---
title: Self-Inductance
module: Electromagnetic Induction
moduleNumber: 8
lessonNumber: 6
order: 806
summary: >
  A coil resists changes to its own current. Drive current through it and the flux it
  produces threads its own turns; change that current and Faraday's law turns the coil
  against the source with a back emf $\mathcal E_L=-L\,\d I/\d t$. We define
  self-inductance as the flux linkage per ampere fixed by winding and core geometry,
  derive the long-solenoid value $L=\mu_0 N^2A/\ell$, and follow the consequence that
  dominates circuits: because a finite voltage can only sustain a finite $\d I/\d t$,
  an inductor's current cannot jump — which is why opening a switch on a live coil
  throws a spark.
topics: [Electromagnetic Induction]
draft: false
sources:
  - book: Tipler & Mosca
    ref: "Ch. 28 — Magnetic Induction; §28-6 Inductance"
---

A current in a circuit produces magnetic field throughout the surrounding space. Part
of that field threads the circuit's own turns. When the circuit current changes, its
self-produced flux linkage changes and Faraday's law produces an emf in the same
circuit. The induced emf opposes the current change. This self-response is called
self-inductance.

With fixed circuit geometry and a linear magnetic environment, flux linkage is
proportional to current:

$$
\Lambda=LI.
$$

Here $\Lambda$ is the signed flux linkage and $L$ is the self-inductance. A tightly
wound coil with $N$ identical turns has $\Lambda=N\Phi_{\rm turn}$ when each turn links
the same flux. The unit is the henry,

$$
1\ \mathrm H=1\ \frac{\mathrm{Wb}}{\mathrm A}.
$$

Inductance is a flux-linkage coefficient set by the circuit's shape, size, winding
pattern, and magnetic material. A large inductance gives a large self-induced emf
for a given current-change rate.

The coefficient is defined for a stated geometry, a stated current path, and a
stated magnetic state. Changing the core, opening a magnetic gap, moving a nearby
conductor, or rearranging the winding can change the linked flux and therefore the
measured inductance. In a linear range the coefficient remains constant as current
changes; outside that range, a current-dependent linkage curve must replace one
constant value of $L$.

## Flux Linkage and Back EMF

Consider a solenoid carrying conventional current $I$. The current produces an axial
magnetic field inside the winding. Every turn is threaded by that field, so the total
linkage grows with both current and turn count. A larger coil radius increases the
threaded area; more turns increase field strength and the number of linked turns;
greater length spreads the turns apart and reduces the interior field for the same total
turn count.

$$
% caption: A current-carrying solenoid links its own magnetic flux. The winding current produces an axial B through the interior cross section. Each turn contributes to the total flux linkage, so changing the current changes the coil's own linkage and induces a back emf.
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$$

The linkage convention uses one chosen traversal direction around every turn. The
right-hand rule gives the associated normal. Reversing the current reverses the magnetic
field and the signed linkage. Reversing the chosen traversal reverses both the reported
current and linkage signs, leaving the physical state unchanged.

A long solenoid of length $\ell$, cross-sectional area $A$, and turn density
$n=N/\ell$ has approximately uniform interior field

$$
B=\mu_0nI.
$$

The total flux linkage is

$$
\Lambda=N(BA)
=N\left(\mu_0\frac{N}{\ell}I\right)A,
$$

so the self-inductance is

$$
L=\mu_0\frac{N^2A}{\ell}=\mu_0n^2A\ell.
$$

The formula assumes a long thin air-core solenoid. Fringing at the ends, nonuniform
winding, and magnetic-core effects change the linkage. Within those limits,
$L\propto N^2A/\ell$: doubling $N$ quadruples $L$, doubling area doubles $L$, and
doubling length with $N$ fixed halves $L$.

$$
% caption: Geometric parameters of a long solenoid. The self-inductance increases with cross-sectional area and with the square of turn count, while a greater length at fixed turn count reduces the turn density and the linked magnetic field.
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$$

### Back emf from a changing current

For fixed inductance, differentiation gives

$$
\mathcal E_L=-\frac{\d \Lambda}{\d t}
=-L\frac{\d I}{\d t}.
$$

The negative sign carries Lenz's law. When a source attempts to increase positive
current, the self-induced emf has a polarity that resists the increase. When the source
current decreases, the induced emf has a polarity that supports the former current
direction. The term _back emf_ refers to this opposition to current change.

$$
% caption: Current rise and fall in one coil. During a positive current increase, the self-induced emf points opposite the source-driven current change. During a positive current decrease, its polarity reverses and tends to maintain the existing current direction. The response concerns \d I/\d t, not the current value alone.
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$$

The magnitude is proportional to $|\d I/\d t|$. A coil carrying a large steady current has
zero self-induced emf if the current is constant. A small current interrupted rapidly
can generate a large emf because the rate of change is large. The distinction explains
why opening a switch in an inductive circuit can create a spark across the separating
contacts.

