Formula and Equations For Inductor and Inductance

Electrical Technology

Inductor and Inductance Formulas and Equations

The following formulas and equations can be used to calculate the inductance and related quantities of different shapes of inductors as follow.

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Inductance of Inductor:

The inductance of the inductor from the basic formula of inductor:

Voltage across Inductor:

Current of the Inductor:

Where

V is the voltage across inductor
L is the inductance of the inductor in Henry
Di/dt is the instantaneous rate of current change through the inductor.
i_to = current at time t = 0.

Reactance of the Inductor:

Inductive reactance is the opposition of inductor to alternating current AC, which depends on its frequency f and is measured in Ohm just like resistance. Inductive reactance is calculated using:

X_L= ωL = 2πfL

Where

X_Lis the Inductive reactance
f is the applied frequency
L is the Inductance in Henry

Quality Factor of Inductor:

The efficiency of the inductor is known as quality factor & its measured by:

QF = X_L/ESR

Where

X_Lis the Inductive reactance
ESR is the equivalent series resistance of the circuit.

Dissipation Factor of Inductor:

It is the inverse of the quality factor and it shows the power dissipation inside the inductor & its given by:

DF = tan δ = ESR/X_L

Where

DF is the dissipation factor
δ is the angle between capacitive reactance victor & negative axis.
X_Cis the capacitive reactance
ESR is the equivalent series resistance of the circuit.

Energy Stored in an Inductor:

The energy E stored in inductor is given by:

E = ½ Li²

Where

E is the energy in joules
L is the inductance in Henry
i is the current in Amps

Average Power of Inductor

The average power for the inductor is given by:

P_av = Li² / 2t

Where

t = is the time in seconds.

Inductor Current During Charge / Discharge:

Just like capacitor, the inductor takes up to 5 time constant to fully charge or discharge, during this time the current can be calculated by:

During Charging:

Instantaneous current of the inductor during charging is given by:

During Discharging:

The current during the discharging at any time t is given by:

Where

I_C is the current of the inductor
I₀ is the current at time t=0
t is the time passed after supplying current.
τ = L/R is the time constant of the RL circuit

Helical Coil Inductance “Wheeler’s Formula”

Where:

L is the inductance
R is the radius
n is the number of turns
h is the height

Spiral Coil Inductance Formula

Where:

OR is the outermost radius in inches
IR is the innermost radius in inches

Spiral Coil Wire Length Formula

Conical Coil Inductance Formula

Where:

θ is the angle outside the cone, (assume θ ≈ 15°)