Inductance

Discussion

introduction

Ready? Here we go.

Start with a solenoid. Run current through it and you've got yourself an electromagnet. The field inside is given by the formula …

B = μ0nI = μ0  N  I
ℓ

At the same time, a solenoid is also a device for capturing flux.

ΦB = NBA

The static situation is certainly interesting enough, but when it comes to flux, what we really care about is the time rate of change. This is what gives us electromagnetic induction, or an induced electromotive force, or whatever you want to call it. This situation is described by Faraday's law.

ℰ = −  dΦB
dt

Let's walk through these equations again, but with a time-varying twist. A solenoid with a changing current running through it will generate a changing magnetic field.

dB  = μ0  N   dI
dt ℓ dt

This changing magnetic field is then captured by the very solenoid that created it. A captured field is called flux and a changing flux generates an emf — in this case, a self-induced or back emf.

ℰ = −  dΦB  = − N  ⎛
⎝		 μ0 N dI ⎞
⎠		  A
dt ℓ dt

Rearranging things a bit gives us this equation …

ℰ = −  μ0AN2   dI
ℓ dt

which may not look like much, until you realize that the terms in the first fraction are largely determined by the geometry of the solenoid. Had we chosen a different configuration of wires, the same basic thing would have happened.

ℰ = − L  dI
dt

The self-induced emf in a circuit is directly proportional to the time rate of change of the current (dI/dt) multiplied by a constant (L). This constant is called the inductance (or more precisely, the self inductance) and is determined by the geometry of the circuit (or more commonly, by the geometry of individual circuit elements). For example, the inductance of a solenoid (as determined above) is given by the formula …

L =  μ0AN2
ℓ

The symbol L for inductance was chosen to honor Heinrich Lenz (1804-1865), whose pioneering work in electromagnetic induction was instrumental in the development of the final theory. If you recall, Lenz' law states that the induced current in a circuit always acts in a manner that opposes the change that created it in the first place. This observation is why there's a minus sign in all the different versions of Faraday's law. Lenz' gave us the minus sign and we honor him with the symbol L.

Inductance is best defined by its role in the equation derived from Faraday's law of induction. Some people don't like this and prefer definitions written in the subject-verb-object form of a simple sentence.

L =  ℰ
dI/dt

In English, we would read this as "self inductance (L) is the ratio of the back emf (ℰ) to the time rate of change of the current producing it (dI/dt)." As I already said, I don't particularly like this kind of definition, but it does help us to determine the appropriate units.

⎡
⎣		 H =  V  =  J/C  =  (kg m2/s2)/(A s)  =  kg m 2 ⎤
⎦
A/s A/s A/s s2 A

The unit of inductance is the henry, named after Joseph Henry (1797-1878), the American scientist who discovered electromagnetic induction independently of and at about the same time as Michael Faraday (1791-1867) did in England. Faraday published his findings first and so gets most of the credit. Henry also discovered self inductance and mutual inductance (which will be described later in this section) and invented the electromechanical relay (which was the basis for the telegraph). A circuit with a self inductance of one henry will experience a back emf of one volt when the current changes at a rate of one ampère per second.

…

Inductance is something. Inductance is the resistance of a circuit element to changes in current. Inductance in a circuit is the analog of mass in a mechanical system.

ℰ = − L  dI ⇔ cause of
change		  =  resistance
to change		  ×  rate of
change		 ⇔ F = m  dv
dt dt

inductive loop detector

Traffic at some intersections is controlled with the aid of inductive loop detectors (ILD). An ILD is a loop of conducting wire embedded just a few centimeters below the pavement. When a vehicle passes through the field, it acts as a conductor, changing the inductance of the loop. A change in the loop's inductance indicates the presence of a car above. This information can then be used to activate traffic signals, monitor traffic flow, or issue automated citations.

examples

inductance is a function of geometry

solenoid (A cross sectional area, N number of turns, ℓ length, n number of turns per length)

ΦB  = N  B   A  
 
ΦB  = N  µ0NI A  
ℓ
ΦB  =  µ0AN2   I  
ℓ
dΦB  =  µ0AN2 dI  
dt ℓ dt  
L  =  μ0AN2    =  μ0Aℓn2
ℓ

coaxial conductors (a inner radius, b outer radius , ℓ length)

ΦB  =  ⌠
⌡		 B · dA
b b
ΦB  =  ⌠
⌡		 µ0I ℓ dr  =  µ0Iℓ ⌠
⌡		 dr
2πr 2π r
a a
ΦB  =  µ0ℓ ln ⎛
⎝		 a ⎞
⎠		 I
2π b
dΦB  =  µ0ℓ ln ⎛
⎝		 a ⎞
⎠		 dI
dt 2π b dt
L  =  µ0ℓ ln ⎛
⎝		 a ⎞
⎠
2π b

toroid (A cross sectional area, R radius of revolution, N number of turns)

ΦB  =  N  B   A
 
ΦB  ≈  N  µ0NI    A 
2πR
ΦB  ≈  N  µ0NA    I 
2πR
dΦB  ≈    µ0AN2   dI
dt 2πR dt
L  ≈  μ0AN2  
2πR

rectangular loop (w width, h height, a wire radius)

ΦB  =  N 
⌠
⌡		 B · dA
 
 
ΦB  =  N 
⎡
⎣		 x w x w ⎤
⎦
µ0NI   ⌠
⌡		 y dr  +  ⌠
⌡		 x dr  +  ⌠
⌡		 y dr  +  ⌠
⌡		 x dr
2π r r r r
a a a a
 
 
ΦB  =  2 µ0N2   ⎡
⎣		 y ln  ⎛
⎝		 x ⎞
⎠		  +  x ln  ⎛
⎝		 y ⎞
⎠		 ⎤
⎦		   I  
2π a a  
dΦB  =    µ0N2   ⎡
⎣		 y ln  ⎛
⎝		 x ⎞
⎠		  +  x ln  ⎛
⎝		 y ⎞
⎠		 ⎤
⎦		   dI  
dt π a a dt  
L  =    µ0N2   ⎡
⎣		 y ln  ⎛
⎝		 x ⎞
⎠		  +  x ln  ⎛
⎝		 y ⎞
⎠		 ⎤
⎦		      
π a a      

This formula doesn't quite work since it ignores edge effects. You can find the exact formula (as well as scripts that will calculate inductance for you) online at several electrical engineering websites.

 
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