## Inductor Voltage and Circuit Behaviour

Connect a coil of inductance $L$ and resistance $r$ in series with a resistor $R$ and a
battery. Immediately after a switch closes, the current is zero if it was zero before
closure. The inductor's back emf initially balances most of the battery emf, so the
current begins at a finite rate and approaches its final resistive value continuously.
The circuit equation is

$$
V_{\rm source}-(R+r)I-L\frac{\d I}{\d t}=0.
$$

At the instant of closure, $I=0$, giving

$$
\left.\frac{\d I}{\d t}\right|_{t=0}
=\frac{V_{\rm source}}{L}.
$$

The initial slope becomes smaller for a larger inductance. As current rises, the ohmic
drop $(R+r)I$ takes more of the source voltage and the magnitude of the back emf falls.
At steady current, $\d I/\d t=0$, the self-induced emf is zero, and the current is set by
the total resistance.

$$
% caption: Closing a series resistor–coil circuit. Immediately after closure, the coil back emf opposes the source and holds current near zero. As current rises, the back emf falls and the resistor drop grows. At steady current, the self-induced emf is zero and the circuit behaves resistively.
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Opening the switch reverses the situation. Before opening, the coil carries current and
has linked magnetic flux. The circuit current attempts to fall rapidly when the conducting
path is broken. The self-induced emf develops a polarity that drives conventional current
in its original direction. A large voltage can appear across the widening switch gap.
If the electric field in air exceeds breakdown strength, a spark provides a temporary
conducting path and allows stored magnetic energy to leave the coil.

$$
% caption: Opening an inductive circuit. Before the switch opens, current flows through the coil. The falling current induces a back emf that maintains the original current direction. A sufficiently large contact voltage can ionize the gap and form a spark, transferring the coil's magnetic energy to the circuit and surroundings.
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The response is often described as current continuity. In a finite-inductance circuit,
an instantaneous finite change in current would require an unbounded emf from
$-L\,\d I/\d t$. Ordinary sources and insulating gaps cannot supply that idealized emf, so
the current evolves continuously. A large but finite voltage can still arise during a
fast interruption, limited by parasitic capacitance, arcs, insulation breakdown, and
the circuit's distributed electromagnetic fields.

## Measuring Inductance

Inductance can be inferred from a controlled current ramp. If a coil's resistance drop
has been measured or is small over a short interval, the self-induced emf gives

$$
L=\frac{|\mathcal E_L|}{|\d I/\d t|}.
$$

For example, a coil whose current changes by $0.80\ \mathrm A$ in
$4.0\times10^{-3}\ \mathrm s$ while producing a $6.0\ \mathrm V$ self-induced emf has

$$
L=\frac{6.0}{0.80/(4.0\times10^{-3})}
=3.0\times10^{-2}\ \mathrm H.
$$

The calculation uses the current rate through the coil. A current sensor and a
voltage probe across the coil can supply the required records.
The probe polarity must be interpreted with the selected current direction: a rising
positive current produces a voltage polarity opposing the source that drives the rise.

Two coils can have identical resistance and different inductance. Resistance depends on
wire length, cross section, and resistivity. Inductance depends on flux linkage produced
per unit current. Adding turns can raise both quantities, but their scaling differs:
resistance grows roughly with wire length, while the ideal solenoid inductance grows with
the square of turn count. A separate resistance measurement is therefore required when
using voltage and current data to infer inductance.

### Current rise in an RL circuit

The resistor–inductor circuit after switch closure has the differential equation

$$
L\frac{\d I}{\d t}+RI=V_0,
$$

where $R$ includes every series resistance and $V_0$ is a constant source emf. The
solution with $I(0)=0$ is

$$
I(t)=\frac{V_0}{R}\left(1-e^{-t/\tau}\right),
\qquad
\tau=\frac{L}{R}.
$$

The time constant $\tau$ has unit second. At one time constant, the current reaches
$1-e^{-1}\approx0.632$ of its final value $V_0/R$. The current approaches the final
value asymptotically in this ideal model. Larger $L$ lengthens the rise. Larger $R$
shortens the rise but lowers the final current.

$$
% caption: Current rise in a series RL circuit after a constant source is connected. The initial slope is V0 divided by L, the current reaches about 63 percent of its final value at one time constant, and the self-induced emf declines as the current slope declines.
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$$

At $t=0$, the resistor drop is zero and the entire source voltage appears as an
opposing inductor voltage. At late time, the inductor voltage is zero and the entire
source voltage appears as resistor drop. The voltage partition evolves continuously:

$$
V_L=L\frac{\d I}{\d t}=V_0e^{-t/\tau},
\qquad
V_R=RI=V_0\left(1-e^{-t/\tau}\right).
$$

The sum $V_L+V_R=V_0$ is a circuit-energy statement at every instant. Early in the
transient, the source changes the coil current and builds magnetic energy. Late in the
transient, nearly all input power becomes resistor heating because the current is nearly
constant. Magnetic-energy storage is developed separately; the voltage equation already
shows why the current cannot jump.

$$
% caption: Source-voltage partition during RL current rise. The inductor voltage starts at the full source value and decays as current growth slows. The resistor voltage starts at zero and rises toward the source value. Their sum remains equal to the source voltage throughout the transient.
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$$

An initially energized RL circuit has a complementary current decay after the source is
removed while the resistor and coil remain connected. With $I(0)=I_0$,

$$
I(t)=I_0e^{-t/\tau}.
$$

The coil polarity reverses relative to the rising-current case and drives current through
the resistor in the original direction. The current path is essential. If the source
switch opens a path without providing a decay route, the coil can create a large contact
voltage and form an arc. A resistor, diode, or other protective path can give the current
a controlled route while the magnetic linkage falls.

