of coil in inches
H= height of coil in inches
Here it is in terms of D, the
diameter of the coil:
(This formula was corrected April
9, 2006 thanks to KB!)
Wheeler's formula does not take
into account wire diameter, and spacing between the turns. In the
Wheeler formula, the turns are touching each other, but some insulation
is assumed to prevent shorting out. In practice, some spacing between
turns is necessary to reduce the inter-turn capacitance and increase
the operating frequency. Let's face it, Wheeler was not interested
in the accuracy of nano-Henry coils for microwave hardware.
A supposedly more-accurate method
of calculating inductance of single-layer air-core inductors for
microwave components can be found on Microwave
Components Incorporated web site:
where:
L = inductance in nano-Henries
N = number of turns of wire
D = inside diameter of the coil (inches)
D1 = bare wire diameter (inches)
S = space between turns (inches)
Using the MCI formula, applied
to 47 gauge wire (1.2 mil bare wire diameter), and 0.5 mil spacing
between turns, wrapping the turns around a 20 mil pin vice, you
can make the following air-coil values:
1 turn= 2 nH
2 turns= 5 nH
3 turns = 8 nH
4 turns = 12 nH
5 turns= 16 nH
6 turns = 20 nH
7 turns = 25 nH
8 turns= 30 nH
9 turns = 35 nH
10 turns = 40 nH
Click here
to go to our American Wire Gauge (AWG) chart.
Spiral
inductor (wire)
This formula and graphic was
also contributed by Sebastiaan (units are also micro-Henries):
Toroid
inductance
A toroid is similar to a solenoid,
but is donut shaped. More to come!
DC
and RF resistance of inductors
Computing the DC resistance of
a spiral inductor is simple, and is often overlooked by designers
until they build an amplifier circuit and the part doesn't bias
up correctly on the first iteration. You need to know the sheet
resistance of your metalization, in ohms per square, and compute
the number of squares in the inductor. The number of squares is
the total length (if you unwound it) divided by the width, and can
easily run into the hundreds for a large inductor.
Computing the RF resistance,
you may have to consider the skin depth
effect.
The model shown below is the
classic model for spiral inductors. Computing the elements is not
as straightforward as you might hope.
Inductor
resonances
More to come!
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