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Inductors & Inductance Calculations

Inductors are passive devices used in electronic circuits to store energy in the form of a magnetic field. They are the compliment of capacitors, which store energy in the form of an electric field. An ideal inductor is the equivalent of a short circuit (0 ohms) for direct currents (DC), and presents an opposing force (reactance) to alternating currents (AC) that depends on the frequency of the current. The reactance (opposition to current flow) of an inductor is proportional to the frequency of the current flowing through it. Inductors are sometimes referred to as "coils" because most inductors are physically constructed of coiled sections of wire.

The property of inductance that opposes current flow is exploited for the purpose of preventing signals with a higher frequency component from passing while allowing signals of lower frequency components to pass. This is why inductors are sometimes referred to as "chokes," since they effectively choke off higher frequencies. A common application of a choke is in a radio amplifier biasing circuit where the collector of a transistor needs to be supplied with a DC voltage without allowing the RF (radio frequency) signal from conducting back into the DC supply.

Series LC circuit - RF CafeParallel LC tank circuit - RF CafeWhen used in series (left) or parallel (right) with its circuit compliment, a capacitor, the inductor-capacitor combination forms a circuit that resonates at a particular frequency that depends on the values of each component.  In the series circuit, the impedance to current flow at the resonant frequency is zero with ideal components. In parallel circuits (right), the impedance to current flow is infinite with ideal components.

Real-world inductor model with resistance, inductance, and capacitance - RF CafeReal-world inductors made of physical components exhibit more than just a pure inductance when present in an AC circuit. A common circuit simulator model is shown to the right. It includes the actual ideal inductor with a parallel resistive component that responds to alternating current. The DC resistive component is in series with the ideal inductor, and a capacitor is connected across the entire assembly and represents the capacitance present due to the proximity of the coil windings.

Equations (formulas) for combining inductors in series and parallel are given below. Additional equations are given for inductors of various configurations.

The HamWaves.com website has a very sophisticated calculator for coil inductance that allows you to enter the conductor diameter.

Parallel Inductors
Inductors in parallel combine in the same manner as parallel resistors.
Parallel inductor equation - RF Cafe

Parallel inductor drawing - RF Cafe
Straight Wire

Inductance of a straight wire - RF Cafe

These equations apply for when the length of the wire is much longer than the wire diameter.

 

For lower frequencies - up through about VHF, use this formula:

Inductance of a straight wire low frequency - RF Cafe

Above VHF, skin effect causes the ¾ in the top equation to approach unity (1), so use this equation:

Inductance of a straight wire high frequency - RF Cafe
L = inductance (μH)

l = length (mm)

d = wire diameter (mm)

The ARRL Handbook presents this equation for units in inches:

Inductance of a straight wire low frequency (inches) - RF Cafe
Use the same VHF frequency point as above for changing the 0.75 to a 1
L=inductance (μH)

b = length (in.)

a = wire radius (in.)

Look up wire diameter here.

Coaxial Cable
 Inductance of coaxial cable - RF Cafe D = outer radius
d = inner radius
z = length (ft)
z = length (m)
Closely Wound Toroid
Inductance of a toroid - RF Cafe

Toroid inductance formula - RF Cafe
(units in inches)

Single-Layer Air-Core

Inductance of a single-layer air-core - RF Cafe

Wheeler's
Formula

Inductance of a single-layer air-core - RF Cafe
(units in inches)
 Inductance of a single-layer air-core - RF Cafe
Where: d and z are in inches
N = number of turns
Note: If lead lengths are significant, use the straight wire calculation to add that inductance.
Multilayer Air-Core Coil
Inductance[μHenry] = Wheeler multi-layer solenoid formula - RF Cafe1

L = inductance (µH)
r = mean radius of coil (in)
z = physical length of coil winding (in)
N = number of turns
d = depth of coil (outer radius minus inner radius) (in)

1: Thanks to Wayne H. for correcting the 0.8 factor, which used to be 0.5
Inductive Reactance

                            XL=jωL

Series Inductors
Inductors in series combine in the same manner as series resistors.
Ls = L1 + L2 + ··· + Ln

Series inductor drawing - RF Cafe

W = 1/2 Li2

Voltage due to inductance - RF Cafe
Current in inductor - RF Cafe
XL = 2 π f L

L = inductance (H)
v = voltage (V)
W = energy (J)

"Q" Factor
  • Q = F0/F3db
  • Q = Estored / Eloss_per_cycle
  • Parallel Circuit:
        Q = R/(2*π*F0*L)
  • Series Circuit:
        Q = (2*π*F0*L)/R
  • 1/Qload = 1/Qext + 1/Qtank
Where: L = inductance
E = energy
ext = external
Finding the Equivalent "RQ"

Since the "Q" of an inductor is the ratio of the reactive component to the Parallel RL circuit for equivalent Rq - RF Caferesistive component, an equivalent circuit can be defined with a resistor in parallel with the inductor. This equation is valid only a a single frequency, "f," and must be calculated for each frequency of interest.

 
Equivalent resistor in parallel with inductor for "Q" factor - RF Cafe

Straight Wire Parallel to Ground Plane w/One End Grounded

Straight Wire Parallel to Ground Plane w/One End Grounded - RF Cafe

The ARRL Handbook presents this equation for a straight wire suspended above a ground plane, with one end grounded to the plane:

Equation for Inductance of Straight Wire Parallel to Ground Plane w/One End Grounded - RF Cafe

L = inductance (μH)
a = wire radius (in.)
b = wire length parallel to ground plane (in.)
h = wire height above ground plane (in.)
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