## Solenoid and Material Inductance

The air-core solenoid expression uses the vacuum permeability $\mu_0$. A magnetic core
can increase flux linkage for the same current by concentrating magnetic field in the
core material. In a linear approximation, replace $\mu_0$ with an effective permeability
$\mu$, giving

$$
L\approx\mu\frac{N^2A}{\ell}.
$$

The approximation becomes limited when a ferromagnetic core approaches saturation.
Then flux is no longer proportional to current, the linkage-current graph bends, and a
single constant $L$ cannot describe the entire current range. The differential slope
$\d \Lambda/\d I$ determines the incremental induced response about a selected operating
point.

$$
% caption: Flux linkage versus current for an air-core and a saturating magnetic-core coil. The air-core relation is approximately linear, so one inductance describes the slope. A magnetic core can produce a larger initial slope, then its linkage growth decreases as the material approaches saturation.
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$$

Core gaps also change inductance. An air gap increases magnetic reluctance and reduces
the flux for a given current, lowering inductance. It can prevent deep saturation and
make the linkage-current relation more nearly linear over a specified current interval. Power
inductors often use a controlled gap for that reason. Core selection therefore balances
inductance magnitude, current range, frequency, temperature, and loss under operating load.

> **Worked example (Air-core solenoid inductance).** Take an air-core solenoid with
> $N=400$ turns, length $\ell=0.120\ \mathrm m$, and radius
> $r=1.50\times10^{-2}\ \mathrm m$. Its cross-sectional area is
>
> $$
> A=\pi r^2=7.07\times10^{-4}\ \mathrm{m^2},
> $$
>
> and the long-solenoid approximation gives
>
> $$
> L=\mu_0\frac{N^2A}{\ell}
> =(4\pi\times10^{-7})
> \frac{(400)^2(7.07\times10^{-4})}{0.120}
> =1.18\times10^{-3}\ \mathrm H.
> $$
>
> If the current rises at $50\ \mathrm{A\,s^{-1}}$, the back-emf magnitude is
>
> $$
> |\mathcal E_L|=L\left|\frac{\d I}{\d t}\right|
> =(1.18\times10^{-3})(50)
> =5.9\times10^{-2}\ \mathrm V.
> $$

The value is modest because the coil is short, air-cored, and driven at a moderate
current rate. More turns or a higher-permeability core raise the inductance sharply,
which is why compact air coils are rated in millihenries while large or magnetically
cored coils reach henries.

The dimensions also matter for the long-solenoid approximation. The formula assumes the
length is several times larger than the diameter and evaluates flux using the nearly
uniform interior field. A short coil has substantial fringing; magnetic field extends
outside the winding and varies across its turns. The true linkage can be computed by
integrating the coil's self-field through every turn, but that calculation is more
involved than the long-solenoid estimate.

Winding layout changes effective area and leakage flux. Closely packed turns couple
strongly to one another. A wide single-layer coil and a compact multilayer coil made
from the same wire can have different inductance because their field distributions and
turn-to-turn linkages differ. A circuit symbol omits this geometry, but the inductance
value records its electromagnetic consequence.

### Self-inductance as a circuit element

An ideal inductor has no resistance and obeys

$$
V_L=L\frac{\d I}{\d t}
$$

under the passive sign convention, where the terminal voltage is measured in the
direction associated with current entering the positive terminal. The induced emf around
the coil has the opposite sign, $\mathcal E_L=-L\,\d I/\d t$. Both statements refer to the
same physical response; they use different circuit sign conventions.

A physical coil combines inductance, wire resistance, inter-turn capacitance, and core
loss. At low frequency, a series model with inductance $L$ and resistance $r$ often
captures the main behavior. At higher frequency, capacitance between turns and
frequency-dependent resistance can create resonances or redistribute current. A single
constant inductance then represents only a limited operating range.

Inductors appear in power supplies, filters, actuators, relays, and electromagnetic
sensors because their back emf limits abrupt current changes. A relay coil stores linked
flux while energized. A switching converter uses an inductor to transfer energy between
intervals of a switching cycle. A sensor can infer a nearby magnetic material or
conducting object from its effect on coil inductance or loss. The circuit behavior always
traces back to the current-dependent flux linkage of the coil.

## Switching and Protection

The post-switch current path sets both peak voltage and release time. With a coil
resistance $r$, an added decay resistance $R_p$, and a protection element having
current-dependent drop $V_p(I)$, the discharge model is

$$
L\frac{\d I}{\d t}+(r+R_p)I+V_p(I)=0.
$$

A freewheel diode has a small forward drop while it conducts. It remains reverse-biased
during source operation, becomes forward-biased after switch opening, and provides a
low-voltage current loop. A resistor, diode--resistor network, or controlled transient
suppressor permits a larger coil voltage and a faster decay. The intended current path
must receive the magnetic energy present immediately before interruption,

$$
U_L(0^-)=\frac12LI_0^2.
$$

$$
% caption: A freewheel-diode path across a relay coil. During source operation, the diode blocks and the coil current rises. When the switch opens, coil polarity reverses, the diode conducts, and current decays through a controlled loop that limits switch voltage.
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  \node[anchor=west] at (0.95,0.4) {diode};
  \node[above] at (3.02,0.5) {switch};
\end{tikzpicture}
$$

Select protection from the electrical and mechanical requirements of the switched
assembly.

| protection path | initial voltage stress | current-decay behavior | primary check |
| --- | --- | --- | --- |
| freewheel diode | near diode forward drop plus winding drop | comparatively slow | release time and diode pulse current |
| resistor or diode--resistor path | set by $I_0R_p$ and winding drop | faster exponential decay | resistor pulse energy and voltage rating |
| transient suppressor | approximately clamped over its operating range | voltage-limited decay | clamp energy, current, and repetitive duty |

- **Map the active loop.** Draw the coil, protection component, switch, and return
  conductor for the turn-off interval. A component connected outside the actual
  high-current loop can leave lead inductance and switch terminals exposed.
- **Specify the initial state.** Determine $I_0$ from the pre-switch circuit, then
  calculate $U_L(0^-)$ before selecting a clamp. Repetition rate converts the
  per-event energy into an average thermal load.
- **Check terminal polarity.** The coil reverses its terminal polarity after a falling
  current begins. The protection element orientation follows this polarity and the
  desired continuation of conventional current.
- **Verify the required endpoint.** A relay release criterion may be a current
  threshold; a switch rating may be a peak voltage; a thermal criterion may be average
  protection-element power. One topology rarely optimizes all three.

### Terminal polarity and current direction

The self-induced emf sign can be found without memorizing a coil terminal rule. Select
a positive current traversal and the associated surface normal. Determine whether the
self-produced flux linkage is increasing or decreasing. A growing positive linkage
requires an induced emf in the negative traversal direction. A falling positive linkage
requires an induced emf in the positive traversal direction. Circuit terminal labels
then follow from the direction in which the induced emf would drive conventional current
through a completed test loop.

A source-driven coil carries increasing current during turn-on. The coil terminal at
which the source current enters has a self-induced voltage that opposes the source
polarity under the passive convention. During turn-off, the coil
polarity reverses to support the old current direction. Labeling a voltage only as
“positive across the coil” without a current reference loses this change of sign.

The polarity also gives an instrument check. A voltage probe across a coil during a
current ramp reports the sum of the inductive and resistive contributions according to
its lead orientation. Reversing the probe leads reverses the reported sign. A current
probe reports the actual current direction. Comparing both records with a stated passive
sign convention allows the measured $L\,\d I/\d t$ term to be separated from $Ir$.

> **Worked example (RL turn-on and turn-off).** A $12\ \mathrm V$ source drives a
> series circuit with $L=0.60\ \mathrm H$ and $R=3.0\ \Omega$. The time constant and
> final current are
>
> $$
> \tau=\frac{L}{R}=0.20\ \mathrm s,
> \qquad
> I_\infty=\frac{V_0}{R}=4.0\ \mathrm A,
> $$
>
> so the turn-on current is
>
> $$
> I(t)=4.0\left(1-e^{-t/(0.20)}\right)\ \mathrm A.
> $$
>
> At $t=0.20\ \mathrm s$ the current is $2.53\ \mathrm A$, and the inductor voltage is
>
> $$
> V_L=12e^{-1}=4.41\ \mathrm V,
> $$
>
> with resistor voltage $V_R=RI=7.59\ \mathrm V$; their sum is $12.0\ \mathrm V$ within
> rounding. The instantaneous current slope is $V_L/L=7.36\ \mathrm{A\,s^{-1}}$, smaller
> than the initial slope $12/0.60=20\ \mathrm{A\,s^{-1}}$.
>
> Now remove the source at steady state, leaving the coil and resistor as a closed decay
> path with initial current $4.0\ \mathrm A$:
>
> $$
> I(t)=4.0e^{-t/(0.20)}\ \mathrm A.
> $$
>
> The current at one time constant is $1.47\ \mathrm A$. The resistor keeps heating as
> the coil current falls; the self-induced voltage holds current in its original
> direction through the resistor, while the coil terminal polarity is opposite its
> turn-on polarity.

The solution assumes a constant $L$ and $R$. A coil resistance rises as it warms, which
changes the final current and time constant. A magnetic core can make inductance vary
with current. A diode protection path can replace the resistor-only decay equation with
another voltage-current relation. The exponential form remains the basic result for a
linear series RL circuit with constant parameters.

### Flux linkage, current, and material limits

The relation $\Lambda=LI$ is a linear model. It holds accurately for an air-core coil
and for a magnetic core over a current range where permeability is approximately
constant. A nonlinear core has a linkage curve $\Lambda(I)$ whose slope varies with
current. The induced emf is always

$$
\mathcal E_L=-\frac{\d \Lambda}{\d t}
=-\frac{\d \Lambda}{\d I}\frac{\d I}{\d t}.
$$

The differential inductance $\d \Lambda/\d I$ replaces a single constant $L$ at a specified
operating point. It can decrease near saturation, so the same current ramp produces a
smaller self-induced voltage than predicted by a low-current inductance measurement.
Power electronics and magnetic actuators often need this nonlinear information to predict
current slopes accurately.

Temperature also changes a physical coil. Copper resistance rises with temperature,
increasing resistive voltage drop. Core permeability and loss can change with
temperature. Mechanical vibration can alter a core gap. A component specification may
therefore state inductance at a test current, frequency, dc bias, and temperature.
Copper resistance, core permeability and loss, and core-gap geometry then identify
the operating configuration to which the measured inductance applies.

### Source ramps and current-slope control

An inductor can be driven by a deliberately shaped voltage source. In an ideal coil,

$$
\frac{\d I}{\d t}=\frac{V_L}{L}.
$$

A constant coil voltage therefore gives a linear current ramp. A positive voltage raises
current at constant slope; a negative voltage lowers current at constant slope. A
resistor in series changes the available coil voltage as current changes, so a constant
source voltage gives an exponential current trace.

A current controller seeking a fixed slope $s$ requires approximate coil voltage

$$
V_L=Ls.
$$

If the coil has resistance $r$ and carries current $I$, the source must also provide the
resistive drop:

$$
V_{\rm source}=Ls+rI.
$$

The required voltage rises as current rises under a fixed positive slope. This relation
appears in actuators and switching converters, where a controller applies a voltage for a
specified interval to change current by a target amount. The coil's self-induced voltage
is the physical response that sets the attainable current slope.

The energy supplied during a current ramp divides among resistor heating and magnetic
storage. The source power is $V_{\rm source}I$. The resistor receives $I^2r$. The
remaining inductive power is $L I\,\d I/\d t$, which integrates to the magnetic energy
associated with the final current. A current-ramp controller must supply this energy
even if the average terminal current is small.

During a decreasing current ramp, the inductor returns energy to the circuit or absorbs
it in a protection path. A diode across a coil returns current through the coil and diode
while resistance converts the stored energy to heat. A switching converter can route the
energy to a capacitor, supply rail, or another magnetic element. The direction of energy
flow depends on the external circuit, while the coil emf always opposes the imposed
change in linkage.

### Inductor voltage across a changing source

The source voltage itself can vary in time. With a prescribed source $V_s(t)$ and series
resistance $R$, the current obeys

$$
L\frac{\d I}{\d t}+RI=V_s(t).
$$

The inductor response depends on both the source history and the present current. A
short voltage pulse changes current by approximately

$$
\Delta I\approx\frac{1}{L}\int V_L\,\d t
$$

when the resistive voltage is small over the pulse. The time integral of coil voltage is
the change in flux linkage:

$$
\int_{t_i}^{t_f}V_L\,\d t=\Lambda_f-\Lambda_i.
$$

Pulsed magnets and current probes rely on this relation. A voltage pulse of known
area changes current in proportion to the inverse inductance. A higher-inductance
coil needs a larger voltage-time area to achieve the same current change.

The approximation fails for long pulses when the resistor drop becomes comparable with
source voltage. Then the current slope decreases as current grows. It also fails near
core saturation when linkage is nonlinear in current. A current trace and coil-voltage
trace together determine whether the constant-inductance model remains appropriate.

## Circuit Models and Configurations

Inductors connected far enough apart that their mutual flux is negligible combine like
independent flux-linkage elements. In series, the same current passes through each and
the total inductance is

$$
L_{\rm series}=L_1+L_2+\cdots.
$$

In parallel, the same voltage appears across each branch. For independent linear
inductors, the reciprocal relation applies,

$$
\frac{1}{L_{\rm parallel}}
=\frac{1}{L_1}+\frac{1}{L_2}+\cdots.
$$

The relations require negligible mutual inductance. Coils placed close together can link
each other's fields. Their series behavior then depends on winding orientation: linked
flux can add or subtract. That coupling belongs to the mutual-inductance treatment and
should be marked before using the independent-inductor formulas.

$$
% caption: Independent inductors in series and parallel. The series connection carries one current through both coils and adds their voltage responses. The parallel connection shares one voltage across branches and divides current. Coils must be magnetically separated for these simple combination rules to apply.
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Series inductors share current, so a rapid current change produces the sum of their
back-emf magnitudes. Parallel inductors share voltage, so branch currents depend on
their inductances and resistances. Ideal equal inductors can divide current equally;
real coils can have tolerance differences, core nonlinearities, and unequal resistance
that make sharing uneven. Current balancing sometimes requires deliberate series
resistance or coupled winding design.

### Inductance from magnetic geometry

The inductance relation can be derived directly from flux linkage. Begin with a trial
current $I$ in a specified coil. Calculate the magnetic field produced by that current.
Integrate the normal field through each turn surface to obtain flux. Sum the turn fluxes
with their winding orientations to obtain $\Lambda$. The ratio $\Lambda/I$ is the
inductance when the geometry and magnetic properties are linear.

Current reversal separates a coil's own field from an externally applied field. A current
in the coil can link its own turns and create self-inductance. An external current in a
second coil can link the first coil and create mutual inductance. The self calculation
uses field produced by the same circuit current. The mutual calculation uses field from
another circuit. Both use flux linkage, but they belong to different circuit relations.

A toroidal winding with mean magnetic path length $2\pi r_m$, cross-sectional area
$A$, and $N$ turns has the common approximation

$$
L\approx\mu\frac{N^2A}{2\pi r_m}.
$$

The toroidal geometry confines much of the magnetic field to the core and reduces
external leakage. Increasing mean radius lengthens the magnetic path and lowers
inductance. Increasing the core cross-sectional area raises linkage. A gap in the
toroid reduces the effective permeability and can dominate the inductance.

$$
% caption: Toroidal-coil geometry. Winding current produces a mostly circular magnetic field within the core. The linked flux depends on turn count, core cross section, and mean magnetic path length. A small air gap interrupts the high-permeability path and reduces inductance while improving linearity.
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$$

Magnetic leakage can reduce the flux shared by all turns. A short solenoid has field
outside its winding; some external field lines do not thread every turn in the same way.
A toroid reduces that leakage. Core shape, winding placement, and nearby magnetic
materials can change the leakage path. Inductance measurements therefore describe a
complete physical assembly, including its core, mounting, and any movable magnetic
parts near the coil.

### Inductance, current continuity, and idealizations

An ideal inductor permits any steady current with zero terminal voltage because
$\d I/\d t=0$. It does not permit an arbitrary instantaneous current jump under finite
voltage because the required $L\,\d I/\d t$ would be unbounded. An ideal capacitor has the
opposite constraint: its voltage is continuous under finite current, while its current
can change abruptly. These complementary constraints determine initial conditions in
transient circuits.

Current continuity applies to current through the inductor's series path. A branch
current in a larger circuit can redistribute through capacitors or parallel paths while
the coil current remains continuous. Circuit diagrams should distinguish the coil branch
from total source current. A disconnected inductor can still retain a current briefly if
a parasitic capacitance or arc provides a path; the resulting high voltage is a sign that
the current path and electric-field energy cannot be ignored.

The ideal model also assumes no magnetic saturation and no energy loss in the core. A
real core can dissipate energy through magnetic processes and eddy currents. A real wire
has resistance. A real winding has capacitance. These effects modify current and voltage
records, particularly during fast switching. The basic local statement remains that a
changing self-linkage produces an emf opposing that change.

### Step-by-step analysis of a coil transient

An inductive transient can be organized with a fixed sequence.

1. **Choose the coil current direction and terminal-voltage convention.** Keep that convention through the calculation.
2. **State the circuit condition immediately before the switch action.** A long-standing dc circuit has $\d I/\d t=0$ and can be replaced by its steady behavior for the initial condition.
3. **Apply current continuity.** Set the current immediately after switching equal to the current immediately before switching for the inductor branch.
4. **Write the loop or node equation.** Include source, resistive voltage, and $L\,\d I/\d t$ with consistent signs.
5. **Solve for the initial slope, time response, or required voltage.** Check that the source and resistor limits agree at early and late times.
6. **Check energy and polarity.** A rising current requires a back emf opposing the source; a falling current requires a coil emf that supports the old current direction.

In a series RL turn-on, step 2 gives $I(0^-)=0$ and step 3 gives $I(0^+)=0$.
In a series RL turn-off from steady current, step 2 gives $I(0^-)=V_0/R$ and step 3
gives the same current just after the switch changes. The subsequent circuit path determines
whether current decays smoothly through a resistor, commutates through a diode, or
creates a high-voltage discharge.

## Model Scope and Verification

Self-inductance gives the current-rate contribution to the coil emf for a specified
coil state:

$$
\mathcal E_L=-\frac{\d \Lambda}{\d t},
\qquad
\Lambda=LI
\quad\hbox{for a linear fixed geometry}.
$$

Current requires a complete circuit model: source waveform, active resistance, initial
current, and switch topology. A terminal-voltage measurement also includes wire
resistance and whatever external path is connected to the coil. The physical coil and
the circuit interval must be declared together.

- **Magnetic declaration:** winding, core material, air gap, position of any movable
  part, bias-current range, and frequency range used to assign $L$.
- **Circuit declaration:** current reference direction, terminal-voltage polarity,
  series resistance, source model, and the post-switch current path.
- **State declaration:** the measured or calculated $I(0^-)$, coil temperature, and
  whether the quoted inductance is a small-signal, differential, or large-excursion
  value.
- **Validation record:** compare initial slope, late-time current, and energy release
  with the same parameter set. A disagreement identifies an omitted resistance,
  nonlinear linkage, parasitic capacitance, or a changed protection path.

The linkage rate fixes the back-emf sign. Geometry sets linkage per ampere; the circuit
sets the current history. A large steady current can coexist with zero self-induced emf
because the linkage rate is zero.

> **Worked example (Solenoid inductance and switching transient).** Take an
> $N=800$ turn air-core solenoid of length $0.200\ \mathrm m$ and radius
> $1.00\times10^{-2}\ \mathrm m$. Its area and long-solenoid inductance are
>
> $$
> A=\pi(1.00\times10^{-2})^2=3.14\times10^{-4}\ \mathrm{m^2},
> \qquad
> L=\mu_0\frac{N^2A}{\ell}
> =(4\pi\times10^{-7})
> \frac{(800)^2(3.14\times10^{-4})}{0.200}
> =1.26\times10^{-3}\ \mathrm H.
> $$
>
> Place the coil in series with a $6.0\ \Omega$ resistor (which already includes the
> coil wire resistance) and a $9.0\ \mathrm V$ source. The final current and time
> constant are
>
> $$
> I_\infty=\frac{9.0}{6.0}=1.50\ \mathrm A,
> \qquad
> \tau=\frac{1.26\times10^{-3}}{6.0}
> =2.10\times10^{-4}\ \mathrm s.
> $$
>
> The small time constant reflects a millihenry inductance and several ohms of series
> resistance. At switch closure the current slope is
>
> $$
> \left.\frac{\d I}{\d t}\right|_{0}
> =\frac{9.0}{1.26\times10^{-3}}
> =7.14\times10^3\ \mathrm{A\,s^{-1}}.
> $$
>
> This large initial slope lasts only a fraction of a millisecond. At one time constant
> the current reaches $0.948\ \mathrm A$, the coil back-emf magnitude is
> $9.0e^{-1}=3.31\ \mathrm V$, and the resistor drop is $5.69\ \mathrm V$; the sum
> matches the source voltage. A digital instrument with insufficient time resolution
> could miss most of this transient and report only the final dc current.
>
> Remove the source while the resistor remains as a decay path, and the current falls
> from $1.50\ \mathrm A$ with the same time constant. The initial coil emf magnitude is
>
> $$
> |\mathcal E_L(0^+)|
> =L\frac{I_\infty}{\tau}
> =9.0\ \mathrm V.
> $$
>
> The coil terminal polarity reverses relative to turn-on and drives current through the
> resistor in its prior direction. The equality with the source voltage holds only
> because the initial decay current equals the previous steady source current; a
> different decay resistance would change both the time constant and the initial
> terminal voltage.

### Inductance measurement by current decay

An experimental current-decay trace can measure $L/R$ without directly measuring coil
voltage. Disconnect the source while retaining a known total decay resistance $R$.
Record the coil current or resistor voltage as a function of time. A linear RL model gives

$$
\ln\left(\frac{I(t)}{I_0}\right)=-\frac{R}{L}t.
$$

A plot of $\ln(I/I_0)$ against time has slope $-R/L$. With independently measured
resistance, the inductance follows from the slope. This approach averages over many
samples and can be less sensitive to one noisy voltage measurement than a single
current-ramp estimate.

The method assumes a known resistance over the measurement interval. Coil heating,
nonlinear core behavior, and a changing protection-path voltage can distort the
exponential trace. A curvature in the log-current plot indicates that a constant $L/R$
model is incomplete. The data may then require temperature-dependent resistance,
core-saturation, or distributed-capacitance terms.

### Common circuit configurations

Several standard coil connections illustrate self-inductance in practice.

- **Relay coil:** A dc source establishes current and an iron armature moves. Opening the driving switch produces back emf; a diode or transient suppressor protects the switch.
- **Solenoid actuator:** Current rise creates magnetic force on a movable plunger. Motion can change the inductance because the core position changes the magnetic reluctance.
- **Filter choke:** A series inductor resists rapid current variation and attenuates unwanted high-frequency current components while passing a dc component.
- **Switching inductor:** A controlled voltage pulse changes current by a known amount. The inductor transfers energy between intervals of the switching cycle.

In a movable-core actuator, inductance can depend on position $x$: $L=L(x)$. A current
through the coil then produces force toward a geometry of higher inductance when current
is maintained by an appropriate source. The motion changes linkage as well as current,
so a complete actuator model includes electrical, magnetic, and mechanical equations.
The fixed-geometry relation $\mathcal E_L=-L\,\d I/\d t$ describes the current-change part;
position-dependent linkage adds another term.

With changing geometry, write the linkage as $\Lambda(I,x)$. Its total rate is

$$
\frac{\d \Lambda}{\d t}
=\frac{\partial\Lambda}{\partial I}\frac{\d I}{\d t}
+\frac{\partial\Lambda}{\partial x}\frac{\d x}{\d t}.
$$

The first term is the ordinary self-inductive response. The second is a motional
linkage-change term associated with changing coil geometry. Separating the terms keeps
self-inductance distinct from motional emf while allowing both effects in an actuator or
moving-core device.

### Unit and sign audit

The unit check for $\mathcal E_L=-L\,\d I/\d t$ is

$$
\mathrm H\,\frac{\mathrm A}{\mathrm s}
=\frac{\mathrm{Wb}}{\mathrm A}
\frac{\mathrm A}{\mathrm s}
=\frac{\mathrm{Wb}}{\mathrm s}
=\mathrm V.
$$

The sign audit begins with current, not with an isolated terminal label. Mark the
positive current direction, determine whether that current and its linked flux are
increasing or decreasing, then draw the induced emf that opposes the change. The same
procedure applies to a solenoid, relay coil, transformer winding, or arbitrary loop.

An inductor's back emf resists a change in current. A constant current gives zero
self-induced emf even if the magnetic field and flux linkage are large. A small current
with a rapid interruption can give a substantial coil voltage. Those two limiting cases
organize most sign and magnitude checks in self-inductance problems.

### Practical error checks in inductance problems

Several numerical checks expose common calculation errors. The long-solenoid formula
contains $N^2$, not $N$. One factor of $N$ appears because more turns raise the magnetic
field; the second appears because more turns link that field. The cross-sectional factor
is area $A$, not radius $r$. Replacing $A$ by $r$ gives incorrect dimensions and misses
the quadratic radius scaling.

Inductance has unit henry, while resistance has unit ohm. Their ratio $L/R$ has unit
second and is the RL time constant. A result reported in henries per second for a time
constant has mixed the circuit quantities. The self-induced emf has unit volt and must
be compared with source and resistor voltages. Current and magnetic field have
different physical dimensions.

Sign errors often come from treating the coil back emf as a voltage with a fixed terminal
polarity. Its polarity changes when $\d I/\d t$ changes sign. At turn-on, back emf opposes
the applied source. At turn-off, it supports the pre-existing current path. Draw the
current arrow and a loop traversal before assigning terminal signs. A passive voltage
label across the inductor can then be converted consistently to the Faraday emf sign.

An ideal inductor in steady dc operation has zero voltage and can carry a nonzero
current. An ideal inductor immediately after an attempt to change current can have a
nonzero voltage while its current remains at its previous value. Both statements follow
from $V_L=L\,\d I/\d t$. They do not contradict one another because the first describes
$\d I/\d t=0$ and the second describes a transient current slope.

Inductor values measured with an ac bridge or impedance meter can depend on test
frequency and ac amplitude. A magnetic core may have different incremental permeability
at different dc bias currents. A data-sheet inductance should therefore be read together
with its measurement conditions. Circuit calculations that use the nominal value outside
that range require a tolerance or nonlinear model.

### Scope of the self-inductance model

Self-inductance describes a circuit's response to its own current-dependent flux linkage.
Nearby circuits introduce mutual inductance, where changing current in one coil induces
an emf in another. Magnetic energy quantifies the work required to build current in an
inductor. Alternating-current circuits combine inductance with capacitive and resistive
impedances. Those topics use the same linkage and Faraday-law foundation, while their
additional circuit relationships require separate analysis.

For a fixed coil, the causal chain is current, self-field, self-flux linkage, and
self-induced emf opposing the current change. Geometry determines the linkage per
ampere; the circuit determines the current history. Together they fix the voltage and
energy records.

A calculation should state whether a quoted inductance is self-inductance of one winding,
an equivalent series or parallel combination, or an incremental value about a biased
operating point. It should also state whether coil resistance is included in the circuit
resistance. These declarations fix the model before numerical substitution and prevent
an inductive voltage, a resistive drop, and a mutual-coupling voltage from being merged
into one unexplained term.

In a time-varying current record, retain the sign of $\d I/\d t$ until the voltage polarity
is interpreted. A positive current can have a positive, zero, or negative self-induced
emf depending on whether it is rising, steady, or falling. The same current value at two
different times can therefore correspond to different coil voltages and different
external circuit behavior.

The current path after a switch action determines the inductance transient. A closed
decay path converts magnetic energy gradually to heat. A broken
path raises coil voltage until a parasitic capacitance, protective component, or arc
provides a route. Drawing that path identifies the relevant resistance and determines
the current-decay time scale. It also identifies the component into which the energy is
released as the coil linkage returns toward zero.

In every case, the sign of the induced emf follows the linkage rate, while the magnitude
also depends on the coil geometry and the external circuit conditions.